Proceedings of the 22 - SAARMSTE

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Proceedings of the 22nd Annual Meeting of the

Southern African Association for Research in

Mathematics, Science and Technology Education

(SAARMSTE)

13 – 16 January 2014

Nelson Mandela Metropolitan University

Port Elizabeth, South Africa

New Avenues to Transform Mathematics, Science

and Technology Education in Africa

LONG PAPERS

Editors: Paul Webb, Mary Grace Villanueva, Lyn

Webb

ISBN: 978-0-9869800-9-1

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SAARMSTE COMMITTEES:

SAARMSTE Executive 2013/2014

President Prof Mellony Graven (Eastern Cape Chapter)

Secretary/Treasurer Prof Margot Berger

Chapter Representative Prof Keith Langenhoven (Western Cape Chapter)

Chapter Representative Prof Lyn Webb (Eastern Cape Chapter)

Research Capacity Building Committee (RCBC) Representative

Prof Hamsa Venkatakrishnan

AJRMSTE Journal Editor Prof Fred Lubben

President Elect Prof Mercy Kazima (Malawi Chapter)

Manager of the SAARMSTE Secretariat

Ms Caryn (Caz) McNamara

SAARMSTE Local Organising Committee (LOC) 2014

Conference Chair: Dr Tulsi Morar

Secretary/Treasurer: Ms Carolyn Stevenson-Milln

Fundraising Prof Paul Webb

Proceedings Chair: Prof Paul Webb

Marketing: Ms Debbie Derry

Programme Chair: Dr Mathabo Khau

Deputy PC Chair: Mr Vuyani Matsha

Logistics Chair: Prof Hugh Glover

Members: Dr Kathija Adams

Dr Andre Du Plessis

Ms Kelley Felix

Ms Marilyn Gibbs

Ms Thandi Hlam

Mr Sherwin King

Ms Elsa Lombard

Ms Joy Turyagyenda

Ms Pam Roach

Ms Gishma Daniels Smith

Dr Lyn Webb

Dr Mary Grace Villanueva

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Acknowledgements

We take great pleasure in thanking our sponsors for their generosity and support for this our 22nd

SAARMSTE Annual Conference:

CASIO

Nelson Mandela Metropolitan University

Parrot Products (Pty) Ltd

ROUTLEDGE, Taylor & Francis Group

Van Schaik Bookstores

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Forward LOC Chairperson – Dr Tulsi Morar

I am delighted to welcome you to the 22nd Annual

SAARMSTE conference held at Nelson Mandela

Metropolitan University in Port Elizabeth. This great

institution on Higher Education has been named after the

Father of the Nation, Nelson Rolihlahla Mandela and at this

time we remember him with sadness while celebrating his

contribution to peace, social justice, democracy and

Education in South Africa.

It is with immense satisfaction that I write this Foreword to

the Proceedings of the 22nd Annual SAARMSTE conference

held at Nelson Mandela Metropolitan University in Port

Elizabeth. The high quality papers presented at the

conference make the SAARMSTE conference the ideal platform for researchers to debate,

challenge and inspire fellow Mathematics, Science and Technology Researchers, while at the

same time establish new contacts both nationally and internationally.

We welcome our International and National key note speakers and look forward to their

presentations. Over the next four days there will be a total of 198 long, short, poster papers

presented. To those who submitted a paper for presentation, the LOC appreciates your

commitment to MSTE research. Without your contribution, we would not have a conference!

The LOC has work tirelessly in putting together a conference for you to remember and I thank

each LOC member for their valuable contribution. Thanks also goes to the SAARMSTE executive

for their ever-willing support and finally my appreciation to Ms Carolyn- Stevenson-Milln, our

conference organiser for all her hard work.

To each one of you enjoy the conference and take time out to enjoy Port Elizabeth.

LOC Chairperson – Dr Tulsi Morar

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Message from SAARMSTE President

It is a great pleasure to welcome you all to the 22nd

annual SAARMSTE Conference held at the Nelson

Mandela Metropolitan University of the Eastern Cape in

this beautiful city of Port Elizabeth.

Our annual SAARMSTE conference is the key event in our

SAARMSTE calendar. It is a wonderful and critically

important opportunity for our mathematics, science and

technology research community to come together to

engage and deliberate with others across our continent

on possible ways to overcome the many educational

challenges facing us. We come together to both

challenge and argue with each other but most importantly to inspire each other to continue to

strive for excellence in this field. We are an established community with a rich history of

supporting each other and we trust that this conference will be an opportunity for strengthening

these networks of support.

Our conference theme this year is: New Avenues to Transform Mathematics, Science and

Technology Education. The focus on finding new avenues of transformation is particularly

important given that Africa’s crisis discourse tends to point to decades of failure in addressing

post-colonial education equity issues, particularly in these subjects. The theme highlights the

need to acknowledge that ‘more of the same’ is unlikely to lead to changing this critical situation.

Organising a SAARMSTE conference is an enormous responsibility and takes an enormous amount

of commitment and passion, often largely behind the scenes, and on top of heavy workloads. I

would like to thank the members of the LOC and many others who have given so generously of

their time and energy to organise this event. In particular I would like to thank the Chair Tulsi

Morar for his leadership of his team and his willingness to lead the hosting of this event. I would

also like to thank the Nelson Mandela Metropolitan University for hosting this conference.

Thank you to all of you, and to all of our supporters, for your ongoing commitment to SAARMSTE

and to the imperative that as members of SAARMSTE we contribute positively to finding new

avenues of transformation in this field. Finally, I wish to thank our funders who have given so

generously in supporting SAARMSTE to achieve its aims and objectives. Without their assistance

this conference and this publication would not have been possible.

Enjoy the conference!

Mellony Graven SAARMSTE President January 2014

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22nd

SAARMSTE ANNUAL CONFERENCE PAPERS

Review Policy of SAARMSTE 2014 Conference Papers Nelson Mandela Metropolitan University, Port Elizabeth 13-16 January 2014

Long Papers

All long papers were reviewed in their entirety by at least two external reviewers.

Reviewers were selected from amount the list of SAARMSTE Journal Reviewers (African

Journal for Research in Mathematics, Science and Technology Education) all of whom are

internationally known in their field. The reviewers‟ suggestions were considered by

members of the Programme Committee, who made final decisions. Where there was

agreement among two reviewers, their recommendations were generally accepted by the

Programme Committee. Where there was disagreement, the Programme Committee

appointed one other reviewer, whereupon the committee took account of the new review

together with the first two reviews and made a final decision. In cases where papers were

accepted with conditions, authors were advised to make changes in order to have their

papers accepted, or provide a compelling argument as to why the conditions were not

adhered.

Short Papers, Posters, Snapshots, Round Tables and Symposia

For these presentations, only an extended abstract (1-2) pages was reviewed. Reviewers

were drawn from SAARMSTE members and authors of long papers. Agreement was

reached by consensus on each abstract and the Programme Committee informed authors of

the decision. Authors were given opportunities to rework their abstract according to the

reviewers‟ suggestions.

Professor Paul Webb

SAARMSTE 2014 Proceedings Chair

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SAARMSTE Reviewers 2014

Kathija Adam Sarah Bansilal Bongani Bantwini Margot Berger Lynn Bowie Marie Botha Deonarain Brulall Charles Chinduka Marc J de Vries Andre du Plessis Anthony Essien Clyde Felix Nosisi Feza Hugh Glover Leo Goosen Mellony Graven Mishack Gumbo Mercy Kazima Hemoine Kemp Bill Kyle Elsa Lombard Caroline Long Fred Lubben Tulsi Morar Nkosinathi Mpalami Audrey Msimanga Vimolan Mudaly Jayaluxmi Naidoo Kenneth Ngcoza Emilia Afonso Nhalevilo Thulisile Nkambule Helena Oosthuizen Tom Penlington Umesh Ramnarain Marissa Rollnick Duncan Samson Marc Schafer Venessa Scherman Gerrit Stols Peter Taylor Thabo J Tholo Rochelle Thorne Zena Scholtz Hamsa Venkatakrishnan Mary Grace Villanueva Hannatjie Vorster Lyn Webb Paul Webb

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TABLE OF CONTENTS

SAARMSTE COMMITTEES........................................................................................................................................ 1

SAARMSTE EXECUTIVE 2013-2014 ..................................................................................................................................... 1

SAARMSTE LOC 2014 .............................................................................................................................................................. 1

ACKNOWLEDGEMENTS ............................................................................................................................................ 2

FORWARD LOC CHAIRPERSON ............................................................................................................................. 3

MESSAGE – SAARMSTE PRESIDENT..................................................................................................................... 4

REVIEW POLICY STATMENT .................................................................................................................................. 5

SAARMSTE REVIEWERS 2014 ................................................................................................................................................. 6

LONG PAPERS .......................................................................................................................................................... 10

MATHEMATICS (ALPHABETISED BY AUTHOR SURNAME) ........................................................ 10

A potential interpretive framework for exploring mathematics teachers’ narratives of parental support Clyde Felix & Marc Schäfer ..................................................................................................................................................... 11

The perceptions of BEd (FET) mathematics students concerning their training

Owen Hugh Glover ...................................................................................................................................................................... 25

Primary learner descriptions of a successful maths learner

Mellony Graven & Einat Heyd-Metzuyanim ..................................................................................................................... 40

Exploring the potential of using cultural villages as instructional resources for connecting mathematics education to learners’ cultures Sylvia Madusise & Willy Mwakapenda .............................................................................................................................. 52

Toward an understanding of authentic assessment: A theoretical perspective Duncan Mhakure .......................................................................................................................................................................... 68

Shifts in practice of mathematics teachers participating in a professional learning community

Nico Molefe & Karin Brodie ..................................................................................................................................................... 80

Learning to teach mathematics by means of concrete representations Nkosinathi Mpalami ................................................................................................................................................................... 94

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Comparing strategies of determining the centre and radius of a circle using repeated measures

design

Eric Machisi, David L. Mogari & Ugorji I. Ogbonnaya ............................................................................................... 103 On South African primary mathematics learner identity: A Bernsteinian illumination

Pausigere Peter.......................................................................................................................................................................... 115

The allure of the constant difference in linear generalisation tasks

Duncan Samson ......................................................................................................................................................................... 131

Surveying the distribution and use of mathematics teaching aids in Windhoek: A Namibian case

study

Duncan Samson & Tobias Munyaradzi Dzambara ..................................................................................................... 145

SCIENCE AND TECHNOLOGY (ALPHABETISED BY AUTHOR SURNAME) ........................... 157

Exploring educators’ perceptions on how SIKSP seminar-workshop series prepared them to use

dialogical argumentation instruction to implement a science-IK curriculum

Senait Ghebru & Meshach Ogunniyi ................................................................................................................................. 158

The effect of an argumentation model in enhancing educators’ ability to implement an

indigenized science curriculum

Meshach Ogunniyi .................................................................................................................................................................... 173 A case study on the influence of environmental factors on the implementation of science inquiry-based learning at a township school in South Africa Umesh Ramnarain .................................................................................................................................................................... 187

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Mathematics

Long Papers

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A potential interpretive framework for exploring mathematics teachers‟

narratives of parental support

Clyde Felix1

& Marc Schäfer2

1 School for Continuing Professional Development, Nelson Mandela Metropolitan

University, South Africa. 2 FRF Mathematics Education Chair, Rhodes University, South Africa.

1 [email protected],

2 [email protected]

In order to understand how narratives shape a professional identity the career stories of

seven experienced mathematics teachers in the Eastern Cape Province were collected. The

issue of family support for their career aspirations came up as a recurrent theme in all of

their stories. This paper suggests that Sfard & Prusak‟s (2005a) operational definition of

identity; Wenger‟s (1998) notion of communities of practice; and Bourdieu‟s (1986)

notion of social capital can be combined into an interpretive framework to explore these

narratives of parental support. As a demonstration, two different stories will be explored –

one of parental support, the other of lack of parental support – both with positive

outcomes. In conclusion, it is suggested the above three theoretical constructs, viz.,

identity; communities of practice; and social capital, might be combined into a viable

interpretive framework for the narrative exploration of teachers‟ career stories.

Introduction/ Background

Family plays an important role in narrative expressions of life experiences. As Denzin

(1989) observed, “It is as if every author of an autobiography or biography must start with

family, finding there the zero point of origin for the life in question” (p. 18). This paper

explores the narratives of parental support of two experienced mathematics teachers from

different sociocultural backgrounds: one an isiXhosa-speaking, black, male, mathematics

teacher in his forties, and the other one an Afrikaans-speaking, white, female, mathematics

teacher, in her mid-sixties. One told a story of parental support (P1), the other told a story

of lack of parental support (P2).

This study is framed by socioculturalism (Cobb, 2006; Lerman, 2000; 2001; 2006; Kieran,

Forman, & Sfard, 2001/2002; Goos, 2008; Goos, Galbraith, & Renshaw, 2004) which

promotes the idea of human thinking as social in its origins, and therefore dependent on

historical, cultural, and situational factors (Kieran, Forman, & Sfard, 2001/2002). Through

a narrative inquiry the two participating teachers were afforded the opportunity to tell their

own professional life stories; to „give voice‟ to their personal experiences and the

meanings drawn from these (Foster, 2006).

One of the themes evident in the narratives of both teachers, parental support, is the focus

of this paper; specifically in view of the different sociocultural contexts in which these

narratives arise and how differently they shape the professional identities of the teachers.

As Burns and Pachler (2004) aptly pointed out, “valuable professional learning is based on

mailto:[email protected]


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experience, that learning is „situated‟ and that it relates to specific contexts of place and

time and the social interactions which occur within them” (p. 153).

As in all narrative studies, context is important, therefore the reader needs to be reminded

of the sociocultural context of the participants‟ stories of parental support. Because of the

ages of the two teachers, 47 years (P1) and 65 years (P2); their reflections on parental

support in their own lives are set within the sociocultural context of race and gender

discrimination of the „old‟ Apartheid South Africa before the onset of the New Democracy

in 1994. The narratives show that Black families often put their hope in a tertiary

education for their children; while in White families, most male children were conscripted

into the army when they should have been at university or entering the job market, while

female children often were faced with little prospects of a tertiary education and limited

career opportunities due to gender discrimination.

A sociocultural perspective focuses on interactions between individual, culture, and

society; it locates identity both within and external to the individual, and identity formation

in social and cultural practices (Grootenboer, Smith, & Lowrie, 2006). Hopefully, as the

ensuing discussion unfolds, it will become clear to the reader how this sociocultural

context, interwoven into the narratives of the two participating mathematics teachers,

continues to shape their professional identities.

Theoretical framework

A combination of three theoretical constructs were used to interpret the two teachers‟

stories of parental support, viz., Sfard & Prusak‟s (2005a) operational definition of identity

(extrapolated to include teacher professional identity); Wenger‟s (1998) notion of

communities of practice; and Bourdieu‟s (1986) notion of social capital. The aim of this

paper is to demonstrate that these three theoretical constructs might be combined into a

viable interpretive framework for the narrative exploration of teachers‟ career stories.

The sociocultural turn (Gee, 1999; Lerman, 2000; Sfard & Prusak, 2005a; Sfard, 2006b)

marked a shift in focus from personal identity to social identity (cf. Parekh, 2009); where a

“social identity is defined as representing the set of values internalized from groups to

which one belongs, as well as the affective valence assigned to membership in the group”

(Schwartz, Zamboanga, & Weisskirch, 2008, p. 631). Sfard and Prusak‟s (2005a)

operational definition of identity marked an attempt to purge the notion of all essentialist

connotations. They define identity as “a set of reifying, significant, endorsable stories

about a person” (p. 14). Reification, here, refers to the act of replacing sentences about

processes with propositions about states and objects (Sfard, 2008); in other words, reifying

the actions of a person and attributing the person with certain identifying qualities (Heyd-

Metzuyanim & Sfard, 2012). According to Sfard & Prusak (2005a; 2005b) reification can

also be linked to the use of verbs such as, be, have, or can rather than do, and with the

adverbs always, never, usually, and so forth, that stress repetitiveness of actions.

Furthermore, a story is significant if “any change is likely to affect the storyteller‟s

feelings about the identified person” (Sfard & Prusak, 2005a, p. 17). Heyd-Metzuyanim

and Sfard (2012) added that: “Operationally, this means that an alteration or removal of

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any of the main elements of the story would change how the author feels about the

protagonist” (p. 132). Finally, for a story to be endorsable the person who the story is

about must agree that it is a true story. Heyd-Metzuyanim and Sfard (2012) pointed out

two features of an endorsed, subjectifying utterance that will reveal whether it counts as

identifying or not: firstly, its power to reify; and secondly, its significance for the speaker.

As first-person accounts, by default, all the stories presented in this paper are already

endorsable; and therefore, in order to count as an identifying story, only needs to be

reifying and significant.

From a sociocultural point of view, learning can be linked to participation in the practices

of a community of practice (Lave & Wenger, 1991; Brown & Duguid, 1991; 2001;

Wenger, McDermott, & Snyder, 2002; Goos, Galbraith, & Renshaw, 2004; Wenger,

1998). Fernandez, Ritchie, & Barker (2008) maintained that engagement in social practice

“is the fundamental process by which we learn, and we become who we are” (p. 190). Our

identities are shaped by our engagement in the practices of the communities of practice of

which we are members. According to Wenger (1998), communities of practice draw their

coherence from joint enterprise, mutual engagement, and shared repertoire (pp. 73-85).

Joint enterprise refers to members communally negotiating understandings and responses,

taking ownership of responses to situations beyond their control, and being mutually

accountable for their actions; mutual engagement refers to participation in mutually

negotiated communal activities; and shared repertoire are communal resources for

negotiating the meanings that the community had developed over time; reflecting a history

of mutual engagement (pp. 82-84). Furthermore, Wenger (1998) also distinguished

between three modes of belonging to social learning systems which are useful in making

sense of identity formation: engagement; imagination; and alignment (pp. 173-181).

Engagement is all about active involvement in the mutual processes of negotiating

meaning; forming trajectories; and, unfolding of histories of practice; which, together,

“becomes a mode of belonging and the source of identity” (p. 174). Imagination is about

“creating images of the world and seeing connections through time and space by

extrapolating from our own experience” (p. 173). Alignment is all about “coordinating our

energy and activities in order to fit within broader structures and contribute to broader

enterprises” (p. 174). Imagination and alignment are necessary to make sense of the

shaping of an identity in contexts where engagement in social practice is not possible; for

example, when individuals see themselves “as participants in social processes and

configurations that extend beyond their direct engagement in their own practice” (p. 173).

Wenger (1998) claimed that each of the three modes of belonging, as discussed above,

create relations that expand identity through space and time and even beyond the confines

of the notion of communities of practice: “With engagement, imagination, and alignment

as distinct modes of belonging, communities of practice are not the only kind of

community to consider when exploring the formation of identities” (p. 181).

In addition to Sfard & Prusak‟s (2005a) operational definition of identity and Lave &

Wenger‟s (1991) notion of communities of practice, Bourdieu‟s (1986) notion of social

capital presents a useful complementary theoretical tool to make sense of the teachers‟

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narratives of family support. The notion of social capital refers to how some individuals

are privileged due to their membership in a social network and is defined as follows:

Social capital is the aggregate of the actual or potential resources which are linked to possession

of a durable network of more or less institutionalized relationships of mutual acquaintance

and recognition – or in other words, to membership within a group – which provides each of its

members with the backing of the collectivity-owned capital, a ‘credential’ which entitles them to

credit, in the various senses of the word. (Bourdieu, 1986, pp. 248-249)

From this perspective, the support of their families that the participants enjoy can be seen

as a form of social capital derived from the privilege of being the first generation of

tertiary students in their families. Furthermore, the amount of social capital individuals

possess hinges on the size of their network (Bourdieu, 1986); as well as their economic

and cultural standing (Gasman & Palmer, 2008).

In combination, these three theoretical constructs: identity (Sfard & Prusak, 2005a),

communities of practice (Wenger, 1998), and social capital (Bourdieu, 1986) make up the

interpretive framework for the subsequent narrative exploration of the two teachers‟ stories

of parental support.

Narrative methodology

First a few words on narrative methodology (Mishler, 1986b; 2006; Cortazzi, 1993b;

Riessman, 2006; Polkinghorne, 1988) – which is relatively new in Mathematics Education

research. It has a longer tradition in other academic disciplines, e.g., Anthropology,

Psychology, and Sociology, etc., where it has been used for many different purposes and

where many different approaches have emerged (Daiute & Lightfoot, 2004; Riley &

Hawe, 2005; Gergen & Gergen, 2006). In recent years, however, narrative has been used

increasingly as a tool to access identity (Clandinin & Connelly, 2000; Smith & Sparkes,

2006; Kaasila, 2007; Eaton & O Reilly, 2009; Lewis, 2011; Slay & Smith, 2010; De Fina,

2006; Søreide, 2006). As Lewis (2011) pointed out: “Story is central to human

understanding – it makes life livable (sic), because without a story, there is no identity, no

self, no other” (p. 505). As research into narrative has shown, individuals do not only

know themselves in the form of stories; their stories also frame and guide the ways in

which they understand and act on new information (Bruner, 1990; McAdams, 1993;

Drake, 2006). The mathematics teachers‟ stories were collected using narrative inquiry

(Clandinin & Connelly, 2000; Daiute & Lightfoot, 2004; Kramp, 2004; Connelly &

Clandinin, 1990), which seemed to be the most appropriate method to collect narrative

data.

Collecting the data

Stories of family support, which is the focus of this paper, emerged as a theme during a

horizontal analysis of the narratives of seven mathematics teachers in the original study on

which this paper is based. A horizontal or “cross-case” (Miles & Huberman, 1994)

analysis looks for common patterns or recurring themes across the narratives of the

different participants. The narratives of the participating mathematics teachers were

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collected in four semi-structured interviews, each about 40 minutes long, with each of the

teachers between 2012 and 2013. All the interviews were digitally recorded, transcribed,

and the transcripts returned to the participants for verification and clarification as analysis

and interpretation formed an integral part of the data-gathering process.

This paper, however, is based on the narratives of only two of the participants in the

original study: an isiXhosa-speaking, black, male, mathematics teacher in his forties (P1),

and an Afrikaans-speaking, white, female, mathematics teacher, in her mid-sixties (P2).

Their narratives were purposively selected (Polkinghorne, 2005) for further analysis in this

paper: firstly, because of their markedly different sociocultural backgrounds; and secondly,

because their stories of parental support are so different – one is a story of unconditional

support; the other is a story of no support at all.

Analysing the data

In this paper, vertical or “within-case” (Miles & Huberman, 1994) analysis was used to

explore the narratives of the two selected mathematics teachers. In line with the holistic-

content mode of analysis (Lieblich, Tuval-Mashiach, & Zilber, 1998), the narratives of the

two selected participating teachers were analysed separately. From a holistic perspective,

the life story of a person is taken as a whole, and sections of it are interpreted in the

context of other parts of the narrative. This perspective is preferred when the whole

person, that is, his or her development into being the current person, is the object of

interest.

Riessman (1997) cautioned that, with this method of analysis, there is always a risk of

losing some meanings due to an overemphasis on fabula (the content) at the expense sjuzet

(the form). This risk is especially acute when the analysis is based on a single interview

with each participant. In this study, however, follow-up interviews were conducted with

each participant in order to tease out more details in greater depth; and in the process,

potential loss of meaning through lack of attention to sjuzet (form) was minimized. The

importance of telling the researcher‟s story alongside those of the participants is well

documented in the literature (e.g., Marshall & Rossman, 1999; Casey, 1995-1996; Patton,

2002; Foster, 2006; Polkinghorne, 2007) and acknowledged here. Due to spatial

constraints the full life story of the researcher cannot be included here. It must be noted,

however, that narratives of family support resonate very strongly with the researcher‟s

own life history in which family support also played a significant and enabling role.

What the narrative data shows, is that where families have limited financial and academic

resources, they would support the participants with alternative forms of social capital; for

example, parents of first-generation tertiary students, while lacking academic resources,

“can instill (sic) in their children the expectation of attending college and can provide

encouragement and emotional support” (Dennis, Phinney, & Chuateco, 2005, p. 224).

Findings/Discussion

The recurrent theme of parental support while they were still studying is prevalent in the

narratives of almost all the mathematics teachers who participated in the original study (P2

was the only exception). Their narratives of parental support, in turn, showed several

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recurrent sub-themes, for example: participants were often first generation tertiary

students; parents and siblings often had limited formal education or none at all; the family

support was never of an academic nature; limited financial resources; participants often

depended on bursaries due to limited financial resources; and lastly, but most importantly,

families understood the importance of a good education and supported the participants‟

efforts to educate themselves in whichever way that they could afford in terms of available

resources.

The phenomenon of family support can be explained by drawing on Bourdieu‟s (1986)

notion of social capital as defined in the Theoretical Framework above. Social capital

refers to how some individuals are privileged due to their membership in a social network.

From this perspective, the family support that the participants enjoyed can be seen as a

form of social capital derived from the privilege of being the first generation of tertiary

students in their families. The narrative data showed that, with limited financial and

academic resources, families had to support the participants with alternative forms of

social capital. Just to reiterate, all the stories reported here are the participating teachers‟

first-person accounts of themselves, and therefore considered endorsable.

Participant 1 (P1) is an isiXhosa-speaking, black, male, mathematics teacher in his forties.

His father divorced his mother when he was still very young, leaving the family to survive

on their own and with very little material resources. In the second interview he was asked

about the role that his family played in supporting his studies. (I = interviewer; P1 =

Participant 1)

I: And your family? What role did your family play?

P1: My babes, my loving thando’s (loved ones, reference to wife and daughters) were just supporting. My mother was not learned. She was the one that made sure that at least I got the necessary needed things, but in terms of the family you’ll find that...

I: Can you talk a little more about the role that your mother played?

P1: Because my father divorced my mother at an early age I was very, very young and that in itself made me to have certain vows that I have with me. My mother was able to take us... we were two from my mother... was able to take us from... I don’t know whether you have seen a dog taking its puppy from one place to another? Though she was a domestic worker, she made sure that at least there is something for us for our education as well, and that to me... even in difficult times, she will make sure that we have at least money to go and study. I studied at different universities and before I could do that I had to go back and work myself before I can again go. So it was her influence that has made where I am or what I have become, meaning the support she has given me whilst growing up in her father’s house. That’s also one of the great things that I’ve learned as a father-figure in the house. I’ve learned it from grandfather – that is [name of grandfather]. They usually called him [nickname of grandfather] and even now I have a shop that is called [name of shop named after grandfather] because of thinking about him and honouring him of what he has done in my life. My family, the ones, they have been there for me, meaning they did not trouble me or even having problems with me when I say I must go and study. Study was number one because they know that I was the leader in the study groups. I love study groups because I love to share, I love to impart knowledge, as I’ve indicated. So it doesn’t matter… Whenever they call me my wife understands that maybe I’ve got to leave you, I’ve got to make sure that we go and prepare this for whether assignment or for the next day. So that understanding came from both my wife and my two little babes as well. Both of them have been supporting me and also I’ve seen them developing to be the leaders in their respective schools

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because they’ve seen how involved I was in my studies and also in the people that I was studying with. Yes, that’s how involved my family was (P1, 2nd interview).

In the opening line, and again at the end, P1 refers to the support of his wife and daughters (“loving

thando‟s”) which he enjoyed throughout his post graduate studies. The interest in this paper,

however, lies in the narratives of family support that precedes his wife and children. In terms of

Sfard & Prusak‟s (2005a; 2005b) operational definition of identity the above story contains

reifying elements, for example, about the mother, “so it was her influence that has made where I

am or what I have become, meaning the support she has given me whilst growing up in her father‟s

house”; the grandfather, “that‟s also one of the great things that I‟ve learned as a father-figure in

the house”; and the family, “I was the leader in the study groups. I love study groups because I

love to share, I love to impart knowledge, as I‟ve indicated” The story is significant, because any

changes in the events would probably have influenced P1 differently. For example, if his father did

not divorce his mother, he would not have grown up in his grandfather‟s home where he learnt how

to be a father-figure and care for his “loving thando‟s” who supported him throughout his

postgraduate studies. From this perspective, the divorce of his parents seems like a turning point in

his life-story. For example, he said: “Because my father divorced my mother at an early age I was

very, very young and that in itself made me to have certain vows that I have with me” He did not

elaborate further, so one is left to infer then from the rest of the narrative what these vows might

be. Perhaps it has something to do with making sure that his own children would be educated,

which he learnt from his mother; perhaps being there for his own family as “a father-figure in the

house”, which he learnt from his grandfather.

Although hardly a traditional community of practice, as originally conceived by Lave & Wenger

(1991), the family as a unit in this case still show elements of joint enterprise, mutual engagement,

and shared repertoire. However, as pointed out before, three additional modes of belonging, viz.,

engagement, imagination, and alignment, allows the exploration of identity beyond the traditional

notion of communities of practice (Wenger, 1998). For this family, however, their joint enterprise

was the education of P1, “study was number one because they know…” Perhaps they were

imagining (Wenger, 1998) a better future through the education of P1. The family members were

all mutually engaged in working together towards that goal. For example, the mother “made sure

that at least I got the necessary needed things… made sure that at least there is something for us for

our education as well”; the grandfather provided a home and stood in as father-figure; the family

“did not trouble me or even having problems with me when I say I must go and study” The shared

repertoire of the family unit is less obvious, but might relate to Bourdieu‟s (1986) notion of social

capital vested in a tertiary education for P1 as the first one of the family to gain a tertiary

education. Perhaps, the family viewed a tertiary education for P1 as a communal resource for the

family, some sort of shared repertoire which might be a useful resource for following generations.

If so, then perhaps they were right, because speaking about his daughters, P1‟s said, “I‟ve seen

them developing to be the leaders in their respective schools because they‟ve seen how involved I

was in my studies and also in the people that I was studying with” Significantly, he concluded his

story with: “Yes, that‟s how involved my family was” The love and support of his mother and

family became a form of social capital, a narrative resource continuously shaping his identity as a

father-figure and as a professional mathematics teacher, as he explained, “I love study groups

because I love to share, I love to impart knowledge, as I‟ve indicated”

In contrast, Participant 2 (P2), an Afrikaans-speaking, white, female, mathematics teacher, in her

mid-sixties responded quite differently to the matter of family support for her studies (she was the

exception alluded to earlier). However, given the sociocultural context in which she grew up, this

comes as no surprise. Her parents were more supportive of her elder brother than of the two sisters.

18

While her younger sister followed her father‟s wishes by accepting a job in the bank after school,

P2 rebelled; she wanted to prove that she could go to university and follow her dream of becoming

a graduate teacher. Using her responses in the first interview as a prompt, she was probed in the

second interview about the family support for the children in terms of their tertiary studies. (I =

interviewer; P2 = Participant 2)

I: My second question here, uh, uh... is also on the first page (referring to the transcript of the first interview) [P2: Yes...] where you speak about your brother who was two years older than, than what you were, and who studied Medicine, and then further down, you talk about your own degree in Mathematics and the Higher Diploma and, [P2: Uhm...] ... Were you only two children? [P2: No, three!] Okay!

P2: My other sister is 6 years younger than me and she never went to study. She, she went into the bank where my father was, and that’s where she … worked.

I: Because I wanted to ask you about your family support, uh... you know, looking at your brother doing Medicine and yourself doing a degree in Mathematics. I got a feeling, that, that you know, there was a lot of support for children to go and study. So, I wanted you to talk about, about that.

P2: Uh... Ja, uhm... there weren’t so many opportunities, because … and the types of jobs, you either became a doctor, or a nurse, or a teacher, or a reverend, or secretary ... where it wasn’t necessary ,!- to go to University for it, but the type of jobs (laughter) that were available weren’t so many *I: mm+ and, uh, ... the bursaries, uh ... also not! So ... in our case, my brother … and, and that’s also… the teachers, when I came to the High School, they always said, “Your brother did so, and so, and so ... so we are going to ... watch you ,!-, what you are going to do” And the same thing, you see, I had this competition in me. And then the 1st year, you know, I was just looking things through and I proved to the teachers that I will do better (emphasizing the words) than him (her brother), so I, I really …I was always motivated to, to learn. My parents never, ever had to tell me... that I had to do my homework. That’s just how I was. Now, uhm... my parents ... they had to pay for him (her brother), [I: Uhm] because there were no bursaries and, uh... then they told me, “If you want to go and study, you’ll have to find ... somewhere, the money. You’ll have to find it” And teaching, uhm... was an opportunity, I could go there and I could pay for myself. In any case, that wasn’t, that wasn’t because of just that, I feel that if I wanted to become a doctor, my parents would have made a plan [I: Uhm] but I wanted {!} to teach. Uh ... on the other hand, [name of brother] also, he wanted to become a woodwork teacher ... woodwork teacher, or a doctor, and in the end he, he became a doctor and, uhm... up till now he, he makes beautiful {!} things, he has all the appliances and whatever, woodwork things, and he, he makes ,!- furniture ... unbelievable *I: It’s a hobby now?+ It’s, it’s a hobby. It was a hobby but, now he’s also getting older, and older, so he doesn’t do it so often anymore. But in their house there’s lovely, lovely furniture. So, that was a hobby that’s also, I think, his first love (P2, 2nd interview).

Significantly, P2 starts by contextualising her story – specifically focussing on career

limitations – in the sociocultural context of the time, “there weren‟t so many opportunities,

because … and the types of jobs, you either became a doctor, or a nurse, or a teacher, or a

reverend, or secretary ... where it wasn‟t necessary {!} to go to University for it, but the

type of jobs (laughter) that were available weren‟t so many” Throughout the story she

narratively positions herself in opposition to the constraints imposed by sociocultural

context. She positions herself as an independent and ambitious woman. For example, she

emphasised that in order to get a job, a university education was not “necessary”; yet, she

wanted a university education. The narrative contains several reifying elements, for

example, the way in which she positions herself as very competitive “I had this

competition in me” and “I proved to the teachers that I will do better (emphasizing the

19

words) than him” Here the use of the verbs “had” and “proved” is in line with Sfard &

Prusak‟s (2005a; 2005b) claim that reification is linked to the use of verbs such as, be,

have, or can rather than do as previously discussed. Furthermore, the story is significant

because any changes in the events would probably have influenced P2 differently. For

example, if the high school teachers had not constantly compared her school results and

career ambitions with those of her brother, she probably would not have competed so

vigorously with him academically; probably would not have been so motivated to learn, to

prove to them “that I will do better (emphasizing the words) than him”; probably would

not have sought a bursary to put herself through university; and, probably would have

followed her father‟s wishes and worked at the bank like her sister. In contrast with the

narratives of family support of the rest of the participants, this narrative shows how lack of

family support actually motivated P2 to gain a professional qualification as a mathematics

teacher.

Despite being a classical example of a nuclear family, all the elements of coherence of

community of practice, as suggested by Wenger (1998), viz., joint enterprise, mutual

engagement, and shared repertoire, are missing; and so are the three modes of belonging,

viz., engagement, imagination, and alignment; as illustrated in the following examples.

P2‟s parents wanted her elder brother to become a doctor, and he did; but, he really wanted

to be a woodwork teacher, and now has to practice woodwork in spare time as a hobby

instead of as a career. In this case there is no joint enterprise, and the apparent alignment,

the elder brother becoming a doctor, is superficial. P2‟s parents wanted her to abandon her

plans for a tertiary education and get a job in the bank like her father, citing that there were

no money for her to study further, “then they told me, „If you want to go and study, you‟ll

have to find ... somewhere, the money. You‟ll have to find it‟” She refused, and found a

teaching bursary instead. Again, there is no joint enterprise, and no imagination, and no

alignment. Her sister, however, succumbed to their parents‟ wishes and took up a position

in the bank. It is not clear from the interview if this really is the career that her sister

wanted for herself. From a social capital perspective, however, it could be construed that

the parents were more willing to invest in the only son in the family, the elder brother of

P2, and that money spent on the education of their daughters, from their perspective, were

not regarded as highly as a form of social capital. Perhaps, because of the perceived lower

economic and cultural standing of females in that social network (Gasman & Palmer;

2008). In this case, the patriarchal nature of white Afrikaner culture may have dictated the

obvious disparities in the amount of social capital invested the first born son as opposed to

the two younger daughters in P1‟s family. This motivated P2 even more to prove that she

too could make it.

In both of the narratives above, despite apparent differential social capital investments,

there were positive outcomes; suggesting that the lack of financial and academic resources

did not prevent investment in alternative forms of social capital. Portes (2000) explained

that: “What families do, above all, is to facilitate children‟s access to education and

transmit a set of values and outlooks” (p. 2). In the above narratives this is evident in the

participants‟ increased motivation to study.

20

Concluding remarks

As mentioned before, narrative methodology is relatively new in Mathematics Education

and many different frameworks for the interpretation of narrative data will undoubtedly

still emerge in this discipline. What is suggested in this paper, is a novel way of combining

Sfard & Prusak‟s (2005a) operational definition of identity; Wenger‟s (1998) notion of

communities of practice; and Bourdieu‟s (1986) notion of social capital into an interpretive

framework fit for exploring mathematics teacher‟s narratives of parental support.

Acknowledgements

The work of the FRF Mathematics Education Chair, Rhodes University is supported by the

FirstRand Foundation Mathematics Education Chair Initiative of the FirstRand Foundation, Rand

Merchant Bank and the Department of Science and Technology.

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Note: Coding notations used in transcripts

24

... pause in speech

{!} emphasis placed on preceding word in midsentence

(clarification) actions/comments/notes of clarification

[I: Uhm...] midsentence interruption by the other party, e.g., to make a comment,

affirm something, or to encourage further elaboration, etc.

[name of teacher] where a name has been left out for the sake of anonymity

25

The perceptions of BEd (FET) mathematics students concerning their

training

Owen Hugh Glover

Faculty of Education, Nelson Mandela Metropolitan University, South Africa

[email protected]

The effective teaching of mathematics at all levels of national educational systems is of

great concern, given the increasing importance of mathematics in 21st century life. This

concern is heightened in countries which perform poorly in international comparisons.

South Africa is one such country. As a consequence there is an increasing attention on

teacher education. This study seeks to contribute to an understanding of pre-service

mathematics teacher training in South Africa through analyzing the perceptions of two

cohorts of students at NMMU on the BEd (FET) programme. These perceptions relate to

their knowledge levels and experience of the subject knowledge and pedagogic content

knowledge components of the programme. The study finds that students overall believe

the programme can be improved. They are motivated to become good teachers but feel

only moderately prepared to teach mathematics in the secondary school and value their

Pedagogic Content Knowledge (PCK) experiences more than their Subject Knowledge

(SK) experiences. They feel the pure mathematics content of the curriculum needs

revisiting, require more method time and seek lecturers and role models who exemplify

good teaching.

Introduction

In our rapidly changing and globalizing world there is the ongoing challenge of preparing

young people for meaningful global citizenship. This includes preparing them to meet the

growing mathematical demands and challenges of modern citizenship (Edge, 2001; Steen,

1999). Ideally this includes acquiring the core mathematical knowledge and skills

necessary to live full and meaningful lives as local and global citizens. Furthermore if

nations are to ensure global competitiveness as many of their citizens as possible should

have the capacity to solve problems, develop and use technology and reason, amidst

complexity, with insight and logic (Guile, 2006; Victor & Boynton, 1998).

These requirements underline the importance for a nation developing mathematical

competencies amongst its citizens. Faced with this need, most developed and developing

countries are grappling with the challenge of declining mathematical performance at

school In particular poor, less developed countries seem to be most at risk (Coben,

Confrey, diSessa, Lehrer & Schauble, 2003; Coben, 2006; FitzSimons, Coben &

O‟Donaghue, 2003). Many of their young people are failing to realize their mathematical

potential, which with the widespread flow of mathematical ideas into many dimensions of

business and social life places both nations and individuals at risk. The risk includes, for

affected nations, the possibility of stagnating economies which may retard

democratization, increase social instability and increase their dependence on nations with

better educational standards (Coben, 2006).


26

For the individual his/her career and employment opportunities could become more

limited, possibly excluding him/her from meaningful work, economic opportunities and

personal growth (Pajares. & Miller, 1994; Stajkovic & Luthans, 1998).

Given this context the central role of the future teachers‟ corps of any nation in achieving

quality improvements seems obvious. As a result increasing research is being done on

teacher education‟s effectiveness in order to better direct national efforts to combat the

scenario of declining learner performance in mathematics. Furthermore the worldwide

shortage of mathematics teachers means that teacher training institutions, faced with

increasing demands, will need to ensure the maintenance of high quality programmes, if

they are to play their role in raising levels of learner performance at school.

The situation in South Africa is possibly more critical than most countries of the non-

developed world. Firstly, it is estimated that we are currently graduating 9 000 teachers per

year (Bertran, 2006) whilst the national requirement might be closer to 30 000 (Crouch,

2001). Given this high demand the realization of the desired quality levels at teacher

training universities will be difficult. This problem is likely to be exacerbated by the lack

of academically well prepared students entering teaching, since the teaching professions

seems to have a weak attraction for strong academic students (Adler, 2002; Duthilleul,

2005). Clearly we are faced with a growing shortage of teachers in general and

mathematics teachers in particular. Secondly, despite being one of the richest countries in

Africa and one that makes the highest investment in education, South Africa performs,

according to recent studies, poorly relative to international and African countries (Van Den

Berg & Louw, 2006).

Background to the problem

Within the current South African context universities are responsible for the quality of

their own programmes, subject to regular audit and review (Kistan,1999; Strydom &

Strydom, 2004). It is thus important that they ensure their mathematics teacher training is

effective and efficient. At the Nelson Mandela Metropolitan University (NMMU) one of

the newer programmes, the integrated Bachelor of Education, Further Education and

Training, (BEd (FET)) was introduced. An increasing number of pre-service mathematics

teachers, many of whom receive bursaries, are following this four year BEd (FET) route.

A core requirement of any such integrated degree is that graduating students should

possess a minimum competency level in the teaching of secondary school mathematics.

Consequently part of the programme requirement is that students pass six modules of

undergraduate mathematics. This requirement has proved to be most demanding, with high

first attempt failure rates. Students can often take an extra year to complete the

qualification and a number either drop out or convert to another phase of teacher training.

Both faculty and students‟ concerns about the design and implementation of the program

have prompted various discussions aimed at remedying the situation.

27

It is within this context that this paper seeks to contribute to the need for solutions through

analysis of the perceptions of a cohort of BEd (FET) students, prospective secondary

school mathematics teachers, about the quality of their training. This research aims to

support the improvement in the training of such prospective teachers, a step which is vital

for the NMMU and which may contribute to other South African and international teacher

education reform initiatives.

Over the past forty years the knowledge base in mathematics education research has grown

significantly (Kilpatrick, 1992). Consequently the training of teachers has advanced well

beyond ensuring that teachers simply master subject content (Cooney, 1994). Today the

majority of nations would require their prospective secondary school mathematics teachers

to have undergone training in at least four broad areas. The first three, namely the

acquisition of subject (content) knowledge (SK), pedagogic content knowledge (PCK) and

education studies or general pedagogical knowledge are frequently discussed in literature

(Schmidt, 2011) and form key components of the broader notion of teacher competency.

The fourth would be sufficient practical classroom experience to ensure that the minimum

skills are in place.

The notions of SK and PCK are usually credited to Shulman (1987). SK means the

knowledge of mathematics and is the domain of the professional mathematicians. Whilst

the selection of content might be nuanced, engineers, doctors, scientists, teachers and other

similar professions would have mastered mathematics content in this area. PCK means

pedagogical knowledge related to mathematics and includes instructional planning

knowledge, the knowledge of student learning and curricular knowledge (Schmidt 2007).

In order to access this knowledge it is often necessary to „decompress‟ or „unpack‟ the

content knowledge, whilst mathematicians, in advancing mathematical knowledge move

towards compression (increasing abstraction) (Ball, 2003). PCK is the exclusive domain of

the teacher who would build his/her professional knowledge over time in the field.

Whilst the four areas are all important for the preparation of teachers there is a growing

body of evidence (Chisolm & Baloyi, 2009; Schmidt, 2011; Taylor & Vinjevold 1999) that

suggests that the quality, depth and robustness of a teacher‟s mathematical knowledge,

both SK and PCK play a critical and vital role in determining the effectiveness of a

mathematics teacher. Consequently this paper focuses on the SK and PCK elements of the

BEd (FET) program at NMMU.

The focus of this research paper

Within the context described the key question for this paper is: „To what extent do BEd

(FET) students at NMMU (intending to teach secondary school mathematics) perceive

their mathematical experience of the BEd (FET) program to have been suitable and

relevant to developing their own mathematical knowledge levels and teaching

effectiveness?

28

This central question requires an exploration of the views of students about their own

mathematical knowledge levels (SK and PCK) attained as a result of the program and their

perceived readiness to teach mathematics at the secondary school level.

The first three years of the four year BEd (FET) progam focuses on students being

equipped with the necessary knowledge to teach effectively with some, but limited, school

based experience. The fourth year is an intensive practical year. The knowledge categories

upon which this study focuses are SK and PCK, which would be unique to the training of

specialist FET mathematics teachers. The focus of this study does not incorporate their

training in educational theory.

For the purposes of operationalizing SK and PCK in this study SK is defined as the

knowledge acquired through completing the six mathematics modules offered by the

Department of Mathematics while PCK is defined as the knowledge developed through the

two 3rd

year method courses and, where appropriate, the 4th

year practical teaching

component. Both are offered by the School of Initial Teacher Education in the Faculty of

Education.

Research Design

Sample

The study focuses on the perceptions of the 3rd

and 4th

year BEd (FET) students registered

in 2010. These students have largely finished their theoretical training, which is the focus

of this research. There were nineteen 3rd

year student and ten 4th

year students. In the final

analysis fifteen 3rd

year and three 4th

year students participated in the research.

Broad approach

The study used a mixed methods approach. Both quantitative and qualitative

methodologies were employed, aiming at a balanced analysis of the central question. It

was decided to use a questionnaire containing two sections to be completed by

participating students. The first section (Section A) of the questionnaire gathered

quantitative data using 26 items, each requiring a response on a five point Likert scale. A

second section (Section B) was designed to allow for open-ended written responses related

to their reasons for choosing teaching and their views on the strongest and weakest aspects

of the SK and PCK program elements. Two focus group interviews were conducted, one

for three third year students and one for two fourth year students. These students were

purposively selected as they had previously been identified as being constructive and

thoughtful in their views on the program.

Questionnaire design

The questionnaire items in Section A of the questionnaire were linked to 13 categories,

two questions per category. The categories emerged by focusing the research question on

the key elements of the BEd (FET) program. The researcher used his knowledge and

29

experience of the program to initially construct categories and supporting questions that

would best operationalize the research question. Through a process of synthesis and

analysis 13 categories and twenty-six questions emerged.

The 13 categories fell into one of four main dimensions, namely the SK, the PCK, the

Readiness and the Overall dimension. The first two dimensions had four sub-dimensions,

namely belief, training, knowledge level and modeling. These sub-dimensions were

defined as follows:

Belief: A persons deeply held convictions or view on a matter;

Training: The training received through the completion of certain modules;

Knowledge level: The students‟ perception of his/her knowledge level of a particular

category;

Modeling: The demonstration of professional competencies by a university

approved lecturer or teacher;

The two categories with no sub-dimensions were defined as follows:

Readiness: The student‟s perception of her readiness to teach mathematics to

high school learners;

Overall: The global measure of the extent to which students perceive the

program needs to change.

The SK and PCK dimensions when analyzed against with the four sub-dimensions

produced 11 categories. One would expect 8 in a neat arrangement, but for SK knowledge

levels were further subdivided into a general high school level and a matric level. In the

case of PCK it was recognized that both the method module and the practical teaching

experience contributed to PCK knowledge levels. Finally the PCK modeling sub-

dimension was further sub-divided into the role of the method lecturer and the assigned

mentor teacher at school. The resulting 11categories and their category numbers are as

follows:

1. SK Belief: The conviction that a good mathematics teacher possesses the key

subject knowledge areas covered in the high school curriculum,

2. SK training: The subject knowledge training received during the course, with the

Mathematics course delivered by the mathematics departments seen as the primary

provider,

3. SK level: The mathematical knowledge levels as measured by the requirements of

the high school curriculum,

30

4. SK level (Matric): The mathematical knowledge levels as measured by the

demands of the Grade 11/12 mathematics curriculum,

5. Modeling by Mathematicians: The teaching received from professional

mathematicians provided an example of good teaching approaches,

6. PCK Belief: The belief that a good mathematics teacher possesses a range of

effective ways in helping learners master the high school mathematics curriculum,

7. PCK through training: The training received in transforming the subject knowledge

into a form that is relevant and appropriate for the school curriculum,

8. PCK through practical teaching: The learning opportunities offered through

participation in practical teaching,

9. PCK (Matric): The knowledge possessed to effectively support and guide matric

students in preparing for their matric mathematics examination,

10. Modeling by Mathematics Educators: The teaching received from mathematics‟

educators provided an example of good teaching approaches,

11. Modeling by classroom teachers: The teaching received from or witnessed in the

classroom of a mentor teacher provided an example of good teaching approaches.

Section B of the questionnaire required students to complete open-ended questions. The

first two dealt with the underlying reasons why they had chosen to specialize in

mathematics teaching. The third question dealt with their perception of their readiness to

teach.

The final six questions were divided into 3 categories, namely student‟s perceptions of the

most valuable and least valuable aspects of their Mathematics undergraduate courses and

their method courses as well as their practical teaching experience.

The responses to the 26 questions were statistically analyzed by individual question. The

average (arithmetic mean) and a five-number (minimum, quartile 1, median, quartile 3,

maximum) summary were obtained for each question. The questions linked to category 13,

the overall category, were designed to be reported separately. For the other twelve

categories each pair of questions was designed to measure the same construct. The

reliability for each pair of questions was then determined by direct comparison of the

statistics for each pair. 11 of the 12 categories matched, allowing for a score to be obtained

for each of these categories. The mean and five number summaries were then recalculated

through combining the paired data

The only paired questions whose statistics were out of line was category 2, „subject

knowledge training.‟ In seeking out possible reasons for the discrepancy it was concluded

that Question 15 was poorly formulated. Whilst its paired question, Question 3, referred to

31

the „mathematics courses I did in the mathematic department‟, Question 15 stated „the

subject knowledge training received‟. After triangulation of this data with the open-ended

questions posed in both the questionnaire and interviews it was decided that the phrase

„subject knowledge‟ had been interpreted wider than its definition and so only the score for

Question 3 was used to determine the score for Category 2.

Each of the responses to the open-ended questions was recorded as a line in a spreadsheet

and one worksheet was assigned to each question. Each response was given an initial

coding. Through repeated sorting, re-reading of certain responses and then further coding

or re-coding response themes emerged. These themes, reported in the results, give insight

into the ranking given to the various categories in the quantitative analysis.

Results

Reasons for choice to teach mathematics

More than 50% of the respondents had chosen teaching as a career since they wanted to

make a difference in society and in the lives of individual students. Four of the respondents

specifically mentioned they chose teaching because of the role of a teacher or teachers in

either explicitly encouraging them to teach or through being a significant role model in

their lives.

Their decision to specialize in mathematics seemed motivated by either the desire to make

a change in society, their own enjoyment in doing mathematics, the positive affirmation

received as high school mathematics‟ students or because they saw mathematics as a

highly relevant and useful subject.

A third of the respondents wished to change negative perceptions towards mathematics as

a „hard and difficult subject‟, „boring‟ or one only for „brilliant people‟ and two

respondents, both women, raised the need to specifically improve girls‟ performance in

mathematics. More than half chose to teach the subject because they enjoyed it.

They liked the challenge and satisfaction in doing mathematics, two found it fascinating

and 4 of respondents wanted to share their love of mathematics with their future students.

Just more than a third (six of the 15) chose to specialize in teaching mathematics since

they had performed well in mathematics at school, whilst five mentioned the value and

importance of mathematics in societal development.

Some of the group had specifically mentioned that the demand for mathematics teachers

and the opportunity to receive a bursary and obtain a useful qualification was an important

factor in their decision. Furthermore teaching was attractive to them as they saw the act of

teaching as enjoyable and one where they would continue to learn for the rest of their

lives. Learning mathematics, a subject in which a number of them had excelled was also a

benefit.

32

The possession of certain personal attributes or personality was also given as a reason for

teaching. Statements like „I like to work with people‟, „I am patient‟ or „I am friendly‟

were some of the reasons giving for teaching.

Perceptions of the overall knowledge training received in BEd FET

Table 1 and its accompanying bar chart, Figure 1, show the results for each category.

Overall (Category 13) there was strong agreement (4.64) that the programme needed

modification and overall students were only marginally positive about being ready to teach

secondary school mathematics (3.28). This suggests that the program is falling short of its

goals.

Table 1: Category scores for various aspects of the BEd (FET) program

Category scores

Group Number Category Score Rank

SK

1 SK belief 4.23 2

2 SK training 2.97 10

3 SK level 3.28 7

4 SK level (Matric) 2.97 10

5 Model-Mathematician 2.33 12

PCK

6 PCK belief 4.67 1

7 PCK training 3.63 5

8 PCK thru prac teaching 3.70 4

9 PCK level (matric) 3.63 5

10 Model – Method lecturer 3.83 3

11 Model – Teacher 3.00 9

Readiness 12 Readiness 3.28 7

Overall 13 Need to change 4.64

Through direct comparison of the related SK and PCK categories, student perceptions

rated the PCK dimension higher than the SK dimension in all instances, namely belief,

training, level and modeling.

33

Figure 1: Average ranking by category of the BEd (FET) program.

Students clearly believe that possessing subject knowledge and pedagogic content

knowledge is very necessary to become an effective teacher. This combined with the

largely positive reasons given for deciding to teach mathematics suggests that student‟s

would generally be reasonably committed to this program.

Table 2: Comparing the SK and PCK dimensions

SK PCK

Belief 4.23 4.67

Training 2.97 Method 3.63

Prac Teaching 3.70

Level General 3.28 3.63

Matric 2.97

Modeling Mathematician 2.33 Method Lecturer 3.83

Teacher 3.00

The ratings for the training received suggest that the PCK training, which includes both

method (3.63) and practical teaching (3.70) was more effective and relevant than the

mathematics training (2.97). When asked about the aspects of the undergraduate

mathematics courses (Mathematics I and II) offered by the Mathematics department that

were of most value it was clear that students only valued topics that related directly to the

matric curriculum, such as differentiation. One fifth of the group was positive about

having done mathematics at a higher level. However this is offset by the strong feelings

that many of these courses were too difficult, that the failure rate was too high and that the

only way one could pass such courses was to resort to rote learning.

The aspect of the mathematics studies that was of least value was the inclusion of

Mathematics II modules and even aspects of Mathematics I, such as learning integration.

0.002.004.006.00

Sco

re

Category

Average Ranking by category

34

Two thirds of those questioned felt that the modules in Mathematics II were irrelevant.

Other content areas mentioned as not relevant, although only by a minority, were Linear

Algebra (which spans 1st and 2

nd year), vector calculus (three students) and integration (2

students). It was also felt that these courses were intended for science and engineering

students. A number of students were not satisfied by having to be incorporated in large

classes with Engineering, BSc Mathematics and other students majoring in mathematics.

The open ended questions showed that the majority feeling was that the requirement to do

six modules in pure mathematics is excessive and not entirely relevant or necessary to their

future careers. Some reported that they had lost motivation to do these courses, a possible

contributing factor to the high failure rate in these modules. In both in-depth interviews the

students expressed great concern about this. One male student stated that there must be

something wrong if „more than 60 students start a programme and only four reach their 4th

year on time and more than half leave the programme before completion.‟

More than half the students mentioned that the greatest value obtained from the method

courses was learning different ways to teach, present or manage the learning of key topics

or areas of the curriculum. Four of the students mentioned that more time should be

assigned to the method courses which should start much earlier than the 3rd

year.

Students were neutral about having the required level of subject knowledge for teaching

high school (3.28) and clearly felt they lacked or were below par with respect to their

matric subject knowledge (2.97). They rated their PCK level slightly higher at 3.63.

Consistent with the ratings given in the training categories the method lecturers modeled

teaching better than the professional mathematicians and five students specifically

complemented one of the two method lecturers who they felt modeled good teaching. One

student said he modeled „technique and passion‟ and another stated „I enjoyed it, the

lecturer made it interesting and got us to work together‟. The lecturers from the

mathematics department were overall negatively rated on modeling good teaching. When

probed in the focus interviews it seemed that most lecturers engaged minimally with

students in lecturing situations. It must be noted that two mathematics lecturers were

specifically mentioned as counter-examples. They were seen as more caring and

approachable, with one being mentioned for using good analogies and making complex

aspects more understandable.

Discussion

Over the past 30 years in South Africa it seems that the primary way universities have

trained secondary school mathematics teachers has been through a 1 year post degree

certificate where most students had least two years of credit in mathematics.

With the closing down of all South African training colleges in mid-1990s it is at present

the role of universities to train all teachers. With the increasing demand for the supply of

qualified teachers in this category and the need to address imbalances of the past

universities are experiencing pressure on course quality and outcomes in two ways. Firstly

35

many of the students entering the BEd (FET) are ill prepared for the demands of

undergraduate mathematics courses with high dropout rates and failures. This had a

negative effect on both student and staff morale. Secondly society, especially government

and the private school sector, it seems, will expect an increase in the supply of

mathematics‟ teachers over the next 10 years.

This study shows a gap between students‟ unsatisfactory experience in their preparation to

become competent high school mathematics teachers and the goals of the programme, with

a large proportion of the students feeling they were required to do too much pure

mathematics. This is contrary to the view that future teachers need to be trained

mathematically. Schmidt (2010, online) states “Our USA teachers are getting weakly

trained mathematically and not prepared to teach the demanding curricula needed for our

students to compete internationally”. Considering that Schmidt is referring to elementary

and middle school teachers the argument to limit or reduce coverage or complexity of the

mathematics done by pre-service mathematics educators seems unwise. A similar

argument is advanced by Posamentier (2003, p.37) who believes that a teacher‟s training

must “include a strong component in mathematics – one that stresses its beauty and

motivates its learners.”

However Ferrini-Mundy (2004) argues that the traditional approach of teachers doing

whatever mathematics is done by students intending to major in mathematics and to

supplement this with a methodology component (or mathematics education) fails to

provide students with the substantial knowledge necessary for effective secondary school

teaching.

Whilst it seems reasonable to accept that no beginning teacher can ever be fully prepared

for the demands of the modern day mathematics classroom, it seems equally true that it is

not desirable that students receive accreditation with knowledge gaps directly related to

content sections of the FET curriculum, and the associated PCK.

If the findings of this research are to be accepted then serious attention will need to be

given to adapting the curriculum to close the knowledge gaps the students perceive (Hodge

& Staples, 2005). It seems that certain matric topics should be handled explicitly and in

more depth – for instance Financial Mathematics, Statistics and Trigonometry. More time

needs to be made available for the acquiring of PCK – either through holding SK and PCK

in tension in an integrated more in-depth approach or through the study of the knowledge,

skills and behavior of exemplary teachers (Ball, 2003).

It also seems important that these students are educated by lecturers, tutors and mentors

who themselves model excellent teaching. Here it might be wise for institutions to define

or describe excellence within the content of mathematics teacher training and seek the

buy-in and commitment of lecturers who are competent and motivated to work in teacher

development. It seems clear that students are strongly opposed to lecturers who spend

most of their time simply writing up their lecture or presenting them on powerpoint but

engage minimally with the students and fail to provide the types of explanations and

36

support that assist students in their own mathematical growth. It has been suggested that

teachers often perform poorly in the classroom because through school and college they

have never seen any kind of mathematics other than the approach of their teachers and

professors as “stilted, constricted and rigid.” (Wu, 1999, p. 2).

To summarize, if the student perceptions are to be taken seriously then an immediate

action plan could include:

1. Ensuring that there is an increased coverage of the topics covered in the FET

syllabus, which include statistics, financial mathematics and other important topics

2. Revisiting the modules offered by the mathematics department. At the planning

level there needs to be a reduction of content with a stronger focus on topics that

are strongly linked to the FET curriculum or which take students to deeper

understanding.

3. Purposively selecting those who work with the education students and giving

attention to the class formations in which they find themselves. Assuming the large

majority of such students will be teaching high school mathematics for many years

they need to be exposed to outstanding teachers who offer a rich and relevant

curriculum in ways that model good classroom practice.

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39

Primary learner descriptions of a successful maths learner

Mellony Graven1

&EinatHeyd-Metzuyanim 2

1

SA Numeracy Chair Project, Rhodes University, South Africa 2Technion – Israel Institute of Technology, Israel

[email protected],

[email protected]

In this paper we present the findings of Grade 3 and 4 learners across twelve schools in the

Eastern Cape area in relation to how they described a good, successful mathematics

learner. An instrument containing several questions and „complete-the-sentence‟ items was

designed in order to elicit data on mathematics learning dispositions. Dispositions for our

purposes are broadly taken to be a tendency to perceive and respond to mathematical

situations in a certain way. The disposition instrument was orally administered to 1208

learners in 38 grade 3 and grade 4 classes across different types of schools including fee

paying and non-fee paying, historically White, Coloured and township schools. Questions

(or complete the sentence items) were explained to learners with translation into Afrikaans

and isiXhosa where required and learners provided written responses on the instrument.

Items investigated aspects of learner mathematical dispositions. This paper focuses on the

findings in relation to one question on the instrument – describing a good strong

mathematics learner. All responses were translated and coded and checked for inter-rater

reliability. The paper interrogates the findings in relation to learner descriptions of an

effective mathematics learner. The low percentage of responses indicating active

participation, sense making or steady effort is argued to be a possible cause for concern.

Introduction

South Africa‟s mathematics education has been described by many to be „in crisis‟ (e.g.

Fleisch, 2008). Several years of mathematics intervention projects and curriculum change

aimed at improving South Africa‟s poor performance in regional and international

comparative studies have done little to shift learner levels of proficiency. Along with our

recently implemented curriculum in the form of the Curriculum and Assessment Policy

Statements (CAPS) (Department of Basic Education, 2011) Annual National Assessments

(ANAs) in Grades 1-6 & 9 have been introduced (Department of Basic Education, 2012).

While their introduction indicates increased monitoring of the „crisis‟ in mathematics

education it does little to support the improvement of learners‟ performance. The results

show alarmingly poor mathematics skills across learners in the primary grades with

average performance steadily declining by about 10% each year from 68% in Grade 1 to

27% in Grade 6 and then to 13% for Grade 9s (DBE, 2012).

A wide range of research (Fleisch, 2008; Spaull, 2011; Carnoy et al, 2011) highlights

several factors as impacting on learner performance, including: social disadvantage;

teachers‟ subject knowledge; teaching time; teacher absenteeism; lack of resources; poorly

managed schools; and poverty effects, including malnutrition and HIV/AIDS. What is not

explained in this research is why South Africa performs even worse in mathematics than

our neighbours with much less wealth and why we perform lowest of all countries



40

participating in TIMSS which includes several developing countries (Reddy, 2006).

Fleisch (2008) proposes that perhaps the dependency and profound disempowerment

experienced by South Africa‟s poor needs consideration. In this respect as mathematics

education researchers we need to begin to research the role of learning dispositions

promoted and or developed within our mathematics classrooms in relation to this „crisis‟.

This paper emerges from a broader study aimed at researching learner mathematical

dispositions and the evolvement of these dispositions within the South African Numeracy

Chair Project work in the broader Grahamstown area. In this work the first author runs

various development projects with twelve schools which include a teacher development

program called the Numeracy Inquiry Community of Leader Educators (NICLE), after-

school maths clubs (see Stott & Graven, 2013), and a range of community based events

(see Graven & Stott, 2011) including family based activities. Across this work the aim is

to support learners in developing mathematical proficiency.We consider this in terms of

Kilpatrick, Swafford, & Findell's (2001)conceptualisation of five interrelated strandsof

mathematical proficiency namely: conceptual understanding, procedural fluency, strategic

competence, adaptive reasoning and productive disposition. For Kilpatrick, Swafford and

Findell (2001, p.116) all strands are equally important as mathematical proficiency „cannot

be achieved by focusing on only one or two of these strands‟. Across our projects we have

used several instruments to annually monitor learner evolving levels of proficiency, in

terms of the first four strands (e.g. instruments adapted from Askew, Rhodes, Brown,

Wiliam, & Johnson, 1997; Wright, Ellemor-Collins, & Tabor, 2012; Wright, Martland, &

Stafford, 2006). However,we realised after the first year that we did not have an

instrument that specifically engaged with the fifth strand of proficiency, i.e. productive

disposition.Productive disposition, as Kilpatrick, Swafford &Findell, 2001, p.131) define

it:

refers to the tendency to see sense in mathematics, to perceive it as both useful

andworthwhile, to believe that steady effort in learning mathematics pays off, and to

see oneself as an effective learner and doer of mathematics. If students are to develop

conceptual understanding, procedural fluency, strategic competence, and adaptive

reasoning abilities, they must believe that mathematics is understandable, not arbitrary;

that with diligent effort, it can be learned and used; and that they are capable of figuring

it out.

In earlier work we have elaborated on the evolution of an instrument for the purposes of

researching learner dispositions (Graven, Hewana& Stott, 2013) and motivated for the

importance of researching this key aspect of mathematical proficiency (Graven, 2012). In

this paper we present the findings of Grade 3 and 4 learners across twelve schools in the

Eastern Cape area in relation to how they described a good successful mathematics learner.

An instrument containing several questions and complete-the-sentence items was designed

in order to elicit data on mathematics learning dispositions. Dispositions for our purposes

are broadly taken to be a tendency to perceive and respond to mathematical situations in a

certain way. The instrument was designed for use as both a questionnaire and interview

and is included in Figure 1 below:

41

Figure 1: An instrument for accessing mathematical learning dispositions (Taken from:

Graven, 2012, p.55)

In earlier work Graven(2012) argued that accessing learner mathematical dispositions can

be difficult, especially with young learners who struggle to articulate their stories. Graven,

Hewana & Stott (2013)describe the evolution of the above instrument and explain how

following the piloting of an earlier instrument it was noted that some learners answered

questions about their relationship to mathematics largely in terms of what they perceived

to be a correct or positive expected response.

The complete-the-sentence items about Mpho and Sam were thus introduced to provide

learners with the opportunity to describe how they envisioned a successful or unsuccessful

learner of mathematics without having to consider their own dispositions or what they

thought they should write about themselves so as to cast themselves in a positive light.

Perspectives on dispositions

Aside from Kilpatrick, Swafford and Findell’s(2001) inclusion of a productive disposition as a key

aspect of mathematical proficiency, other work that foregrounds the importance of learning

dispositions more generally than within mathematics education includes for example that of Carr

and Claxton (e.g. Carr & Claxton, 2002; Claxton & Carr, 2004). More recent work within

mathematics education that highlights the importance of researching learning dispositions is that

of Gresalfi & Cobb(2006) and Gresalfi (2009). While it is beyond the scope of this paper to

conduct a thorough literature review of the emerging field of literature on learning dispositions

42

we briefly note the importance of the above works and their relationship to the earlier definition

of a ‘productive disposition’.

Carr and Claxton (2002), drawing on Wenger’s (1998) perspective of learning and the centrality of

identity as ‘ways of being’ in the world define learning dispositions as a tendency to respond or

learn in a certain way. In this respect they emphasise that:

not all dispositions are equally relevant to learning power. The inclination to bebossy, for

example, is probably less crucial to learning in general than the tendency to persist with

learning in the face of confusion or frustration (p. 12).

They identify three key learning dispositions, namely: resilience, playfulness and reciprocity in

their work that draws on research with early learners. The aspect of resilience connects well with

Kilpatrick, Swafford and Findell’s indicator of seeing steady effort as paying off. Carr and Claxton

(2002:14) explain resilience as:

the inclination to take on (at least some) learning challenges where the outcomeis

uncertain, to persist with learning despite temporary confusion or frustrationand to

recover from setbacks or failures and rededicate oneself to the learning task.

Similarly Gresalfiand Cobb (2006) and Gresalfi (2009) note that learning involves a process of

developing dispositions.Thus Gresalfi (2009: 329) drawing on her earlier work with Cobb writes:

Thus, learning is a process of developing dispositions; that is, ways of beingin the world

that involve ideas about, perspectives on, and engagement with information that can

be seen both in moments of interaction and in more enduring patterns over time

(Gresalfi&Cobb, 2006).

These perspectives on dispositions link with Kilpatrick et al.’s (2001) notion of habitual behaviours

or dispositions that should be attended to, both by practitioners and researchers as a component

of learning. Our research questions thus ask: What is the nature of Grade 3 and 4 learners’

mathematical dispositions in the schools that we work with and in the after school mathematics

clubs that we run? How might these dispositions evolve over time (if at all)? How might these be

accessed across a large number of learners? While we gather in depth case study research on

learner evolving dispositions of learners in our club through a combination of methods including

observation and interviews the focus of this paper is on data emerging from our gathering

dispositional data from a large number of learners in written questionnaire form.

Methodology

The methodology of the broader research combines qualitative and quantitative research

methods. In our work with learners in clubs we gather data via interviews and transcribed

club sessions in order to analyse the nature of learner dispositions and the possible

evolution of these dispositions within our clubs. The data that forms the focus of this paper

is quantitative in nature having been derived from use of the above instrument as an orally

administered questionnaire given to Grade 3 and 4 classes in twelve schools. The

disposition instrument was orally administered to 1208 grade 3 and grade 4 learners in 38

43

classes across twelve schools including fee paying and non-fee paying, historically White,

Coloured and township schools. Questions (or complete-the-sentence items) were

explained to learners with translation into Afrikaans and isiXhosa where required and

learners provided written responses on the instrument. Learners were encouraged to write

in whichever language they were most comfortable with. Permission for research was

obtained from the department of education, parents, teachers and principals.

All 1208 learner responses were transcribed (without changes to spelling or grammar),

translated where necessary and coded. We developed a coding system for each item on the

questionnaire that was informed by examining a portion of responses. Numerous revisions

of our coding system took place before the final coding system was agreed upon. This

coding system was checked for consistency on 40 learner responses across the authors.

Following this the first author trained a „coder‟ to code all responses. While the vast

majority of learners only provided single code responses 76 learners provided responses

that required two codes. For example „Sam is a good girl. She does her homework‟

received two codes, one for each part of the response. Thus the total number of codes

derived from the 1208 learner responses was 1284. No learner provided a response

requiring more than two codes for this item. 290 learner responses (24% of all learner

responses), across a range of classes and languages, were coded by the first author in order

to assess the level of inter-rater reliability with the trained coder. Across all items coding

was more than 90% in agreement. For the item under discussion in this paper,(i.e. Sam

is…), coding differed on only 19/290 learner responses (i.e. 93.4% reliability).

Additionally more than half of these 19 responses included two coded responses per

learner of which only one response differed across coders.

The complete-the-sentence items „Mpho is…‟ and „Sam is…‟ were introduced to provide

learners the opportunity to describe how they viewed an unsuccessful and a successful

mathematics learner respectively. These items were introduced since our earlier

experiences of other instruments we piloted seemed to indicate that if learners were asked

about their own mathematical participation they tended to answer what they thought we

wanted to hear (Graven, Hewana & Stott, 2013). These items thus allowed them to

describe an unsuccessful or successful mathematics learner without referring to

themselves. The „Sam is…‟ item provides particularly rich information in relation to

learner dispositions as it elicits a description of imagined participation that learners

perceive would lead to successful mathematics learning. We thus have chosen to focus on

this item for the purposes of this paper.

Findings

A finding revealed by the instrument was the weak literacy levels of learners across grade

3 and grade 4. For the „Sam is…‟ item only 770 out of the 1284 codes provide data

relevant to the question. 19% of responses were illegible or incomprehensible (for example

a learner wrote: „msts is mtseay‟) and another 2 % did not respond to the item (i.e. they did

not write anything). These percentages are similar to the proportion of „illegible/

incomprehensible‟ and „unanswered‟ responses on other items on the instrument. This

44

finding concurs with wider research that points to a crisis in literacy levels of South

African learners beginning in the foundation phase (e.g. Fleisch, 2008). The recent

National Education Evaluation and Development Unit‟s 2012 National Summary report

notes that foundation phase learners receive insufficient opportunity for writing and

practice in the writing of „original consequential thinking‟ (NEEDU, 2013, p.12). The

instrument used in its written form requires the writing of such „original thinking‟.

Another 19% of coded responses indicated a repeat of what they were told by the

facilitator administering the instrument. That is during the oral administration of the

instrument facilitators tell learners that Sam is good/strong at maths and point to the figure

to the right of the spectrum of learners and to where it says Sam is the strongest learner in

the class. Learners were then asked to „describe how Sam is in the maths class‟. This

sentence is repeated and or translated into isiXhosa for learners. This 19% of learners

responded with either Sam is… „good at maths‟ or „strong at maths‟ which while being

perhaps an appropriate answer provided little in terms of how learners perceived a

successful or strong mathematical learner to be or what dispositions they thought such a

learner had. The pie chart in Figure 2 below shows the breakdown of answers in terms of

those that provided us with relevant dispositional data and those that did not.

Figure 2: Learner responses to the „Sam is…‟ item

Despite the limitation of only 60% of coded responses providing data in relation to our

disposition related research questions, and that these responses are likely to be from a

more literate portion of learners, the responses provide interesting results. We discuss this

in the following section.

45

Learner descriptions of a strong maths learner

The pie chart in Figure 3 below shows the proportional distribution of the 770 codes

derived from learner responses.

Figure 3: Learner descriptions of a strong maths learner

The high percentage (22%) of descriptions of Sam as being innately clever, gifted or bright

contrasts with the low percentage (1%) of learners who indicated that Sam worked/

practiced or tried hard at maths and did homework (engaged in steady effort). Some

examples of learner responses in these categories are given below. Responses have been

provided as learners wrote them and thus no grammatical or spelling corrections have been

made.Translations are given in italics.

Table 1: Examples of indicators *‘Innate’ and ‘Effort’+

Indicators of innate characteristics

Indicators of effort

* Sam is gifted

* Usemngovenomfundiubalaseleyo (Sam is the

gifted learner)

* Sam ukleva (Sam is clever)

* always practice maths and listen to the teacher

* he is good because he does his homework

* Sam is a good girl she does her homework

The instrument deliberately chose the names Mpho and Sam as these could be interpreted

to be either male or female. The examples given in the right hand column of the table

above show that, as intended, some learners assumed Sam to be male while others

assumed Sam to be female.

46

Some examples of learner responses in these categories are given in the table below. The

responses are written exactly as learners wrote them and thus have not been edited for

spelling errors.

Table 2: Examples of indicators [„Active participation‟ and „good behaviour‟

(including listening)]

Active participation and/or

thinking/ sense making

„good‟ behaviour or „listens‟

* I'm thinking

* uyablalaisamu (write sums)

* uyabalaimathsmameleimithetho (he

counts maths and listen to the instructions)

* Sam is boy behave wele

* uyamamelakhakhuleeclass(listening

carefully in the class)

* uyabalaimathsmameleimithetho (he

counts maths and listen to the instructions)

Similarly, interviews with a smaller number of learners indicated views that passive

listening and compliance are the reason for Sam‟s mathematical competence (see Graven,

2012; Graven, Hewana, Stott, 2013). Tirosh, Tsamir, Levenson, Tabach, &

Barkai(2012)cite a range of research where young learners incorrectly associate effort with

competency. Within the data of this study it seems that rather than associating competence

with steady effort many learners associate it with passive listening and teacher compliance.

Discussion

If we are to consider steady effort and resilience to be a key mathematics learning

dispositions as argued by Kilpatrick, Swafford and Findell (2001) and Carr and Claxton

(2002) then the above data suggests perhaps restricted learning dispositions for these

learners. The large contrast between the high percentage of learners who identify innate

characteristics as a descriptor for Sam and the low percentage of learners who identify

steady effort as a descriptor is perhaps cause for concern.

Additionally if only 15% of our learners indicate some level of active mathematical

participation, thinking/sense making and effort then this could indicate a problematic in

relation to our assumptions about learning that foreground participation and sense making.

This contrasts with the 32% of learner responses which foreground „good‟ and mostly

passive behaviour as a key descriptor of Sam.

While, as researchers with teaching experience, we do not wish to underestimate the

advantage of respectfully behaved learners who listen when the teacher is talking, we are

aware that within our perspective on learning such behaviours do not in and of themselves

result in mathematical learning. Thus, we consider that „the development of individual‟s

reasoning and sense-making processes cannot be separated from their participation in the

47

interactive constitution of taken-as-shared mathematical meanings‟ (Yackel & Cobb, 2013,

p.460). The extent to which learners‟ foregrounding of listening, behaving well and

complying with teacher instructions indicates an absence of learner independence and

agency for these learners would require further investigation. Earlier research based on

interview responses of six learners at the start of their participation in a Grade 3

mathematics clubs of the South African numeracy Chair Project provided some interesting

insights relevant to the findings discussed above. We share this briefly as it supports our

sense that our findings on this item point to a concern for learner dispositions being not as

productive as one would hope. In this club learners perceived Sam to be a compliant

worker who did what he was told. So for example, Graven (2012, p.56) writes:

the learners viewed Sam in terms of doing the work he was told to do and

writing what was required. For example one learner explained: “He takes

everything he needs when the teacher tells him to and he writes all the things

she writes and he finishes it.

Similarly teacher dependent comments emerged from interview responses with these club

learners on the final question on the instrument: „What do you do if you don‟t know an

answer?‟:

In this club all of the six learners suggested asking someone. For example, five of the

six learners suggested drawing on the teacher: “Ask your teacher”, “put up your hand

and the teacher will explain”, “stick up my hand. Have to wait”, while one learner said

“I must ask someone – I‟ll ask my friend”. While one might of course expect such

answers, and of course in many cases I have given this advice to learners that I have

helped with mathematics, the absence of utterances that indicate that one might find a

way forward by drawing on one‟s own resources is significant (Graven, 2012, p.57).

Graven contrasts these responses with some interview responses of a few learners in

another club where responses suggested a greater degree of independence and sense

making. The Maths Clubs are conceptualized as „informal, extra curricula clubs focused on

developing a supportive learning community where learners active mathematical

participation, engagement and sense making are the focus. Individual, pair and small group

interactions with mentors are the dominant practices with few whole class interactions

(Graven & Stott, 2012). Learners in this club said for example: “I thought in my mind”, “I

work it out”, “I take scrap paper or counters or my brain” (Graven, 2012, p.57).

Concluding remarks and implications

In the paper we presented the findings of how 1208 Grade 3 and 4 learners across twelve

schools in the Eastern Cape area describe a good successful mathematics learner. These

descriptions provided us with insight into an aspect of learner dispositions. Our broader

research provides further data on other aspects of learner dispositions. Our notion of a

productive disposition drew on Kilpatrick et al (2001) and Carr and Claxton‟s (2002)

indicators which include, seeing mathematics as sensible and useful, believing in steady

effort, belief in one‟s own ability to do maths, resilience, resourcefulness and willingness

to engage with others. Particularly the almost absent (only 1% of learners) description of a

strong learner as someone who puts in steady effort (a Kilpatrick et al (2001) indicator)

and/or doesn‟t give up (a Carr & Claxton (2002) indicator) raises cause for concern.

48

Similarly the low frequency of descriptions that indicate thinking and/or sense making (2%

of learners) is worrying. Instead the most common descriptor (22%) was that a good

learner had innate talent for mathematics – a view that is unhelpful if one is considered not

to have that talent.

Thus we have argued from our data that the low percentage of responses indicating active

participation, sense making, resilience or steady effort is a cause for concern in relation to

these Eastern Cape learners‟ mathematical learning dispositions. Our data leads us to

consider that perhaps a key aspect of South Africa‟s problematic in relation to our

comparatively weak mathematics performance across assessments is related to an absence

of productive mathematics learning dispositions. This might be as a result of our legacy of

restricted, passive and compliant learning dispositions promoted under apartheid

education. Perhaps we need to take seriously what Mamphele Ramphele said (Ramphele,

2013) in her speech Rekindling the South African Dream. She argued that we must shift

our mind-sets from „compliant subjects‟ to actively participating dignified citizens if we

are to rekindle the South African dream. This call particularly resonates in relation to the

discussion of the findings from our data on Grade 3 and 4 learning dispositions discussed

above.

Our concern for possibly restricted learning dispositions resonates with our experiences

and observations of working with learners in our maths clubs. Our observations and

preliminary analysis of transcripts of learner interactions in clubs point to learners being

confused, compliant and „facilitator pleasing‟ behaviour as synonymous with mathematical

competence and success.

Finding ways to support the development of more effective learning dispositions across the

South African landscape will require further research. We need to find ways to shift

classroom practices in order to shift learners‟ dispositions in positive ways that enable and

support mathematical learning and the development of all five strands of proficiency.

Kilpatrick, Swafford and Findell (2001, p.131) note in this respect that:

Developing a productive disposition requires frequent opportunities to make sense of

mathematics, to recognize the benefits of perseverance, andto experience the rewards of

sense making in mathematics.

Indeed across the work of the various projects that we run providing these opportunities

for learners is a key focus. Our broader research will continue to explore learner

dispositions and the possible evolution of these dispositions given access to the above

learning opportunities.

References

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of Numeracy: Report of a study carried out for the Teacher Training Agency.

London: Kings College.

49

Carnoy, M., Chisholm, L., Addy, N., Arends, F., Baloyi, H., Irving, M., Raab, E., et al.

(2011). The process of learning in South Africa: The quality of mathematics teaching

in North West Province. Technical report 11th June 2011. Pretoria: HSRC.

Carr, M., & Claxton, G. (2002). Tracking the Development of Learning Dispositions.

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Claxton, G., & Carr, M. (2004). A framework for teaching learning: the dynamics of

disposition. Early Years, 24(1), 87–97.

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Grades 1-3: Mathematics. Pretoria: Department of Basic Education, South Africa.

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Grades 1 to 6 & 9. Pretoria: Department of Basic Education.

Fleisch, B. (2008). Primary education in crisis: Why South African schoolchildren

underachieve in reading and mathematics. Johannesburg: Juta.

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South African Journal of Childhood Education, 2(1), 49–62.

Graven, M.,& Stott, D. (2011). Exploring online numeracy games for primary learners:

Sharing experiences of a Scifest Africa Workshop. Learning and Teaching

Mathematics, 11, 10–15.

Graven, M., & Stott, D. (2012). Design issues for mathematics clubs for early grade

learners. In D. Nampota & M. Kazima (Eds.), Proceedings of the 20th Annual

Meeting of the Southern African Association for Research in Mathematics, Science

and Technology Education (pp. 94–105). Lilongwe: University of Malawi.

Graven, M.; Hewana, D. & Stott, D. (2013) The evolution of an instrument for researching

young mathematical dispositions. African Journal for Research in Mathematics

Science and Technology Education, 17, 26-37.

Gresalfi, M. S. (2009). Taking up opportunities to learn : Constructing dispositions in

mathematics classrooms, Journal of the Learning Sciences (April 2013), 37–41.

Gresalfi, M. S., & Cobb, P. (2006). Cultivating students‟ discipline-specific dispositions as

a critical goal for pedagogy and equity, Pedagogies (April 2013), 37–41.

Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding It Up: Helping Children Learn

Mathematics. Washington DC: National Academy Press.

National Education and Evaluation Development Unit. (2013). National report 2012.

Pretoria: Department of Basic Education.

Ramphele, M. (2013). Rekindling the South African Dream. Retrieved from

http://www.citypress.co.za/politics/full-speech-mamphela-ramphele-rekindling-the-

south-african-dream/ 18th Feb 2013

Reddy, V. (2006). Mathematics and science achievement at South African schools in

TIMSS 2003. HSRC Press: Cape Town.

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Stellenbosch University. Stellenbosch: BER University of Stellenbosch.

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Tirosh, D., Tsamir, P., Levenson, E., Tabach, M., & Barkai, R. (2012). Exploring young

children‟s self-efficacy beliefs related to mathematical and nonmathematical tasks

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teaching and intervention. London: Sage.

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in mathematics.Journal for Research in Mathematics Education, 27(4), 458–477.

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Acknowledgements

The work of the SA Numeracy Chair, Rhodes University is supported by the FirstRand Foundation

(with the RMB), Anglo American Chairman’s fund, the Department of Science and Technology and

the National Research Foundation.

We thank the broader team of researchers within the South African Numeracy Chair Project,

namely: Varonique Sias, Olivia Penehafo Kaulinge and Peter Pausigere for their support in the

data collection reported on in this paper.

51

Exploring the potential of using cultural villages as instructional

resources for connecting mathematics education to learners‟ cultures

Sylvia Madusise &Willy Mwakapenda

Department of Mathematics, Science and Technology Education, Tshwane University of

Technology, Soshanguve North Campus, South Africa [email protected]; [email protected]

This article examines the potential of using a South African cultural village as a site for

mathematisation. Mathematics and culture are often interconnected, making school

mathematics intimately linked to the society in which it is taught. However, teaching in

schools rarely brings the interconnection between mathematics and culture in

pedagogically informed ways. Connections are often done superficially because of

teachers‟ inexperience in ways of connecting. Also, the curriculum in schools lacks

content and specific strategies that enable the making of the connections explicit in the

context of teaching. The study from which this paper emerges worked with three

mathematics teachers in an attempt to teach mathematics in ways that connect key

concepts with culture. Through mathematisingculturally-based activities performed at a

cultural village, two Grade 9 mathematics topics in the South African curriculum were

indigenised. A teaching unit on the indigenised topics was designed and implemented in

five Grade 9 classes at the same school. The paper demonstrates that the experience of

designing, implementing, and reflecting on the intervention study had some positive

contribution to the participating teachers‟ pedagogical repertoire. We argue that cultural

villages can be used as instructional resources for connecting mathematics education to

learners‟ cultures in the South African curriculum.

Key words: culturally-relevant pedagogy, mathematisation, indigenisation

Background to the study

South Africa has embarked upon a curriculum that strives to enable all learners to achieve

to their maximum potential (Revised National Curriculum Policy, Department of

Education, 2002).Policy statements for Grades R-9 Mathematics envisage learners who

will “be culturally and aesthetically sensitive across a range of social contexts”

(Department of Education, 2002, p.2). In this regard, the curriculum promotes knowledge

in local contexts, while being sensitive to global imperatives (Department of Education,

2011). Interestingly, some assessment standards expect learners to be able to solve

problems in contexts that may be used to build awareness of social, cultural and

environmental issues. The National Curriculum Statement (NCS) challenges educators to

find new and innovative ways to reach learners from diverse cultures in their mathematics

classrooms. Valuing indigenous knowledge systems is one of the principles upon which

the NCS is based. Part of the teacher‟s work involves coming to an argument for

ethnomathematics as a cultural way of doing mathematics. The NCS calls for radical

teaching practice changes on the part of some teachers in order to see mathematics

incorporated in the real world as a starting point for mathematical activities in the



52

classroom. Therefore, for there to be a real possibility of implementing such kind of

classroom activity, there is need to investigate the mathematical ideas embedded in

cultural practices, ethnic and linguistic communities of the learners. Khisty, (1995) argues

that learners of all background would benefit from the opportunity to learn about and

identify with their rich mathematics heritage and on-going cultural practices.

Implementation problems

Although these new understandings of mathematics teaching and learning may sound very

appropriate, the implementation and impact of explicit instructional strategies may not be

widespread and unproblematic. Teaching in schools rarely brings the interconnection

between mathematics and culture in pedagogically informed ways (Mosimege, 2012).

Mosimege (2012) reiterated that mathematics teachers lack the ability to make

connections in their mathematics classrooms; their indigenous content knowledge is

shallow. Also, a report from the Task Team for the review of the implementation of the

NCS (Department of Education, 2009)revealed that teachers had problems of converting

the vision of mathematics teaching from the written into the taught curriculum. From the

Task Team report, some teachers face mathematisation (mathematisation here is denoted

as the activity or process of representing and structuring real world artefacts and/or

situations by mathematical means) challenges when using social/cultural contexts to

reveal the underlying mathematics while simultaneously using the mathematics to make

sense of the contexts themselves.In so doing they are hindered from developing in their

learners the ability to read and understand their world mathematically. We argue that this

stagnancy in classroom pedagogy maybe in part related to the failure of educational

research to adequately investigate and promote the relationship between teacher

professional development and enhanced understanding of the espoused pedagogical shifts.

There is widespread agreement that improving teaching and learning requires that teachers

participate in high-quality professional development (Elliot&Kazemi, 2007). Such

professional learning communities may be linked to teacher learning in and from practice

where mathematics education is connected to indigenous knowledge systems.

Study focus

The study from which this paper emerges mathematised cultural activities being performed

at a cultural village, interrogating connections between mathematics and indigenous

knowledge systems. A cultural village is a tourist establishment where tourists can view

aspects such as: the homestead, traditional clothing, food and food related practices,

societal structures as well as song and dance routines of one or more of South Africa‟s

cultures (Mearns& Du Toit, 2008). The aim of the study was to determine how

mathematised cultural activities could inform the teaching and learning of Mathematics in

Grade 9 classrooms. The study sought to assist teachers in terms of where to access the

indigenous mathematical content knowledge and how to integrate the extracted indigenous

mathematical ideas in their mathematics lessons.Mathematics teachers were then engaged

in a school-based professional learning community, basing the teaching of mathematics on

the cultural background of the learners, using out-of-school, culturally-based activities.

53

The major aim was to extract mathematical ideas from the environment and embed them

within mathematical instruction.

Through mathematising culturally-based activities performed a ta cultural village, the

research team indigenised (transformed to suit learners‟ cultures) two Grade 9

mathematics topics in the South African curriculum. A teaching and learning unit on the

indigenised topics was designed and implemented in five Grade 9 classes at the same

school. This paper addresses the following central research question: What is the potential

of mathematical ideas associated with activities at a cultural village for influencing

teachers’ pedagogical repertoire?


The study was guided by Ladson-Billings‟ (1995) culturally-relevant pedagogy theory.

Ladson-Billing asserts that culturally-relevant teaching is designed to use students‟

cultures as the basis for helping students understand themselves and conceptualize

knowledge. Culturally-relevant pedagogy has been defined as a means to use students‟

cultures to bridge school knowledge and cultural knowledge (Boutte & Hill, 2006) to

validate students‟ life experiences by utilizing their cultures and histories as teaching

resources, thus connecting home with school experiences (Boyle-Baise, 2005). Therefore,

culturally-relevant pedagogy is a teaching style that validates and incorporates learners‟

cultural background, ethnic history, and current societal interests into teachers‟ daily

instruction. Many mathematicians, mathematics teachers and students possess “only a

limited understanding of what and how [cultural] values are being transmitted” through the

discipline (Bishop, 2001, p.234). Culturally relevant mathematics lessons work against this

ignorance by reversing the trend in traditional mathematics curricula to divorce

mathematics from its cultural roots (Troutman &McCoy, 2008).

Ladson-Billings (1995) documented the success of innovative lessons that appeal to

diverse cultures in improving students‟ attitudes towards classroom subject matter.

Teachers who participated in her study developed lessons that incorporated the knowledge

students gained from their lives outside of class and demonstrating the value of students‟

home cultures and languages. By so doing the participating teachers positively influenced

student test scores, engagement in the classroom community, and overall attitude towards

school and learning.

Methodology

Samples and sampling procedures

The sample in this qualitative case study consisted of three mathematics teachers from one

rural school in the North West Province of South Africa and their Grade 9 learners.

Purposive and convenience sampling was used to select the research sites (Patton, 1990).

Purposive sampling is based on the assumption that one wants to discover, understand,

gain sight; therefore one needs to select a sample from which one can learn the most. In

this case, a cultural village was identified as the research site and mathematics teachers

who teach at a school very close to the selected cultural village were sampled. A cultural

54

village was selected with the belief that it is where the community‟s indigenous knowledge

is preserved. There is tremendous potential for cultural villages to act as custodians of

indigenous knowledge (Mearns, 2006). Visitors and workers at cultural villages

interviewed by Mearns (2006) expressed that cultural villages conserve respective cultures

they are representing. Itwas considered that activities at a cultural village could assist

teachers and learners in understanding condensed cultural ways of living. The intention

was to use the cultural village as an instructional resource for connecting mathematics to

culture. A school close to the cultural village was chosen with an assumption that its

members (including learners) could be quite familiar with the activities taking place at the

cultural village.

Nature of data

The data collected in the study on which this paper is premised included seventeen video-

recorded culturally-based lessons from five Grade 9 classes (these lessons were co-taught

by the researcher and the class teachers), learners‟ responses from pre- and post-

questionnaires, learners‟ lesson journal entries, audio-recordings from learners‟ group

post-lesson interviews, audio-recordings of teachers‟ pre- and post-interviews, notes from

post-lesson reflective meetings with teachers and teachers‟ lesson reflections. These data

served as corroborating evidence to enrich the picture of teaching practices presented in

the study. The multiple sources of data provided convergent lines of evidence to enhance

credibility of assertions (Yin, 2003). Lessons were collaboratively planned. However, for

the purpose of this paper only data from participating teachers are used; triangulating data

from pre- and post-interview transcripts, lesson reflective meetings, comments from

teachers‟ lesson reflections and lesson observations.

Data presentation

Analytic induction involved reading and re-reading interview transcripts and notes from

reflective meetings to unveil different subject issues. Responses were then classified on the

basis of the formed subject issues (units of analysis).

The three participating teachers are referred to as Teacher A, Teacher B, and Teacher C for

confidentiality reasons. All the teachers had a minimum of seventeen years teaching

middle grades (Grade 7 to 9) mathematics, which means they should have gained

substantial experience of teaching mathematics up to Grade 9.

Four issues emerged in the analysis of the interview data relating to teachers‟ existing

practices. They were linked to: Coverage of indigenous mathematical knowledge in the

textbooks; Improvising teaching materials on indigenous knowledge; Instructional

strategies, and Learners‟ role. These issues are exemplified below.

With respect to coverage of indigenous mathematical knowledge in the textbooks, the

teachers made the following remarks in the interviews:

Teacher A: There is not much really.

55

Teacher B: There isn‟t much.

Teacher C: It is confusing because the children come from different cultures.

It can be seen from the above remarks that Teacher A and Teacher B believe the textbooks

they are using are not covering much of indigenous mathematical knowledge. Teacher C

thinks what is in the textbooks confuses her; it is not representing all the learner‟s cultures.

All the teachers were not improvising teaching materials on indigenous mathematical

knowledge. They said they used textbooks recommended by the Department of Education,

which from above, they had evaluated as not covering much on indigenous mathematical

knowledge. This is illuminated by the following:

Researcher: Do you sometimes improvise teaching materials on indigenous

mathematical knowledge?

Teacher A: To improvise! No I find it difficult. I find it difficult really. I always refer to

what is in the textbooks.

Teacher B: I can improvise materials for other aspects. For cultural mathematical

knowledge, we use recommended textbooks and other textbooks as

references.

Teacher C: No, I don‟t improvise.

If the textbooks the teachers and learners are using do not cover much on indigenous

mathematical knowledge and the teachers are not improvising teaching materials, the

conclusion one can draw is that there is limited link of mathematics education to learners‟

cultures.

The teachers were also asked to describe the different instructional strategies which they

employed in the teaching/learning process in their mathematics classrooms. They were

also asked to describe the usual activities which they undertook in their mathematics

lessons. The teachers gave the following remarks:

Teacher A: I start by explaining using the chalkboard, chalkboard explanations. Using

what they know then I can introduce new work using examples from the

textbook.

Teacher B: I mainly use question and answer method. I give instructions; tell them what

they should do, what the topic is all about and then ask them and they give

me answers. I also show them how to get to the answer using chalkboard

demonstrations.

Teacher C: I usually use question and answer, explanatory, chalkboard demonstrations.

Learners must know the formula where it is required. I sometimes use

practical work like drawings depending on the topic.

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Teacher B uses question and answer to check if learners got the instructions. She uses

demonstration to show learners what they are expected to do. Teacher A uses chalkboard

explanations and textbook examples to facilitate understanding and to illustrate an idea.

Teacher C uses question and answer to check if learners are following the formula. She

uses explanations maybe to emphasise and summarise important ideas. From the above

remarks it can be observed that the case teachers‟ roles in the classroom can be classified

into helping learners to remember what was learnt previously, checking if learners are

following the lesson, helping learners to check misconceptions and conveying information.

None of them indicated reference to learners‟ out of school experiences. This indicates that

the mathematical knowledge which learners‟ may develop out of school is not taken into

consideration when planning. Teachers were not connecting mathematics education to

learners‟ cultures. When the teachers were asked to describe the usual activities their

learners engaged as they learnt mathematics in their mathematics classrooms, they gave

the following statements:

Teacher A: Individual classwork, copying homework, oral work just to check how

much they understand. In fact I believe if they all participate, they must be

involved.

Teacher B: Oral work, written work and I sometimes allow them to ask questions and

work in groups. The activities I give them are guided by the teacher‟s guide.

Teacher C: Writing corrections, explanations and asking questions.

Teacher A gives individual work to make sure all learners participate in his lessons. To

him active participation means engaging in individual classwork, oral work and copying

homework. Teacher B allows learners to ask questions to help them to identify

misconceptions and misunderstandings. In Teacher C‟s classes learners write corrections

to identify misconceptions. It was also observed that their instructional strategies were not

based on a clear theoretical framework as they could not clearly explain the learning

theories which they were engaging.

Researcher: Is your teaching based on any learning theory or generalised ideas on how

mathematics can be taught or learnt?

Teacher A: Hmm…m theories? No…In fact I believe when they all participate they

must be involved. Yeah they must be involved so that they can understand.

If you are quiet we never know whether you are with us or not.

Teacher B: Yes I try by all means to make my learners understand; I do not base my

teaching on one method, but use different approaches.

It seems the teachers did not quite answer the question on teaching/learning theories they

were engaging.

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Intervention teaching context

Two Grade 9 topics were co-taught by the researcher and the participating class teachers

using culturally-based activities in five Grade 9 classes. The lessons were collaboratively

planned by the researcher and the class teachers. A group of Grade 9 learners (these

learners had previously participated in the cultural dances at the cultural village)

demonstrated a Setswana step dance, a cultural dance practiced at a cultural village near

the school. It was observed that the dancers were following a certain dancing style where

each dancer was making five footsteps forward, backward and sideways. The modelling of

the dancing style through class discussions produced a number pattern involving the

number of dancers and the number of cumulative foot-steps made in one direction before

change of direction (see Table 1 below).

Table 1. A number pattern derived from the dancing style.

Number of dancers 1 2 3 4 - - - - n

Number of foot-steps 5 10 15 20 - - - - nx5

The second row was used to introduce a sequence. Through deductive reasoning the rule

connecting the terms of the sequence was generalised. Learners managed to explain their

understanding of a sequence leading to its definition. However, there was a heated

argument on whether „n‟ could take any value. Realistic considerations were recruited.

Making „n‟ =0 meant no dancer, therefore no dance and making „n‟ too large meant too

many dancers dancing at the same time making it difficult to follow the dance. At higher

levels the depicted scenario can be used to introduce bounded sequences. Given the

periodic nature of the cultural dance – going forward, backward and sideways, the implied

mathematics involved is periodic in motion since the steps were repeated over time. This

led to another sequence - a constant sequence: 5, 5, 5,5, …whose nth

term is 5.

In Teacher B‟s observed lessons, the same dancing context was used to introduce plotting

of linear graphs. The number of dancers represented the independent variable x and the

number of footsteps represented the dependent variable y. In her other lesson on „input‟

and „output‟, the number of dancers represented the input the footsteps represented the

output. In another topic, artefacts from the Ndebele paintings and beadings, collected from

the cultural village (see Figures 1 and 2) were used to teach properties of shapes and

transformations.

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Figure 1. Ndebele paintings

Figure 1. Ndebele beadings

59

Perceived benefits of the intervention study on the teachers‟ practices

For the purpose of this paper we chose to focus on Teacher B. We chose to focus on this

particular teacher because of her commitment and participation in the activities of the

intervention study. She even used cultural contexts in some of her observed lessons. Also

her espoused claims of how her participation changed her thinking about connecting the

teaching of mathematics to learners‟ cultures led us to focus our analysis on her practices.

Three issues emerged in the analysis of Teacher B‟s perspectives of the intervention

teaching. The issues were categorised into:

Cultural village as a mathematics instructional resource.

Use of connections in mathematics education.

Effects of the intervention on teachers‟ pedagogical repertoire.

Cultural village as a mathematics instructional resource

Teacher B: What I have gained is that I can use resources like culture….from cultural

villages, like dancers (pause) to create, plan a lesson.

Researcher: Do you think the way you are thinking about setting homework, class

exercises, tests, is different from the way you were thinking before the

project?

Teacher B: We have all the tasks included in the assessment programme. The problem

is when we research learners go to ……maybe the library, but there is no

library which is nearer for the learners to use. They have to go to town for

the library. And also if they want to research using the computers it is a

problem as we do not have computers at our school. But we will now think

of ….the cultural village….of using the cultural village and ask the learners

to go to the cultural village as it is nearer to them.

Teacher B: Yeah….you can give the learners a task which needs them to go to the

cultural village so that they can research more.

Teacher B: Or to take them to some places, like to take them to a museum or a cultural

village where they can see all these things.

From the above comments, Teacher B approves the possibility of using the cultural village

as a context for mediating culture and mathematics. She contends that in the project all the

required resources were available but all the used materials were designed using the

cultural village as a resource. To her the cultural village can play the role of a library. She

believes learners can use the cultural village as a research centre to assist them to answer

given mathematics tasks. She now sees the richness of the cultural village in terms of

mathematical knowledge. According to Teacher B, one advantage of using the cultural

60

village when doing research is visualisation. “…take them to the museum or cultural

village where they can see all these things”.

According to Teacher B, before the intervention the educational value of the cultural

village was only attached to Arts and Culture.

Teacher B: During Arts and Culture festive we use dancers from the cultural village and

these dancers are our learners who practice there during their own time.

Therefore that means they are important but they were only important for

Arts and Culture.

Researcher: But not for mathematics?

Teacher B: We haven‟t linked them to mathematics at all. This is the first time that the

cultural dancers or cultural activities were linked to mathematics teaching

and learning.

Now that in the intervention cultural activities were linked to mathematics education,

Teacher B sees the need to consider the educational value of the cultural village across all

the subjects in the curriculum.

Teacher B: I think there is need to consider the educational value of the cultural village

across all the school subjects. Also the school should work hand in hand with the

community.

Use of connections in mathematics education

Researcher: What do you think about the preparation required?

Teacher B: Just to link the mathematics and the culture so that the children can see

where these two topics, the culture and….how their culture integrates with

mathematics.

Researcher: You mean they need something like an educational tour.

Teacher B: Yeah …..that is also important because I realised that most of them learn

better when they see something and they can make connections.

Researcher: What advice would you give to other mathematics teachers in general?

Teacher B: I think we need to look at our environment and identify the places where

learners can learn on their own using the environment, where they can gain

more using our own environment, places like the cultural village and they

may also be some other resources here in their community which they can

use and can benefit them in their learning.

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Teacher B now sees the value of connecting mathematics to learners‟ cultures in her lesson

preparations. To her, use of environment can assist learners to learn on their own, they can

gain more using their own environment. By conducting educational tours which take

learners to places where mathematics is being used, Teacher B believes learners can make

connections. She now sees the need to link mathematics and culture in mathematics

education. Teaching and learning resources can also come from learners‟ communities; she

believes mathematical knowledge learnt from outside school can be transferred to the

classroom. Teacher B also wants mathematical content in the textbooks they use to reflect

the everyday. She wants textbook writers when coming up with Learning Outcomes (LOs)

to give more information that refer to cultural activities.

Teacher B: About topics in the textbooks, I think their content must be in relation to t

learners‟ cultures. That will be the most important thing for our learners. These authors

should give us more information on the LOs that refer to cultural activities. This

information is awash at cultural villages.

Effects of the intervention teaching on teachers‟ pedagogical repertoire

Researcher: Do you think the way we used activities at the cultural village will shift the

way you will see these activities when you visit the cultural village today or say in future?

Teacher B: Yes now we are going to can see activities differently, because we are now

going to see different kinds of shapes, number patterns, colours, different colours used and

all these are included into mathematics.

Researcher: Do you think the way you are thinking about assessment is now different

from the way you were thinking about assessment before?

Teacher B: Before the project yes, it is different because we didn‟t prepare our lessons

like we usually did in the project. For this project we had all the resources we needed.

Learners were actively involved and were able to answer tasks on their own.

Teacher B: Yeah….you can give the learners a task which needs them to go to the

cultural village so that they can research more.

Despite her comment in the first meeting that she was familiar with activities at the

cultural village, Teacher B contends that the way she sees these activities is completely

different now. She is now going to look for the mathematics being used- a perceived

change. Another perceived change is on her lesson preparation. She now knows where to

search for the cultural mathematics content- at the cultural village. She is also thinking of

setting tasks which can be answered using the mathematical content to be extracted at the

cultural village. Teacher B also reiterated that for mathematics to be interesting to teach

and learn there is need for adequate resources, resources that learners can visualise.

62

Learners must be interested in the learning and to her learners enjoy use of cultural

examples.

Teacher B: I think by using the dancers this made the lesson funny for them and they

enjoyed the lesson. The more they enjoy the more they learn. Then it has

more impact on them than when they just read from the book. Learners

were able to make their own explanations.

Teacher B: In my other lessons only three or four learners participate, but in these

lessons almost all learners participate. In groups I could see they were

sharing ideas. I also observed that almost all the learners submitted the

given tasks unlike in my previous lessons where most learners do not write

home work.

It is observed that Teacher B was ready to critique her own lessons due to the perceived

benefits of using culturally-based activities in the mathematics lessons. According to her,

learners were involved in mathematical thinking because they could come up with their

own explanations.

Discussion

The presented analysis of the case teachers‟ existing practices before the intervention

revealed that teachers base their teaching on recommended textbooks and other supplied

curriculum materials. Their pedagogical strategies are influenced by instructional

approaches of the materials (Reys, et al., 2003). Learners‟ out of school mathematical

experiences were not taken into consideration when planning to teach. Teachers were not

connecting mathematics education to learners‟ cultures as they reiterated that the textbooks

they were using were not covering much on local cultures and they were not improvising

teaching materials on cultural knowledge. The curriculum in schools lacks indigenous

content and specific strategies that enable the making of the connections explicit in the

context of teaching. Studies based on the concept of cultural differences make an

assumption that learners coming from culturally diverse backgrounds will achieve

academic excellence if classroom instruction is conducted in a manner responsive to the

learners‟ home culture (de Beer, 2010).

Contrary to her contention in the first interview that she could not improvise teaching

materials on indigenous mathematical knowledge, but used recommended textbooks,

Teacher B, now affirms the cultural village as an instructional resource. She contends that

in the project all the required resources were available and all the used materials were

designed using the cultural village as a resource. She believes learners can use the cultural

village as a research centre to assist them to answer given mathematical tasks. Going

through the learning outcomes in the materials supplied by the Department of Education,

Teacher B cannot see the link between mathematics and culture but in the study the used

teaching materials clearly linked the two. She now sees the richness of the cultural village

in terms of mediating culture and mathematics education. Some of Teacher B‟s observed

63

lessons indicate some possible shifts in her instructional practices. Her engagement in the

study impacted positively on her teaching practices. This is in line with Vescio et al‟s

(2008) argument that well-developed professional learning communities can positively

improve teachers‟ teaching practices.

In her remarks Teacher B notes that by using resources from the cultural village, learners

were actively involved and were able to answer tasks on their own. Her exemplification

provides important insights into the authentic activities of the members of the cultural

village which learners need. When such authentic activities are transmitted to the

classroom, their context is inevitably transmuted; they become classroom tasks. Resnick

(1988) proposes bridging of the gap between the theoretical learning in the formal

instruction of the classroom and the real-life application of that knowledge.

It now seems clear that Teacher B now wants classroom mathematics to be connected to

mathematics in the learners‟ communities. This view embraces the practice of

mathematics, its history and applications, the place of mathematics in human culture.

Lerman (1990) sees mathematical knowledge as a library of accumulated experiences, to

be drawn upon and used by those who have access to it. According to the study findings

these accumulated community experiences can be studied at cultural villages. Teaching

and learning resources can also come from learners‟ communities; Teacher B believes

mathematical knowledge learnt from outside school can be transferred to the classroom.

Transfer can occur when the transformed situation contains similar constraints and

affordances to the initial context that are perceived as such by the learner (Bracke, 1998;

Corte, 1999, cited in Bossard et al. 2008).

When describing the effects of the intervention study on her learners, Teacher B

emphasised that she had noticed a change in her learners‟ attitude towards mathematics.

The use of culturally-based activities made learning interesting. What learners find

interesting is relevant to them, and what is interesting to learners is also motivating to them

and therefore relevant to teaching and learning (Kazima, 2013). Teacher B claims the more

learners enjoy the more they learn. It is observed that Teacher B was ready to critique her

own lessons due to the perceived benefits of using culturally-based activities performed at

the cultural village in the mathematics lessons. According to her, learners were involved in

mathematical thinking because they could come up with their own explanations. Learners

also made an effort to complete and submit given tasks, even the tasks given in her

observed lessons. Shannon (2007) posits that a realistic context will facilitate student

success by intrinsically motivating students and thus increasing the likelihood that they

will make a serious effort to complete given problems.

In addition Teacher B contends that before the project she could not see the mathematical

educational value of the cultural village since it was never linked to mathematics

education. The educational value of cultural villages was only attached to Arts and

Culture, but not to other subjects. Mosimege (2004) argues that cultural villages could

serve more educational purposes than being merely tourist centres. Teaching maybe

informed not only by the content of the discipline but also by the lives of the learners. An

64

ethnomathematical or cultural view of mathematics argues that mathematics is an intrinsic

part of most people‟s cultural activities (Ernest, 2001). By attending to ethnomathematics,

one can identify the broad and living informal cultural presents of mathematics. However,

teachers themselves need to be professionally empowered to have the confidence to work

in such ways. In some of her observed lessons, Teacher B used the cultural dance context

to introduce linear graphs and to teach the aspects of „input‟ and „output‟. This shows that

the pedagogical intervention had positive effects on Teacher B‟s pedagogical repertoire.

Conclusion

After mathematising cultural activities, teachers saw the possibility of improvising

teaching materials on indigenous mathematical knowledge through using cultural villages

as resource centres. Instead of focusing on the fact that textbooks are unrepresentative of

many of the cultural backgrounds of learners in the classrooms, teachers recognised that

they can also bring in articles and resources that represent the knowledge, to supplement

that presented in the textbooks. The teachers emphasised the need for connecting

mathematics education to the environment. A new value is attached to the cultural village

(by the participating teachers) – it is rich in mathematical knowledge and can play the role

of a library as a research centre. The use of culturally-based activities can make

mathematics more interesting to learn and to teach, teachers reiterated. Based on the above

analysis we argue that cultural villages are highly-useful yet underutilised contexts for

connecting mathematics education to learners‟ cultures in the South African curriculum.

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67

Toward an understanding of authentic assessment: A theoretical

perspective

Duncan Mhakure

Department of Academic Development Programme, University of Cape Town, South

Africa.

[email protected]

One of the major challenges in education reform is that there exists a gap between

instruction in learning institutions and practical experience in the world of work, and

between assessment tasks and professional practice. In the last two decades, through

educational reforms, the goals of global education have shifted towards competency-based

education in order to cater for the needs of diverse student bodies and the growing demand

of developing students with skills and competencies that are relevant in the real world. The

construct of „authentic assessment‟ is a criterion-referenced assessment, which includes

authentic tasks that have been designed for students to demonstrate the competencies,

attitudes and skills that they need in their everyday professional practice. This paper offers

a theoretical perceptive on how authentic assessment is a new assessment approach that is

different from the traditional assessment practices, which are largely decontextualized and

reliant on proxy items. In addition, the paper illustrates how the five-dimensional

framework of authentic assessment can be operationalized by using an example from a

university Mathematical Literacy course. Drawing on the conceptual differences between

authentic assessment and traditional assessment, the paper concludes by giving

suggestions for further research that are intended to expand on the positive implications of

the five-dimensional framework of authentic assessment within educational reforms.

Introduction

There is a general acknowledgement by education practitioners and researchers in

education reform that instruction and assessments, with the view of addressing the ever-

increasing diversity of curriculum changes and students, have become more complex.

Increasingly, students from diverse backgrounds and with different learning capabilities

are enrolling in institutions of learning, thus posing many new challenges for teachers,

who must manage effectively all the various dispositions from this diverse student

population. These different and complex dispositions include “prior experience and

knowledge, cultural and linguistic capital, and sources of identification and opposition”

(Darling-Hammond & Snyder, 2000, p. 524). A lack of understanding on the part of

teachers with regard to the influence of culture, readiness, experience and context on the

cognitive development of students, limits the opportunities to make meaningful decisions

on how instruction and assessment should be carried out in practice.

I posit that, in order for learning goals to be met, there needs to be a constructive

alignment of instruction, learning and assessments. In traditional classroom learning

environments, assessment usually involves short answers or multiple choice questions,


68

while instruction is characterised by rote memorisation and the transmission of knowledge

from teacher to student. Assessment within a so-called „testing culture‟ occurs primarily

by means of “decontextualized, psychometrically designed items in a choice-response

format to test for knowledge and low-level cognitive skill acquisition” (Gulikers et al.,

2004a, p.67). This type of assessment is summative, in addition, it is not context sensitive,

and thus tends to be inadequate and ineffective when assessing learners from diverse

backgrounds (Haertel, 1990; Shulman, 1987, 1992). A small but growing body of research

in education has found that there is little evidence that traditional assessment improves

teachers‟ classroom effectiveness (Andrews et al., 1980; Ayers & Qualls, 1979; Wiggins,

1989; Wolf, 1989), the main reason being that these traditional testing methods are very

different from the teaching scenarios that are actually taking place in the classroom, where

multiple strands of knowledge and skills are integrated. Teaching scenarios that support

the integration of multiple knowledge strands are a result of new education initiatives.

Their new educational goals focus on developing competent employees and citizens that

will be capable of adapting and/or using their acquired skills to solve problems they

encounter in their everyday lives and in their workplaces even when they have not been

taught specifically how to solve those particular problems. So an ability to adapt the skills

that they do have to new scenarios is envisaged (Darling-Hammond & Snyder, 2000).

The question we should be asking ourselves is: Do traditional assessments, using pen and

paper, meaningfully support the life-long development and acquisition of skills and

competencies, either as summative and/or formative assessments? In brief, a summative

assessment is intended to make judgements about individual student achievement and

assign grades at the end of a teaching phase, whereas, a formative assessment takes place

during teaching, and is intended to inform and improve learning – it is used as a feedback

device (Angelo & Cross, 1993). From what has been said already in this section, the

answer to the latter question is no – the reason being that teaching complex content to

diverse students and afterwards assessing their knowledge and insight demands that

teachers understand the effects of context. Throughout this paper, and in seeking an

answer to this question, the paper advocate for the use of so-called „authentic assessment‟

as a substitute for traditional assessment – the reason being that authentic assessment

fosters the development of skills and competencies relevant to students‟ disciplines of

studies and future world of work (Cumming& Maxwell, 1999; Gulikers et al., 2004a;

Messick, 1994). Therefore, this paper highlights, from a theoretical perspective, the

importance of authentic assessments in aligning the educational goals of learning,

instruction and assessment. More specifically, the paper seeks to illustrate how the tenets

of authentic assessments, as represented by the theoretical five-dimensional framework

(FDF),can be used in learning and teaching environments to develop the skills and

competencies required in everyday life, in the workplace, and in disciplinary studies.

The paper seeks to demonstrate the usefulness of the FDF in developing best assessment

practices by using an example of how an authentic assessment can be operationalized in a

Mathematical Literacy (ML) course – viz. an undergraduate course intended to develop

quantitative reasoning skills for humanity students at university.

69

The next section thus describes the rationale for promoting contextualised authentic

assessments, explaining why this is expected to have such a positive effect on student

learning – and particularly the development of skills and competencies required in the

workplace. The FDF of authentic assessment is discussed next, and an example

demonstrating the operationalization of FDF in a ML course is given. Finally, areas of

further research, and the implications of using authentic assessment in education and

training, are highlighted in the concluding remarks.

Rationale for promoting contextualised authentic assessments

The paper presents a number of ways in which authentic assessment is conceptualised. In

order to do so, the meanings of several key concepts, such as „authentic tasks‟ and

„authentic learning‟, must be defined first, before we can focus on the construct of

authentic assessment.

Authentic tasks

Cognitively, an authentic task has three characteristics. Firstly, it is grounded in the

learner‟s prior knowledge base. Secondly, authentic tasks require solutions that are

generated from integrated abilities and performances. Lastly, knowledge production of a

discipline using authentic tasks requires a deep understanding of the issues, techniques and

purposes relevant to the field in which the activity occurs (Archbald, 1991).

The term „authentic tasks‟ has been variously interpreted; the definition offered in this

paper is that an authentic task has the potential to “inspire student engagement in academic

work and non-school activities” (Archbald, 1991, p. 279). The emphasis here is on the fact

that the knowledge produced through engaging with authentic tasks in institutions of

learning should be easily transferrable to new and unfamiliar settings outside the

institution. In addition, authentic tasks, I believe, help students to understand the relevance

and meaning of learning activities, especially if these learning tasks mirror real-life

experiences, as they are implemented by practising professionals (Newmann, 1991;

Nicaise et al., 2000). In my opinion, this conception of knowledge transferability is vividly

absent in psychometric and standardized tests.

Authentic learning

Before discussing the concept of authentic learning, I refer to this quote from an

anonymous writer in Nicaise et al (2000, p.79), which says: “Education is what‟s left over

when you subtract what you have forgotten from what you have learned”. This statement

underscores an important notion when pursuing the goals of education: teaching and

learning for meaning making can be facilitated by authentic learning (Perkins & Blythe,

1994; Resnick, 1989).

The term „authentic learning‟ is subject to broad interpretations,it has been defined in

various ways, and is not identified with a specific instructional model. Authentic learning

involves several elements, one of the key elements is that learning must be foregrounded

in authentic tasks – real-world, complex problems and their solutions that are related to the

field of study (McKenzie et al. 2002; Rule, 2006). Renzulli et al. (2004) argue that

authentic learning thrives better in environments that are inherently multidisciplinary,

70

where learners participate in communities of practice. I use communities of practice to

refer to learning environments where students with diverse expertise meet, share

knowledge skills, put emphasis on learning how to learn, and share what is learned (Lave

& Wenger, 1991; Wenger, 1998; Collins; 2006). Such environments can mimic real world

situations, for example: An assessment in the form of a project in a ML course at

university, where students investigate the extent of drug and alcohol abuse in their own

community–illustrates that authentic learning can occur at the “intersection of workplace

information problems, personal information needs, and academic information problems or

tasks” (Rule, 2006, p.2).

In addition, Callison and Lamb (2004) posit the following: Authentic learning fosters

greater student learning, it facilitates access to multiple resources beyond and outside the

school environment, it creates opportunities to gather original data, and it encourages

lifelong learning beyond the learning task. All these facets show that authentic learning

goes beyond ensuring that students are simply technically or academically competent:

instead, it develops a deeper understanding of how knowledge can be produced, and how it

can be used within communities of practice, where students, according to Lave and

Wenger (1991), are recognised as legitimate peripheral participants.

The role of the teacher changes, when students participate in communities of practices

during authentic learning. Learning activities become learner centred, and whilst the

teachers continue to provide the students with information, they do so as guides,

scaffolders and task presenters.

It is important that, during the facilitation of authentic learning processes, teachers reflect

on and re-think their assessment strategies. A constructive alignment of instruction,

learning and assessment is required, if the goals of education are to be met. At the

beginning of this section, we looked at the construct of authentic learning, which should be

aligned intentionally to contextualised assessments that are embedded in interesting, real-

life and authentic tasks. These contextualised assessments are referred to as authentic

assessments in this paper.

Authentic assessment

Authentic assessments typically include authentic tasks, where students are required to

“demonstrate the same (kind of) competencies, combinations of knowledge, skills,

attitudes, that they need to apply in criterion situations in professional life” (Gulikers et al.,

2004b, p.5). The term „authentic‟ in the literature is viewed as a relative concept, that is,

authentic in relation to something else. On the one hand, assessments could be authentic

with respect to course- or institution-based problems or tasks that have been obtained from

learning and teaching materials, such as course readers and textbooks, or on the other

hand, they could be authentic in terms of real world contexts (Honebein et al., 1993;

Messick, 1994).

In this paper, the authenticity of an assessment resides in its resemblance to the real world

of work. The rationale here is that, if goals of education around the world are to produce

competent professionals, then assessment practices should take into account the

71

environments with which students are likely to be confronted in future, as professionals in

particular careers or places of work. Through authentic assessment, apart from developing

cognitive skills, such as problem solving and critical thinking skills, students develop

abilities to integrate knowledge, skills and attitudes, and also meta-cognitive and social

competencies, such as communication and collaboration – all key facets of communities of

practice (Birenbaum, 1996). On a cautionary note, however, proponents of authentic

assessment must remember that authenticity is subjective, meaning that what teachers and

assessment developers deem as authentic may not necessarily be perceived as authentic by

the students. This leads to the two-fold concept of pre-authentication, which alludes, it is

very difficult to design an authentic assessment, and it is necessary to consider the prior

knowledge and experiences of the students when designing an assessment. I think these are

two equally valid factors that must be borne in mind when designing authentic assessments

and I posit that authentic assessments are not necessarily difficult to design, however, the

challenge lies in the fact they are time-consuming and expensive(Gulikers, 2005; Gulikers

et al., 2004b; Lajoie, 1991). Proponents of authentic assessments, including the author of

this paper, are in favour of the latter.

Traditional assessment versus authentic assessment

Throughout the initial sections of this paper, indirect comparisons have been made

between traditional assessments and authentic assessments. It is helpful at this stage to

summarise the differences between these two approaches, with reference to Wiggins

(1990) in Herrington and Herrington (1998, p.308).The defining attributes of these two

types of assessments ,as listed in Table 1,are by no means exhaustive, but they do lead to a

better conceptual understanding of the differences between the assessments.

Table 1: A comparison of authentic and traditional assessment

Authentic assessment Traditional assessment

Directly examines student performance on

worthy intellectual tasks.

Relies on indirect or proxy items.

Requires students to be effective performers

with acquired knowledge.

Reveals only whether students can recognise,

recall, or „plug in‟ what was learned out of

context.

Presents the student with a full array of tasks. Conventional tests are usually limited to pencil-

and-paper, one-answer questions.

Attends to whether the student can craft

polished, thorough and justifiable answers,

performances or products.

Conventional test typically only ask the student

to select or write correct responses –

irrespective of reasons.

Achieves validity and reliability by emphasising

and standardising the appropriate criteria for

scoring varied products.

Traditional testing standardises objective

„items‟ and the one „right‟ answer for each.

Test validity should depend in part upon

whether the test simulates real-world „tests‟ of

ability.

Test validity is determined by matching items to

the curriculum content.

Involves ill-structured challenges that help Traditional tests are more like drills, assessing

72

students rehearse for the complex ambiguities

of professional life.

static and too-often arbitrary elements of those

activities.

Source: Wiggins 1990 in Herrington & Herrington (1998, p. 308)

According to the reviewed literature, authentic assessment is preferred to traditional

assessment; however, I posit that a teacher/facilitator of a course does not have to choose

between the two models of assessment: both can be applied jointly and to complement

each other, depending on the goals that need to be accomplished. In mathematics

education, for example in ML at school level, students could benefit from undergoing both

traditional and authentic assessments at the end of each learning phase. In higher

education, authentic assessment could be used as the main assessment model, even though

this approach too is criticised, the reason being that it is too expensive and time-

consuming.

One of the main criticisms of authentic assessment “is that validity is achieved at the

expense of reliability” (Herrington & Herrington, 1998, p.308). Dimensions of validity,

such as fairness, transfer of knowledge, skills and experience, content coverage, cognitive

complexity and meaningfulness of task, among others, can be ensured under authentic

assessment; however, the same cannot be said about the reliability of an authentic

assessment. Reliability, which is about repeatability, becomes less problematic, if it is

applied in large-scale assessment environments, similar to those in high schools, where the

emphasis is on standardized assessments. But it becomes more difficult to attain reliability

in tertiary institutions, where learning and assessment cannot be easily separated (Gipps,

1995;Lajoie; 1991; Linn et al., 1991; Reeves &Okey, 1996). Another criticism of authentic

assessment is that it does not permit students‟ results to be compared, as it is difficult to

find general principles that apply to more than one context (Reeves &Okey, 1995). In

addition, critics of authentic assessment fear that its emergence would render traditional

assessments less favourable, as compared to it, because credence lies in its real-world

usefulness.

A five-dimensional framework of authentic assessment: A description

In this section, a five-dimensional framework (FDF) of authentic assessment, as proposed

by Gulikers et al (2004b, p.7) is described. The FDF is rooted in its construct validity (i.e.

whether the assessment measures what it is intended to measure) and its consequential

validity (i.e. what the intended and unintended effects of the assessment are). There are

five main dimensions of authenticity to the FDF: the task; the physical context; the social

context; the assessment result or form; and the criteria. It is important to view each one of

these five dimensions as part of a real-life situation that students could encounter in the

workplace, whether as interns or as practicing professionals (Darling-Hammond & Snyder,

2000; Gulikers et al., 2004a). More importantly, the five dimensions proposed in the FDF

for authentic assessment are not static, but may vary, depending on the levels of the

students‟ education.

Table 2: Overview of the five-dimensional framework of authentic assessment (Source:

Gulikers et al., 2004b, p.7)

73

Dimensions of

authentic

assessment

Descriptions

Task Should be perceived as representative, relevant, and meaningful to a

professional practice. Degree of complexity should match students‟ level

of education.

Physical context Environment should be similar to the professional space, with similar

professional resources, and similar professional time frames.

Social context Should reflect the community of practices in professional practices –

where individual decision making is juxtaposed with group work and/or

collaborative work.

Assessment

result/form

Should allow students to demonstrate competencies –with regard to

quality products and valid inferences, for instance –and to reach fair

conclusions using multiple indicators of learning; students should be able

to defend their work (orally or in written form).

Criteria Should be made explicit and transparent in advance; criterion-referenced

judgement leading to profile scores should be used; realistic and

achievable outcomes should be set to a final product, performance or

solution that students need to create.

There is no doubt that these five dimensions play important but different roles when

providing authenticity to an assessment; they may also be given different weighting. In my

opinion, out of the five dimensions described in the FDF, the task, assessment result and

the criteria are pivotal to the implementation of authentic assessment, whilst physical and

social contexts are less important. The reasons for my inference here is that it is often

challenging to assess collaborative activities within a social context, such as group work,

since it is difficult to measure individual contributions. With regard to the physical

context, this could be either the actual physical environment, or an environment that has

been simulated through technology; simulations are generally cheaper to use, and to set up

(Gulikers et al., 2006). However, some researchers have argued that implementing

authentic assessments in a simulated environment misses the intended goals of such

supposedly authentic assessments, and that it moreover does not improve educational

practice (Cumming & Maxwell, 1999; Newmann & Archbald, 1992).

Operationalizing the FDF using a Mathematical Literacy example

The focus in this section is to apply the FDF to an assessment task of a ML course – a

course intended for first year humanities students studying Psychology at a tertiary

institution. As we have already alluded to in the introduction section of this paper, the

objective of this course is to develop quantitative reasoning skills of humanities students to

enable them to cope with the quantitative demands of their discipline, to equip students

with the quantitative skills they require as practising professionals, and to become

quantitatively literate citizens. This assessment task, which is summative, requires students

to use their acquired research skills to investigate alcohol and drug perceptions among the

youth in their own communities. The task is set out below.

74

Introduction: Adapted from: “Factors associated with female high-risk drinking in

rural and urban South African site” (Ojo et al, 2010) - “High-risk drinking by women is

a major problem, especially in the Western Cape. Measures of low socio-economic status

and an alcohol problem in one or more family members were associated with high-risk

drinking. Having parents, siblings or partners who abuse alcohol fosters an environment

where alcohol consumption is modelled, accepted and encouraged. Targeted interventions

are needed especially for women with alcohol problems in the family setting, lower socio-

economic status, and concurrent substance abuse.”

Assessment task - project: As a student of the social sciences, you are concerned about the

impact of alcohol and drugs on young people in your community and believe very strongly

that the community ought to take action in implementing an awareness programme in the

area in which you live. The tasks that must be completed in three months in order to meet

the assessment criteria of this project include: Carrying out a survey to determine how

young people in your community view alcohol and drug use, and/or the kind of experiences

that they may have had with these substances. Writing a report that you would like to

present to the community leaders and that motivates why the best way to fight the problem

is to educate teenagers and young adults about the problems. Your report ought to include

graphical representations of relevant data and clear explanations of these graphs in the text

of the report.

At this point, it is instructive to examine how the FDF dimensions are recognised and

valued, if at all, in this assessment – in other words, is there an alignment between the

objectives of the assessment task and the dimensions of the FDF?

Task

The students are required to investigate the impact of alcohol and drugs on young people

in their own community. Students will find this type of assessment meaningful and

relevant, and representative of their own experiences from their own communities.

Physical context

Upon graduation, students are likely to work in similar professional spaces where they will

deal with behavioural issues at the core of communities, such as alcohol and drug abuse

among the youth.

Social context

The assessment task does not state explicitly that students are required to work with each

other directly – as communities of practice. Nonetheless, it is hoped that students will seek

to work together with their colleagues to ensure that the task is completed timeously.

Assessment result

On completion of the report, the student is required to present it to the community leaders,

thereby defending their own findings.

Criteria

This assessment task is summative, with realistic and achievable outcomes. In addition,

students should be made aware upfront how the criteria for assessment will be

implemented on submitting their work; alternatively, the marking rubric should be given to

the students before they start working on the project.

75

Although the example given here is intended for undergraduate ML students, similar

examples of this nature, albeit with less rigour, could be useful when teaching school

Mathematics and Mathematical Literacy subjects, especially when teaching concepts, such

as data handing techniques. The point here is that, instead of teachers simply assessing

concepts such as central tendencies and measures of spread from proxy items, students

would find these concepts more meaningful, if they could identify real-world contexts in

which these concepts are embedded.

Conclusion

This paper recognises that, whilst there have been major shifts in instructional and learning

approaches, such as the creation of the cognitive apprenticeship model, whose goals are to

develop skills and competencies required in the world of work, the same could not be said

about assessment approaches. The cognitive apprenticeship model, with its emphasis on

solving real-world problems, provides an alternative to conventional approaches to

education and training, and aims to “produce graduates with equal thinking and

performance capabilities” (Bockarie, 2002, p.48).

During the past decade, the goals of education have changed, primarily in response to the

demands of the global job market. Essentially, education now focuses on equipping

students with the skills that enable them to handle with confidence the complexities of ill-

defined real-world problems in the workplace. Throughout this paper, the key argument

was to urge the adoption of authentic assessments in institutions of learning, both at school

level and in higher education, as a way of developing higher order skills among the

students, such as analysis and complex communication, both of which are crucial for

students if they wish to function as effective professionals (Collins, 2006; Lombardi,

2007). Under the authentic assessment model, course facilitators are encouraged to teach

their students to carry out meaningful tasks by using specific rubrics.

In closing, it is important to urge future researchers working in education reform to

explore more thoroughly the use of authentic assessment environments, as many important

questions still need to be answered. Central to these questions is the construct of

authenticity. Although this has been discussed in detail in the literature review, the

question remains: Do the teachers and the students share this view of authenticity? This

could be an interesting research area and one that could potentially provide a deeper

understanding of authentic assessments.

Further research should also be directed at the FDF of authentic assessment. The five

dimensions play different roles in authentic assessment – and research is needed to focus

on the contribution of the physical context and the social context. For example, if the

physical context is replaced by simulation, this will obviously reduce the expenses and

time required to carry out authentic assessments in real-world contexts. However, will

replacing the physical context with simulations in authentic assessments affect the quality

of the development of skills and competencies that are needed in professional practice?

76

Earlier in the paper, the role of the teacher or assessor in the implementation of the FDF in

authentic assessments was discussed. If the authenticity of the tasks in the FDF depends on

the interpretation of the assessor, then perhaps the assessor should be included as a sixth

dimension of the FDF. In fact, this is indeed something I am advocating.

Lastly, the goal of global education should be to educate and train human capital so that

they have the skills and competencies that are required in the actual workplace; employers

often complain about the mismatch between graduate students‟ skills and the skills they

actually require in the workplace. In that case, research and resources should be directed

towards developing and improving the constructs of authentic learning and assessment.

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79

Shifts in practice of mathematics teachers participating in a professional

learning community

Nico Molefe1

& Karin Brodie 2

1 School of Education, University of the Witwatersrand, South Africa.

2 School of Education, University of the Witwatersrand, South Africa


[email protected]

This paper focuses on practices of mathematics teachers who participated in a professional

learning community (PLC) from 2011 to 2012. The PLC was part of the Data Informed

Practice Improvement Project (DIPIP), which involved eight schools across two districts,

and focused on teachers shifting their practices in order to take account of and work with

their learners‟ mathematical errors. The main data source for this paper is the lesson videos

of the two teachers who participated in the project over the two years of intervention. The

paper discusses the shifts in practice of two teachers who participated in the project over a

period of two years. The results reveal how one teacher sustained her shifts and the other

did not.

Introduction

The paper works with a key principle in teacher development: that improved instructional

practice is key to improved learner performance (Earl & Katz, 2006; McLaughlin &

Talbert 2006; Roberts & Pruitt, 2003). This paper reports on how a PLC was used as a

model for teacher development, where a PLC is a professional community of teachers who

“work collaboratively to reflect on their practice, examine evidence about the relationship

between practice and learner outcomes, and make changes that improve teaching and

learning for particular learners in their classes” (McLaughlin & Talbert, 2006, p. 3).

PLCs, as models for teacher professional development, are generally not common in South

African schools because little has been done to create platforms for them to be explored.

However, the Department of Education and The Department of Higher Education and

Training suggest through the Integrated Strategic Planning Framework for Teacher

Education and Development that PLCs should be established in South African schools

(DBE & DHET, 2011).

An important requirement for PLCs is that they should have a “clear, defensible focus”

(Katz, Earl, & Ben Jaafar, 2009). The DIPIP project has its focus on learner errors and

misconceptions. The DIPIP project aims to support teachers to embrace learner errors and

misconceptions in mathematics teaching and learning, helping teachers acknowledge

learners as “reasoning and reasonable thinkers” (Brodie, 2005). A number of activities are

given to teachers participating in the project to work on. As teachers work on the

activities, they also have conversations about how these activities relate to their practice.



80

Project activities

Table 1 shows all the activities that teachers in this study worked on with guidance from

the university-based facilitator. The table also shows the data collection activities. The

activities were designed by the project team and discussed before they were given to the

teachers to work on. Working on these activities is demanding in terms of time and

commitment – teachers met for two hours per week, and they were expected to avail

themselves regularly for meetings despite pressures from their school work.

The test that the learners wrote was an international standardised Algebra test which has

been used for about three decades, and proved to have shown the interesting errors that

learners make in the test (Hart, 1981). Learner responses on this test provided data that

teachers could use for the error analysis activity, and selecting learners for as well as

conducting the interviews. The results of the error analysis activity and the learner

interviews were a resource for teachers to plan lessons. More on these lessons is discussed

in the next sections.

The activities entailed in the table illustrate the full cycle that was followed in working on

the activities in the PLC, with one cycle done in the first year and the other cycle in the

second year. The different PLCs come to a combined meeting to present the results of

their PLC activities. The combined PLCs comprise a Networked Learning Community

(NLC).

Table 1: Project activities – what the teachers worked on in a one-year period

Week Learning Community Activities Data collection activities

1 – 3

Interview teachers and

videotape lessons

4 – 5 Develop (or find) test and test learners Developing (or finding) test

and testing learners

6 – 8 Map test onto curriculum Record PLC interactions

9 – 12 Develop the data wall Record PLC interactions

Analyse test data for learner errors Analyse test data

13 – 14

Select items and learners for learner

interviews and conduct interviews, to

understand learner thinking in more depth

Record PLC interactions

Interview learners

15 - 16 Meeting with other teachers in district to

report back on findings

Record NLC interactions

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17 Based on the above analyses – choose a

leverage concept to work on Record PLC interactions

18 – 19 Read about and discuss research on the

chosen concept Record PLC interactions

20 – 21 Plan a set of lessons on chosen concept Record PLC interactions

22 – 23 Teach the planned lessons Video-record lessons

24 – 25 Reflect on teaching Record PLC interactions

26 – 27 Meeting with other teachers in district to

report back on the activities Record NLC interactions

Theoretical framework and related literature

This paper is informed by a situative perspective on learning, which views learning as

participation in discourse and communities of practice (Borko & Koellner, 2008; Lave &

Wenger, 1991). The perspective is derived from an assertion that learning is situated –

how a person learns and the situation in which a person learns form important parts of that

learning. Teacher learning thus becomes a process of participation in a community, and

increased participation in the community is learning. The study from which this paper is

drawn looks at how teachers use classroom data in their conversations with the guidance of

a facilitator to create new knowledge that they can utilise to improve their teaching

(Brodie, 2013). A professional learning community is thus used in this study as a platform

for mathematics teachers to work together as a community and have conversations about

their practice and the possible ways of improving that practice.

Teachers work with different learners in their classrooms, and these learners bring a

variety of ways of thinking about mathematics. In the process of learning mathematics,

learners provide data sources for teachers to use in learning about learners‟ thinking. The

main sources of data are the errors and misconceptions that learners produce when they do

mathematics. Smith, diSessa & Roschelle (1993) explain a misconception as “a learner‟s

conception that produces a systematic pattern of errors” (p. 119). This paper is premised

on the notion that errors are a result of “a consistent conceptual framework based on

earlier acquired knowledge” (Nesher, 1987, p. 33) and make sense to learners in terms of

their current thinking. Nesher (1987) argues that “a good instructional program will have

to predict types of errors and purposely allow for them in the process of learning” (p. 33).

The teachers in the PLC meetings are encouraged to work with learner errors in ways that

can help shift their instructional practices from correcting or ignoring the errors that

learners produce to working with those errors in innovative ways. What is important is for

teachers to understand the reasons behind these learner errors so that they can plan and

modify their teaching accordingly.

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The teachers‟ classrooms and the PLC provide two environments in which teachers can

learn, and these two environments should complement each other (Kazemi and Hubbard,

2008). In the PLC, teachers can learn with and from each other, and in the classroom they

can learn from the errors that the learners make and how they (the teachers) deal with these

errors as they try to understand the reasoning behind the learners‟ errors. It is for this

reason that the DIPIP project is grounded on the notion that learners‟ learning needs

inform teachers‟ learning needs (Katz, et. al, 2009; Brodie, 2013). It is important for the

teachers to know that learners have learning needs, to understand the difficulties that

learners face in mathematics in order for the teachers to know the learning they need

themselves to address these learners‟ needs. The two environments, the classroom and the

PLC, thus create a pedagogic movement between the classroom and the PLC as illustrated

in the diagram below (fig 1). Kazemi and Hubbard (2008) refer to this movement as

transformation of participation or co-evolution of practice. Through this movement, it is

hoped that the teachers‟ practices can inform the nature of conversations that are held in

the PLC, and the PLC conversations can also inform the teachers‟ instructional practice.

Figure 1: Co-evolution of practice – showing the two environments that contribute to

the teachers‟ professional development on a continual basis.

Kazemi and Hubbard (2008) argue that research often focuses on the unidirectional

process of professional development – emphasis is usually placed on what teachers take

from the professional development to their classrooms, and not the other way round. In

this study, teachers are encouraged to work as a community and bring to the attention of

the PLC, whatever they encounter in their classrooms as they teach. When teachers bring

their classroom data to the PLC, this completes the fully cycle of professional

development.

Research Design and Data Collection

This paper reports on a case study that involves teachers of one district, where the case is a

community of teachers comprising a professional learning community. The teachers work

in a collaborative way to look at their classroom data and use this data to improve their

instructional practices (see Brodie, 2013 and Brodie & Shalem, 2011 for more detail). The

data comprise tests that learners write as well as the errors that learners produce as they do

mathematics in their classrooms.

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Participants

The participants in this study are four mathematics teachers, Dimpho, Funeka, Mapula and

Khumo, who are employed in two neighbouring schools in two districts. All teacher names

are pseudonyms. The four teachers comprise a single professional learning community.

Two of these teachers, Dimpho and Funeka, teach in an FET (Further Education and

Training) school and the other two, Mapula and Khumo, teach in a Junior Secondary

school. This paper will only report on two teachers: Funeka and Khumo. Funeka has been

teaching for eight years, and has been teaching mathematics for seven years. Khumo has

been teaching for 32 years, with 21 years of teaching mathematics. The teachers have

their PLC meetings once a week for two hours after school during school terms. The first

author was the facilitator and the researcher for this professional learning community for

the two years that we collected data for the study.

Data collection

Four types of data were collected from the teachers: biographical data, lesson videos,

interviews, audio recordings of the PLC meetings, and facilitator‟s notes taken during

lesson videotaping and also during PLC meetings. Each teacher selected a class in which

she could be video recorded over an agreed period of time – during the pre-intervention

period, during the two cycles of the intervention period, and also during the post-

intervention period (see Table 2). The teachers reflected on the concept development

lessons that they had taught and focused primarily on the errors that the learners had made

in the lessons and how they, as teachers, dealt with these learner errors. The teachers used

the feedback they got from these reflections with the hope of improving their practices in

their classrooms.

This paper reports on the data from the lesson videotapes. The first author of this paper

coded this data and discussed the results with members of the DIPIP team. We will

discuss more on this coding in the next sections.

Table 2: The three types of lessons that were videotaped in each of the participating

teachers’ classrooms

Pre-intervention lessons Concept development

lessons Post-intervention lessons

Normal teaching lessons

taught at the beginning of

the intervention period

Lessons based on the lesson

plan that the teachers jointly

worked on

Lessons that were taught

at the end of the two

years’ intervention period

Teachers had not started

working on DIPIP activities

Teachers had already

analysed learner errors and

interviewed selected learners

Teachers had completed

all the DIPIP activities

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Lessons followed

individual teachers’

schedules

Lessons followed an agreed

schedule according to the

teachers’ lesson

Lessons followed

individual teachers’

schedules and were not

part of the joint lesson

planning process

It should be noted that there were two concept development lessons: concept development

one and concept development two, with concept development one lessons having two

cycles of teaching. For these concept development lessons, the PLC decided on the

concept that they wanted to teach based on the error analysis and the learner interviews.

The first concept that the PLC agreed on was the notion of a variable and the second

concept was the language of algebra. Brodie and Shalem (2011) detail what counts as a

concept in the DIPIP project. The facilitator then sourced research papers that dealt with

the chosen concepts and gave those to the teachers to read and use in their lesson planning.

For the first concept, teachers taught the lessons and reflected on their teaching of the

concept. This teaching is referred to in tables 3 and 4 as concept development one (CD1),

cycle one. After the teachers had reflected on their teaching, they re-planned the lesson

using their reflection as a community and re-taught the lessons. This teaching is referred in

tables 3 and 4 as concept development one (CD1), cycle two.

Once the second cycle of teaching concept development one lessons had been reflected on,

the teachers returned to the error analysis and selected a new concept, which is referred to

as concept development two (CD2) in tables 3 and 4. Here there was only one cycle of

teaching. The last set of lessons was the post-intervention lessons, and these lessons were

not a result of the joint-planning of the PLC as explained in table 1 above.

Every teacher in the PLC allowed her lesson videotapes to be watched and reflected on by

the PLC. The format of the reflections entailed teachers choosing episodes in which they

thought they dealt well in addressing learners‟ errors and also episodes in which they

thought they did not deal so well with the learners‟ errors. This process of reflection was

intended to help the teachers to inquire into their practice and allow PLC members to

provide suggestions regarding their instructional practice. The process allowed for a

pedagogic movement from classroom to PLC as discussed above. A lesson video provides

a powerful artifact of practice for teacher development (Borko, et al. 2008). Teachers can

use this artifact to reflect on their practices in different ways and learn from that process as

they critique their lessons and give each other feedback on what they think are alternative

strategies that can be used in their classrooms.

Analysis

In order to address the research question: to what extent do mathematics teachers‟

practices shift as a result of learning through a PLC, we share our analyses of the shifts in

the two teachers‟ practices over a period of two years. The analysis is based on five sets of

lessons from each teacher as described above. To analyse the teachers‟ practices, we use

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Hill, Blunk, Charalambos, Lewis, Phelps, & Sleep‟s (2008) Mathematical Quality of

Instruction (MQI) coding scheme to code the teachers‟ lesson videos. The instrument has

five key dimensions: “mode of instruction”, “richness of mathematics”, “working with

students and mathematics”, “errors and imprecision”, and “student participation in

meaning-making and reasoning”. These dimensions fit with the kinds of instructional

improvements DIPIP is interested in.

“Mode of instruction” is used to code how mathematical content is dealt with in the

classroom – is this in a teacher-directed manner where there‟s high amount of teacher talk,

or are there alternative methods used in class to develop content. Alternative methods

would include allowing learners to give explanations of their work and to share their

thinking.

“Richness of mathematics” focusses on the depth of the mathematics offered to learners.

The MQI tool stresses that rich mathematics allows learners to build a conceptual

mathematical base. A strong conceptual mathematical base is what DIPIP hopes to realise

among teachers and learners.

“Working with students and mathematics” captures whether teachers respond to learners‟

mathematical productions or mathematical errors. Learners‟ productions can be in the

form of their utterances or their written work. This is the key dimension that resonates

with what DIPIP emphasises in the teachers‟ professional development. Teachers in the

project are encouraged to embrace learner errors and misconceptions in their teaching in

ways that can help them understand the reasoning behind these errors.

“Errors and imprecision” is a dimension that captures the errors that teachers make as they

deal with content in their classrooms, and it also includes imprecision in language and

notation. The focus in this dimension is on the teacher. Serious errors are coded high and

minor errors are coded low. Uncorrected learner errors and lack of clarity by the teacher

are also coded. However, the errors will be coded low if they are captured and addressed

during the lesson.

“Student participation in meaning-making and reasoning” captures evidence of the extent

to which learners are involved in the lesson, how they participate and contribute to

meaning-making and reasoning in class. Learner participation includes learners asking

mathematically motivated questions.

The above dimensions help us to explain the teachers‟ practices during the videotaped

lessons and give an account of what was happening in the teachers‟ mathematics

classrooms. Each of the five key dimensions has sub-dimensions. Within each sub-

dimension are three levels, 1, 2 & 3, where level 1 is considered as „low‟ and level 3 is

considered as „high‟.

Each lesson video was sub-divided into eight-minute segments, and the coding was done

according to the sub-dimensions for every eight-minute segment. A count was taken to get

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the total number of codes within each sub-dimension for a specific level (1, 2 or 3) and

these codes have been converted to percentages that describe proportions for each level

within a particular sub-dimension.

The first author started coding a few lesson videos with members of the project team for

purposes of inter-rater reliability. After reaching agreement of over 80 percent, he then

continued to code the rest of the lesson videos. This paper focuses on the results of this

coding. A shift in the teacher‟s practice is viewed as improvement from a lower level to a

higher level, for example from level 1 to 2, except in the dimension of “errors and

imprecision”, where improvement is viewed as moving from a higher level to a lower one,

for example from level 2 to 1, i.e. the teacher makes fewer errors or teaches less

inaccurately.

Results

All of the teachers shifted their practices in the concept development lessons. Two of the

teachers, Funeka and Mapula maintained their shifts in the post-intervention lessons. The

other two, Dimpho and Khumo did not maintain the shifts. Tables 3 and 4 show the shifts

of two teachers, Funeka and Khumo. After each table, we give a summary of the shifts we

observed for each teacher, showing how one teacher sustained her shift in the post-

intervention lessons and how the other teacher did not sustain her shift in the post-

intervention lessons. Table 3 shows the coding from Funeka‟s grade 10 lessons and Table

4 shows the coding from Khumo‟s grade 7 lessons. The values given in the two tables are

percentage representation of codes with respect to the episodes within a particular sub-

dimension. This paper focuses on the quantitative results, indicating areas for further

exploration of qualitative data.

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Table 3: Frequency of codes – Funeka, showing percentage records of teachers’ practices according to the three levels of the MQI tool

Dimension Sub-Dimension Pre-int,

(14 episodes)

CD1, Cycle 1

(26 Episodes)

CD1, Cycle 2

(12 episodes) CD2 (9 episodes)

Post-int

(20 episodes)

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

Mode of

Instruction

Direct instruction 79 21 4 96 8 92 100 5 95

Whole class discussion 100 35 65 42 58 22 78 25 75

Richness of

Mathematics

Linking or connections 100 92 8 83 17 100 90 10

Explanations 86 14 81 19 75 25 22 78 12 8

Multiple procedures or

solution methods 100 84 12 4 75 17 8 100 100

Developing

mathematical

generalisations

100 92 8 92 8 100 100

Mathematical language 100 100 100 78 22 90 10

Working with

students and

mathematics

Thorough remediation

of student errors and

difficulties

100 65 31 4 50 50 56 44 35 60 5

Responding to student

mathematical

productions in

instruction

100 54 46 25 75 44 56 30 65 5

Errors and

Imprecision

Major mathematical

errors or serious

mathematical oversights

43 57 54 35 11 75 25 78 22 80 20

Imprecision in language

or notation

(mathematical symbols)

86 7 7 54 42 4 67 33 89 11 75 20 5

Lack of clarity 71 29 42 47 11 50 50 67 33 80 15 5

Student

participation in

meaning

making and

reasoning

Students provide

explanations 71 29 15 81 4 25 67 8 33 67 25 70 5

Student mathematical

questioning and

reasoning

100 73 19 8 58 42 89 11 75 25

Enacted task cognitive

activation 100 69 31 58 42 33 67 90 10

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Funeka‟s results show a significant shift in the dimension Mode of instruction. This shift

is mostly linked to whole class discussion, wherein the teacher gives the learners the

chance to discuss each other‟s ideas regarding their responses to mathematics problems

given to them in class or as homework. Whole class discussion was completely absent in

the pre-intervention lessons of Funeka‟s lessons but appeared in the CD1 lessons and was

maintained in CD2 and the post-intervention lessons. At the same time Direct Instruction

decreased slightly, suggesting an interesting mix of Direct Instruction and Whole Class

Discussion. This shift is an indication of Funeka allowing learners to share their ideas in

her class.

A small shift in the Richness of mathematics is observed, predominantly in the sub-

dimensions of explanations and mathematical language. Mathematical language is

emphasised during DIPIP meetings while multiple procedures and developing

generalisations receive less attention. The small shift suggests little depth of how the

mathematics is dealt with in class.

There is a significant shift in Working with students and mathematics, which increased

over time and reached level 3 in the post-intervention lessons. As discussed above, this is a

key element of DIPIP‟s work. The shift is an indication that Funeka was aware of

learners‟ productions in her teaching and she dealt with those in different ways.

There is an increase in Errors and imprecision from pre-intervention to the CD1 lessons.

This dimension focuses only on the teacher‟s errors as well as how the teacher deals with

learner errors. A similar result was found by Chauraya (2013) and he argued that as the

teachers began to interact more with learners, their mathematical errors and imprecision

increased. In Funeka‟s case we see a decrease in errors and imprecision in the CD2 and

post-intervention lessons suggesting that she was better at co-ordinating her own

knowledge with the learners‟ ideas.

There is a significant shift in Student participation in meaning making and reasoning,

which is mostly linked to learners providing explanations during the lessons. In this

dimension, the coding scheme does not distinguish between the correct and the incorrect

explanations as it does in the dimension of the Richness of mathematics above. This shift

is an indication of learners‟ involvement in their own learning – learners provided

explanations of their work and asked mathematically motivated questions.

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Table 4: Frequency of codes – Khumo, showing percentage records of teachers’ practices

according to the three levels of the MQI tool

Dimension Sub-Dimension Pre-int

(9 episodes)

CD1, cycle 1

(18 Episodes)

CD1, Cycle 2

(19 episodes)

CD2 (9

episodes)

Post-int

(15 episodes)

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

Mode of

Instruction

Direct instruction 11 33 56 5 90 5 95 5 55 45 6 94

Whole class discussion 100 77 23 89 11 89 11 100

Richness of

Mathematics

Linking or connections 100 100 100 100 100

Explanations 100 67 33 63 37 89 11 100

Multiple procedures or

solution methods 89 11 100 89 11 100 100

Developing mathematical

generalisations 100 100 100 100 100

Mathematical language 100 94 6 53 47 100 87 13

Working with

students and

mathematics

Thorough remediation of

student errors and difficulties 100 77 23 63 37 89 11 80 20

Responding to student

mathematical productions in

instruction

100 83 17 68 32 100 87 13

Errors and

Imprecision

Major mathematical errors or

serious mathematical

oversights

44 12 44 77 23 74 2 24 33 55 12 67 6 27

Imprecision in language or

notation (mathematical

symbols)

33 45 22 83 17 58 42 22 56 22 60 40

Lack of clarity 67 33 83 17 53 32 15 22 56 22 53 33 14

Student

participation

in meaning

making and

reasoning

Students provide

explanations 100 60 40 42 58 45 55 73 27

Student mathematical

questioning and reasoning 100 100 100 100 15

Enacted task cognitive

activation 22 78 100 100 45 55 15

90

Khumo‟s results in the Mode of instruction show a shift from the pre-intervention lessons to

the two cycles of CD1 lessons. This shift is related to direct instruction. Direct instruction

shows a slight increase in the CD2 lessons, but this is reduced in the post-intervention lessons

and becomes similar to CD1 lessons. The shift shows a presence of whole class discussion

which was completely absent in the pre-intervention lessons. This shift is a demonstration of

an attempt by Khumo to involve learners in discussions and sharing their ideas in class. The

shift declines in the post-intervention lessons, which also shows a complete absence of whole

class discussion.

There is a slight shift in the Richness of mathematics from the pre-intervention lessons to the

concept development one lessons. This shift is linked to the sub-dimension of explanations,

where Khumo and the learners give correct explanations during mathematics teaching and

learning. The shift shows a decline in the CD2 lessons and a complete absence in the post-

intervention lessons. The sub-dimension of language of mathematics shows a significant

increase in the CD1, cycle two lessons with a complete decline in the concept development

two lessons, and a slight increase in the post-intervention lessons.

The results show a slight shift in Working with students and mathematics in the CD1 lessons,

which demonstrates an attempt by Khumo to apply in her practice what DIPIP encourages

about working with learners. However, this shift is not sustained in the CD2 lessons as well

as in the post-intervention lessons, which suggests that Khumo was not responding

appropriately to learners‟ mathematical productions in these lessons.

Under Errors and imprecisions, Khumo‟s results show prevalent errors in the pre-

intervention lessons, with a slight shift in the CD1, cycle one lessons. CD1, cycle two

lessons show a decline, and this remains low in the CD2 lessons as well as in the post-

intervention lessons. This means Khumo produced errors and also did not address the

learners‟ errors and misconceptions appropriately in these lessons. DIPIP encourages

teachers to embrace learners‟ errors in their teaching and also work in ways in which they can

understand the learners‟ thinking behind these errors.

The results show a slight shift in Student participation in meaning making and reasoning.

This shift is linked to learners providing explanations and, to a lesser extent, linked to the

enacted task cognitive activation. The shift in the sub-dimension of learners providing

explanations is sustained in all the concept development lessons, but shows a decline in the

post-intervention lessons.

Conclusion

This paper has reported on the extent to which two teachers participating in a teacher

development project shifted their instructional practices. We have shown how two teachers,

Funeka and Khumo, shifted their practices. One teacher, Funeka, sustained her shift and the

other, Khumo, did not sustain her shift. Both teachers participated in the same PLC for a

period of two years but show different outcomes in their practices. Key points to be made

from these results are that:

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There was more participation of learners in the teachers‟ lessons after the joint lesson plan

was done by the PLC. This participation of learners was not evident as shown in the pre-

intervention lessons.

There was more engagement with learner errors in the classroom as teachers responded to

learners‟ contributions. The teachers‟ engagement differed from one teacher to the other

and this led to a difference in the observed shifts. We have explained in this paper how

DIPIP encourages teachers to embrace instead of ignoring or avoiding learner errors.

These results came from an analysis of teacher practices. The next step will be to analyse the

data from the professional learning communities to explore whether their participation in the

communities is related to the instructional shifts.

References

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productive discussions in mathematics professional development. Teaching and Teacher

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programs of Professional Development. Paper presented at the Reflecting and Shaping the

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interactions with learner errors. In Chick, H.L. & Vincent, J.L. (Eds.). Proceedings of

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Conference of the International Group for the Psychology of Mathematics

Education, Vol. 2, pp. 177-184. Melbourne: PME

Brodie, K. (2013). The power of professional learning communities. Education as Change,

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Brodie, K. & Shalem, Y. (2011). Accountability conversations: Mathematics teachers

learning through challenge and solidarity. Journal for Mathematics Teacher Education 14,

419-439.

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community. Unpublished PhD thesis submitted at the University of the Witwatersrand,

Johannesburg.

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2011–2025. Pretoria, South Africa.

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school improvement. CA: Corwin Press.

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Murray.

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(2008). Mathematical Knowledge for Teaching and the Mathematical Quality of

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development. Journal of Teacher Education, 59(5), 428-441.

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Lave, J., & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. New

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93

Learning to teach mathematics by means of concrete representations

Nkosinathi Mpalami

Lesotho College of Education, Maseru, Lesotho

[email protected]

The purpose of the study reported in this paper is to explore a student teacher‟s choice and

use of mathematical representations in a Standard 4 class in Lesotho. The participant was a

full time registered Diploma in Education Primary student teacher who was engaged in

teaching practice that takes place in second year of a three year diploma programme at the

Lesotho College of Education. The teaching of mathematics is challenging for novice

teachers due to their underdeveloped Pedagogical Content Knowledge (PCK). The ability to

choose and use effective mathematical representations in lessons is an important component

of teachers‟ knowledge for teaching. By mathematical representations in this study, I refer to

concrete objects, images, and symbolic constructs that are used in teaching to make abstract

mathematical concepts and processes accessible to learners. The student teacher used fake

money to scaffold learners‟ strategies of addition and subtraction of both whole numbers and

decimals. It is concluded that the choice and use of concrete objects (fake money) in a

familiar context for Basotho learners afforded them opportunities to access mathematics.

Introduction

In this paper, I give an account of a study that was carried out in a primary school in Lesotho.

The purpose of the study was to explore mathematical representations that a student teacher

(Thandi) used in a mathematics lesson in a Standard 4 (8-9 years old) class. The lesson lasted

for a period of 40 minutes and it was about the teaching and learning of addition and

subtraction of money. At the time when the study was conducted, Thandi was a second-year

Diploma in Education Primary (DEP) student. DEP is one of the diploma programmes

offered at the Lesotho College of Education (LCE). It is a three-year programme where the

second year is scheduled for Teaching Practice (TP). The lesson referred to in this paper is

one of the lessons that I video recorded, transcribed and analysed as part of data collected for

a broader study, which was funded by the Centre for Global Development through Education

(CGDE).

Background

Thandi was a student teacher who was training to become a primary school teacher. The

current situation in Lesotho is that there is only one institution (Lesotho College of

Education) which has been mandated to train students who have completed high school

education to become primary school teachers. A large population of primary school teachers

are diploma holders and teach all subjects stipulated in the National Curriculum Development

Centre (NCDC)‟s documents. The three-year diploma programme for primary school trainees

is designed in such a way that students spend years one and three at the college studying both

content and methods courses in various learning areas. The findings presented in this paper

emanate from data that were collected during second year when student teachers were on TP.

The findings obtained as a result of Tier 1 data analysis in the bigger study (doctoral research

project), reveal that a large percentage of candidates who enrol for the DEP programme have

weak mathematical background and are generally females. However, Thandi is an exception

in that she obtained a credit in the Form E (final year of schooling) mathematics examination.

This means she is one of the few students who pass mathematics well in Form E and opt to


94

become primary school teachers. She practised teaching in a poorly resourced church school

located about 40 kilometres away from Maseru city centre.

Theoretical orientations

The ability to represent mathematical ideas in various and flexible ways is an important

component of pedagogical content knowledge (PCK). Shulman (1986, p. 9) argues that

“since there are no single most powerful forms of representation, the teacher must have at

hand a veritable armamentarium of alternative forms of representation”. It then follows that

both experienced and novice primary school teachers‟ PCK of mathematics can be made

manifest through their ability to select and use multiple representations in lessons in order to

make abstract mathematical concepts accessible to learners. Back in the 1960s a well known

cognitivist Jerome Bruner in his work on the cognitive development of children proposed and

identified three modes of representations:

• Enactive representation (hands on/action-based)

• Iconic representation (visuals/image-based)

• Symbolic representation (abstract and language-based)

Bruner (1966, p. 10).

In Bruner‟s point of view, modes of representation are the means through which information

or knowledge are stored and encoded in a leaner‟s memory. Mathematics teachers in

particular have to be conscious of these three modes of representations when they plan and

teach mathematics especially at primary school level. In Lesotho, mathematics teachers have

a tendency to focus more on symbolic representations than the other two representations,

perhaps that is why many learners fail mathematics in three national examinations (Standard

7, Form C, and Form E).

Enactive representations involve tasks that call for action (hands on activities) on the part of

learners. In many Lesotho primary schools, teachers use concrete objects such as counters,

matches, sticks, and stones in mathematics lessons to assist learners to do addition,

subtraction, multiplication, and division calculations. These physical objects prove to be

valuable especially in the early years of schooling because learners at this stage are not yet

conversant with the four basic mathematical operations namely addition, subtraction,

multiplication, and division. Therefore, the models play an important role in helping learners

to understand why this sentence is true, 8 – 3 = 5.

Iconic representations are visuals that both learners and a teacher can refer to in class in order

to facilitate effective learning and teaching of certain mathematical concepts. Iconic

representations are resources that act as scaffolds for learners‟ indecisive thinking and

provide strategies for mathematical operations. Examples of iconic representations commonly

used in many primary schools include; number-line, number square, number fan, number

track, place value cards, multiplication square, and multiplication array. While most of these

representations are commercial, student teachers have to be taught, during their training,

ways of improvising and constructing these resources using recycled cardboard, plastic and

waste. Given the economic state of Lesotho, it is logical to assume that many primary schools

cannot afford commercial teaching aids. But irrespective of financial constraints of the

country learners in all schools have an educational right to be taught mathematics well. The

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use of iconic representations is meant to assist learners‟ mathematical strategies to be strong

and independent, and once this stage is reached then an iconic representation can be removed.

The three modes of representations are not only hierarchical in nature but also intertwined

(Rowland, Turner, Thwaites, & Huckstep, 2009). Symbolic representation being at the

highest level and the most abstract representation compared to the other two (enactive and

iconic). This is the most adaptable form of representation than actions and images, which

have a fixed relation to that which they represent. Symbols are flexible in that they can be

manipulated, ordered and classified and as such the learner is not constrained by actions or

images. In the symbolic stage, knowledge is stored primarily as words, mathematical

symbols, or even in other symbolic systems (Bruner, 1966).

Rowland et al. (2009) have cited an example of an empty number line to substantiate their

point that the three modes of representations cannot only be used hierarchically but also in an

intertwined fashion. They argue that an empty number line is an iconic representation by its

nature but as learners make „hops‟ or „jumps‟ on it, they are using it in an enactive way. Yet

the operations demonstrated by such hops and the answer reached are symbolic in nature.

During their training courses, student teachers are expected to acquire and develop a web of

connected representations for various mathematical concepts which they can draw on in

lessons to help learners understand mathematics.

PCK and representations

The ability of any teacher to translate his/her knowledge of the subject matter into something

comprehensible to learners is an important part of that teacher‟s (PCK). When working with

experienced science teachers in Australia, Loughran, Mulhall, and Berry (2004) found that

teachers‟ knowledge of their practice (teaching) is tacit. They found that although teachers

find it challenging to provide reasons for teaching certain scientific concepts in particular

ways, in general, teachers commonly share activities, teaching styles, and insightful thoughts

of how best to teach science. Loughran et al. (2004: p. 374) take a view that researching

teachers‟ PCK requires working at both an individual and collective level because “PCK

resides in the body of science teachers as a whole while still carrying important individual

diversity and idiosyncratic specialized teaching and learning practices”. When exploring the

notion of mathematics teachers‟ knowledge resource, Rowland et al. (2009, p. 14) concur that

within a school context, teacher‟s knowledge resource is “both individual – what each teacher

knows, and collective – what is accessible by reference to colleagues”. In class a teacher

draws from his/her own content knowledge but in situations where teachers work

cooperatively in school they plan lessons together and talk about methods of delivery in class.

In that way each teacher receives necessary assistance from colleagues. In Tier 3 of the

reported study in this paper, I worked with five student teachers as a group in workshops and

with each individual participant in one‟s lesson to explore their understanding and use of

mathematical representations.

When working with science teachers in South Africa Rollnick, Bennett, Rhemtula, Dharsey,

and Ndlovu (2008) argue that the ability to choose an appropriate representation and use it

effectively in lessons reflects that teacher‟s PCK. Shulman (1986 & 1987) also takes a view

that multiple representations that teachers use in lessons are the central part of each individual

teacher‟s PCK. It then follows that being able to choose and use representations efficiently in

lessons is an important component of any teacher‟s PCK.

The Knowledge Quartet

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In Thandi‟s lesson, there were multiple activities taking place, however I needed a theoretical

lens through which I could focus my eye on the teaching/learning of mathematics through

representations. The „Knowledge Quartet‟ proved to be a useful tool in this case. Devised by

Rowland, Huckstep and Thwaites (2005), the Knowledge Quartet (KQ) is a typology that

emerged from a grounded approach to data analysis of primary mathematics teaching in the

UK. The KQ identifies the manner in which the student teacher‟s mathematical knowledge

impacts on a mathematics lesson along four dimensions namely; foundation, transformation,

connection, and contingency. While all four dimensions are interconnected and all are useful

in looking at mathematics teaching, the „transformation‟ dimension lends itself particularly

well to this study in that it focuses the eye of the researcher on the uses student teachers make

of representations when teaching mathematics. I also realise that „connection‟ established

between mathematical concepts is key in Thandi‟s lesson. Therefore, in this section I only

discuss two dimensions that I consider to be of less relevance in this paper namely

„foundation‟ and „contingency‟ however, useful in shedding light on the use of the whole

analytical framework of the Knowledge Quartet.

In the „foundation‟ dimension of the KQ, the teacher‟s background knowledge and beliefs

with regard to the meanings and descriptions on mathematical concepts and practices are

manifested during teaching. A teacher‟s ontological position of mathematics and the rationale

for teaching it at primary school level is noticeable and made manifest in mathematics

lessons. According to Rowland et al. (2009) the codes for this dimension include among

others: awareness of purpose, identifying errors, overt subject knowledge, use of

mathematical terminology, use of textbook, reliance on mathematical procedures, and

theoretical underpinning of pedagogy.

The „contingency‟ dimension calls for a teacher to make sound decisions during the lesson

about learners‟ contributions. Unlike experienced teachers, student teachers lack the ability to

take on and respond immediately to unexpected learners‟ contributions in class. Hume and

Berry (2011) distinguish the most limiting factor as student teachers‟ lack of classroom

experience and experimentation. According to Rowland et al. (2009, p. 126) “there are times

when the teacher is faced with an unexpected response to a question or an unexpected point

within a discussion and so has to make a decision whether or not to explore the idea with the

child”. A teacher has to be always alert for such moments and be ready to react appropriately

to such unexpected situations during the teaching episode. It could be unfortunate if

unexpected learners‟ contributions could pass unnoticed in a class by the teacher because

unpacking such contributions might be of special benefit to that learner or, as Rowland et al.

(2009) put it, might suggest a particularly fruitful avenue of enquiry for others. They identify

the key contributory codes in this dimension of the knowledge quartet as: responding to

children‟s ideas; use of opportunity; and deviation from agenda. In what follows, I address

the ways in which these methodological issues were attended to.

Methodological approach

As mentioned earlier, the reported study here is part of a three-tiered longitudinal research

(doctoral) project with prospective primary school teachers conducted over a period of three

consecutive years (2009 – 2011). For the purposes of this paper, I choose to focus on part of

Tier 3 data. This part of the study is guided by the critical question, How do student teachers

on Teaching Practice use mathematical representations in lessons?

The five student teachers in Tier 3 were conveniently drawn from the previous Tier 2‟s ten

participants. They were a convenient sample in that I invited only those who were practising

97

teaching of mathematics in primary schools located in the Maseru district which I could

easily and economically access. Each of the five participants was observed teaching a

mathematics lesson on arithmetic operations. However, this paper is about one lesson which

was taught by Thandi, one of the five participants in Tier 3. There were eighty (n = 80)

Standard 4 learners in Thandi‟s class. The learners‟ ages varied from 9 to 15 years. Below I

present Thandi‟s lesson synopsis:

Thandi asks learners to take out their money. (On the previous day, Thandi had given learners

home-work in which they had to construct fake money and bring it to class the next day). The

picture below is a sample of the money that learners had constructed:

The teacher placed representations of fruits items on the wall and asked the six (n = 6) chosen

learners to imagine being in a shop to purchase certain fruits. It is a common practice for

learners in this area to buy fruit not only in supermarkets but also from the street vendors at

the school gates. Many people in Lesotho earn a living by selling fruit in various places

including school surroundings. It is reasonable, to conclude that learners in Thandi‟s class are

familiar with buying fruit.

Below are some of the pictures that Thandi placed on the wall:

Thandi mentions that money is subtracted when used to buy items in a shop. She then places

pictures of different fruit on the wall in front of learners. She asks six learners to come to the

front of the class and give her fake money to buy items placed on the wall. She asks each

learner to say how much money they have initially. Learners are then asked to buy items of

their choice and say whether they will be given change or not, and if they will be given

change, to say how much change they will be given. This continues until all chosen learners

have spent their money. Learners are then asked to go back to their seats. Thandi distributed

textbooks to learners and asks the learners to complete a set exercise.

Analysis of Thandi‟s lesson

While the discussion of the analysis of data is organised to revolve around all four

dimensions of the Knowledge Quartet (KQ) special attention is paid to transformation in this

section because it focuses the eye of the researcher on representations. I also focus on

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connections of concepts that came about as a result of the use of manipulatives in the form of

fake money in Thandi‟s lesson.

Connection

If teachers are to make mathematics comprehensible to learners they have to make efforts to

present it as a series of connected concepts, procedures, ideas, and practices. Marshall, Superfine

and Canty (2010) take a view that making connections between multiple representations help

students see mathematics as a web of connected ideas and not as a collection of arbitrary,

disconnected rules and procedures. The connections can be made between concepts, operations,

units, topics, and branches (e.g. Geometry and Arithmetic). The use of an iconic representation such

a number square in class can help learners to recognise the connection that exists between two

operations namely addition and subtraction of whole numbers. Rowland et al. (2009) argue that

teachers have to bear in mind the complexity and cognitive demands of mathematical concepts and

procedures in their attempt to sequence and connect mathematical content. They further identify

contributory codes for this dimension (connection) as: making connections between procedures;

making connections between concepts; anticipation of complexity; decisions about sequencing; and

recognition of conceptual appropriateness. In Thandi’s lesson, the main ‘connection’ that I observed

is the way the use of money as a context enabled the learners to connect operations of addition and

subtraction together. The other ‘connection’ was made between two topics namely whole numbers

and decimal numbers and this occurred as learners were buying fruit and had to determine their

change.

In an incident during the lesson, Thandi made the connection between two representations namely

money and ordinary numbers:

Thandi: He has M2.00 … and from the M2.00 he has ... the banana costs

M2.00

Pupils (chorus): Yes madam.

Thandi: From the M2.00 he has and the banana costs M2.00, is he going to

get the change or not?

Pupils (chorus): No.

Thandi: No, because he has finished the money, ha ke re (isn’t it so)? When

you subtract M2.00 from M2.00 you get zero (0). Ha ke re (isn’t it

so)?

It seems that the connection of zero change and the number zero (0) here is critical. The connection

here is between two sentences: Two Maloti take away two Maloti results in no change and 2 – 2 = 0.

Transformation

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Of the four dimensions of KQ, the key dimension is ‘transformation’ in this study that interrogates

the choice and use of mathematical representations in teaching. When unpacking the notion of

Specialised Content Knowledge (SCK) in mathematics education Ball, Thames and Phelps (2008)

emphasize the key role representations play in making mathematics concepts understandable to

learners. Ball et al. (2008, p. 393) further argue that because some representations are more

powerful than others in affording learners’ access into mathematics concepts, teachers who have

rich SCK choose and use “appropriate representations” that make content comprehensible to

learners.

Every mathematics teacher has to make choices in planning and delivering a lesson. The choices

involve among others selecting a key representation for the concept intended to be taught. For

instance, a number line can be used for teaching addition, subtraction, multiplication, and division.

But a number line is a key representation for addition and subtraction while a multiplication square

or a multiplication array might prove to be key representations for performing multiplication at any

level of the primary school mathematics. During lesson preparation, a teacher has to think carefully

about the examples, illustrations, and analogies that he/she can use in class to make concepts,

procedures, or even core vocabulary comprehensible to learners. Rowland et al. (2009) identify

contributory codes in this dimension of the Knowledge Quartet as: choice of representations,

teacher demonstrations, and teacher’s choice of examples.

In her lesson, Thandi chose to use fake money as a key representation in order to scaffold learners’

skills of addition and subtraction of whole numbers and decimal numbers. When the lesson started

Thandi demonstrated to learners what she meant by decomposing numbers:

Thandi: Yes, when we decompose a number we break it into pieces, ha ke re

(is that so)?

Pupils (chorus): E-ea ‘m’e (yes madam).

Thandi: If you break it up into pieces, we just take out any numbers that can

add up to fifty, ha ke re (is that so)? So my own number… I can

extract M20, Nka etsa (I can make) 20 + 20 + 10 = M50.00. We add

up to M50.00. So which other numbers can we decompose fifty

Maloti into? Which numbers can we decompose fifty Maloti into?

Tefo!

Tefo: M10 + M10 + M10 + M10 + M10 = M50.00

The demonstration that Thandi made appears helpful in helping learners to comprehend the

meaning of decomposing numbers. The evidence is seen by Tefo’s response as shown in the excerpt

above. Later in the lesson when learners were struggling to subtract decimal numbers from whole

numbers, Thandi encouraged learners to use other forms of representations:

Motsamai: The banana is 1 Loti and 50 cents. We subtract M1.50.

Thandi: We subtract 1.50 Loti. It’s 150 lisente (cents), from M5.00 he has,

we subtract 1.50 lisente ha ke re (isn’t it so)?

100

Pupils (chorus): Yes madam.

Thandi: So what is the change? What is Makoro’s change? So how much is

he going to get as the change for Makoro? How much is he going to

get? Use your fingers, use our money … just think, think, use your

fingers, your head, whatever! What do you think is going to be

Makoro’s change? What do you think is going to be Makoro’s

change, when we subtract 1.50 from the M5.00 he has? Nkele!

Nkele: Makoro’s change is going to be M3.50.

Here Thandi asked learners to use fingers and their heads to think about the correct answer for the

change. The excerpt suggests that the use of any of these representations (fingers or/and reasoning)

afforded the learner (Nkele) strategies that made it possible for her to obtain the correct answer

(M3.50). It is possible that Nkele uses these representations in her daily buying to determine her

change. I would like to argue that Thandi’s choice of the selling and buying situation assisted

learners to manage subtracting decimal numbers from whole numbers, which could have been more

cognitively challenging if it was only presented symbolically as 5 – 1.5 = ?

Findings and concluding remarks

Analysis of the data reveals that Thandi chose to use various representations in order to support

teaching and learning of mathematical operations namely addition and subtraction. Thandi had

various resources at her disposal to use in this lesson; however she carefully chose to use fake

money made of paper in order to facilitate the teaching of whole numbers and decimal numbers.

The home-work that was given to learners, namely to construct fake money prior to this lesson, had

the potential of helping learners to represent money in an enactive way (hands on activity) while at

the same time helping them to think more seriously about calculations with real money used in their

community. When learners encountered some subtraction difficulties, Thandi encouraged them to

use the ‘fingers’ on their hands as representations to assist them to get to the correct answer. It is

also noted in the analysis that the effective use of fake money in this lesson afforded learners

opportunities to make crucial connections between subtraction of whole numbers and decimal

numbers. Again, the use of fake money in the lesson encouraged discussion, and I conclude that it

helped Thandi to effectively achieve her lesson objectives.

While the findings of this study cannot be generalised, I feel that this work might give some useful

insights with regard to student teachers’ mathematical understanding and use of concrete

representations in teaching.

References

Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it

special? Journal of Teacher Education, 59(5), 89–407.

Bruner, J. S. (1966). Toward a theory of instruction. Cambridge, MA: The Belknap Press of Harvard

University Press.

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Hume, A. & Berry, A. (2011). Constructing CoRes – a strategy for building PCK in pre-service science

teacher education. Journal of research in science education, 41, 341–355.

Loughran, J., Mulhall, P., & Berry, A. (2004). In search of Pedagogical Content Knowledge in science:

Developing ways of articulating and documenting professional practice. Journal of Research

in Science Teaching, 41(4), 370–391.

Marshall, A. M., Superfine, A. C., & Canty, R. S. (2010). Star students make connections. Teaching

Children Mathematics, 17(1), 38–47.

Rollnick, M., Bennett, J., Rhemtula, M., Dharsey, N., & Ndlovu, T. (2008). The place of subject matter

knowledge: A case study of South African teachers teaching the amount of substance and

chemical equilibrium. International Journal of Science Education, 30(10),1365–1387.

Rowland, T., Huckstep, P., & Thwaites, A. (2005). Elementary teacher’s mathematical knowledge:

The knowledge quartet and the case of Naomi. Journal of Mathematics Teacher Education, 8,

255–281.

Rowland, T., Turner, F., Thwaites, A., & Huckstep, P. (2009). Developing primary mathematics

teaching. London EC1Y 1SP: SAGE Publishers Ltd.

Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational

Researcher, 15(2), 4–14.

Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational

Review, 57(1), 1–22.

102

Comparing strategies of determining the centre and radius of a circle using

repeated measures design

1Eric Machisi,

1David L. Mogari&

2Ugorji I. Ogbonnaya

1University of South Africa

2Tshwane University of Technology, South Africa

[email protected], [email protected], [email protected]

A repeated measures design was employed to compare students‟ achievements in determining

the centre and radius of a circle using two strategies. Twenty-five low-performing Grade 12

students from a secondary school in Limpopo province took part in the study. Data were

collected using an achievement test and were analysed using the Wilcoxon Signed-Ranks

test. Findings indicated significant differences in students‟ scores due to the strategies used.

On average, students scored better using formula strategy than with the strategy of

completing the square. The study therefore recommends that low-performing students should

be exposed to a wide range of mathematical solution strategies for solving mathematical

problems.

Introduction

According to Dakin and Porter (1991, p. 274), “[a] circle is the locus of a point which moves

at a constant distance from a fixed point”. If we let A(a; b) be the fixed point and B(x; y) be

the moving point at a constant distance r from A (as shown below), then the relationship

between (x; y) and (a;b) is which is the equation of the circle with

centre (a; b) and radius r.

Figure 1. Circle illustrating relationships

r

B(x;y)

A(a;b)




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If we let the centre be (-a;-b), then the equation of the circle can be written as

which reduces tothe form , where

. This is calledthe general equation of the circle.

Finding the centre and radius of a circle given the general equation is a commonly examined

mathematical aspect in the South African Grade 12 Mathematics Examination (Paper 2).

However, examiners‟ reports indicate that learners have not been doing well on this

mathematical aspect. The learners‟ difficulties could be as a result of educators who confine

their teaching to only what is in the prescribed learners‟ textbooks. Given that there is only

one approach to finding the centre and radius of a circle presented in the prescribed learners‟

textbooks (according to the authors‟ observation), then learners who fail to understand „the

method of the textbook‟ are likely to be frustrated.

In the present study, we exposed learners to two strategies of determining the centres and

radii of circles given the general equations. The first strategy, which makes use of some

formulae, is not in the prescribed learners‟ textbooks whereas the second strategy, which

involves completing the square, is the one found in the prescribed learners‟ textbooks. This

study therefore compares the students‟ use of the two strategies in determining the centre and

radius of a circle. The research question addressed is: is there any significant difference in

low-performing learners‟ test scores due to using the two strategies to determine the centres

and radii of circles?

Strategies for determining the centre and radius of a circle

Strategy number 1: Using formulae

Suppose we have an arbitrary equation of a circle , where , then the centre of the circle is:

The radius of the circle is:

(Gonin, Du Plessis,Kuyler, De Jager, Hendricks, Hawkins, Slabber, &Archer, 1987)

Example: Determine the centre and the radius of the circle with equation

0384822 yxyx (Department of Education [DoE], 2010)

Solution:

The centre of the circle

The radius 5838()2()4( 22

This strategy seems short but relies heavily on learners‟ ability to memorise the formulae for

the centre and radius of a circle since these formulae are not in the formula sheet used in the

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Grade 12 mathematics examination. Understanding the meaning of the term „coefficient‟ is

also important here since substituting a wrong coefficient leads to a wrong solution. The

formula for finding the radius uses the results obtained for the centre. It therefore implies that

if the coordinates of the centre are wrong, then the result for the radius will also be incorrect.

However, the rule of Consistency and Accuracy (CA) will be applied.

Strategy number 2: Completing the square

By completing the square, we can express the equation of the circle in the form

, where is the centre and is the radius of the circle

Example: Determine the centre and the radius of the circle with equation

0384822 yxyx (DoE, 2010)

Solution:

58)2()4(

4163844168

03848

22

22

22

yx

yyxx

yxyx

Centre = (-4;-2) and radius = as obtained previously

Here, learners should be able to carry out the procedure of completing the square. That is,

dividing +8 and +4 by 2, squaring the results and, adding the squares on both sides of the

equation. Learners are also expected to be able to factorise quadratic expressions, that

is and . In addition, learners should then

be able to rewrite as and as in order to identify

the coordinates of the centre of the circle. Failure to do this will result in learners writing (4;

2) for the centre instead of (-4;-2). Lastly, learners should match their result after completing

the square with the general form of the equation of a circle given in the formula sheet in

order to see that , which implies that .

Objectives of the study

This study sought to compare the two strategies of determining the centre and radius of a

circle presented above. The objectives were to first test whether there are any significant

differences in students‟ achievement scores as a result of the strategy used and secondly, to

determine which strategy is better understood and preferred by low-performing learners.


This study was largely influenced by some aspects of Bruner‟s (1960) cognitive theory and

Van de Walle‟s(2004) constructivist theory of mathematics education. According to Bruner

(1960), any mathematical idea can be taught in a simple form for any student to understand as

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long it is adapted to the student‟s intellectual capacity and experience. Van de Walle (2004)

asserts that all students can learn all the mathematics we want them to learn provided we

offer them opportunities to do so. Based on these two learning perspectives, the researchers

conceived that even low-performing Grade 12 students are capable of learning any

mathematical aspect we want them to learn provided they are offered opportunities to explore

different strategies of solving mathematical problems. As students solve mathematical

problems using different strategies, they are likely to arrive at a strategy they understand

better which they can easily employ to solve such problems in future.

Literature review

South Africa has conducted a number of national assessments of learners‟ achievement (such

as the Systemic Evaluation Study [SES], and the Annual National Assessment [ANA]) and

has also participated in international surveys of learner performance in mathematics and

science (such as the Trends in International Mathematics and Science Study [TIMSS], the

Southern and East Africa Consortium for Monitoring Educational Quality [SACMEQ], and

the Monitoring Learning Achievement [MLA] project). The apparent convergence of

findings from these studies is that learners have been performing far below expectations in

the critical subjects (mathematics and science).

The results from national and international surveys of learners‟ performance in mathematics

and science have prompted research into the reasons for the poor state of mathematics and

science education in South Africa. According to Long (2007) and Mukadam (2009), not all

mathematics teachers are adequately equipped to effectively teach mathematics (and science).

A survey conducted by Rakumako and Laugksch (2010) on the demographic profile of

secondary school mathematics educators in Limpopo indicates that most educators are

“academically under qualified and professionally ill-prepared for their classroom

responsibilities as they have only Standard 10 (Grade 12) as their highest academic

qualification with a three-year teaching diploma” (p.148). Stoffels (2008) asserts that

educators with low knowledge of subject matter tend to teach from the textbook, avoiding

those mathematical aspects in which they are not competent.

Although several other reasons could be drawn to explain why South African learners have

been performing poorly in mathematics (see Van der Westhuizen, Mosoge, Nieuwoudt,

Steyn, Legotlo, Maaga and Sebego, 2002), there is growing consensus among researchers that

what goes on within the classroom outweighs all other factors as a predictor of learners‟

achievement (Arnold & Bartlett, 2010). A view of the present study is that it is the quality of

mathematics teaching that needs to be improved and not just an increase in the allocation of

resources towards mathematics (and science) education. While information obtained from

national and international surveys of learners‟ performance is valuable to educators (Long,

2007), “it does not necessarily provide the means to improve, especially in a conceptually

complex subject like mathematics” (p.3). However, according to the Centre for Teaching and

Learning of Mathematics [CTLM] (1986), even the worst mathematics performance can be

improved considerably if compensatory strategies are put in place to remediate learning

difficulties. Despite a proliferation of studies on the state of mathematics education in South

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Africa, little is known about ways in which secondary school mathematics educators could

enhance the mathematics achievement of low-performing students in some problematic

mathematical aspects.

Discussions with mathematics teachers and analyses of examiners‟ reports confirm that many

Grade 12 students have difficulties in finding the centre and radius of a circle from the

general equation. It is the authors‟ perception that the problem could be as a result of the way

teachers present this mathematical aspect to their students. Due to limited mathematical

knowledge, teachers tend to stick to teaching only what is in the prescribed textbooks (Cai,

Mamona-Downs & Weber, 2005). In many classrooms, mathematics teaching and learning is

confined to strategies that are in the prescribed textbooks and students who do not understand

the solution strategies in the textbook are regarded as unable to learn mathematics (Elmore,

2002). Yet, the growing demand for a mathematically-skilled workforce in South Africa

demands pedagogical reform in mathematics teaching (McCrocklin& Stern, 2006).

It is possible to improve the achievement of low-performing students in mathematics if the

students are exposed to an environment that enables them to explore a wide range of

strategies of solving mathematical problems. Such exposures will likely help the students

understand how and why certain strategies work. Donovan and Bransford (2005) report that

exposing students to a wide range of solution strategies serves as a scaffold to help the

students move from their own conceptual understanding to more abstract approaches of doing

mathematics which involve their own reasoning and strategy development. However, some

teachers argue that exposing students to multiple strategies and heuristics will confuse the

students (Naroth, 2010). The findings of this study could be drawn upon to assess such views.

This study intends to find a way of helping low-performing Grade 12 mathematics students to

achieve better scores in determining the centre and radius of a circle.

Research design

In this study, the repeated measures research design was employed. This research design uses

the same participants for each treatment condition and involves each participant being tested

under all levels of the independent variable (Shuttleworth, 2009). The researchers adopted the

repeated-measures research design because it allows statistical inference to be made with

fewer participants and enables researchers to monitor the effect of each treatment upon

participants easily.

Sample

A purposive sample of twenty-five low-performing Grade 12 students from a secondary

school in the Capricorn District in Limpopo province took part in the study. Low performing

students are students who persistently scored below pass mark in mathematics examinations

for three years before this study. The school and the students were used because they

consented to participate in the study. According to Tabachnick and Fidell (2006), the

minimum sample size for detecting treatment effect(s) in a repeated-measures design is 10

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plus the number of dependent variables (2 in this case). Hence, the recommended minimum

sample size was satisfied.

Instrument

An achievement test was used to collect data to measure students‟ achievement in

determining centres and radii of given circles. The test items were generated based on the

concept and depth of knowledge specified in the National Curriculum Statement,

Mathematics Grades 10-12 (DoE, 2008). The test was made up of essay type questions

designed to allow the students to show their understanding of the two strategies of

determining the centre and radius of a circle. The appropriateness of the test items was

evaluated by six mathematics teachers who had at least five years of mathematics teaching

experience. After the evaluation process, the instrument was pilot-tested on a sample of ten

low-performing students from another school in order to detect and correct any errors and

ambiguities in the instrument before the main study was launched. The final instrument was a

ten-item instrument.

Reliability and validity of the instrument

The reliability of the achievement test was established by calculating the Kuder-Richardson

(KR 20) reliability estimate, using data from the pilot study. From the Kuder-Richardson 20

calculations, a reliability value of 0.91 was obtained meaning that the instrument was reliable

(Gay, Mill & Airasian, 2011).

The test‟s content validity was established through expert judgement. The experts were one

Mathematics subject advisor, one Head of Mathematics Department and four mathematics

teachers who had experience in teaching Grade 12. The experts independently judged

whether the test items reflected the content domain of the study. Based on their judgements,

the content validity ratio (CVR) of each item was calculated using

where

is the content validity ratio for the item; is the number of judges rating the item as

reflecting the content domain of the study and N is the total number of judges (Lawshe,

1975). The mean of the test items‟ CVRi was computed in order to find the content validity

index (CVI) of the test. A CVI value of was obtained which implies that there was

complete agreement among the judges that the test items reflected the content domain of the

study (Wynd, Schmidt & Schaefer, 2003).

Procedures

After the students were exposed to the two strategies of finding the centre and radius of a

circle, the test was administered to assess individual student‟s ability to use each of the two

strategies. Students wrote the test twice, using a different strategy each time. The duration of

the test was one hour fifteen minutes and it was marked out of sixty marks.

Research hypotheses

The following hypotheses were tested:

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There is no significant difference in the two sets of scores.

The two sets of scores are significantly different.

Findings

Table 1 shows the test scores of the learners using the two strategies.

Table 1. Learners‟ percentage scores

Learners (Using formulae) (Completing the square)

98 77

95 97

78 75

70 80

97 65

98 88

83 73

60 38

98 77

78 12

83 75

85 20

70 98

60 95

92 47

90 68

100 75

67 30

100 65

98 90

78 58

109

98 97

83 58

70 38

98 60

Since there were only two levels of data for analysis, a test of normality was performed in

SPSS in order to choose an appropriate statistical test for the data analysis. Table 2 shows the

results of the test of normality.

Shapiro-Wilk‟s test of normality

Hypotheses:

H O: There is no difference between the observed data distribution and a normal distribution.

HA: The data is non-normal.

Since the dataset for Strategy number 1 and Strategy number 2 are smaller than 2000

elements, we are to report the results under Shapiro-Wilk (Zar, 1999).

Table 2. Shapiro-Wilk‟s test of normality

Test of Normality: Shapiro-Wilk

Statistic df Sig.

scores .886 25 .009*

scores .935 25 .114

Note: * significant at p < .05

From Table 2 above, the Shapiro-Wilk‟s significance value for S1 scores (p =. 009) is less

than alpha (.05). Therefore, the null hypothesis is rejected and we conclude that the scores for

Strategy number 1 are not normally distributed. The significance value for S2 scores (p

=.114) is greater than the standard alpha (.05). This result is non-significant and hence we fail

to reject and conclude that the distribution of the S2 scores is normal. Since the distribution

of S1 scores violated the assumption of normality, it was inappropriate to analyse the data

using the ordinary paired-samples t-test. The Wilcoxon Signed-Ranks Test (a non-parametric

test equivalent to the paired samples t-test) which does not assume normality in the data was

used instead (Laerd, 2012).

Results of the Wilcoxon Signed-Ranks Test

The Wilcoxon Signed-Rank Test was performed in SPSS to evaluate the following

hypotheses:

H0: There is no significant difference in the two sets of scores.

110

Ha: The two sets of scores are significantly different.

Table 2 shows the main SPSS output for the Wilcoxon Signed- Ranks Test.

Table 3. Wilcoxon Signed-Rank Test Statistic

Test Statistics

scores – scores

Z -3.176

Asymp. Sig. (2-tailed) .001*

Note: * significant at p < .05

The p-value of the Wilcoxon Signed-Ranks Testis less than alpha meaning that the

difference in the scores for the two strategies is statistically significant. Therefore, we reject

H0 and conclude that the two sets of scores are significantly different

In order to see which scores were better, we analysed the results from the Wilcoxon

Signed-Ranks table.

Table 4. Wilcoxon Signed-Ranks Table

Ranks

N Mean Rank Sum of Ranks

scores - scores

Negative Ranks 21a 13.36 280.50

Positive Ranks 4b 11.13 44.50

Ties 0c

Total 25

a. scores < scores

b. scores > scores

c. scores = scores

The Wilcoxon Signed Ranks table (Table 4) shows that of the 25 participants obtained

higher scores with Strategy number 1(using formulae) than for Strategy number 2

(completing the square). Only participants obtained higher scores with Strategy number 2

than Strategy number 1. The negative mean rank is greater than the positive mean

rank , suggesting that most of the scores for Strategy number 2 were lower than those

for Strategy number 1. Hence, there is a significant difference in low performing learners‟

test scores due to using the two strategies to determine the centres and radii of circles.

Wetherefore conclude that the use of formulae (strategy 1) to find the centre and radius of a

circle could redeem the performance of many learners who might not be good at the strategy

of completing the square.

Discussion

The purpose of this study was to test whether there were significant differences in low

performing learners‟ test scores due to using two different strategies to determine the centres

and radii of circles with given equations and to determine which strategy helped the learners

to achieve better. Results from Wilcoxon Signed-Ranks test indicated that there were

111

statistically significant differences in the two sets of scores due to using two different

strategies to find the centres and radii of the given circles. Results from the Wilcoxon Signed-

Ranks table revealed that 84%of the participants achieved better test scores using formulae

than with the strategy of completing the square. It is important to note that the formulae that

were used by the learners to find the centres and radii of circles with given equations were

not in the recommended mathematics textbooks used in secondary schools. The implication

here for classroom practice is that educators should therefore not confine mathematics

teaching and learning to only strategies found in the prescribed textbooks. While the textbook

strategy of completing the square is a much better strategy (with wide ranging applications,

transferable across all of algebra) than following a formula in the context of mathematics, we

argue here that the students of the study were low-achievers who are unlikely to continue

with mathematics after Grade 12.

Another important implication of the findings of the study is that it is possible for

mathematics educators to improve their learners‟ achievement not only in finding the centre

and radius of a circle but also in other mathematical aspects. By exposing learners to multiple

solution strategies, educators can help many of their students learn and achieve better results

in mathematics, including those who might have lost hope of doing well in the subject.

According to Naroth (2010), exposing learners to multiple strategies of solving mathematical

problems will likely enhance their proficiency in problem solving. This view is also

supported by Donovan and Bransford (2005) who report that giving learners opportunities to

apply multiple strategies serve as scaffold as learners move from own conceptual

understanding to more abstract approaches of doing mathematics. Although it may take

several attempts to see positive results in learners‟ achievement, we should not give up. If one

strategy does not work, we should try another.

Recommendations and conclusion

Based on the findings of this study, we recommend that mathematics teaching and learning

should not be confined to only what is in the prescribed mathematics textbooks. Mathematics

teaching and learning will be enriched by broadening the strategies and processes

encountered in the classroom. Educators have to refer to as many text books as possible and

also use the internet in order to develop their repertoire of solution strategies. Educators can

even approach colleagues in their clusters for more support in teaching mathematical aspects

in which their students have difficulties. The Department of Basic Education could provide

packages of many mathematics textbooks to schools, as resources for educators to refer to.

We also recommend that the Department of Basic Education should appoint qualified

mathematics educators in secondary schools to teach Grades 10, 11 and 12.

Future research should extend this study to other mathematical aspects and Grade levels to

see if similar results are obtainable. Perhaps a similar study with a large randomised sample

of students can provide more definitive evidence to strengthen the present findings. This

study involved only low-performing Grade 12 learners hence, the findings should be

interpreted in that context.

112

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114

On South African primary mathematics learner identity: A Bernsteinian

illumination

Pausigere Peter

South African Numeracy Chair Project, Education Department, Rhodes University, South

Africa [email protected]

This paper is theoretically informed by Bernstein‟s (1975) earlier work on learner positions

and his notion of pedagogic identity (Bernstein, 2000), supplemented by Tyler‟s (1999)

elaboration of the model. The paper analyses key primary mathematics curriculum policy

documents to investigate the official primary mathematics learner identity as constructed by

the current South African education curricula. In order to analyse learner identity we need to

consider their relationship to the promoted primary mathematics teacher identity. In our

earlier study (Pausigere & Graven, 2013) we revealed that the recent South African

curriculum policy changes constructs and promotes a “Market” (Bernstein 2000) primary

mathematics teacher identity and we argue in this paper that this relates with the

“Detachment” (Bernstein, 1975) pupil learner identity position. Drawing on Bernstein‟s

(1975; 2000) work, I construct a theoretical model, that relates the pedagogic identity classes

and the pupil learner positions based on framing elements and the classification concept. I

finally discuss the implications of such primary mathematics identities for the teaching and

learning of mathematics.

Introduction

In this paper I investigate the type of learner identity promoted by the recent changes in South

African primary mathematics education, as revealed in curriculum policy documents. Our

earlier study (Pausigere & Graven, 2013) on how the current CAPS curriculum changes

project a particular primary mathematics teacher identity motivated this research. To help us

explain the construction of the local primary mathematics learner identity we draw both on

Bernstein‟s (1975) earlier work on how pupils position themselves to school work in relation

to the instrumental and expressive orders and his pedagogic identity model(Bernstein, 2000),

which explains how different modalities of curricular reform construct different identities

We supplement Bernstein‟s concept of pedagogic identities with the findings of Tyler‟s

(1999) study, which interprets Bernstein‟s (2000) pedagogic identity categories in terms of

knowledge coding properties (that is, classification and framing) and also extends this

theoretical foundation to learner identity classes (Bernstein, 1975).

Bernstein (1975) presented a framework for analysing how pupils relate to school work; he

expressed the learner positions as a function of both the expressive and instrumental orders.

Bernstein (1975) also introduced the pupil learner identity categories to understand how

British pupils defined their school roles in terms of their social class position. Later in his

career Bernstein (2000) used the concept of pedagogic identity to analyse Britain‟s

contemporary educational reforms. Tyler (1999) also used and extended the pedagogic

identity model in the Australian education context which like the England‟s National

115

Curriculum reforms both began in the late 1980s and were characterised by a common

curriculum framework and the compulsory testing of primary learners in core subjects.

Recently in primary mathematics education, South Africa has also experienced some

curriculum reforms changes, which witnessed in 2011 the introduction of universal

standardised primary learner Annual National Assessment (ANA) tests in numeracy and

literacy and the implementation of a common curriculum framework (Curriculum and

Assessment Policy Statement, CAPS) at the primary level in 2012. This development is

similar to the education reforms experienced in the United Kingdom and Australia in the last

quarter of the century. The question therefore arises of how the South African primary

mathematics learner identity, promoted by the current South African mathematics education

policies, relate to Bernstein‟s (1975) earlier work on pupil learner positions and the

pedagogic identity model (Bernstein, 2000). Following Tyler (1999) and extending both his

scheme and Bernstein‟s work we explain the relationship between the pedagogic identity

categories (Bernstein, 2000) and the pupil learners positions (Bernstein, 1975) and express

these as a function of the framing elements (expressive/regulative and

instrumental/instructional orders or discourse) as well as the classification concept. Our

earlier work, in which we argued that the current CAPS curriculum changes project a

“Market” (Bernstein, 2000) primary mathematics teacher identity (Pausigere & Graven,

2013) also illuminates our interrogation of the South African learner identity.

To investigate the notion of primary mathematics learner identity we analysed key national

curriculum documents, we focused mainly on the CAPS primary mathematics policy

documents. We also draw from policy documentation relating to ANA in our discussion of

CAPS as ANA is part of the interventions associated with CAPS. Embedded in these

curriculum policy documents is an officially sanctioned version of primary mathematics

learner identity (Tyler, 1999, Bernstein &Solomon, 1999). Coupling our theoretical

perspective with our document analysis indicates that the current CAPS curriculum changes

project a “Detachment” (Bernstein, 1975) primary mathematics learner identity that closely

relates with the “Market” (Bernstein, 2000) primary mathematics teacher identity, which we

disclosed in our earlier study (Pausigere & Graven, 2013).We analyse the implications of

such mathematical identities on the teaching and learning of primary maths.

Literature Review

This paper will narrow its literature review to studies that focus on the concept of

(mathematical) teacher and learner identities and those that are theoretically informed by

Bernstein‟s work. There have been both local and international studies drawing upon different

aspects and ideas of Bernstein‟s theoretical concepts in order to study the notion of

mathematical teacher identity. Bernstein‟s (1971, 2000) classification and framing theory and

the pedagogic model have been used to study mathematical teacher‟s official pedagogic

identities within reform contexts in South Africa (Parker, 2006; Graven, 2002; Pausigere &

Graven, 2013), in Britain (Morgan et al, 2002; Morgan, 2005) and Sweden (Johansson,

2010). Bernstein‟s concepts of pedagogic models and pedagogic discourse have also been

used to study official learner identities (Muller, 2000; Bourne, 2006) and primary school

116

learner identities (Hempel-Jorgensen, 2012).Closely related and relevant to this study is

Johansson‟s (2010) paper and our work (Pausigere & Graven, 2013) that has been informed

by Bernstein‟s concept of pedagogic identities to study school mathematics reforms in

Sweden and primary teacher identity in South Africa. There however have been no studies

that have drawn on Bernstein to investigate the notions of mathematics learner identity or

primary mathematics learner identity, furthermore I have not found in published work or

conference proceedings studies that interrelates mathematics teacher and learner identities

using Bernstein‟s theoretical lens. This study thus contributes to these identified gaps in the

literature; firstly of investigating primary mathematics learner identities and secondly of

exploring the relationship between teacher and learner identities informed by Bernstein‟s

constructs of pedagogic identity and pupil learner positions.

Theoretical Framework

In investigating the officially projected South African primary mathematics learner identities

this paper draws on Bernstein‟s (1975) earlier work about how pupils define their school

roles, Bernstein‟s (2000)concept of the pedagogic identities and Tyler‟s (1999) extension of

the model. This paper relates and links Bernstein‟s (2000) four pedagogic identity categories

and four of the “five types of pupil role involvements” (Bernstein, 1975, p. 43). Bernstein

(1975) explains the pupil learner identity positions as a function of the instrumental and

expressive orders. To help us illustrate and explore the interconnectedness of Bernstein‟s

(2000) pedagogic identity classes and the pupils‟ school role categories (Bernstein, 1975), is

Tyler‟s (1999) work, which explains how pedagogic identities and their realisations are

constructed by variations in classification and framing relations. The pedagogic model

(Bernstein, 2000; Tyler, 1999) illuminates our understanding of the South African primary

mathematics learner position. Following Tyler‟s (1999) model I extended Bernstein‟s (1975)

school learner roles and express these as a function mainly of framing properties and relate

these to the classification concept, thereby establishing criteria and a basis on which to

connect the learner‟s positions with the pedagogic identity classes.

Central to Bernstein‟s pedagogic identity model (Bernstein, 2000; Bernstein & Solomon,

1999) is the argument that the official knowledge and pedagogic modalities of curriculum

reforms distributed in educational institutions construct, embed and project different official

pedagogic identities. Bernstein‟s concept of pedagogic identities generated four distinct

pedagogic identity positions, namely Conservative, Neo-Conservative, Therapeutic and

Market, with Tyler‟s (1999) study, explaining how the pedagogic identity categories are

outcomes of classification and framing principles. Key also for this study are Bernstein‟s

(1975) four of the five types of pupil role involvements; Commitment, Detachment,

Deferment, Estrangement and Alienation whose construction are realised by the instrumental

and expressive orders. The Deferment learner position cannot be linked to any of the

pedagogic identity categories as this learner, according to Bernstein, is not involved either in

the expressive or instrumental orders of the school. Bernstein (1975) also expressed his

categories on how pupils relate to the school in relation to social class positions and these will

not be considered in this paper.

117

Before discussing the relationship between Bernstein‟s pedagogic learner positions and the

identity categories, I briefly explain, showing similarities where necessary, between the

framing concept and the instrumental and the expressive orders. The expressive order is

similar to what Bernstein in his later work calls the regulative discourses or social order

rules and these establish the conditions for conduct, character and manner of the school

(Bernstein, 1975) or in the pedagogical relation (Bernstein, 2000; 2003). The regulative

discourse also refers to the “forms of hierarchical relations in the pedagogic relation” and this

can lead to the creation of either explicit hierarchical or implicit hierarchical relationships

(Bernstein, 2000, p. 13; 2003). The instrumental order closely relates to the instructional

discourse or discursive rules and both are concerned with how knowledge is transmitted and

acquired (Bernstein, 1975),in fact it refers to the selection, sequence, pacing and criteria of

knowledge (Bernstein, 2000; 2003). The expressive/regulative discourse/social orders rules

and the instrumental/instructional discourse/discursive rules are a function and elements of

framing with Bernstein defining framing as follows:

Framing = instructional discourse ID

regulative discourse RD

Bernstein (2000, p. 13) distinguishes between the instructional and the regulative discourse,

with the former being “always embedded in the regulative discourse” and the latter being the

“dominant discourse”. It is important to note that the strength of the instructional and

regulative discourses and also the elements of the instructional discourse can vary

independently of each other (Bernstein, 2000).

Classification and framing, according to Bernstein (1971) determine the structure of

curriculum (knowledge), pedagogy and evaluation in any education system. The concept of

the frame “determines the structure of the message system” and refers to the “specific

pedagogical relationship of teacher and taught” (Bernstein, 1971, p. 205). According to

Bernstein (1971; 2000) where framing is strong, there is a sharp boundary between what may

be and may not be transmitted and the transmitter has explicit control over selection,

sequencing, pacing, criteria and social base. Where framing is weak, there is a blurred

boundary between what may be and may not be transmitted and the acquirer has more

apparent control over the communication and its socialbase. Classification on the other hand

is concerned with the organisation of knowledge into curriculum, with strong classification,

areas of knowledge and subject contents are well insulated into traditional subjects

(Sadovnik, 2001; Bernstein, 1971). Weak classification refers to an integrated curriculum

with blurred boundaries between contents (Sadovnik, 2001; Bernstein, 1971). It is important

also to note that evaluation is a function of the strength of classification and framing, yet the

strengths of the classification and framing can vary independently of each other (Bernstein,

1971).

Whilst there are similarities between the expressive order and the regulative discourse and on

the other hand between the instrumental order and the instructional discourse, this study also

argues, following our theoretical underpinnings, that the criteria for linking the pedagogic

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identity categories and the learner positions is based on the connection between the

instrumental/instructional order and classification and expressive/regulative order and

framing, and their respective strengths. Thus a strong regulative discourse or expressive order

(R/E+) leads to strong framing (F+) whilst a weak regulative discourse or expressive

order(R/E-) leads to weak framing (F-). A strong instrumental order or instructional rules (I+)

points towards strong classification (C+), whilst a weak instrumental order or instructional

rules (I-) points towards weak classification (W+).These two propositions benefit from

Bernstein‟s (1975) earlier work and the pedagogic identity model (Bernstein, 2000) which

Tyler (1999) relates to classification and framing principles. In the table below I show

framing, firstly as made up of both the instrumental and the expressive orders and secondly as

resulting from the combined strengths of the instructional and regulative rules. Table 1 below

indicates the interconnectedness of Bernstein‟s pedagogic identity categories and the learner

position classes.

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Table 1. The inter-connectedness of Bernstein‟s pedagogic identity classes and the learner‟s

positions in terms of classification and framing

Pedagogic

identity

classes

Learner

positions

Framing Framing Classification

Instructional

Instrumental

Regulative

Expressive

Conservative Commitment I + R/E + F+ C+

Market Detachment I + R/E - F - C +

Therapeutic Alienation I - R/E - F- C -

Neo-Conservative Estrangement I - R/E + F+ C -

I discuss below the relationship between each of the four learner positions (Bernstein, 1975)

and the four pedagogic identity categories (Bernstein, 2000). I express firstly the pedagogic

identities categories as a function of framing and classification and draw similarities with the

learner positions based on these two key educational knowledge codes properties. The space

to relate and express both identities as outcomes of classification and framing emanates from

our theoretical orientations (Bernstein, 2000; Tyler, 1999). It is also practically impossible to

discuss the learner position without discussing the pedagogic categories for in the

pedagogical relation the „taught‟ co-exists with the „teacher‟.

Conservative Pedagogic identities are “formed by hierarchically ordered, strongly bounded,

explicitly stratified and sequenced discourse and practices” (Bernstein, 2000, p. 67). Tyler

(1999) thus explains that in terms of educational codes this identity position can be described

as having both strong classification and framing properties typical of a collection code, as

was the case with Britain before the 1960s. The Conservative pedagogic identity class relates

to the Commitment pupil position whose, “behaviour is appropriate and committed”. S/he

“spontaneously produces the behaviour accepted by the school in both its expressive and

instrumental orders” (Bernstein, 1975, p.44). The Conservative pedagogic identity exhibits

strong classification and strong framing, which resonates with the explicit and strong

instrumental and expressive orders characterising the Commitment position.

Bernstein (2000) also identified the Market position, which focuses on producing competitive

output-products (students) with an exchange value in a market and constructing an outwardly

responsive identity driven by external contingencies. This identity is also orientated towards

the intrinsic value of the discourse responsible for the serial ordering of subjects in the

curriculum, and has to contend with the possible tension between enhancing learners‟ test

performance and teaching disciplinary knowledge. This pedagogic position according to

Tyler‟s (1999) theoretical scheme is weakly framed but strongly classified. I relate the market

pedagogic identity category with the Detachment learner position. The Detachment learner is

“involved in the instrumental order, but he is cool or negative towards the expressive order”

yet “he is eager to learn and pass examinations” (Bernstein, 1975, p. 45). A weak expressive

120

order leads to weak framing whilst a dominant instrumental order translates to strong

classification, and it is on this basis that I relate and link between the Detachment learner role

and the Market position. Quite common to both positions is their interest in “examinations”

or “tests”. In our earlier work we discussed how the Market primary mathematics pedagogic

identity is promoted in the CAPS curriculum (Pausigere & Graven, 2013). This paper will

explain how these pedagogic and learner identity positions are reflected in the current

changes in the South African primary mathematics education.

Neo-Conservative Pedagogic identities are “formed by recontextualising selected features

from the past to stabilise the future through engaging with contemporary change” (Bernstein,

2000, p. 68). Because of its dual desire to stabilise the past and engage with change, this

teacher identity category exhibits strong framing typical of the Conservative position, yet its

disregard for traditional disciplinary boundaries and academic identities leads to weak

knowledge classification (Bernstein, 2000; Tyler, 1999). The Neo-conservative pedagogic

identity relates with the Estrangement learner position who is “highly involved in the

expressive order” and his behaviour is “consonant with the image of conduct, character,

manner and the moral order of the school” (Bernstein, 1975, p. 46). The high involvement in

the expressive order translate to strong framing, yet the estrangement learner cannot manage

the demands of the instrumental order, “…it is all a bit difficult for him” , thus this learner

prefers weakly classified practices. There is therefore resonance between the Neo-

conservative identity category and the estrangement learner position.

Therapeutic pedagogic identities are “produced by complex theories of personal, cognitive

and social development, often labelled progressive” (Bernstein, 2000, p. 68). The Therapeutic

position projects autonomous, sense-making, integrated modes of knowing and adaptable co-

operative social practices that create internal coherence. Tyler (1999, p. 276) describes the

Therapeutic position as “weakly classified and framed since it exhibits low specialisation and

localised, adaptable practices”. In our earlier work (Pausigere & Graven, 2013) we discussed

how this identity position was promoted through Curriculum 2005(C2005), launched in South

Africa in the late 1990s. The therapeutic pedagogic category relates with the Alienation

learner position where “the pupil does not understand, and rejects both the instrumental and

the expressive orders of the school” and this fits with the weak classification and framing of

the therapeutic identity position (Bernstein, 1975, p.46; Tyler, 1999).

Figure 1 relates and links Bernstein‟s (2000) four pedagogic identity categories and four of

the five pupil learner identity positions (Bernstein, 1975) and expresses these as a function of

framing and classification. The model developed here, whilst informed by Tyler‟s (1999)

scheme, also extends Bernstein‟s (2000, 1975) work on pedagogic and learner identity

classes, and can be used in other studies to investigate national-official learner and teacher

identities.

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Classification

Strong Weak

Weak Market Therapeutic

Detachment Alienation

Framing

Conservative Neo-Conservative

Strong Commitment Estrangement

Figure 1.Bernstein‟s pedagogic identity and learner identity classes repositioned according to

classification and framing properties.

It is this close link and connection between Bernstein‟s pedagogic identity categories and the

pupil learner positions that provides us with exciting possibilities of investigating and relating

the primary mathematics learner position and the pedagogic identity class in the South

African curriculum reform context.

Research Method - Document Analysis

The data collection technique and strategy used for this descriptive qualitative study is

document analysis also called content analysis (Best & Kahn, 2006). The main, primary and

official sources of data analysed for this paper are the South Africa‟s Department of (Basic)

Education‟s curriculum policy documents and statements. Thus the study analysed and

scrutinised CAPS curriculum documents, primary mathematics education subject guidelines

for the Foundation and Intermediate phase, the Foundations for Learning Campaign policy

document and Annual National Assessment reports. Content analysis of official education

policies and curriculum documents is the most suitable and relevant data collection strategy

for interpreting and studying the official projected South African primary mathematics

teacher and learner identities. Some studies cited in the literature review that have

investigated notions of mathematics teacher identity and learner identity drawing on

Bernstein‟s work have also analysed their respective national curriculum and policy

documents (Graven, 2002;Parker, 2006; Johansson, 2010: Muller, 2000; Bourne, 2006;

Morgan et al, 2002; Pausigere & Graven, 2013; Hempel-Jorgensen, 2013). These documents

The

State

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spell out the official teacher and learner identities as perceived and intended by the

Department of Education or the national government.

A deductive data analysis approach that is theory-driven was used to synthesise and make

sense of data obtained from curriculum policy documents and statements and also in

presenting our research findings (Best &Kahn, 2006). Thus the coding and exploration of

data was theoretically guided mainly by Bernstein‟s (1975; 2000) pupil learner positions and

the pedagogic identity model supplemented with Tyler‟s (1999) insightful interpretation of

Bernstein‟s work. Bernstein‟s pupil learner positions and pedagogic identity model provides

an analytic tool that serves as a template, to position the local primary mathematics learners

and teachers in the current education reform and change context. Bernstein (2000) and

Tyler‟s 1999, p. 277) typology of pedagogic identity also provides the “langue of reform” for

describing and explaining firstly the officially projected primary mathematics teachers‟

identities and relating these to learner positions. Such structuring of data places learner and

teacher identity at the centre and assists in explaining how primary mathematics learners and

teachers are projected and constructed through the official educational discourse. The unit of

analysis for this study is “Primary mathematics learner identity”. I focus on how

contemporary resources construct who South African primary learners are, with respect to the

subject of mathematics (Bernstein & Solomon, 1999).

Discussion - CAPS‟ Detachment primary mathematics learner position and the Market

primary mathematics teacher identity

In this part of the paper I discuss the primary mathematics learner identity projected by South

Africa‟s most recent curriculum changes. I explain how the recent curriculum restructuring

projects a Detachment learner position which relates with the Market primary mathematics

teacher identity; both are interpreted in relation to framing elements and the classification

principle.

The CAPS primary mathematics curriculum documents emphasise the need for learners to

acquire key mathematical knowledge and deep conceptual understanding. The main focus

falls on the first of the five content areas, “numbers, operations and relations”, which makes

up half of the foundation and intermediate phase mathematical content. The focus stems from

the intention of ensuring that learners “secure number sense and operational fluency” and

“develop more efficient techniques for calculations” (DBE, 2011a, p. 8; DBE, 2011b, p. 13).

The importance of mental maths initially highlighted in the Foundations for Learning

Campaign, launched in 2008, also features strongly in the primary mathematical curriculum,

which promotes “number bonds”, “multiplication table facts” and “calculation techniques”

(DBE, 2011a, p. 8; DBE, 2011b, p. 35; DOE, 2008). The primary mathematics curriculum

documents also highlight the need for learners to engage in problem-solving activities,

thereby creating a context for the development of higher order mathematical concepts (DBE,

2011a; DBE, 2011b). South African primary mathematics education‟s focus on improving

learners‟ number sense, operational fluency, mental maths and problem solving aligns with

the influential and international primary mathematical studies that have identified these

mathematical activities as central for developing learners‟ mathematical proficiency. The

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resulting primary mathematics teacher and learner identity thus corresponds firstly with

Bernstein‟s Market pedagogic position, which is strongly classified (Tyler, 1999) and the

Detachment learner position under which the pupil is strongly involved in the instrumental

order or the instructional discourse (Bernstein, 1975).

To understand the envisaged primary mathematics learner identity we also look at the

depicted primary mathematics teacher identity in terms of the key instructional elements of

“selection, sequence, pacing and criteria of knowledge” (Bernstein, 2000, p. 13). A strong

instructional discourse or instrumental order is evident in the CAPS primary mathematics

curriculum documents through its specification, clarification, timing and sequencing of

content from grade to grade across the four terms of the year (DBE, 2011a; DBE, 2011b).In

the curriculum strong pacing and sequencing is indicated through grade by grade

“specification of content to show progression” (DBE, 2011a, p. 19; DBE, 2011b).Such

sequencing serves to indicate the “progression of concepts and skills”, how content can be

adequately spread over time and give guidance “on the spread of content in the

examination/assessment” (DBE, 2011a, p. 19, 11; DBE, 2011b).Bernstein‟s (2003, p.

206)elaboration that “with strong pacing, time is at a premium” is also illustrated in the

primary mathematics curriculum documents‟ recommended distribution and allocation of

mathematics teaching topic-cum-time schedules (DBE, 2011a; DBE, 2011b). Furthermore the

CAPS primary mathematics school-based formal assessment tests and examinations(DBE,

2011a, DBE, 2011b), give rise to ordered principles of evaluation which emphasis that the

pupil reveals relatively objective procedures and leads to a strong instructional discourse,

especially on the criteria aspect of the discursive order (Bernstein, 1971; 2000).The listing of

the school-based formal assessments recommended under the new curriculum and the explicit

stating and timing of the mathematical concepts to be relayed and acquired at the primary

level indicates strong instructional discourse elements or an explicit instrumental order.

Foregrounding the instructional discourse resonates with the Market pedagogic identity

which emphasises in this case deep conceptual mathematical knowledge typical of strong

classification and relates with the Detachment learner who engages in the instrumental order.

Whilst the instructional discourse of CAPS primary mathematics is strong there is however

indications that the regulative discourse of the CAPS curriculum carries mixed messages of a

weak and strong social order. A weak regulative discourse is evident in the CAPS curriculum

which, like the previous curricula, is founded on and retains allegiance to the principles of

“social transformation… human rights, inclusivity and social justice” that were fore grounded

in C2005 (DBE, 2011a, 3). Thus the curriculum still emphasises learner-centred approaches

such as “small group focused lessons” or interactive group work sessions in which learners

should be encouraged to “talk, demonstrate and record their mathematical thinking” (DBE,

2011a, p. 9; DBE, 2011b). The new primary mathematics curriculum policy documents

encourage an active and critical approach to learning, under which teachers accommodate

learners‟ computational strategies (DBE, 2011a; DBE, 2011b). This also concurs with

Bernstein‟s (2003, 2000, p. 13) assertion that under an implicit social order the acquirer

“struggles to be creative, to be interactive, to attempt to make his or her own mark”. Such

weak regulative discourse practices consequently impact on the instructional discourse which

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in the local case, has resulted in the CAPS primary mathematics evaluative practices to

monitor learners‟ daily progress through informal assessments, such as observations,

discussions, practical demonstrations, learner-teacher conferences and informal classroom

interactions (DBE, 2011a, 2011b). These informal evaluations of primary mathematics

learners give rise to "multiple criteria of assessment” which emphasis the “inner attributes of

the student” and points towards weak framing (Bernstein, 1971, p. 223, 224).A weak

regulative discourse is also evident in the curriculum‟s subject guidelines which leaves room

for primary mathematics teachers to “sequence and pace the maths content differently from

the recommendations” in the policy documents (DBE, 2011b, p. 32). According to Bernstein

(2000) the Market position radically transforms the regulative discourse of the institution as

this affects its conditions of survival, resulting in both a weak regulative discourse and a

weakly framed transmission (Tyler, 1999). Similarly CAPS did not forego the social

transformation and political pedagogical intentions that initially set the groundwork for

curriculum reform in South Africa and these are carried through. Such a weak regulative

discourse both closely relates with the Detachment learner position which is apparently

uninvolved in the expressive social order and the key feature of the Market teacher identity

which is sustained in weak framing.

On the other hand there is also evidence that the regulative discourse of the primary

mathematical curriculum is strong. The strong regulative discourse might be emanating from

the CAPS‟ primary mathematical curriculum documents‟ emphasis on the need for learners to

acquire key mathematical knowledge and deep conceptual understanding which indicates

strong classification. Secondly the CAPS primary mathematical classroom teaching and

learning practices also emphasis teacher-centred and independent activities that foreground

mathematical concepts and skills. The whole class activity teaching approach is outlined as

the main teaching strategy meant to consolidate key mathematical concepts, promote mental

mathematics and independent activities (DBE, 2011a).The fact that the individual learners

have to engage in independent mathematical activities closely relates with an explicit

regulative discourse or conditions for a strong social order. The emphasis in the primary

mathematics curriculum documents, of the whole class teaching approach and independent

learner activities, indicates that the pedagogical relationship between the primary

mathematical teacher and learner shows some hierarchical relations characteristic of strong

regulative discourse. According to Bernstein (2003; 2000, p. 13) under such explicit rules of

the social order the candidates for labelling the acquirer are such terms as “conscientious,

attentive, industrious, careful, receptive”.

Whilst the CAPS‟ instructional discourse elements are strong, the regulative discourse carries

mixed messages of a weak and strong social order. Because of such mixed transmission

signals the primary mathematical classroom teaching and learning practices, allow for both

learner-centred and teacher-centred activities that foreground mathematical concepts and

skills. This has an impact on framing which is a function of both the instructional and the

regulative discourse; the latter is the dominant discourse which in the CAPS case shows both

a weak and strong social base. In other words the strength of the frame is determined by the

regulative discourse. Analysis of the primary mathematics curriculum documents using

125

Bernstein‟s work (1971; 2000) and Tyler‟s (1999) theoretical insights, indicates that the new

curriculum‟s framing ought to be weak, so as to resonate with the Market pedagogic identity

position which relates with the Detachment learner position that is negative towards the

expressive order or the regulative discourse. However from both a theoretical perspective

(Bernstein, 1971, 1975, 2000; 2003; Tyler, 1999) and an analysis of the primary mathematics

curriculum documents there is evidence that the new curriculum‟s framing is strengthened

and thus stronger than C2005‟s frame. The CAPS‟ strengthened frame results from the strong

instructional discourse elements and some hierarchical pedagogical relations promoted in the

primary mathematics‟ regulative discourse.

The strengthening of the frame under CAPS could also be a result of the type of mathematical

knowledge supposed to be learnt in local primary classes, especially given the fact that the

new curriculum puts emphasis on the learners‟ operational fluency. This argument emanates

from Bernstein‟s (1971) assertion that the form of knowledge transmitted affects the nature of

the framing. It logically follows that the strong CAPS content knowledge classification has

resulted in a strengthened primary mathematics frame. It is also useful to view strengths of

classification and framing along a continuum rather than simply as polar opposites of strong

and weak classification and framing. Because the CAPS primary mathematics curriculum‟s

framing is strengthened, the resultant primary mathematical teacher identity is orientated

towards a strengthened frame and strong classification, a position that we argued for in our

earlier work (Pausigere & Graven, 2013). The strengthening of the framing also impacts on

the Detachment learner position whose expressive order has to align with this new

development resulting in a strengthened expressive order. These findings add a new

dimension and perspective to Bernstein‟s (1975) earlier work on the learner positions and to

the pedagogic identity model (Bernstein 2000; Tyler, 1999). It also shows how the theory

(Bernstein, 2000, 1975; Tyler, 1999) has illuminated my understanding of the local

Detachment primary mathematics learner and the Market primary mathematics teacher

identity positions.

There is a striking similarity between the Detachment learner position (Bernstein, 1975) and

the Market pedagogic identity category (Bernstein, 2000, 2003) concerning their interest and

high regard for (universal standardised learner) tests and examinations. This trend emerged

locally in the form of a national roll out in 2011 of standardised tests that are aimed at

ensuring that 60% of learners achieve 50% and above in literacy and numeracy by 2014

(DOE, 2008). The 2012 ANA national mathematics mean scores reveal that the Grade 1 and

2 learners have achieved above the set targets whilst the Grade 3 to Grade 6 scores are still

far below the desired threshold (DBE, 2012). In fact performance tends to decline as one

moves up the grades with 77.4% of Grade 1 learners achieving over 50% for mathematics

reducing to 67.8%, 36.3%, 26.3%, 16.1% and 10.6% for grades 2 to 6 respectively. Under the

new national monitoring measures all South African primary learners undergo Annual

National Assessments (standardised tests) to monitor, track and improve the level and quality

of their literacy and numeracy (mathematics) levels across Grades 1 to 6 and Grade 9 (DBE,

2008; 2011; 2012). Secondly, the ANA tests are meant to serve as a diagnostic tool for

identifying areas of strength and weakness in teaching and learning, which can ameliorate

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classroom assessment practices and inform the teaching and learning of literacy and

numeracy (DBE, 2011; 2012). Thirdly, from an education policy management perspective,

the ANAs provide credible and reliable information to monitor progress, and guide planning

and the distribution of resources to help improve learners‟ literacy and numeracy knowledge

and skills (DBE, 2011; 2012). Both Bernstein (2003) and Tyler (1999), argue that the

periodic mass testing of learners enables centralised monitoring and the homogenisation of

educational practices, thereby creating performance indicators for accountability,

transparency and efficiency. The fact that the Detachment learner position “wants to do well;

he is eager to learn and pass examinations” (Bernstein, 1975, p. 45) closely relates with the

market pedagogic identity category whose focus is on enhancing learner performance in

national standardised tests. In the same way the South African Detachment primary

mathematics learner and the market primary mathematics teacher identity are both concerned

with performing well in the ANA tests.

The South African primary mathematics detachment learner and market teacher identities

have to meet the dual challenge of teaching and learning key mathematical concepts and

improving their performance in the ANA tests. The teacher identities in this category must

negotiate the tension between “satisfying external competitive demands” and “the intrinsic

value of the discourse” (Bernstein, 2000, p. 71). In the same way the Detachment learner

position is also strained by his “eager to learn and pass examination” and his negative attitude

“towards the expressive order” (Bernstein, 1975, p. 45). Thus both the market pedagogic

identity and the Detachment learner position are in a “Janus-schizoid position” characterised

by conflicting or contradictory ideas (Bernstein, 2000, 2003). The market pedagogic identity

category is “ideologically a much more complex construction” so is the Detachment learner

position which is a “more interesting situation” (Bernstein, 2003, p. 213; Bernstein, 1975, p.

45). Both identities have revealed themselves in the South African primary mathematical

education context in a slightly changed form; they thus currently both exist in strong

instrumental orders, strong classification with a focus on tests - typical of the market and the

detachment positions however their framing and the expressive order has been strengthened.

Concluding Remarks

This paper sought to investigate the type of primary mathematics learner identity portrayed

by the current changes in the South African mathematics education as contained in

curriculum policy documents. It also explains how the promoted South African primary

mathematics learner identity can be linked to a particular teacher identity whose theoretical

genesis is Bernstein‟s (1975; 2000)earlier work on pupil learner positions and the pedagogic

identity model which are expressed as a function of classification and framing elements,

following Tyler‟s (1999) elaboration of the pedagogic identity concept. My findings, which

bear the influence of a particular methodological and theoretical lens, indicate that the new

CAPS curriculum constructs a detachment primary mathematics learner position and a

market primary mathematics teacher identity, which are both interested in the teaching and

learning of fundamental mathematical concepts and partaking in national tests. We

prophetically depict and picture the future South African primary mathematics learner and

127

teacher identities heeded towards a Commitment learner position which is strongly involved

in both the expressive and the instrumental orders and the Conservative pedagogic identity,

characterised by strong classification and framing. Whilst our key findings are applicable to

primary mathematics learners they can also be extended and generalised to understand South

African teacher and learner identities in the new curriculum dispensation and in the future.

The pedagogic-learner identity model outlined in this paper can be used in other countries to

investigate teacher and learner identities.

To conclude this paper I raise critical issues concerning learner and teacher identities, the

teaching and learning of primary mathematics and curriculum and policy development.

Firstly a critical issue raised by Hempel-Jorgensen (2009), which is applicable to the current

local curriculum changes, concerns the focus on learner performance in national assessments

which she argues compromises the development of learning disposition in schools. Similarly

I argue that the focus on primary mathematics learner performance in the ANAs retards the

development of a primary mathematics learner identity that embraces maths learning

dispositions. Secondly the over-prescription of content in local primary mathematics

curriculum subject guidelines can erode primary teachers‟ professional autonomy and

responsibility, thus challenging their professional identity, a point also elaborated by Morgan

(2005) and Hempel-Jorgensen (2009) in Britain‟s National Curriculum reforms context.

Using Dowling‟s (1998) principles, Morgan (2005) argues that over-specification of content

and the concern with assessment leads to specialising-proceduralising strategies that focus on

the procedure required for the construction of legitimate texts for evaluation which distribute

to learners and teachers “dependent” voices. What might be relevant for the local primary

mathematics education are subject guidelines and policy documents that distribute

specialising-principling strategies whereby the understanding, competences and reasoning

behind the mathematics are required for the construction of the legitimate texts for

evaluation. In other words the South African primary mathematics documents and policy

statement must encourage the teaching and learning of mathematics to focus on the how and

why which is generative, and not mainly emphasis on the what, as is the current situation.

Acknowledgements

This work is supported by the South African Numeracy Chair, Rhodes University; as usual

author disclaimer conditions apply.

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The allure of the constant difference in linear generalisation tasks

Duncan Samson

Rhodes University, Grahamstown, South Africa

[email protected]

This paper highlights the allure of the common difference within the context of figural pattern

generalisation. The study centres on an analysis of pupils‟ lived experience while engaged in

the generalisation of linear sequences presented in a pictorial context. The study is anchored

within the interpretive paradigm of qualitative research and makes use of the complementary

theoretical perspectives of enactivism, knowledge objectification and figural concepts. A

micro-analysis of a vignette is presented to support the central thesis of this paper – that in

the context of linear pattern generalisation the inherent allure of the numeric and visual

analogues of the constant difference has the potential to create tension between different

apprehensions or „ways of seeing‟.

Introduction

Generalisation can broadly be described as “deliberately extending the range of reasoning or

communication beyond the case or cases considered, explicitly identifying and exposing

commonality across cases” (Kaput, 1999, p. 136). As such it is “an important aspect in

mathematics that permeates all branches of the subject” (Dindyal, 2007, p. 236).

Generalisation can thus be seen as a core component of mathematical activity. As Mason

(1996) succinctly puts it, generalisation is the “life-blood, the heart of mathematics” (p. 74).

Generalisation can be seen as both a process and concept and is “a critical aspect of algebraic

thinking and reasoning” (Becker & Rivera, 2008, p. 1). The use of number patterns as a

didactic approach to engaging with the concept of generalisation and algebraic reasoning has

become standard practice around the globe. Rather than presenting tasks in a purely numeric

context, pattern generalisation activities are often provided to students in a pictorial or

practical context such as dot patterns, tiling patterns, matchstick patterns, as well as two- and

three-dimensional building block patterns.

Number patterns presented in the form of a sequence of pictorial terms are more than simply

a visual representation of a given numeric pattern. An important difference between numeric

and pictorial patterns is that a pictorial representation is inherently less ambiguous than its

isomorphic numeric counterpart. This can be understood by reflecting on the fact that a finite

numeric sequence can be generated by an infinite number of functions. Thus, although no

finite sequence of numerical terms uniquely specifies the following term in the sequence (see

for example Mason, 2002; Samson, 2012) this is not the case for pictorial sequences, since

the pictorial context suggests a deeper underlying structure. In addition, and perhaps more

importantly from a pedagogical perspective, number sequences presented in a pictorial

context have the potential to allow for a deeper engagement with generalisation, both as a

process and a concept, as the pictorial context allows for a greater scope and depth of

interpretation.

131

By way of example, consider the pictorial sequence shown in Figure 1. Reducing the pictorial

context to its isomorphic numeric analogue, i.e. the numeric sequence 4; 7; 10; …, readily

leads one to the general expression 13 nTn . This can be arrived at by any number of

standard algorithmic approaches. However, by engaging with the pictorial context itself, a

context that is open to multiple visual interpretations, there is a far richer diversity of

potential expressions of generality, where different yet algebraically equivalent expressions

stem from different routes of visual engagement. Duval (1998) makes the pertinent point that

most diagrams contain a great variety of constituent gestalts and sub-configurations.

Critically, it is this surplus that constitutes the heuristic power of a geometrical figure since

specific sub-configurations may well trigger alternative solution paths.

Figure 1. Three consecutive terms of a linear pictorial sequence.

One visual interpretation of the pictorial sequence shown in Figure 1 is to see the nth

term in

the sequence as being composed of a single central dot with three “rays”, each containing n

dots, radiating out from it. This would yield the general expression 13 nTn . Alternatively,

one could interpret the nth

term as comprising three overlapping “rays” each containing

)1( n dots. Correcting for the overcount caused by overlapping dots, this would yield the

general expression 2)1(3 nTn . Rather than seeing each term as rays, one could

subdivide the structure into a vertical column containing )1( n dots and a horizontal row

containing )12( n dots. Correcting for the overcount caused by overlapping dots, this would

yield the general expression 1)12()1( nnTn . Alternatively one could make use of

“ghost” dots in order to see each term as a )12( n by )1( n rectangular array of dots.

Subtracting the two square arrays of “ghost” dots from the total tally yields the general

expression 22)1)(12( nnnTn . These four different visual apprehensions, each of

which yields a different yet algebraically equivalent expression of generality, are shown in

Figure 2.

Term 1 Term 2 Term 3

132

Figure 2. Different visualisations of the pictorial context shown in Figure 1.

Thus, even with a relatively simple pictorial context, engagement with the context itself has

the potential to open up a diverse range of visually mediated expressions of generality. The

pedagogical distinction here is that focusing on the context that gives rise to the sequence

foregrounds the development of a sense of generality rather than merely the construction of

an algebraic relationship (Thornton, 2001). As Hewitt (1992) succinctly remarks, the problem

with divorcing patterns of numbers from their original context is that any generalised

statements become “statements about the results rather than the mathematical situation from

which they came” (p. 7).

A frequent theme in the number pattern generalisation research literature relates to the

tendency of pupils to fixate on the constant difference between consecutive terms and thus to

attempt to generalise recursively, i.e. by relating each term to the preceding term in the

sequence, rather than attempting to use the independent variable to construct an explicit

formula (Hargreaves, Shorrocks-Taylor & Threlfall, 1998; Hershkowitz et al., 2002). English

and Warren (1998) found that once students had established a recursive strategy they were

reluctant to search for a functional relationship, and MacGregor and Stacey (1993) cite one of

the main causes of difficulty in formulating algebraic rules as being pupils‟ tendency to focus

on the recursive patterns of one variable rather than the relationship linking the two variables.

Similar observations have been made by other researchers (Orton, 1997). Interestingly, a

reliance on differencing (i.e. a recursive strategy) has also been found with adults (Orton &

Orton, 1994). Lannin (2004) remarks that there would seem to be a natural tendency for

pupils to reason recursively when engaging with number patterns.

Noss, Healy and Hoyles (1997) note that the tendency of pupils to focus on a recursive

strategy shouldn‟t necessarily be interpreted as pupil failure and remark that strategies are

influenced not only by the nature of the task but also by the presentation of the task. This has

been echoed by Frobisher and Threlfall (1999) who comment that presenting a task in

sequential stages (e.g. asking for the 10th, 20th and 50th terms) often leads pupils to use a

step-by-step recursive approach. Hershkowitz et al. (2002) found that the presentation of

consecutive terms encouraged recursion, while terms presented non-consecutively tended to

n2n2

(2n+1)(n+1)1

(n+1)

(2n+1)

(n+1)

(n+1)

(n+1)n n

n

+1 2

133

encourage generalisation by means of the independent variable. The use of a pictorial

context, particularly if non-consecutive terms were presented, also tended to encourage

generalisation by means of the independent variable.

With these introductory remarks as a contextual backdrop, this paper engages with the

inherent allure of the numeric and visual analogues of the constant difference, in the context

of linear pattern generalisation, and explores the potential this allure has in terms of

obfuscating alternative apprehensions or „ways of seeing‟. The purpose of the study is thus to

gain insight into visually mediated approaches to pattern generalisation tasks set within a

pictorial context with a view to informing classroom practice.

Theoretical background

This paper draws on three key theoretical ideas, enactivism (Maturana & Varela, 1998;

Varela, Thompson & Rosch, 1991), knowledge objectification (Radford, 2003, 2008) and

Fischbein‟s (1993) notion of figural concepts. A brief overview of each theoretical idea is

presented here to provide a theoretical context for the study.

The fundamental feature of enactivism is a blurring of the division between mind and body

and hence between thought and behaviour (Davis, 1997). A direct consequence of this is that

from an enactivist perspective there is no separation between cognition and any other kind of

activity. Cognition is thus viewed as an embodied and co-emergent interactive process, “an

ongoing bringing forth of a world through the process of living itself” (Maturana & Varela,

1998, p. 11) where the emphasis is on knowing as opposed to knowledge. From this

theoretical stance, language and action are not merely outward manifestations of internal

workings, but rather visible aspects of embodied understandings (Davis, 1995). For the

enactivist, the act of perceiving something is not a process of recovering properties of an

external object. Rather, we perceive things in a certain way because of the manner in which

we relate to them through our actions (Lozano, 2005). Thus, how we make sense of our

experiences, and indeed what we are able to experience, is dependent on the kinds of bodies

that we have and the ways that our bodies afford interactions with the world we inhabit and

the various environments in which we find ourselves (Johnson, 1999).

From an enactivist stance, perception needs to be considered as a fully embodied process – a

complex activity related to the manner of our acquaintance with the objects of perception, in

other words the activity that mediates our experience with objects (Radford, Bardini &

Sabena, 2007). Radford (2008) refers to the process of making the objects of knowledge

apparent as objectification, a multi-systemic, semiotic-mediated activity during which the

perceptual act of noticing progressively unfolds and through which a stable form of

awareness is achieved. Importantly, use of the word “objectification” in this context needs to

be interpreted in a phenomenological sense, a process whereby something is brought to one‟s

attention or view (Radford, 2002). Radford‟s (2008) theoretical construct of knowledge

objectification foregrounds the phenomenological and semiotic aspects of figural pattern

generalisation and hence allows one to critically engage with pupils‟ whole-body experience

and expression while they explore the potentialities afforded by a given pictorial pattern

generalisation task.

134

Visually mediated approaches to pattern generalisation tasks set within a pictorial context

provide for an interesting interplay between two different modes of visual perception: sensory

perception and cognitive perception (Rivera & Becker, 2008). These different modes resonate

with Fischbein‟s (1993) theory of figural concepts, and the notion that all geometrical figures

(or figural objects) possess, simultaneously, both conceptual and figural properties. In a

similar vein, the figures such as the pictorial terms shown in Figure 1 could be said to contain

both figural/spatial properties as well as conceptual qualities. What one sees in the individual

images is a result of the Gestalt laws of figural organisation. However, this is further

influenced by the additional conceptual qualities of the image, qualities that have been added

by virtue of the image being contextualised, in this case within a growing sequence of similar

images. Thus, an important aspect of figural pattern generalisation lies in the notion that such

pictorial cues, or rather visual triggers, possess both figural/spatial and conceptual qualities,

each of which resonates with a different mode of visual perception – sensory and cognitive,

respectively. This position perhaps seems somewhat at odds with an enactivist view of

perception as being a fully embodied and co-emergent process. However, it is not being

suggested that these two modes of perception are independent of one another, or that they are

able to occur in isolation. Indeed, one could even argue that sensory perception cannot occur

without cognitive perception – a view that resonates strongly with the mind-body unity that is

the core of enactivism. Nonetheless, the distinction between figural and conceptual properties

provides a useful framework to explore figural pattern generalisation.

For the purposes of this paper the most important terminological distinction is that between

what I have previously referred to as local visualisations and global visualisations (Samson,

2011). Local visualisations have at their heart a recursive or term-by-term visual

apprehension focusing on the local additive unit – i.e. the structural unit which needs to be

added to (or inserted into) one pictorial term in order to form the next term in the sequence.

In contrast, global visualisation represents a more holistic view where each term of a given

pictorial context is visualised in terms of a general structure that does not make use of the

iterative addition or insertion of the local additive unit.

Methodology

This study is oriented within the conceptual framework of qualitative research, and is

anchored within an interpretive paradigm. Research participants (a mixed class of 23 high-

ability Grade 9 learners) were provided with different linear patterns presented in a 2-

dimensional pictorial context. All patterns were presented as two non-consecutive terms.

This was a purposeful decision based on previous research experience (Samson, 2007) as

well as research literature (Healy & Hoyles, 1996; Hershkowitz et al., 2002) which suggested

that non-consecutive terms would be more appropriate with respect to encouraging

generalisation by means of the independent variable, i.e. by encouraging attention to be

focused on the visual stimulus.

In Phase 1, participants were required to provide a numerical value for the 40th

term (along

with a written articulation of their reasoning) as well as an algebraic expression for the nth

term (along with a justification/explanation of their expression). Participants were also

informally interviewed in instances where the written articulation of their reasoning was

135

either ambiguous or required additional explication. This process of member checking

constituted a form of external validation. Participants who were identified as preferring a

visual mode as opposed to a numeric approach when solving pattern generalisation tasks

were invited to take part in Phase 2.

In Phase 2, participants were individually required to provide, in the space of one hour,

multiple expressions for the nth

term of the sequence along with a justification or explanation

of their expression. Tools such as paper, pencils and highlighters as well as appropriate

manipulatives such as matchsticks were provided. The provision of a variety of tools and

manipulatives stems from a sensitivity to the enactivist theoretical framework. Such items

are not mere auxiliary components but open up spaces of possible action and thus have the

potential to shape enactive processes of construction (Lozano, Sandoval & Trigueros, 2006;

Radford, 2003).

Participants were asked to think aloud while engaged with their particular pattern

generalisation task, and the researcher also prompted the participants to keep talking or

provide further explication as and when necessary. Each session was audio-visually recorded

and field-notes were taken. Audio-visual recordings were analysed with specific reference to

how participants made use of multiple means of objectification en route to a stable form of

awareness. These means of objectification included the use of words, linguistic devices,

metaphor, gestures, rhythm, graphics and physical artefacts. These processes of „coming to

know‟ were carefully scrutinised through multiple viewings of the audio-visual recordings of

each research participant, the essential character underpinning the data acquisition and

analysis protocol being the treatment of all responses, particularly those that were unexpected

or idiosyncratic, with a genuine interest in understanding their character and origins.

Not only does enactivism form a crucial ontological backdrop to this study, but enactivist

notions of epistemology also had important implications for the research process itself. From

an enactivist perspective, researchers are seen as “developing their learning in a particular

context” (Lozano et al., 2006, p. 91), a context within which researcher and research

environment are seen to co-emerge (Reid, 2002). This interdependence of researcher and

context was characterised by a flexible and dynamic process of investigation (Trigueros &

Lozano, 2007). This iterative and reflexive process of co-emergence was built on over time

through the use of multiple perspectives and the continuous refinement of methods and data

analysis protocols. Audio-visual data were examined repeatedly in different forms (e.g. video

and transcript) and in conjunction with additional data retrieved from field-notes and

participants‟ worksheets. In addition, nodes of activity which seemed particularly interesting

were identified and meticulously characterised with reference to the various semiotic means

of objectification in the form of descriptive vignettes, thus providing in-depth analyses of

each pupil‟s lived experience.

Results and Discussion

Local visualisation dominated in those contexts where the growth pattern occurred in a single

direction and where progression from one term to the next can be accomplished by the direct

attachment of the additive unit. Local visualisation foregrounds a recursive or term-by-term

visual apprehension focusing on the local additive unit, thus potentially obfuscating other

136

(a) (b)

11

10

9

8

7

6

5

4

3

2

1

1 2 3

4 5 6 7 8 9

10 11

apprehensions or „ways of seeing‟ and thereby diminishing the heuristic potential of the

pictorial context. By contrast, global visualisation dominated in those questions in which the

growth pattern occurred in more than one direction or in which progression from one term to

the next could only be accomplished by the insertion of the additive unit into the previous

term as opposed to the direct attachment of the additive unit onto the previous term. There is

thus evidence to suggest that the nature of the pictorial terms themselves could act as

potential triggers with respect to favouring or supporting specific visual strategies (Samson,

2013).

A vignette is now presented which highlights the central thesis of this paper, that the inherent

allure of the numeric and visual analogues of the constant difference, in the context of linear

pattern generalisation, has the potential to hinder alternative apprehensions or „ways of

seeing‟.

Kelly, a Grade 9 learner, was presented with the two non-consecutive terms shown in Figure

3. When presented with her pictorial pattern for the very first time, Kelly counted the matches

in Term 3 in the manner shown in Figure 4(a). Immediately upon completion of this counting

procedure she double-checked her tally by re-counting the matches. However, she now used a

very different counting technique (Figure 4(b)). In both cases she counted aloud while

pointing to each match in turn with her pencil. She then went on to count the total number of

matches in Term 5 using the second of these two counting procedures.

Figure 3. Pictorial context presented to Kelly.

Figure 4. Kelly’s different counting procedures.

The first of these two counting procedures one could characterise as being economical in the

sense that it utilises the minimum amount of time and energy to count the matches. The

second procedure one could characterise as being uneconomical since it requires significantly

more time and energy to accomplish. This can readily be understood in terms of the overall

path of the pencil as traced in the two counting procedures. The overall path traced for each

of the counting procedures is represented by the dotted lines in Figure 5. The first counting

method traces a continuous zigzag path through the 11 matches from left to right. The second

Shape 3 Shape 5

137

(a) (b)

111

11

1

counting method requires counting from left to right along the base of the structure, then

returning to the far left to count the central matches, and then once again returning to the left

to count the top row of matches.

Figure 5. Traced paths of Kelly’s different counting procedures.

Since the second counting procedure is uneconomical, I would argue that it must then be

systematic – i.e. from the counter‟s perspective it must represent an efficient way to

accomplish the task of counting. I would argue further that a necessary condition for a

counting method to be systematic and/or efficient is a perceived sense of structure, whether

conscious or unconscious, on the part of the person performing the counting operation. It is

this perceived sense of structure that then guides the systematic counting procedure. In

Kelly‟s second counting method the perceived structure seems to be in terms of a bottom row

of horizontal matches, a central zigzag of oblique matches, and a top row of horizontal

matches. This is confirmed by her later verbal commentary after completing the counting:

Figure 6. Kelly’s verbal commentary upon completion of her counting.

At this point, based on her second counting procedure, it would have been possible for Kelly

to construct the following expression for the nth

term: )1(2 nnnTn . However, instead

of doing this she continued to interact with the pictorial context with hardly a pause.

Um, 5 triangles in Shape 3 [pointing to each in turn] and 1, 2, 3, 4, 5, 6, 7, 8, 9 triangles

in Shape 5 [pointing to each in turn]. Okay, so I’m guessing that you’re adding on 1

there [creates an extra triangle by adding 2 lines onto Term 3 – the first two dashed

“Okay, so whatever the nth

term is that’s the number of

lines at the bottom.”

“And then (…) the lines in

the middle are twice that...”

“…and the lines at the top is

1n .”

138

lines shown in Figure 7] which would give you 1, 2, 3, 4, 5, 6 [counting the 6 triangles

but then adding on another 2 lines (the second two dashed lines shown in Figure 7) to

create a 7th

triangle]. Okay, so I’m guessing that that’s Term 1 [indicating the 5-unit

structure shown by matches a – e in Figure 7] and then you’re adding on, okay no hang

on. (…) Hang on, there’re 11 in Term 3 and 19 in Term 5, and 4 [i.e. Term 4] has to

come somewhere in between those two numbers [pointing to the numbers 11 and 19

which she had written down earlier]. So you’re adding on, you’re either adding on 2

matchsticks [indicating the first two dashed lines shown in Figure 7] or you’re adding

on 1, 2, 3, 4 matchsticks [indicating all 4 dashed lines shown in Figure 7]. And if you’re

adding on 4 matchsticks that would make that 15 [referring to Term 4] and then it

would go, and then it would plus 4 each time [indicating the jump from 15 to 19, i.e.

from Term 4 to Term 5]. Ya, that’ll work. Hmm, but, if this is Shape 3 and you’re

adding on 4, [Kelly then started counting backwards in multiples of 4 matches to arrive

ultimately at Term 1], 1, 2, 3, 4 [counting off the right-most multiple of 4 matches in

Term 3] 1, 2, 3, triangles in Shape 2 [pointing to each of the 3 triangles], then 1, 2, 3, 4

[counting off the next group of 4 matches from the right in Term 3], and 1 triangle in

Shape 1. Okay, that makes sense. Okay, so you’re adding on 4 each time. Um, so the

difference is 4 so that makes it n4 , um and n4 will give me 12 [indicating Term 3] and

n4 will give me 20 in Term 5, so I’m gonna minus 1 to get n4 is 12 minus 1 is 11 [for

Term 3], n4 is 16 minus 1 is 15 [for Term 4], n4 is 20 minus 1 is 19 [for Term 5], 4

times 6 is 24 minus 1 is 23 [for Term 6] and the difference between 5 [i.e. Term 5] and 6

[i.e. Term 6] is 4. Okay, so the first one is 14 n .

Figure 7. Kelly’s augmentation of Term 3.

From her initial perceptual apprehension of the figural cue – i.e. two horizontal rows of

matches with a zigzag of matches between them – Kelly very quickly changed her

apprehension by becoming aware of the total number of triangles (upward pointing and

downward pointing) in each pictorial term. This new apprehension led her to „guess‟ the

number of matches that one would need to add to Term 3 in order to construct Term 4. Her

guess was that it would be either 2 or 4 matches, which would respectively create either 1 or

2 additional triangles. At this point she reverted to a numeric argument. Since Term 3

contained 11 matches and Term 5 contained 19 matches she reasoned that Term 4 had to fit

somewhere between these two terms. Sensing that the addition of 4 matchsticks was more

likely to be correct (perhaps because of the difference between 11 and 19) she added 4 to 11

to arrive at 15 (Term 4) and was satisfied with the veracity of her conjecture when she

realised that the addition of another 4 would give the 19 matches required for Term 5. She

then returned to the pictorial representation of Term 3 and worked backwards in multiples of

4 matches to determine that Term 1 was in fact a single triangle and not a 2-triangle structure

as she had initially thought. This visual appreciation of the structure of Term 1 was the final

e

d

c

ba

4

32

1

139

→ →

component in the development and ultimate stabilisation of a new apprehension of the

pictorial context. Happy that a common difference of 4 matches made sense both visually and

numerically she returned to a final numerical argument using a rate-adjust strategy to arrive at

a final formula of 14 nTn .

Interestingly, if one looks back at Kelly‟s very first counting procedure (as shown in Figure

4(a)) then one could perhaps argue that right from the beginning there seems to be fleeting

evidence of this final apprehension. Although her counting procedure does seem to have

some semblance to this final apprehension, the rhythm in her counting suggests that this

similarity is merely coincidental. She counted the first 5 matches slowly and deliberately, as

if establishing a counting strategy, after which she counted the remaining 6 matches more

rapidly (Figure 4(a)). The rhythmic gaps between each count, although shorter in the case of

the remaining 6 matches, were nonetheless constant. This rhythm suggests that after the

counting strategy had been established, i.e. after counting the first 5 matches, all further

matches were seen to be equivalent. This suggests that the counting procedure was used for

its economy rather than as a result of an unconscious apprehension based on perception of the

4-match additive unit.

Kelly‟s gradual growing awareness, as different structural aspects of the pictorial terms were

brought forth, shows a transition between three different apprehensions (Figure 8). Kelly‟s

initial apprehension (two horizontal rows of matches with a zigzag of matches between

them), which was on the verge of being stabilised in the form of a general algebraic

expression, was rapidly replaced with an apprehension that brought forth the gestalt of the

triangle. This triangular feature in turn led to the gradual development of the 4-match unit

that represented the constant difference, a process that incorporated both visual and numeric

elements. The foregrounding of the visual analogue of the numeric constant difference, along

with a retro-synthesis of the growth pattern to determine the visual structure of Term 1,

finally led to a new apprehension – a single triangle for Term 1 with multiples of the 4-match

additive unit.

Figure 8. Kelly’s transitioning between 3 different apprehensions.

Images such as the pictorial terms presented to Kelly contain both figural/spatial properties as

well as conceptual qualities. What one sees in the individual images is a result of the Gestalt

laws of figural organisation. However, this is subtly influenced by the conceptual qualities of

the image brought about through the image being contextualised within a sequence of related

images. Kelly‟s initial apprehension was arrived at through a visually mediated global

structural awareness – i.e. the perceptual organisation of the matches into different groups as

supported by the Gestalt laws of figural organisation, in particular the laws of similarity and

proximity (Katz, 1951; Wertheimer, 1938). The transition between the first and second

140

apprehension was very rapid and one can only conjecture that the transition was once again

supported by the Gestalt laws of figural organisation, in this case the laws of good

continuation and closed forms (Katz, 1951; Spoehr & Lehmkuhle, 1982; Zusne, 1970), which

led to the structural unit of the triangle gaining prominence. In spite of these two highly

visually mediated apprehensions, it was nonetheless the gradual foregrounding of the

constant difference with its numeric as well as visual recursive allure that led to Kelly‟s final

apprehension and her final algebraic expression for nT .

As a final aside it is worth noting that in the space of one hour Kelly managed to arrive at

seven different algebraic expressions for nT , of which the above vignette describes the first.

Five of these algebraic expressions were based on different visual apprehensions of the

pictorial context, while two were arrived at through numerical considerations. However, it

was the inherent allure of the constant difference in the form of the 4-match additive unit that

led Kelly to her first stable apprehension ( 14 nTn ), and where she would have stopped

had she not been prompted to seek alternative algebraic expressions for the general term.

Interestingly, both of the fleeting apprehensions that she passed through en route to this first

stable apprehension resurfaced again later. The subdivision into two horizontal rows of

matches with a zigzag of matches between them led to her third algebraic expression (

nnnTn 21 ), while her apprehension of overlapping triangles eventually led to her

fifth algebraic expression ( )22()1(3 nnnTn ). Thus, in this particular case, because

of the requirement for Kelly to determine multiple expressions of generality, the potential in

these earlier transitional apprehensions was still able to be realised in spite of the allure of the

constant difference.

Concluding Comments

It was the purpose of this paper to explore the notion of figural concepts (Fischbein, 1993),

i.e. objects (or visual triggers) with both figural/spatial properties and conceptual qualities,

within the context of linear pattern generalisation activities presented pictorially. These

spatial properties and conceptual qualities each resonate with a different mode of visual

perception – sensory and cognitive, respectively. Although figural cues contain

simultaneously both spatial and conceptual properties, and while it is acknowledged that

perception is at once both sensory and cognitive, what is important is the nature of the spatial

and conceptual properties of pictorial cues within the context of pattern generalisation. In

order to unambiguously present a pictorial sequence, at least two terms of that sequence need

to be shown. Such a visual stimulus or trigger can be perceived in any number of different

ways suggested by the Gestalt laws of perceptual organisation. However, by being visually

anchored within the context of a sequence of images which provides a sense of sequential

growth from one term to the next, the pictorial trigger can also be perceived on the basis of

this conceptual quality, an aspect of the pictorial context which seems to have an inherent

allure.

This paper focused on an analysis of one pupil‟s lived experience while engaged in the

generalisation of a linear sequence presented in a pictorial context. A critical aspect of the

analysis focused on the interplay between the different apprehensions of the pictorial context

141

brought about through the spatial and conceptual properties of the visual stimulus. A micro-

analysis of a vignette was presented to support the central thesis of the paper – that tension

between different visual apprehensions is likely to pervade generalisation strategies applied

to linear pictorial pattern generalisation tasks as a result of the relationship between the

spatial properties and conceptual qualities of the given images. In addition, there is evidence

to suggest that the nature of the pictorial terms themselves could act as potential triggers with

respect to favouring or supporting specific visual strategies. More critically, the inherent

allure of the numeric and visual analogues of the constant difference could potentially result

in obfuscating other apprehensions or „ways of seeing‟, particularly in those contexts where

progression from one term to the next can be accomplished by the direct attachment of the

additive unit, thereby diminishing the heuristic potential of the pictorial context. Since one

would want pupils to be able to experience a range of visual strategies, a range of pictorial

patterns should be included in patterning tasks. These should include (i) questions where the

growth pattern occurs in a single direction and where progression from one term to the next

can be accomplished by the direct attachment of the additive unit, (ii) questions in which the

growth pattern occurs in more than one direction, and (iii) questions in which progression

from one term to the next can only be accomplished by the insertion of the additive unit into

the previous term as opposed to the direct attachment of the additive unit onto the previous

term. Sensitivity to the type of visual apprehensions that different pictorial contexts are likely

to evoke has direct pedagogical application and importance within the context of the

mathematics classroom.

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144

Surveying the distribution and use of mathematics teaching aids in

Windhoek: A Namibian case study

Duncan Samson

1 & Tobias Munyaradzi Dzambara

2

1Rhodes University, South Africa

2University of Namibia, Khomasdal Campus, Namibia

1 [email protected]

2 [email protected]

This paper investigates the types of mathematics teaching aids available at both public and

private secondary schools in Windhoek. The paper characterises their use and source as well

as teachers‟ perceptions towards the use of such teaching resources in the Mathematics

classroom. The study is grounded in an interpretive paradigm and employed a mixed methods

approach. 75 teachers from 25 secondary schools in the Windhoek metropolitan area took

part in the study. The majority of the participants were found to have a positive outlook

towards the importance and role of teaching aids in Mathematics, seeing them as promoters

of hands-on engagement, visual reasoning, active participation and learner motivation.

Nonetheless, some schools were found to be under-resourced with respect to certain types of

teaching aids. A need for appropriate in-school support on the use of teaching aids was also

identified.

Introduction

The educational system in Namibia went through a process of transformation after the

country obtained political independence in 1990. At the heart of the new educational system

lies the concept of learner-centred education (LCE). This resonates with numerous national

education policies of developing countries where LCE has become a “recurrent theme”

(Schweisfurth, 2011, p. 425). The Namibian Ministry of Education and Culture (Namibia.

MEC, 2003) highlights the importance of the learners‟ active participation and meaningful

contribution in a learner-centred approach:

Learning is seen as an interactive, shared and productive process where teaching creates learning

opportunities which will enable learners to explore different ways of knowing and developing a

whole range of their thinking abilities both within and across the whole curriculum. (Namibia.

MEC, 2003, p. 8)

The use of appropriate teaching aids and resources has been encouraged as a potential means

to help learners to develop positive attitudes towards Mathematics, to acquire basic

mathematical concepts, as well as to develop a “lively, questioning, appreciative and creative

intellect” (Namibia. Ministry of Education and Culture [MEC], 1993, p. 56). The Broad

Curriculum document of the Basic Education Teacher Diploma (BETD) in Namibia clearly

outlines the Ministry of Education‟s expectations from teachers with regard to LCE and the

use of teaching aids in schools:

Teachers should be able to select content and methods on the basis of the learners‟ needs, use

local and natural resources as an alternative or supplement to readymade study materials and thus

develop their own and the learners‟ creativity. A learner-centred approach demands a high degree

of learner participation, contribution and production. (Namibia. Ministry of Education [MoE],

2009, p. 2)

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Teachers thus not only play a pivotal role in the teaching of mathematics in a learner-centred

environment but need to facilitate the active participation of all learners through the use of

appropriate teaching resources. With these introductory observations as a contextual

backdrop, this paper explores the availability and use of teaching aids at secondary schools in

the Windhoek metropolitan area along with the perceptions of teachers towards the use of

teaching aids in their Mathematics lessons. Specifically, the study seeks to answer the

following questions:

(i) What is the availability and use of teaching aids in secondary school Mathematics

classrooms in Windhoek?

(ii) What are teachers‟ perceptions of the use of teaching aids in their lessons?

Teaching Aids

Mathematicians have used a variety of tools and teaching aids to support the learning of

mathematics throughout history (Durmus & Karakirik, 2006). Teaching aids can be broadly

categorised into three groups: (i) those encouraging concrete experiences, (ii) those

encouraging iconic experiences, and (iii) those encouraging abstract experiences (van der

Merwe & van Rooyen, 2004).

Examples of teaching aids that encourage concrete experiences are physical manipulatives.

These represent a variety of physical objects that have the potential to develop learners‟

understanding of concepts and relations and can be used to explore mathematical ideas by

using a hands-on approach. Such manipulatives include base ten blocks, algebra tiles, fraction

pieces, pattern blocks as well as geometric solids (Durmus & Karakirik, 2006). Such

manipulatives are objects that can be “handled by an individual in a sensory manner during

which conscious and unconscious mathematical thinking will be fostered” (Swan & Marshall,

2010, p. 14). In addition to physical manipulatives there are also virtual manipulatives, i.e.

computer generated images that represent concrete objects and which can be manipulated

directly through a technological interface (Lee & Chen, 2010).

Examples of teaching aids that encourage iconic experiences include pictures, drawings and

posters, while teaching aids that encourage abstract experiences include maps and diagrams.

Developments in modern technology have greatly enhanced access to these types of teaching

resources. In addition, the availability of powerful ICT tools such as digital cameras, scanners

and the internet has the potential to provide Mathematics teachers and learners with an

abundant variety of digital images to harness and integrate everyday experiences from the

outside world into the Mathematics classroom (Ahmed, Clark-Jeavons & Oldknow, 2004).

The teaching aids that will be specifically focused on in this particular study include

mathematical instruments for the chalkboard, charts and posters, geometric models, graph

boards, mathematical instrument sets, geoboards, overhead projectors, computers, interactive

whiteboards, improvised teaching aids made using local resources, as well as any other

physical manipulatives or artefacts that facilitate teaching and learning. Although the

terminology used in the research literature in relation to teaching aids is often used somewhat

idiosyncratically, this study uses the term „teaching aid‟ as an all-encompassing umbrella

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term referring to all teaching resources found and used in the Mathematics classroom that

have the potential to enhance teaching and learning.

Theoretical Background

Having interrogated the notion of a teaching aid we now need to provide a theoretical

rationale for how the learning theory underpinning LCE relates to the use of teaching aids in

the Mathematics classroom at secondary school level. A number of related theoretical ideas

are explored, including constructivism, the zone of proximal development (ZPD), the notion

of scaffolding, as well as aspects of Kilpatrick, Swafford and Findell‟s (2001) framework of

teaching for mathematical proficiency.

Cornelius-White and Harbaugh (2010) characterize learner-centred instruction as an approach

to teaching and learning that “prioritizes facilitative relationships, the uniqueness of every

learner, and the best evidence on learning processes to promote comprehensive student

success through engaged achievement” (p. xxvii). The main indicators of successful LCE

include, among other factors, an acknowledgement of prior knowledge, skills and interests,

the desire and eagerness to learn and the learners‟ active involvement in learning (Namibia.

Ministry of Basic Education and Culture [MBEC], 1999). LCE is concerned with intrinsic

motivation such as a “development of curiosity or interest in the subject matter or wanting to

become proficient to the best of one‟s ability” (Cornelius-White & Harbaugh, 2010, p. 59).

The Mathematics teacher in the Namibian educational context is thus encouraged to develop

among learners a culture of curiosity and an eagerness to learn and investigate through the

use of teaching aids.

The concept of LCE has its roots in social constructivism which stresses the importance of

the nature of the learner‟s social interaction with knowledgeable members of the society. The

Piagetian roots of constructivism lie in the notion that “children construct their own

understanding through interaction with their environment – that is, through their actions on

objects in the world” (McInerney & McInerney, 2006, p. 37). Constructivism is thus an active

process in which “learners construct and internalize new concepts, ideas and knowledge

based on their own present and past knowledge and experiences” (Cohen, Manion &

Morrison, 2010, p. 181). Constructivism therefore suggests that the learners construct

knowledge out of their own experiences, rather than through a process of passive reception,

and this active and participative notion of learning is in turn a fundamental precept of LCE.

The Namibian Ministry of Education and Culture (Namibia. MEC, 2003) points out that in a

learner-centred approach there should be a strong triangular relationship between three

elements: teachers, learners, and the teaching aids. The teacher in a learner-centred approach

therefore serves as a facilitator in the learning process, promoting co-operative learning

which develops learners‟ thinking through stimulating reflection, comparison and exploration

as well as continually improving on their acquired knowledge (Namibia. MEC, 2003).

Mathematics teachers are therefore encouraged to identify and use teaching aids to promote

co-operative learning activities in their lessons in which learners at different levels assist each

other in the learning process. The use of teaching aids to promote interaction and

collaboration between less competent learners and more competent peers in classroom

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activities resonates strongly with Vygotsky‟s (1978) concept of the zone of proximal

development (ZPD). The ZPD can be defined as:

…the distance between the actual development level as determined by independent problem

solving and the level of potential development as determined through problem solving under

adult guidance or in collaboration with more capable peers. (Vygotsky, 1978, p. 86)

Reid-Griffin and Carter (2004) define the ZPD as “a zone of possibilities, what the individual

is able to accomplish when assisted by more capable others in the presence of mediating

tools” (p. 496). This reasoning strongly supports the use of teaching aids in facilitating the

learning process by broadening the range of experiences and increasing the opportunity for

learners to develop their own understandings (Reid-Griffin & Carter, 2004). Teaching for

conceptual understanding in Mathematics is thus likely to be promoted through the use of

teaching aids because in the learning process “tools can shape the students‟ performance and

understanding of the task in terms of key disciplinary content and strategies and thus

problematize this important content” (Reiser, 2004, p. 273).

Kilpatrick et al. (2001) argue that when teaching aids are used well in Mathematics lessons

they enable learners and teachers to “have a conversation that is grounded in a common

referential medium, and they can provide material on which the learners can act productively

provided they reflect on their actions in relation to the mathematics being taught” (p. 354).

Teaching aids in Mathematics are likely to have a positive effect on the learning process if

they meet certain criteria. According to Ahmed et al. (2004) the following characteristics are

important:

They must allow for learner-centred activity with the learner being in charge of the process.

They must utilize the learners‟ current knowledge and must also help develop links between

learners‟ current mental schemata while interacting with the tools. They must reinforce current

knowledge and assist future problem solving through enhancing future access to knowledge.

(p. 319)

Methodology

This paper explores the availability and use of teaching aids at secondary schools in the

Windhoek metropolitan area along with the perceptions of teachers towards the use of

teaching aids in their Mathematics lessons. The study is grounded in the interpretive

paradigm and makes use of a mixed methods approach (Creswell, 2003) in which both

quantitative and qualitative empirical data was collected in two sequential phases.

In the first phase, which took the form of a survey, mostly quantitative data was collected by

means of a structured questionnaire to provide information on the overall availability and use

of teaching aids in Mathematics. The questionnaire also made use of a Likert scale rating

system to measure and help quantify the responses of the participants in relation to their

attitude towards the use of teaching aids. In the second phase the statistical data from the

survey was used to purposefully select teachers from five secondary schools from whom

qualitative data was collected by means of semi-structured interviews. These schools were

chosen such that they represented a broad spectrum in terms of teaching aid availability. The

second phase of the research process thus took the form of an instrumental case study, the

unit of analysis being teachers‟ perceptions towards the use of teaching aids in their

Mathematics lessons. Although the first phase of the study centred primarily on quantitative

148

data, the second phase was firmly rooted in the interpretive paradigm. The purpose of the

second phase was to corroborate and elaborate on the findings of the first phase by interacting

with selected teachers in order to gain a more nuanced insight into the nature and use of

teaching aids in schools.

The data analysis process was also carried out in two phases. In the first phase, the

quantitative data collected from the questionnaires was analysed using spreadsheet software

to characterise the availability, source and frequency of use of teaching aids in schools. In

addition, a Likert scale rating system was used to quantify participants‟ attitude towards the

use of teaching aids. In the second phase, qualitative data from the interviews was transcribed

and coded. Themes that emerged from this process were gradually grouped to provide a rich

characterization of teachers‟ experiences and perceptions of the use of teaching aids in the

Namibian Mathematics classroom at secondary school level. The second phase of the data

analysis process was used to provide a more nuanced understanding of the quantitative data

analysed in the first phase, and as such acted as a form of methodological triangulation

(Cohen, et al., 2007; Oliver, 2010).

Results and Discussion

Teaching aid survey

The first phase of the data acquisition process took the form of a questionnaire.

Questionnaires were delivered to 100 secondary school Mathematics teachers at 30 different

secondary schools. 75 teachers at 25 of the schools completed the questionnaire. The school

response rate was thus 83.3% and the total teacher response rate was 75%.

In terms of the overall availability of the twelve different types of teaching aid audited, 53%

of the responses indicated that the particular teaching aid in question was available (Table 1).

This suggests that the different types of teaching aids surveyed are reasonably available in the

schools that were audited. With respect to the use of teaching aids, the survey shows that only

a limited number of teaching aids are used on a daily basis (11%) while 41% are used as

frequently as possible. 48% of the teaching aids were indicated as being never used (Table 1).

On a note of clarity, careful analysis of the data revealed that all respondents who indicated

that a particular teaching aid was “not available” also indicated that the teaching aid was

“never used”. The teaching aids that are available but which are never used can thus be

determined by subtracting the tally for “not available” from the tally for “never used”. A total

of 428 teachers‟ responses indicated teaching aids that were never used compared to a total of

425 responses for teaching aids not available. This suggests that only 3 responses indicate a

teaching aid that is available but is never used. With respect to the source of those teaching

aids surveyed (Table 1), the major source was indicated as being school purchase (50%)

followed by personal purchase (31%). The Ministry of Education was indicated as the source

of only 12% of the teaching aids surveyed, while 4% of the teaching aids were indicated as

having been donated. The availability, frequency of use and source of each of the twelve

types of teaching aids audited are summarised in Table 1.

The types of mathematical teaching aids most readily available in the 25 secondary schools

surveyed include: chalkboard set squares; chalkboard protractors and compasses; charts and

posters; mathematical instrument sets; overhead projectors; and improvised teaching aids. For

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each of these teaching aid categories teachers indicated an availability of greater than 60%.

The availability of physical objects (other than geometric models), geometric models and

computers was calculated as being in the 40%60% range, and such teaching aids were thus

classified as being only moderately available. Graph boards, geoboards and interactive

whiteboards were the least available items with availability scores of 21%, 3% and 12%

respectively.

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Table 1. Overview of availability, frequency of use, and source of teaching aids.

Availability Frequency of

use Source

Teaching Aid

Av

aila

ble

in s

choo

l

No

t av

aila

ble

in

sch

oo

l

Use

d d

aily

Use

d a

s o

ften

as

po

ssib

le

Nev

er u

sed

Min

istr

y o

f E

duca

tio

n

Sch

oo

l p

urc

has

e

Per

sonal

pu

rch

ase

Do

nat

ed

Oth

er

Chalkboard set squares 66 9 25 39 11 27 39 5 0 0 Chalkboard protractors & compasses 63 12 20 44 11 13 53 5 0 0

Charts/posters 60 15 12 50 12 1 38 29 1 0 Physical objects (not geometric models) 31 44 5 27 43 2 13 20 2 0

Geometric models 31 44 1 29 45 1 19 10 2 1 Graph boards 16 59 1 14 60 2 10 2 0 1

Mathematical instrument sets 59 16 7 57 11 2 5 43 5 11 Geoboards 2 73 1 14 60 0 1 1 0 0

Overhead projectors 47 28 6 23 46 7 33 3 3 0 Computers 42 33 9 30 36 2 15 26 5 1

Interactive whiteboards 9 66 5 3 67 2 6 1 0 0 Improvised teaching aids 49 26 7 42 26 0 5 4 1 0

TOTAL 475 425 99 372 428 59 237 149 19 14

Computers for classroom-based learning were classified as being only moderately available.

Given that some of the schools surveyed had more than four Mathematics classes per grade,

with around 35 learners per class, and given that 33 of the 75 teachers surveyed indicated that

they had no access to computers for teaching, this translates to a very high number of young

learners who are not being exposed to computers and modern technology in their

Mathematics lessons.

In terms of low-tech resources, the study unearthed that only two out of 75 secondary school

teachers surveyed had geoboards available in their classrooms, and only 16 teachers had

access to graph boards. The poor availability of graph boards and geoboards, and the

relatively poor availability of geometric models, is somewhat disturbing since not only are

these items useful and effective tools for the teaching and learning process, but they can be

easily and cheaply manufactured from available raw materials.

An encouraging finding of the survey was evidence that teachers in many instances had gone

an extra mile to contributing to the supply of teaching aids by personally financing resources.

It is noteworthy that the Ministry of Education is responsible for supplying far fewer teaching

aids than those that are personally financed by teachers. Although the major source of

teaching aids was indicated as being school purchase (50% of the total responses), it is

interesting to note that personal purchases accounted for a surprising 31%. This makes

personal purchase the second-highest source of teaching aids in the schools surveyed. This

151

highlights the readiness and willingness of some teachers to take responsibility for sourcing

and financing personal teaching aids for their classroom teaching and learning. By contrast,

the Ministry of Education was indicated as being responsible for only 12% of the teaching

aids surveyed.

Much research in mathematics education has given credence to the importance of having a

variety of teaching aids available for use in the classroom as such resources are likely to

promote learners‟ understanding of mathematical concepts. As Nool (2012) remarks, the use

of teaching aids has “...been found to yield positive outcomes for learners‟ understanding in

different levels of Mathematics learning from elementary to college levels” (p. 309).

Furthermore, research on the importance of using teaching aids has shown that learners

develop visualisation skills through hands-on experiences in Mathematics lessons (Obara &

Jiang, 2011). Teaching aids can be used to promote not only active participation of learners in

lessons but also to support multiple representational accesses to mathematical concepts

thereby foregrounding the goals of LCE. As this survey clearly highlights, learning resources

are still insufficiently provided for in many Namibian secondary schools.

Likert scale attitudinal responses

Table 2 provides a summary of the responses of the 75 teachers who took part in the survey.

The Likert scale rating system helped quantify participants‟ attitudes towards the use of

teaching aids.

Table 2. Overall responses to Likert scale questions.

Statement

Str

ongly

agre

e

Agre

e

Neu

tral

Dis

agre

e

Str

ongly

dis

agre

e The use of teaching aids in Mathematics classes promotes learners‟

participation and interest in Mathematics 52 19 3 0 0

Teaching can only be effective when adequate and relevant

teaching resources are used in Mathematics lessons 34 24 11 4 1

Mathematics teachers have enough time to prepare teaching aids

for most of their lessons 8 12 33 19 2

Using teaching aids in Mathematics lessons promotes the teacher‟s

programme to complete the syllabus in time 18 23 18 9 6

The use of teaching aids in Mathematics is made difficult because

resources are not available in schools 23 24 13 11 3

The use of teaching aids promotes good academic performance of

learners in end-of-year Mathematics examinations 28 28 17 1 0

Teachers should be given more in-service training on the use of

teaching aids in Mathematics 35 20 13 5 1

Mathematics teachers can easily improvise effective teaching aids

that help learners grasp important concepts using local resources 17 26 20 9 2

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Teachers graduate from university and college with adequate

knowledge on the use of teaching aids in Mathematics 16 21 20 12 5

The use of teaching aids in Mathematics promotes the ministry‟s

policy of learner-centred education in schools 23 34 10 7 0

A holistic analysis of the general opinions elicited from the teachers indicates that the

majority of teachers value and appreciate the role that teaching aids play in teaching for

mathematical proficiency at secondary school level. 96% of the teachers believed that

teaching aids promote learners‟ active participation and interest in Mathematics while 76%

agreed that using teaching aids leads to better academic performance. There was also a 77%

consensus that using teaching aids helps in delivering learner-centred lessons in Mathematics.

Research has shown that teachers‟ perceptions influence the selection of instructional

strategies when delivering lessons (Ball, 1996). With this in mind it is an encouraging

observation that the general attitude of the research participants clearly demonstrates a

positive perception towards the use of teaching aids in their lessons.

Particularly in under-resourced schools there is a pressing need for teachers to be able to take

the initiative and come up with simple but innovative ways of improvising appropriate

teaching aids. Approximately 58% of the participants agreed (23% strongly so) that

Mathematics teachers were in a position to easily improvise effective teaching aids from

limited resources. However, the availability of time to prepare teaching aids appears to be a

problem for the teachers who participated in this study. Only 23% of the teachers indicated

that they have time to make teaching aids despite of the fact that the majority acknowledged

the importance of using such tools in their classes.

50% of the teachers said that they left teacher training institutions with sufficient and

appropriate knowledge with respect to the use of teaching aids. However, 74% of the

participants felt that they needed in-service training on the use of teaching aids to supplement

the skills acquired at training institutions. It is clear that additional in-service training on the

use of mathematical teaching aids would not go amiss.

Qualitative data

Qualitative data was derived from an open-ended question in the initial questionnaire as well

as subsequent semi-structured interviews. Through repeated engagement with the qualitative

data a number of themes gradually emerged. These themes include: hands-on nature of

concrete objects; reality and visualization; enhanced teaching of concepts; active participation

and interest; inadequate resources and the need to improvise; motivation and learner

performance; and time and support from the ministry. Although these themes are presented

individually, it is however acknowledged that they are interrelated and overlapping.

The teachers who participated in this study were of the opinion that the use of teaching aids

in Mathematics lessons at secondary school should be encouraged because their use provides

learners with concrete examples to explore concepts while at the same time promoting

empirical reasoning. The teachers felt that the use of tangible, hands-on objects in their

153

Mathematics lessons help them to explain abstract mathematical ideas more effectively.

Some teachers felt that this hands-on physical contact with manipulatives was particularly

beneficial for weaker learners as well as those learners with partial visual impairment.

Teachers were unanimous in their view that teaching aids provide learners with experiences

of reality that support the visualisation of mathematical concepts. Teachers in this study held

the opinion that teaching aids assist learners to relate real-world situations to the mathematics

being taught, and visual teaching aids are therefore important during the teaching and

learning process. In the words of one teacher, “The use of teaching aids in Mathematics

brings reality to the classroom!”

The majority of the teachers in this study were of the opinion that the use of teaching aids at

secondary school level promotes long-term understanding of mathematical concepts, and that

teaching aids enhance the learning and teaching of Mathematics because the learners build a

better conceptual understanding of the topic. In addition, some teachers showed a striking

sensitivity towards the notion that different learners learn in different ways (Presmeg, 1986).

Some learners may be more visual (as opposed to analytic) in terms of their cognitive

processing and reasoning, and the use of teaching aids that promote visual reasoning would

resonate very strongly with such learners. To deprive such learners of appropriate teaching

aids would be, in the opinion of one participant, tantamount to constraining their learning

potential. One critical aspect which a number of teachers highlighted is that teaching aids

provide tangible referents that assist learners with remembering key concepts.

The approach to teaching and learning of Mathematics in Namibian schools is focused on

LCE, the roots of which are found in social constructivism. The social constructivist

perspective of learning posits that “learning involves the active construction of knowledge

through engagement and personal experience” (von Glasersfeld, as cited in Kaminski, 2002,

p. 133). Teachers were generally of the opinion that the use of teaching aids promotes interest

among learners and this encourages the promotion of LCE in which learners are encouraged

to participate meaningfully.

An analysis of the quantitative data in the first phase showed that many schools are under-

resourced when it comes to teaching aids. In spite of this, the vast majority of teachers in this

study acknowledge the potential benefits that teaching aids can bring to the classroom. The

juxtaposition of these two observations raises an important question: How do teachers who

do not have adequate teaching resources in their schools operate? Some teachers found a

partial solution to the problem by borrowing important pieces of equipment from

neighbouring schools. Although this is not an ideal scenario, and certainly not a long-term

solution, it does nonetheless show a level of collegiality amongst teachers from different

schools, and the mutual sharing of teaching aids thus seems to be one way of increasing

access to important pieces of equipment and other scarce teaching resources. In addition,

there is also evidence that some teachers have tackled this problem by improvising and

making their own teaching aids. As one teacher remarked, “We are depending on what we

can produce by ourselves.”

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According to Sowell (1989), studies on the effectiveness of the use of teaching aids revealed

that learners‟ academic performance in Mathematics “is increased through the long term use

of concrete instructional materials and that students‟ attitudes toward mathematics are

improved when they have instruction with concrete materials provided by teachers

knowledgeable about their use” (p. 498). Köller, Baumert and Schnabel (2001) further

underscore that “mathematics is often seen as a very difficult subject in which motivational

factors are particularly important for the enhancement of learning” (p. 452). The general

response from participants in this study was that teaching aids tend to encourage motivation

as well as positive attitudes towards the teaching and learning of Mathematics. In addition,

many teachers felt that the use of teaching aids contributed to good academic performance in

examinations. Teachers were also of the opinion that teaching aids have the potential to

motivate bored learners to engage more in the Mathematics classroom. Teaching in a learner-

centred environment should be made easier if learners are not only interested in the

mathematics activities but are self-motivated, and teachers acknowledged that appropriate

teaching aids are a means to supporting this objective.

A common theme running through the data related to the time commitment needed to prepare

teaching aids. Many teachers felt that the preparation of teaching aids, as well as their

subsequent use in lessons, takes a lot of time. One teacher remarked that “The packed

mathematics school syllabus leaves teachers with less time to make teaching aids especially

at grade 10 and 12 levels”. Nonetheless, there were also some who felt that the use of

teaching aids has the potential to save time in the long run by allowing for the timely

completion of the syllabus. Closely linked to these views relating to time constraints was a

general feeling amongst the participants that the Namibian Ministry of Education needed to

take responsibility for supplying teachers with appropriate teaching resources.

Concluding Comments

Mathematics teachers in the Namibian context, where LCE is strongly advocated by the

education policy, are encouraged to ensure that their teaching methods promote “the active

participation of the learners in the learning process” (Namibia. MEC, 1993, p. 60).

Furthermore, teachers are encouraged to embrace a learner-centred approach in the teaching

and learning process by developing a culture of curiosity and eagerness to learn amongst the

learners through the use of appropriate teaching aids. It was the purpose of this study to

explore the availability and use of teaching aids at both public and private secondary schools

in the Windhoek metropolitan area along with the perceptions of teachers towards the use of

teaching aids in their Mathematics lessons.

In general, this study shows that the different types of teaching aids surveyed are reasonably

available in the schools that were audited, although in many instances schools are under-

resourced with respect to specific types of teaching aids. The poor availability of graph

boards and geoboards, and the relatively poor availability of geometric models, is particularly

disturbing given that not only are these items useful and effective tools for the teaching and

learning process, but they can be easily and cheaply manufactured from readily available raw

materials.

155

In terms of computer technology, 33 of the 75 teachers surveyed indicated that they had no

access to computers for teaching. Given that some of the schools that formed part of the

survey had more than four Mathematics classes per grade, with around 35 learners per class,

this represents a very high number of young learners who are not being exposed to computers

and modern technology in their Mathematics lessons. Furthermore, bearing in mind that the

schools taking part in this survey were located in a metropolitan area, the situation for rural or

remote schools is likely to be significantly worse.

This study has shown that the majority of teachers at secondary schools in Windhoek have a

positive attitude towards the importance and role of teaching aids in Mathematics. While

teachers value the importance of using appropriate resources and teaching aids in their

lessons, time constraints prevent many teachers from preparing their own teaching aids.

Furthermore, there is a widespread expectation from teachers that the Ministry of Education

should take responsibility for supplying schools with appropriate resources. Nonetheless,

despite many schools being under-resourced in terms of teaching aids, the analysis revealed

that some teachers were prepared to take ownership of this problem and either borrowed

resources from neighbouring schools or improvised with home-made teaching aids.

Based on the results of this study, the following recommendations are put forward: (i) the

Ministry of Education should be encouraged to strengthen the teaching and learning of

Mathematics at secondary schools by providing adequate teaching resources in the form of

teaching aids, (ii) the institutes of higher learning tasked with producing secondary school

teachers should work in collaboration with the Ministry of Education to provide in-service

support on the use of teaching aids to practicing teachers, and (iii) instances of teacher

creativity and resourcefulness need to be shared more broadly within the Namibian

educational landscape as examples of best practice and as examples of what is possible with

limited resources.

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Science and Technology

Long Papers

158

Exploring educators‟ perceptions on how SIKSP seminar-workshop series

prepared them to use dialogical argumentation instruction to implement a

science-IK curriculum

Senait Ghebru 1

& Meshach Ogunniyi 2


[email protected]

1, 2 School of Science and Mathematics Education, University of the Western Cape

Abstract

This study focuses on how the Science and Indigenous Knowledge Systems Project (SIKSP)

seminar-workshop series prepared educators‟ to use argumentation instruction to implement

science-IK curriculum. It draws on perceptions and experiences from 21 teachers and teacher

educators who were actively involved in a series of workshops and seminars organized by

SIKSP. A predominantly qualitative research approach was used to gain insight into the

educators‟ and teacher educators‟ perceptions with regard to the potential of Dialogical

Argumentation Instructional Model (DAIM) in implementing an integrated Science–IK

curriculum. Data drawn from a reflective diary questionnaire was analysed using the

Contiguity Argumentation Theory (CAT). CAT categories were used to describe the

perceptual shifts that the teachers and teacher educators experienced. The findings show that

the experiences gained from workshop-seminar series facilitated their efficacy (ease) in

which they used DAIM in teaching controversial issues such as the integration of science and

IK. The findings also suggested that the experiences gained helped them to make

considerable cognitive shifts from their initial opposition to the integration of science and IK

to the point they became more aware of the value and the relevance of IK to own lives as well

as the lives of their learners. The implications of these findings are discussed in the study.

Introduction

Understanding the social context of learning, as well as the effect of learners‟ socioeconomic

and cultural backgrounds in the teaching of science is of primary importance if a firm

foundation is to be laid for successful student achievement and positively affect outcomes of

science teaching and learning process (Aikenhead & Huntley, 2002; Cobern, 1994; Ogunniyi,

1988). An Indigenous Knowledge (IK) responsive curriculum is likely to fulfil the

millennium development goals which point towards „education for all‟ and „science for all‟

(Dziva, Mpofu & Kusure, 2011). Emeagwali (2003) reiterates that a science curriculum that

is responsive to IK promotes the grounds for sustainable development, environmental

responsibility, and cultural survival. Within this perspective, in the last decades increased

emphasis has been placed in different regions of the world (e.g., Australia, Canada, India, US

and many African countries) on the need to include IK in the science curriculum (Emeagwali,

2003; Nichol & Robinson, 2000; Odora-Hoppers, 2002).

For the same reason, the new South African National Curriculum Assessment Policy

Statement mandates teachers to teach IK alongside canonical school science (Ogunniyi, &

Hewson, 2008a) in order to make science more relevant to learners‟ life worlds (Odora-



159

Hoppers, 2002; Ogunniyi, 2004). The general assumption is that science teachers have the

necessary knowledge and pedagogical skills to bring about the integration of the two systems

of thought (Ogunniyi, & Hewson, 2008a). However, Aldous and Rogan (2009) reported that

teachers have difficulty in attaining the goal stipulated in South African curriculum where,

among others, students are required to understand the relationship of science and Indigenous

Knowledge System (IKS). In the same vein, Pillay, Gokar and Kathard (2008) and Amosun

(2010) found that many South African teachers still operate in the expository mode (in which

a teacher presents information without overt interaction taking place between the teacher and

the students) and therefore require adequate preparation for the required paradigm shift for

implementing a socio-culturally relevant classroom practice. In response to these difficulties,

the Science Indigenous Knowledge Systems Project (SIKSP) was established in the School of

Science and Mathematics Education of a South African university.

Since its inception in 2004 the SIKSP has been employing dialogical argumentation

instructional model (DAIM) to equip both prospective and practising teachers with content

and pedagogical content knowledge that would enable them to implement the science-IK

curriculum in their classrooms. Within this broader aim, the SIKSP offers a Practical

Argumentation Course (PAC) to post-graduate science education students. PAC provided the

needed opportunity for students, researchers and other participants to explore the Nature of

Science (NOS), Nature of Indigenous Knowledge System (NOIKS) along with Toulmin‟s

(1958) Argumentation Pattern (TAP) and Contiguity Argumentation Theory-CAT (Ogunniyi,

2007a & b). Regular lectures and seminar-workshop series enabled the participants to share

ideas about their research related to the NOS, IKS and the integration of both knowledge

corpuses. Dialogical argumentation was used as a framework to scaffold discussions.

The effectiveness or otherwise of DAIM in preparing experienced teachers to appreciate and

implement aspects of the new South African curriculum has been examined in several

instances by science education researchers using an argumentation model (e.g., Ogunniyi

2005, 2006 a & b, 2007a & b, Ogunniyi & Hewson, 2008a). This is because argumentation

instruction has been found to be very effective for resolving controversial issues (Erduran,

Simon & Osborne, 2004) e.g. the integration of science and IK (Ogunniyi, 2004; Ogunniyi

and Hewson, 2008 a). For instance, Ogunniyi& Hewson (2008a) investigated whether or not

a curriculum involving argumentation would allow teachers to: develop a sense of acceptance

of the new South African curriculum C2005, particularly the mandate to integrate IKS into

school science curriculum; distinguish between science and indigenous knowledge and select

appropriate instructional methods to integrate IKS into the science classrooms. The same

authors, Hewson and Ogunniyi (2008a) have also examined: (a) teachers understanding about

argumentation strategies and IK, (b) their perception on how argumentation helps in teaching

IK, (c) examples of how argumentation had or not had worked in their classroom and (d)

what they thought was still needed to make this approach successful. The focus of this study

is to examine how the experiences from the SIKSP seminar-workshop series prepared

educators to use argumentation instruction in implementing a science-IK curriculum. The

study also explores the type of perceptual shifts occurred as the result of their involvement in

the SIKSP seminar-workshop series. More will be said in this regard later on.

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Argumentation as a teaching strategy

Argumentation has its roots in ancient times and is associated with philosophers such as

Socrates and Aristotle who were primarily interested in the study of thinking. They posited

that the formation of reasoned argument was central to the act of thinking and their research

was guided by a desire to improve or change discourse in a society (McDonald, 2008). The

concept of argument has been defined in a variety of ways. Generally argumentation is

usually construed as distinct differences in the viewpoints of two persons or parties on a

subject matter. The view of argument underlining current studies gives the word argument

two distinct meanings (Leitao, 2000). According to Billig (1987) argument has both an

individual and a social meaning. The individual meaning refers to any piece of reasoned

discourse, whereas, the social meaning entails the dispute or debate between people opposing

each other with contrasting sides to an issue. In this perspective individuals who hold

contrasting positions attempt to convince each other of the acceptability of each adopted

opinion (Leitao, 2000). On the other hand, VanEemerson & Grootendorst (2004) focused

only on the social meaning of argumentation. They view argumentation as a verbal, social

and rational activity aimed at convincing a reasonable critic of the acceptable of a standpoint

by putting forward a constellation of propositional justifying or refuting the proposition

expressed in the standpoint. Kuhn (1993) asserts that there is a link between the individual

and the social meaning of argumentation. An argument can be either an inner chain of

reasoning (i.e. what Ogunniyi (2007a) designate as intra-argumentation) or a difference of

positions between people.

Duschl and Osborne (2002), Duschl (2008) and Newton, Driver & Osborne (1999) have

indicated that argumentation helps students to develop complex-reasoning and critical-

thinking skills, understand the nature and development of scientific knowledge, and improve

their communication skills. In the same vein, Billig (1996) and Kuhn (1992) elucidate that

lessons involving arguments will require learners to externalize their thinking. Such

externalization requires a move from intra-psychological plain and rhetorical argument to the

inter-psychological and dialogical argument (Ogunniyi, 2007a & b; Vygotsky, 1978). Similar

views have been expressed by Quinn (1997). The author indicates that when learners engage

in the process of argumentation, the interaction between the personal and the social

dimension promote reflexivity, appropriation, and the development of knowledge, beliefs and

values. In addition, students grasp the connection between evidence and claim, understand

the relationship between claims and warrants and promote their ability to think critically in a

scientific context. In view of this Billig (1996) and Kuhn (1992, 2010) have seen learning to

argue as a core process both in learning to think and to construct new ideas. Erduran (2006)

Ogunniyi (2007a &b) and Zeidler, Waker, Ackett and Simmons (2002) attempted to explain

the importance of interactive classroom arguments and dialogues from socio-cultural and

psychological perspectives. To them interactive classroom arguments and dialogues can help

teachers and students to clear their doubts, upgrade current knowledge, acquire new attitudes

and reasoning skills, gain new insights, make informed decisions, and even change their

perceptions.

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According to Ogunniyi (2007a), TAP is more appropriate to straight forward rational

arguments but is less appropriate to conductive arguments which deal with non-logical and

value-laden socio-scientific issues. For this reason this study is underpinned by the

Ogunniyi‟s (2004) Contiguity Argumentation Theory (CAT). The CAT is a dialogical

framework that offers explanation for both rational and non-rational interpretations made by

people in general and learners in particular because of what they encounter in their daily

experiences (Hewson & Ogunniyi, 2008). CAT holds that claims and counterclaims on

science and IKS can only be justified if neither thought system is dominant. There must also

be valid grounds for juxtaposing the two distinctive worldviews within a given dialogical

space. The dialogical space facilitates the process of re-articulation, appropriation, and/or

negotiation of meanings of the different world views. According to Hewson & Ogunniyi

(2008) students must therefore be able to negotiate the meanings across the two distinct

thought systems in order to integrate them.

CAT recognizes five categories that describe the movement of conceptions within students‟

minds when involved in dialogues warranting the mobilization of scientific and/or IKS-based

conceptions which are: dominant conceptions, suppressed conceptions, assimilated

conceptions, emergent conceptions, and equipollent conceptions (Ogunniyi, 2007a).

Ogunniyi (2007a) goes on to refine the conceptions as: A conception becomes dominant

when it is the most adaptable to a given context. However, in another context the same

dominant conception can become suppressed by, or assimilated into another more adaptable

metal state. An emergent conception arises when an individual has no previous knowledge of

a given phenomenon as would be the case with many scientific concepts and theories (e.g.,

atoms, molecules, evolution, etc). An equipollent conception occurs when two competing

ideas or worldviews exert comparably equal intellectual force on an individual. In that case,

the ideas or worldviews tend to co-exist in his/her mind without necessarily resulting in a

conflict e.g., religious thoughts and scientific thoughts. The context of a given discourse

plays an important role in the amount or intensity of emotional arousal experienced by the

participants in such a discourse. All these cognitive categories are in a state of dynamic and

are likely to change from one form to another depending on the context in question.

Purpose

The purpose this study was to examine how the experiences from SIKSP seminar-workshop

series prepared educators‟ to use argumentation instruction in teaching a science-IK

curriculum. The study also explores the type of perceptual shifts occurred as the result of

their involvement in the SIKSP seminar-workshop series. This study is guided by the

following questions.

How have the experiences gained from the SIKSP seminar-workshop series prepared

teachers and teacher educators to use argumentation instruction in teaching a science-

IK curriculum?

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What type of perceptual shifts occurred in the mindset of teachers and teacher

educators about the integration of science and indigenous knowledge as the result of

their involvement in the SIKSP?

Method

A group of 21 science teachers and teacher educators were exposed to a series of bi-weekly

three-hour lectures, workshops, advanced seminars underpinned by the TAP and CAT for a

period of four years (2008-2012). Each lecture lasted for one and half hours followed by

another hour for arguments and discussions. The last 30 minutes was used for recapitulation

and summary. The initial lectures of SIKSP focused on the nature of science (NOS) and the

role of argumentation in scientific practice. Teachers and teacher educators were also

introduced to the controversies that ensued among early natural philosophers and scientists

from the 20th

century to the present period with respect to the nature of the atom, etc. They

were confronted with tasks to brainstorm individually, in pairs and in smaller groups and

design a lesson and then present to the whole class group. Participating teachers and teacher

educators attended and participated in vigorous sessions of practice in argumentation,

involving argument-based tasks, practical design of instructional materials and vigorous

SIKS research. These educators were asked to complete a six item open-ended questionnaire

that engendered a reflective diary of their experience of practical Argumentation Course and

subsequent growth. This paper is limited only to the teachers‟ and teacher educators‟

responses to item 4 of the questionnaire, which posed the following question:

“How have the frames constructed from your experiences in the lectures,

seminars, and workshops prepared you to use argumentation instruction in

teaching a controversial subject such as the integration of science and IK?”

A predominantly qualitative research approach was used to gain insight into the teachers‟ and

teacher educators‟ perceptions on how the experiences from the SIKSP seminar-workshop

series prepared them to use argumentation instruction in teaching a science-IK curriculum

and in changing their views about the integration of science and IK. Data drawn from a

reflective diary questionnaire was analysed using Contiguity Argumentation Theory (CAT)

and CAT categories were applied to describe the perceptual shifts that the teachers and

teacher educators experienced.

Findings

We report our findings from a reflective diary in terms of the research questions. The

responses of teachers and teacher educators (subjects) presented in Table 1 below are quoted

verbatim in relation to research question 1. To ensure confidentiality participant teachers are

designated as T1, T2 etc, teacher educators as TE1, TE2 etc and researchers as Res 1, Res 2

etc.

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Research question one:

How have the experiences gained from the SIKSP seminar-workshop series prepared teachers

and teacher educators to use argumentation instruction in teaching a science-IK curriculum?

Table 1. List of constructed frames emanating from the experiences in the SIKSP seminar-workshop

series

Frames constructed from the experiences gained in the SIKSP

seminar-workshop series

No. of

references

%

…have played a role in making me more comfortable and at ease

in using argumentation as a teaching strategy when it is

expected to set up a lesson using this framework.”

9 34.62

…convinced me that Science–IKS integration is possible. …“I feel

confident to talk about inclusion of IKS

5 19.23

equipped me to handle my Science-IK argumentation lessons

with confidence

4 15.38

Personally I have grown in the way I dialogue with others. I listen

and think about how best to argue without being

confrontational

3 11.54

to define more clearly the interconnectedness of

dichotomized knowledge and worldviews

1 3.85

The CAT framework contributed as an analytical frame of

reference to understand the flexible state of the mind of people

as they navigate through belief, culture and religion

1

3.85

gave me insight on how to design materials for the learners. 1 3.85

enabled me to appreciate the rich diversity of cultures and

points of view within our South African environment

1

3.85

Helped me to better control the argumentative discourse

among students

1 3.85

Total number of response references 26 100

Ranked and coded in descending order. (N=21)

The analysis of participating teachers’ and teacher educators’ responses depicted in Table 1 above

identified nine frames constructed from the experiences in the SIKSP seminar-workshop series that

prepared them to use argumentation instruction in teaching a controversial subject such as the

integration of science and IK. Similar frames constructed are then grouped and ranked in descending

order of occurrence.

Of the 26 responses listed in Table 1, nine (34.9%) responses showed that subjects involvement in the

seminars and workshops have played a role in making them more comfortable and at ease in using

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argumentation as a teaching strategy when it is expected to set up a lesson using this framework. The

following excerpts derived from the reflective diaries of some of the subjects are representative:

T2: I now understand how to use DAIM in the classroom with the learners

TE2: The experiences from lectures, seminars and workshops have prepared me to use

argumentation instruction in the sense that when dealing with science concepts there should always

be a reason why certain things have to be perceived in a certain [manner].

T6: The seminars and workshops contributed a great deal in understanding the theoretical

framework of argumentation. … I actually attempted this methodology with a grade 11 class and

they were over the moon.

However, a science educator with 25 years of teaching experience had cautioned that there are

situations where argumentation did not seem to work. He pointed out that “although it has to be

appreciated that argumentation might not work in every kind of lesson and that we need to use

multiple strategies within the same lesson to keep learners interested and stimulated.”

5(19.3%) of the responses showed that overtime the subjects are convinced that integration of science

and IK is possible and they now feel confident to talk about inclusion of IKS, though with some

preconditions: The following excerpts derived from the reflective diaries of some of the subjects are

representative:

TE8: “My experiences with SIKS group in 2010 and 2011 convinced me that Science – IKS integration

is possible and comes with many benefits to the teacher and learners, however a lot of preparation

is required. I am actually looking to incorporate the SIKSP model in my teaching at undergrad level

and also in my discussions with WITS IKS group.”

T2: My experiences with SIKS group empowered me to tap into IK-knowledge, because I know now

what to look for and how to use (IKS) to explain science concepts to the learners.”

5 (15.38%) of the responses indicated that the seminars and workshops equipped them to handle the

Science-IK argumentation lessons with confidence. In view of its importance subjects are of the view

that such seminars and workshop should be provided to all science teachers. The following excerpts

derived from the reflective diaries of some of the subjects are representative:

T9: The constructed frames on argumentation have really helped and made it easy to teach the

integration of science and IK.

T11: The exposure of lesson demonstrations done by educators and myself made it easy for me to

teach IK

3(11.54%) of the responses showed that subjects had gained much in interpersonal communication

skills, in engaging in argumentation in many social issues and increased their confidence to support

their claims with evidence. The following excerpts derived from the reflective diaries of some of the

subjects are representative:

165

T7: “I am still learning how to use argumentation instruction model and I am looking forward to

trying it. Personally I have grown in the way I dialogue with others. I listen and think about how best

to argue without being confrontational.”

T10: “The exposure has instilled and increased confidence in personality, when I make claims I always

support them with evidence, in my academic work and professionally.”

The remaining 5(19.25%) of the responses showed that the workshops and seminars enabled teachers

to define more clearly the interconnectedness of dichotomized knowledge and worldviews,

understand the flexible state of the mind of people as they navigate through belief, culture and

religion, design materials for the learners, to better control the argumentative discourses among

students and to appreciate the rich diversity of cultures and points of view within South African

environment. The following excerpts derived from the reflective diaries of some of the subjects are

representative:

TE3: The frames have helped to define more clearly the interconnectedness of dichotomized

knowledge and worldviews. The CAT framework contributed as an analytical frame of reference to

understand the flexible state of the mind of people as they navigate through belief, culture and

religion.

TE4: Overall it has been an enriching experience that has enabled me to appreciate the rich diversity

of cultures and points of view within our South African environment.

Our findings attest that there are several phrases within the subjects’ responses that show

progressive perceptual shifts occurred in their mindset. For example, “I now understand’, “I now feel

confident”, “define more clearly”, “Personally I have grown” just to mention but a few. Thus, it is

worthwhile to further analyse the type of the perceptual shifts occurred in their mindset, which is

the concern of the second research question.

Research question two:

What type of perceptual shifts occurred in the mindset of teachers and teacher educators

about the integration of science and indigenous knowledge as the result of their involvement

in the SIKSP?

In order to determine the type of the perceptual shifts occurred as the result of teachers‟ and

teacher educators‟ involvement in the SKIPS seminar-workshop series, the initial and final

stances of each subject with regard to the implementation of Science-IK curriculum was

carefully scrutinized. The change of views of each subject was analyzed using CAT

categories and similar trajectories were then grouped resulting three patterns. Table 2 below

depicts the trajectories of the cognitive shifts.

166

Table 2. Perceptual shifts among teachers and teacher educators concerning implementing a science-IK

curriculum

Group Perceptual shifts Teachers Teacher

educators and

researchers

Total

Group 1 [EQ] [E] [EQ] 5 3 8

Group 2 [SD&IKS] [E] [EQ] 4 6 10

Group 3 [SD&IKS] [E] [SD&IKS] 2 1 3

N.B.: “SD” stands for Science Dominance, “IKS” for Indigenous Knowledge Suppressed, “E” for

Emergent and “EQ” for Equipollence

The discussion will proceed in three sub-sections. The first sub-section will look at teachers

and teacher educators who were initially in favour of the integration of science and IK but

lack the required cognitive knowledge and pedagogical skills to implement it, which we call

Group 1. The second sub-section will look at those teachers and teacher educators who were

initially opposed, but who subsequently changed their views overtime to being in favor of the

integration as the result of their participation in the SKIPS seminar-workshop series, which

we call Group 2. The third sub-section will see those educators who were initially opposed,

and whose views remained opposed at the end of the programme, which we call Group 3.

For all these groups CAT categories were used to indicate the type of the perceptual shifts.

Group 1: teachers and teacher educators in favor initially and remained in favor at the

end of the programme

As Table 2 indicates, of the 21 teachers and teacher educators involved in the SIKSP

seminar-workshop series 5 teaches and 3 teacher educators and researchers were initially in

favor of the inclusion of Science-IK curriculum. According to CAT categories they are in a

state of equipollence i.e a state where two competing ideas of comparable equal intellectual

force, in this case western science and indigenous knowledge could co-exist without resulting

in conflict

However, it seems that they lacked sufficient cognitive knowledge and pedagogical skills to

implement it. From the excerpts presented below there is sufficient evidence to show that

after their involvement in the SIKS project the subjects had acquired the necessary skills to

implement a science-IK curriculum. As alluded to earlier, the SIKS project encouraged the

subjects to (a) focus their research projected on NOS, NOIkS and integration of Science and

IKS (b) situate their research projects within argumentation theoretical framework and (c) use

dialogical argumentation as a teaching strategy in their respective schools. Consequently, a

more cohesive understanding of the domains of NOS, NOIKS and the use of argumentation

as a teaching strategy was gradually assimilated and new concepts emerged as there was no

167

well-formed prior knowledge or understanding of the concepts. This was concisely

articulated by:

TE1: My position on IK knowledge has always been that I valued the role that it played in society…..I

have always been in favour of this integrated curriculum as it encourages us to look back at how

practices from the past have informed modern day practices….. the SIKSP activities had enabled me

to master the use of argumentation. “The lectures and experiences have indeed played a role in

making me more comfortable and at ease in using argumentation as a teaching strategy when it is

expected to set up a lesson using this framework.”

TE3: I was not opposed to the trajectory of the new curriculum because it addressed the issue of

relevance, in that science was seen in context and provided a reason for knowing about science and

its’ impact on everyday living. …..The frames have helped me to define more clearly the

interconnectedness of dichotomized knowledge and worldviews…and the CAT framework

contributed as an analytical frame of reference to understand the flexible state of the mind of people

as they navigate through belief, culture and religion.”

T6: I did not oppose it, … Always had misconceptions about it and that IKS was not part of my culture.

I certainly changed my view as I have gained more knowledge about IKS. I am a fully believer of the

integration of IKS and Sciences. I believe that it is important to make the Sciences students learned in

school their own…. The seminars and workshops … gave me a deeper understanding of how to apply

argumentation in order to integrate the IKS with the Sciences. I actually attempted this methodology

with a grade 11 class and they were over the moon….

Group 2: Opposed educators changing in an in-favor view at the end of the SKIPS

Perusal of the trajectories displayed in Table 2 show that before involvement in the SIKS

project 4 teaches and 6 teacher educators and researchers have opposed the integration of IKS

in the science curriculum. According to CAT categories acceptance of IKS conception was

suppressed and modern science prevailed as the dominant mindset. However, after

participation in the SKIPS project they developed new understanding and insights of the

nature of science and Ik, argumentation and what the science-IK curriculum entails. This

resulted in progressive shift in mindset of their perceptions about the integration of science

and IK. According to the CAT, these teachers and teacher educators have developed an

emergent view having been exposed and having developed new understandings on a

topic/concept of which they had no prior experience. There is also sufficient evidence from

subjects‟ responses that after extended participation they had reached a state of equipollence,

i.e a state where two competing ideas of comparable equal intellectual force, in this case

western science and indigenous knowledge could co-exist without resulting in conflict

(Ogunniyi, 2004, 2008). This was succinctly articulated by:

Res 1: At the beginning: When I started, I felt that IKS and science where two different knowledge

and that it was almost impossible to combine the two… My experiences in workshops have shown

that preparation in terms of worksheet preparation, lesson plans and notes with introductory case

studies were important in making the integration of science and IK possible. Over time my

understanding grew to a point where I also understood that the present day or modern science was

168

just a conglomeration of knowledge adopted from indigenous people all over the world. This

realization convinced me that there was a real need to mitigate by bringing IK in or to integrate the

two so as to enrich our experiences of the world we are living in.

T1: I was opposed to the new curriculum because it carried new content which required more

training, research and new thinking. I wasn’t so much equipped to face it. Now I have changed my

view because my participation in SIKSP has prepared me to handle my Science / IK argumentation

lessons with confidence. “This should be done to all the other teachers”. I can now see the new

curriculum as an innovation for better education

Group3: Educators remained opposed at the end of the SKIPS program

Analysis of the trajectories indicated in Table 2 reveal that two teachers and one teacher

educator have initially opposed the implementation of science IK curriculum. According to

CAT categories acceptance of IKS conception was suppressed and modern science prevailed

as the dominant mindset. Nonetheless, there is an indication that these educators have been

exposed to the same programme and gained similar experiences as those of Group 1 and

Group 2 educators. Consequently, a more informed understanding of the domains of NOS

and NOIKS were gradually assimilated and new conceptions emerged. Yet they remained

opposed after the end of the programme and attempted to justify their claim by providing

practical reasons, which is an indication that there was no cognitive shifts in their mindset.

This was succinctly articulated by:

T2: Yes, I oppose the integration of science and IK b/c the NCS (2003) curriculum depends on kind of

resources that were not readily available to teachers at schools….. I now understand how to use the

(DAIM) model in the classroom with the learners. It also has empowered me to tap into IK-

knowledge, because I know now what to look for and how to use (IKS) to explain science concepts to

the learners.”……I have not change my view because … the CAPS (2011) policy statement document

… include IKS under specific aim 3.2 to be implemented in the assessment of learners’ again without

any proper guidelines and resources materials.

Discussion

The science teachers and teacher educators involved in this study were of the opinion that the

seminar-workshop series enhanced their understanding of argumentation. Looking at the data

displayed in Table 1, there is sufficient evidence which indicate that the experiences gained

from the SIKSP seminar- workshop series prepared subjects to use DAIM in teaching

controversial issues such as the integration of science and IK. This implies that the explicitly

reflective argumentation-based instructional approach employed in the SIKSP seminar-

workshop series enabled them to develop their argumentation skills. This finding is resonant

with the general assertion that scientific argument needs to be explicitly taught if students are

to enhance their argumentation skills (Osborne, Erduran & Simon, 2004a). Our findings

confirm that DAIM had played a great role in subjects‟ intellectual, professional and personal

growth. These findings seem to be consistent with the results of previous studies that show

the potential of argumentation in knowledge building and in enhancing students‟ and

169

teachers‟ conceptual understanding of scientific concepts (e.g., Leita, 2000; Venville &

Dawson, 2010; Zohar & Nemet, 2002).

Our data showed a change in the teachers and teacher educators understanding about the

possibility of an integrated science-IKS school curriculum. Our selected verbatim quotes

revealed changes that we categorized according to Contiguity Argumentation Theory (CAT).

It can be deduced from the data displayed in Table 2 that the SIKSP seminar-workshop series

have played a great role in making a cognitive shift for ten out of thirteen teachers and

teacher educators who initially opposed the notion of integrating science and indigenous

knowledge. There are also sufficient evidences that these subjects have changed their views

and reached to a state of acknowledging the benefits of an integrated science and indigenous

curriculum. This group of teachers and teacher educators can be placed under the equipollent

category of CAT. The equipollent category refers to a situation where competing

explanations are judged to be equally powerful and convincing by the individual (Ogunniyi,

2004, 2008). The group of teachers and teacher educators then holds the two apparently

opposing positions side by side without any apparent cognitive dissonance.

Conclusion

In our analysis of subjects‟ reflections on how the SIKSP seminar-workshop series prepared

them to use argumentation instruction in teaching a science –IK curriculum, two important

points relating to the use of DAIM in science education surfaced. On one hand, there is

sufficient evidence that subjects appeared to deepen their understanding of the use of

argumentation through research and classroom practice. On the other hand, there is an

indication that the communal sharing of ideas during workshops and seminars resulted in the

professional, intellectual and personal growth of the subjects. This implies that interactive

classroom arguments and dialogues employed in the SIKP helped the subjects to clear their

doubts, upgrade current knowledge, acquire new attitudes and reasoning skills, gain new

insights, make informed decisions, and even change their perceptions (Erduran, 2006;

Ogunniyi, 2006a,b; Zeidler, Waker, Ackett & Simmons, 2002).

The fact that 10 out of 13 teachers and teacher educators have changed their initial views

about the integration of IKS into school science after their involvement in the SKIPS project

may possibly bring a paradigm shift in the teaching and learning of science. In view of the

findings, we posit that creating contexts in which learners can argue about socio-cultural

issues creates an environment where they are encouraged to reveal their lived experiences

from which they could extract valid scientific concepts. However, as stated earlier recent

studies revealed that South African teachers still operate in the assimilatory mode and

therefore require adequate preparation for the required paradigm shift for implementing a

socio-culturally relevant classroom practice (Amosun, 2010; Pillay, Gokar & Kathard, 2008).

Further exploration in this regard would yield meaningful insights into how teachers

operationalize innovations in curricula. These findings have implications for curriculum

development and instructional practice which the Department of Education(DOE),

curriculum designers and teacher educators in South Africa and perhaps other countries

implementing science IK curricula could find informative and useful.

170

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173

The effect of an argumentation model in enhancing educators‟ ability to

implement an indigenized science curriculum

Meshach Ogunniyi,

School of Science & Mathematics Education, University of the Western Cape, South Africa

[email protected]

In the pursuit of relevance of education, one of the goals of the new South African curriculum

is for teachers to integrate science with the indigenous knowledge. However, since its

inception in 1997, the new curriculum has generated a lot of controversy. This study, a part of

a larger project, stemmed from our belief that argumentation as a dialectical tool could be

used effectively to resolve the controversies surrounding the curriculum. Based on this belief,

a cohort consisting of 23 teachers and teacher-educators (henceforth subjects) were exposed

to dialogical argumentation instructional model (DAIM) for a period of two years. Data based

on their responses to an open-ended questionnaire, completed worksheets and video

recordings showed that the subjects had over the period developed the ability to: (1) use valid

arguments in classroom discourse as basis for their dispositions towards the new curriculum;

(2) mobilize argumentation skills to reach collaborative consensus; and (3) value the

scientific and indigenous ways of knowing and interpreting experience.

Introduction

Since the last decade there has been an increased interest in the field of science education to use

argumentation as a dialectical tool to achieve learners‟ conceptual understanding and to resolve

controversial socioscientific issues. Concomitantly, there has been considerable effort to

improve science teachers‟ professional practice in organizing argumentation-based classroom

discourses (e.g. Erduran, Osborne & Simon, 2004; Sampson & Grooms, 2009; Simon, Erduran

& Osborne, 2006; Simon & Johnson, 2008). However, what seems lacking in all these attempts

is a systematic account of how and why teachers change views or practice e.g. as a result of

being exposed to a sustained participative experience in a project in which they played an

active role. Often, teachers are at the consumers‟ end of the curriculum development process

rather than being active co-participants. As a result there is usually a chasm between

curriculum idealization and implementation on the one hand and between teacher education

and professional practice on the other.

Ebenezer (1996), citing Fenstermacher argues that change in teacher professional practice is a

holistic affair which involves a change in the truth value of the premises and assumptions

underlying their beliefs and convictions. In other words, to bring about a change in teachers‟

professional practice one must first challenge their beliefs about practice, as well as their

understandings, dispositions, intellectual interests and value orientations. However, teachers‟

willingness to participate in an experience that could lead to change in practice depends on the

role they are allowed to play in the process of change. This is because change is not usually a

comfortable experience for anyone, not least teachers (Ebenezer, 1996). It involves moving

from the zone of actual development where a teacher can handle instruction efficaciously to the

zone of proximal development where they are confronted with tasks that require assistance

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from the knowing individual e.g. teacher educator to accomplish the desired success (Parson &

Brown, 2002; Vygotsky, 1986).

It is important to point out that it will be preposterous to assume that teachers would easily

change their instructional practice on account of a few weeks or even months of exhortation

about the value of argumentation instruction. Rather, what tends to occur when people are

exposed to a new instructional approach such as argumentation is that they first need to

undertake activities that enable them to understand essential argumentation protocols such as

the elements of an argument, the series of back-and forth intra-locution or ontological

reflexivity as well as the critical self-reflection on one‟s beliefs, biases, viewpoints,

predispositions, preferences and value orientations. Indeed, as shown in a number of earlier

studies, attempts to attain change in teacher professional practice have tended to result in a sort

of Heisenberg‟s “uncertainty principle” thus indicating the complex nature of the whole

process (e.g. Ogunniyi, 2004, 2007a & b; Ebenezer, 1996; Sampson & Groom, 2009).

In recent years teachers have been given more voice and communicative freedoms than before

to express their views on programmes that have been set up to improve their practice. It is light

of this that the study explored the potential of argumentation instruction for building the

teachers‟ knowledge about practice and consequently strive towards change in their beliefs and

professional practice, especially while attempting to implement an arguably controversial

science curriculum. However, in view of the holistic nature teacher change, and as revealed in

previous studies, it would be unreasonable to expect change in the Vygotskian sense of moving

from the zone of actual development to proximal development without a sustained programme

of support (e.g. Erduran, Simon & Osborne, 2004; Newton, Driver & Osborne, 1999;

Ogunniyi, 2004, 2007a & b; Simon & Johnson, 2008).

It is apposite to mention upfront that neither the teachers nor the teacher educators involved in

this study were well equipped to implement an indigenized science curriculum. As would be

shown in the result section both groups were in a learning situation. It is often not realized that

in certain contexts (e.g. integrating canonical school science with indigenous knowledge) even

teacher educators might lack the necessary facility to bring about what is required to effect

teacher change in practice. Most teacher educators like the teachers they teach have been

schooled in western science and not in the indigenous ways of knowing (Ogunniyi, 2013). For

the same reason the study explored how the teachers and teacher educators perceived DAIM to

which they had been exposed.

As stated earlier, the value that teachers place on an education programme aimed at improving

their practice lies in determining what is required to modify the truth value of their

epistemology, metaphysical beliefs, value orientations and intellectual interests. This implies

affording them access to communicative freedoms beyond their personal preferences to

consider what they can learn from their peers as would be the case while working in an

interactive social context (Ogunniyi, 2007a & b). In the same vein, the change introduced into

their underlying epistemology and metaphysical beliefs must be compatible with the postulates

of the model of practice that is worthy emulation. Also, teachers must be willing participants in

determining the change in their practical arguments to accommodate a instructional practice or

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perspective that they ultimately would own. This is akin to Habermas‟ (1971) notion of moving

from basic techinicist understanding of issues to acquiring the emancipatory knowledge or

Freire‟s (1993) libratory pedagogy critical to their ontological, epistemological and axiological

sense of self or ownership of what or not to share, believe or do to transform their practice.

Model professional practice through argumentation

A model of argumentation that has attracted increasing usage in science education is the

Toulmin‟s (2003) Argumentation Pattern-TAP (Erduran et al, 2004). Essentially TAP consists

of a claim-an assertion awaiting validation, data-evidence in support of that claim, warrant-a

general statement which links the claim with the evidence, backing-the underlying assumption

to the claim, qualifier-the contingent conditions in which the claim holds, and the rebuttal-

possible exception to the claim. The merits and demerits of TAP have been well rehearsed in

the literature (e.g. Erduran, Simon & Osborne, 2004; Simon, Osborne & Johnson, 2008) and do

not need being repeated here.

One limitation of TAP which is of interest in this study is that TAP is restricted only to

syllogistic form of argument i.e. the deductive-inductive form of argument. It does not apply to

conductive arguments commonly encountered in meta-ethical discourses i.e. value-laden

discourses relating metaphysical issues (Ogunniyi, 2004, 2007a & b). It was in light of this that

this study deployed the Contiguity Argumentation Theory (CAT) form reasoning which draws

on the Aristotelian association by contiguity i.e. a type of association whereby elements of two

mental states tend to combine and recombine to attain a form of harmony in conformity with

the laws of association of ideas (Dunes, 1975). Guthrie (as cited by Hilgard & Bower, 1975)

extends the same notion of contiguous association as key to learning. In other words, as

contiguous ideas interact in the mind, a state of equilibrium or harmony is reached which in

itself is a new form of learning and a cue or template for subsequent learning episodes. In the

same vein CAT construes the resolution of conflicting worldviews in terms of a cognitive

process of accommodation, assimilation, integrative reconciliation and adaptation (Ogunniyi,

1988, 2004, 2007a).

Essentially, CAT consists of five main cognitive states namely: dominant- i.e. the worldview is

the most suited for a given context; suppressed- a subordinated worldview; assimilated- a

subsumed worldview that is so incorporated into the dominant worldview to the extent of

losing most of its original characteristics; emergent-a newly developed worldview (scientific or

indigenous) resulting from a new experience; and equipollent-an amalgamated worldview

which draws on the elements of conflicting worldviews. These cognitive states are in a state of

dynamic flux and may change from one context to another depending on the nature of

interacting constituent elements (Ogunniyi, 2007a; Ogunniyi & Hewson, 2008).

An assumption underpinning the study is that exposing the subjects to Science and

Indigenous Knowledge Systems Project (SIKSP) dialogical argumentation instructional

model (DAIM) would provide them the necessary intellectual space or what (Bhabha, 1994)

calls hybrid space to argue, externalize their thoughts, clear their doubts and express their

views freely about a science curriculum that accommodates some aspects of IK in the

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classroom context. In this regard, DAIM proved handy for exposing the subjects to

argumentation protocols. Although findings on argumentation studies vary widely,

researchers in the area are in agreement that argumentation is an effective dialectical tool for

attaining collaborative consensus building especially on a controversial subject (e.g. Berland

& Lee, 2012; Leitao, 2000; Sampson & Grooms, 2009). But for a fair and meaningful

argumentation to take place, the conflicting views must be considered on an equal footing.

However, this is often not the case when the two thought systems are compared (e.g. Diwu &

Ogunniyi, 2012; Nichol & Robinson, 2000; Ogunniyi, 2004). According to Habermas

(2001) four basic features of a fair argument must include at least four basic features such as:

(i) no person who could make a relevant contribution may be excluded; (ii) all participants

have equal opportunities to make contributions; (iii) participants are truthful in what they say;

and (iv) the contributions are freed from internal or external coercion i.e. participants‟ stances

are open to criticizable validity claims motivated solely by the rational force of better

reasons. In light of this principle discourses on science and IKS were considered in the study

as equally legitimate systems of thought.

Purpose of the study

The aim of the study was to determine the effectiveness of dialogical argumentation

instructional model (DAIM) in enhancing the subjects‟ ability to implement an inclusive

science-indigenous knowledge curriculum. More specifically, the study explored to what

extent DAIM enhanced the subjects‟ ability to: (1) use argumentation to resolve conflicting

ideas in their own minds about the new science-IK curriculum; (2) attain collaborative

consensus on controversial issues surrounding the curriculum; and (3) use argumentation as

basis for evaluating the scientific or indigenous ways of knowing that the new curriculum seeks

to promote. In pursuance of this aim answers were sought to the following questions: Q1: What

narratives can you tell about your experiences in the SIKSP and your evolving stance (position)

on the issue of implementing a science-IK curriculum in your classroom? Question 2(a): In

what ways have your experiences in the SIKSP informed the way you frame the issue of

integrating school science and IK? (b) Were you once opposed to the new inclusive curriculum

demanding the integration of the two? Please express your view. Question 3: How has your

ability to leverage or reflect your frames about integrating science and IK in your instructional

practice helped you to value the scientific and indigenous ways of knowing and interpreting

experience?

Methods

The study involved the selection of a purposive sample of 23 subjects i.e. 10 science educators

and 13 practising teachers enrolled on a Masters course. The merit of using a purposive sample

lies in selecting information-rich cases or sources for in-depth study (Patton, 1987) as was the

case of these subjects with mutual regarding the integration of science and IK in the classroom

(Ogunniyi, 2011). The subjects participated in a series of weekly three-hour argumentation

sessions which lasted for a period of two months. But even before this, they were exposed to

the Toulmin‟s (2003) argumentation pattern (PAT) which helped them to argue logically and

recognize the different aspects of an argument such as: claim-an assertion in need of data or

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evidence; a warrant-statement used to justify a claim or link a claim to an evidence; backing-

supporting evidence; qualifier- the contingent condition in which a claim holds; and a rebuttal-a

contrary or an exception to the claim (Erduran, Simon & Osborne, 2004; Leitao, 2000).

Further, these argumentation sessions helped the subjects on how to mobilize arguments to

resolve conflicting viewpoints and in reaching consensus without feeling offended. During the

period the subjects agued in pairs on various controversial issues before the whole group on

various topics such as: differential electricity rates on the basis of socio-economic statuses;

building a five-star hotel in a wetland near metropolis; replacing coal with clean power sources

e.g. wind, atomic and solar power and so on.

After the preliminary activities the subjects were exposed to a series of three-hour lectures and

seminars per week underpinned by dialogical argumentation instructional model (DAIM) for

six months. The first one and half hours were devoted to lectures followed one hour session of

argumentation evinced by the by tasks in the worksheets. The last half was used to recapitulate

the important points and to give the next reading assignments (Ogunniyi, 2007a). To prepare

for each lecture or seminar, the subjects read several papers on NOS as espoused by renowned

scholars e.g. Popper, Hempel, Kuhn, Merton, Kline, Habermas, Toulmin, Ziman, pre-Socratic

debates on the nature of matter; Ptolemaic geo-centric versus Copernican helio-centric systems;

Semmelweis‟ childbed fever case; controversies that surrounded the structure of the atom and

so on. After each lecture, the subjects mobilized individual (intra-), small-group (inter-) and

whole-group (trans-) argumentation protocols to carry out specified tasks in the worksheets for

the next one-and a half hour. The goal to reach consensus (cognitive harmonization) where

feasible accords with the community practice of ubuntu (the indigenous concept for

togetherness, relatedness communality or mutuality) in the process of working on assigned

tasks. It was also at the trans-argumentation stage that the CAT cognitive categories were

determined. The last 30 minutes was used for recapitulation and for giving reading assignments

for the next lectures and seminars.

After the lectures the subjects were exposed to three-hour weekly lab-based workshops

underpinned by DAIM coupled with assigned cognitive tasks in the work sheets for a period of

six months. The remaining 10 months were devoted to a combination of lectures, seminars,

material development workshops and classroom visits to see to what extent the subjects were

able to implement an indigenized curriculum based on AIM in their classrooms. During the

same period the drafts of three resource books on various topics (including detailed exemplary

lesson plans for integrating science and indigenous knowledge) were developed. More details

about the lectures, seminars and the workshops have already been published (Ogunniyi, 2004,

2007a & b; Ogunniyi & Hewson, 2008).

The data were collected from four instruments: a five open-ended questionnaire; worksheets;

video recordings of classroom interactions; and interviews. Details of the development of these

instruments have already been published (Ogunniyi, 2007a, 2011; Ogunniyi & Hewson, 2008).

The data collected were analysed qualitatively and quantitatively. However, due to space

limitation only snippets of the findings are reported in the paper.

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Findings and discussion

Q1: What narratives can you tell about your experiences in the SIKSP and your evolving stance

(position) on the issue of implementing a science-IK curriculum in your classroom?

This open-ended question expected the subjects to reflect on their SIKSP experiences during a period

of two years. Table 1 below, based on ATLAS software analysis of their responses, and provides the

following emerging themes:

Table 1. Frequency of evolving stances on implementing a science-IK curriculum

Emerging Themes f

1. SIKSP dialogical argumentation model enlightened me to integrate science

and IK in my classroom

38

2. SIKSP provided a rich environment for sharing knowledge and feelings about IKS 37

3. SIKSP argumentation model removed my ignorance about IKS 36

4. Science and IKS are interwoven and sometimes compatible 36

5. Before SIKSP I did not recognize IK as valid and legitimate knowledge like science 30

6. South Africans generally have a nonchalant attitude towards IKS 11

Total 188

An examination of Table 1 shows 188 instances of themes emerging from the subjects‟

responses to question 1. It should be noted that some subjects contributed more than one theme.

Of these themes: 38 (20%) relate to how SIKSP argumentation model showed them a way to

integrate science and IK in a classroom context; 37(approx. 20%) are concerned about how it

provided them a rich environment to share their knowledge and feelings about IKS; 36 (19%)

deal with how it removed their ignorance about IKS; another 36 (19%) deal with how SIKSP

was instrumental in convincing them that the two knowledge systems are interwoven and

sometimes compatible while only 11(6%) are concerned with the nonchalant attitudes of South

Africans to IKS. The italicized phrases (i.e. themes 1-4) in Table 1 in terms of the Contiguity

Argumentation Theory (CAT) categories show discernible perceptual shifts of the subjects

from a predominantly scientific worldview to an emergent one in knowledge, attitudes,

awareness and favourable disposition towards IKS most probably as a result of their SIKSP

experience. The excerpts below show the nuanced nature of such perceptual changes:

Estralita, a science teacher educator stated that: Being involved with SIKSP has

been an inspirational journey. The fact that the group is heterogeneous (students

and lectures with different perspectives and knowledge about teaching IKS) has

provided a very rich environment for sharing knowledge and feelings about IKS.

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Alan, a science teacher turned weather engineer in a private firm said: At

first I did not understand what IKS means…But after being exposed to

various SIKSP-workshops and different form of IK and how it will help to

understand the science curriculum, I was amazed to see how much IK-

knowledge we can tap into to implement a science-IK curriculum in the

classroom.

Sammy, a bio-informatics/teacher educator said: It has equipped me with all the

necessary skills that I need to implement the new curricula, especially “IKS-

science.” For me, I now appreciate my socio-cultural attributes and also view

them as essential for my day to day life. Whenever I am involved in curricula and

instructional development related activities, I consider societal implications and

integration of IKS.

Dana, a physical science teacher added: From the beginning, I never

thought of any science in IK. I believed in the science that I was taught from

the western point of view which regarded anything different from what the

textbook said as superstition. After my experiences in the SIKSP I now

began reflecting on the practices of my people and saw that many of them

were in fact congruent with science.

Diamond, a science/mathematics teacher educator said: Before being part of the

SIKSP group I was a bit skeptical about the role which IKS can play in our everyday

life situations later on to be integrated as part of science curriculum. Having

attended the workshops and seminars I have grown to understand that

knowledge from both IKS and modern science are all the same. Further

reflections have led me to conclude that two knowledge systems can actually co-

exist.

An analysis of the excerpts above shows that SIKSP argumentation instructional model did

impact positively on the subjects‟ perceptions about the relationship between science and IK.

In terms of CAT it is safe to state that the AIM model assisted them to reflect on their

experience in the project and consequently to shift from their initial opposition or scepticism to

a more accommodating stance of IK as a legitimate way of knowing that is compatible with

that of science. Reflection, a critical aspect of learning, seems to have enabled the subjects

undertake a sort of intra-argumentation, introspection and/or retrospective analysis of their

experiences in the project and to furnish them with the intellectual tool to improve their

instructional practice concerning the new inclusive curriculum.

Question 2(a): In what ways have your experiences in the SIKSP informed the way you frame

the issue of integrating school science and IK? (b) Were you once opposed to the new inclusive

curriculum demanding the integration of the two? Please express your view.

The themes in Table 2a indicate how SIKSP argumentation model might have impacted on the

way the subjects framed their perceptions about the issue of integrating school science with IK.

The pattern of this impact in a descending order in are: science dominated worldviews (18%);

favourably disposed to the integration of science and IK as a result of SIKSP (17%);

180

unawareness about IK before SIKSP (16%); valuing IK as a result of SIKSP (14%); looked

down on IK and regarding it as unscientific (14%); dialogical argumentation as a useful tool for

integrating science and IK(9); and lastly, knowledgeable about IK before participating in

SIKSP (7%). The italicized phrases are suggestive of perceptual shifts in terms of CAT

categories ranging from a scientifically dominant or suppressed IK worldview to an emergent

and/or an equipollent worldview perspective. Overall, as in Table 1 the subjects became more

favourably disposed to the new curriculum than was the case before they were exposed to AIM.

Table 2. Frequency of emerging perceptual shifts towards a science-IK curriculum

Emerging Themes f

2a: SIKSP and the subjects’ framing of a science-IK curriculum

Science alone shaped my worldview before I joined SIKSP 16

I now support the integration of science and IK 15

I was ignorant or unaware of IK before joining SIKSP 14

SIKSP has enabled me to value IK now 12

I looked down on IK and regarded it as unscientific before joining SIKSP 12

Dialogical argumentation is an enabling tool for science/IK integration 8

I had knowledge of IK before joining SIKSP 6

Sub-total 88

2b: Reasons for opposition the indigenization of science and indigenous knowledge

Opposed new curriculum because teachers were not well trained or equipped 13

I was opposed to IKS 10

Still oppose to new curriculum because of inadequate training and provision made 8

IK is valuable 7

I was opposed to IKS before, but now want to try it 6

Sub-total 44

The themes emerging from question 2b reveal the reasons why some of the subjects were

opposed to the integration of science and IK. Compared to question 2a, only a relatively

small number of themes emerged from question 2b. But even these as in earlier studies (e.g.

Author, 2004, 2007a &b; Author & Associate, 2008) relate mainly to the inadequate training

that the subjects received. As the excepts below show, some of the subjects indicated that

they valued IK while others were initially opposed to the idea but later became willing to

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integrate the two knowledge corpuses in their classrooms. Yet others were not really opposed

to the introduction of IK into science but were unfamiliar with the term. Nevertheless after

their experience with AIM their knowledge and awareness about IK increased.

Pavi, a physical science teacher said: No, I was never opposed to the new

curriculum and its IKS aspects. Although I did not know the term “IKS” at the

beginning I could easily recognize and realize that the science we have today have

its feet entrenched in indigenous practices.

Ruti, an active retired teacher educator said: I was never against the integration

of indigenous knowledge into the science curriculum. I grew up with a familiarity

with many cultural practices embedded in African storytelling and through

interacting with children in the farm environment in which I grew up...With the

insight gleaned from SKISP workshops particularly argumentation exercises my

approach would be different.

Lora, a primary school science teacher said: Yes, I was in opposition

towards the new curriculum, as mentioned previously, but after the Science-

IKS curriculum was unpacked during workshops, I tried to convince myself

that this curriculum can work if teachers are trained properly… The more

workshops I attended, the more, the terminology made sense, and the more I

convinced myself that, perhaps this is the way forward…I realized that IKS

is not something new to me, perhaps the terminology. I grew up in a

multicultural house.

Similar views to those above were expressed by several other subjects whether they are

science teachers or teacher educators. Whatever the case, there is sufficient evidence to show

that their experiences with AIM helped them to be more favourably disposed to the new

curriculum than was the case in earlier studies (e.g. Author, 2004, 2007a & b, 2011; Author

& Associate, 2008). Also, these findings corroborate earlier findings showing the

effectiveness of argumentation as a rhetorical or dialectical tool for knowledge building and

attitudinal changes (e.g. Erduran, Simon & Osborne, 2004; Leitao, 2000; Author, 2004,

2007a & b; Osborne, 2010; Sampson & Grooms, 2009; Simon & Johnson, 2008)

Question 3: How has your ability to leverage or reflect your frames about integrating science

and IK in your instructional practice helped you to value the scientific and indigenous ways of

knowing and interpreting experience?

Table 3. Frequency of emerging awareness about the value of science and indigenous ways of

knowing and interpreting experience

Emerging Themes f

I now believe that IK does involve the use of scientific knowledge and skills 22

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Integrating science and IK through argumentation does enrich the experiences of

both teachers and learners.

Integrating science and IK makes science teaching and learning more relevant to

learners’ experiences

I have integrated science and IK successfully in my classroom

I now support and look forward to introducing IK into my classroom

19

14

12

2

Total 76

Like the data in Tables 1 and 2 the subjects made considerable perceptual shifts from their

initially negative or sceptical disposition towards IK or its integration with science to a more

positive stance. The word “now” occurring in 22 (29%) instances is characteristic of this

perceptual and attitudinal shift. The affirmative claims such as: “...argumentation does enrich

the experiences of both teachers and learners (25%)... I now support and look forward to

introducing IK into my classroom” are further indications of such perceptual shifts. The

statement, “I have integrated science and IK successfully in my classroom” (16%) is further

evidence of the subjects‟ transformation of practice or Frèire‟s (1993) notion of “libratory

pedagogy which occurs when a teacher shifts from a traditional form of pedagogy to a more

radical self-propelled instructional approach.

Of course, it is worth mentioning that the function of argumentation is not limited to the

integration of science and IK alone. It is a useful means for resolving conflicting perspectives

on any subject matter. It is a dialectical tool which people use to externalize their thoughts,

clear their doubts, and even change their minds in light of a more convincing argument.

Likewise, argumentation is means for constructing or co-constructing knowledge in an attempt

to attain collaborative consensus on a controversial subject matter. But despite its benefits, the

actual process by which people change their viewpoints (as the excerpts below would show), is

fully known (Erduran et al, 2004; Leitao, 2000; Simon, Erduran & Osborne, 2006). What

seems to emerge from this study, however, is that perceptual changes are as a result of a

combination of diverse factors. The following excerpts derived from the reflective diaries of

some of the subjects are representative of this standpoint:

According to Dana, a physical science teacher: Many activities in the SIKSP helped

me to begin to see more value in IK ... I saw that I held both scientific as well as

IKS conceptions concerning the origin of life, the world, the rainbow, and so on.

Thus I began to see that both views had their place in life and hence can become a

subject of interesting debate in the science class... As we learned during the

workshops, I could reflect back and see that I had looked down on many

indigenous practices from my background as unscientific just because of the Euro-

centric way in which I was taught science; not that there was no science in these

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methods...In conclusion, I would say that several factors contributed to my

change of perspectives as far as science-IK integration is concerned.

Sena, a teacher educator summarized her experience in the project as follows: In

a nutshell the lectures, series of advanced seminars and workshops, the tasks

given, Book One and Two were instrumental in building my capacity and played a

central role in changing my worldview. At the moment, the two competing

worldviews (science and IKS) exert equal cognitive force in my mind, which could

be considered as considerable shift of view. This stance in terms of CAT unit of

analysis could be regarded as an equipollent.

Noni, a life science teacher said: My exposure to IKS and argumentation

instruction model in the SIKSP has prompted me to have an inquisitive mind and

to always listen to other peoples’ viewpoints. Argumentation instilled in me to be

a critical thinker. It made me become appreciative of differences between people

and not to view these differences as antagonism. Personally, I have grown in the

way I discuss issues with my family and friends. I also tend to do intra personal

argumentation when I have challenges and in doing so I come out better that way

in problem solving.

Chis, a physical science teacher with a chemical engineering background asserted

that: Having gone through all the course work, workshops and seminars, I came

to understand that without the understanding of the NOSIKS [nature of science

and IKS] and argumentation ...it would be almost impossible to integrate the

two...IK was not just knowledge of the past, but many people are still using it

nowadays and hence is just knowledge which is authentic to a particular people’s

experiences and by no means inferior to present day technologies.

Diamond, a science/mathematics teacher educator said: The workshops which I

have attended have, in a lot of ways, helped me to move away from the idea that

IKS is not concrete knowledge which can be tested and validated. There has been

talk among teachers that IKS is associated with myths and witchcraft and that

there was no way science and IKS could co-exist.

Brenda, a physical science teacher said: My experiences in the SIKSP activities

have reformed my way of thinking, doing things completely. When I started I did

not appreciate indigenous knowledge nor did I realize its richness until I matured

in these workshops...I have now reached a level where I am confident of

integrating these two worldviews harmoniously.

The excerpts above and similar ones are indicative of how SIKSP, particularly its

argumentation model contributed to the perceptual and attitudinal changes observed among the

subjects with respect to the implementation of an indigenized curriculum in their classrooms.

The common themes resonating through these excerpts range from the subjects‟ changing

perspectives about IK over time starting from a state of unawareness to a state in which they

were much more informed of what the term stood for. In the latter they had begun to value it as

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a legitimate way of knowing and interpreting experience. In terms of CAT categories the

perceptual shifts range between a scientifically dominant or suppressed IK worldview to an

emergent and finally an equipollent worldview.

Conclusion

If the enthusiasm shown by the subjects in the SIKSP lectures, seminars and workshops is

anything to go by, then the prospect of the new curriculum being successfully implemented

by these subjects in their classrooms using AIM should be much higher than was previously

the case (e.g. Ogunniyi, 2004, 2007a & b). As the subjects got involved in argumentation and

reflected on their experiences in the project, constructed and co-constructed their ideas, they

gradually revised their perceptions about IK and consequently their negative dispositions

towards the new inclusive curriculum changed (e.g. Berland & Lee, 2012; Leitao, 2000;

Osborne, Erduran & Simon, 2004; Simon & Johnson, 2008). Further, their positive comments

certainly have implications not only for their pedagogical beliefs but also their instructional

practice as well. Another important glimpse from the study, which require further

investigation is the potential of argumentation for promoting the subjects‟ self-understanding,

awareness, value orientations as well as their willingness to revise their own perceptions in

terms of reciprocal perspective–taking in an uncoerced joint-acceptance setting (Habermas,

1999) provided by SIKSP. Reflexivity normally entails a process of self evaluation in the

face of new experiences. When this is followed with reflexive action i.e. praxis, it becomes

an emancipatory or libratory pedagogy (Freire, 1993).

Many of the subjects attested to the fact that their changed views towards the new inclusive

curriculum were not unrelated to their experiences with AIM to which they were exposed. To

some extent they were able to: (1) provide valid reasons why they initially opposed the new

curriculum; (2) demonstrate an increased awareness or understanding of what an integrated

IKS or science-IK curriculum stood for as a result of being exposed to an argumentation

model; (3) attain collaborative consensus on the various controversies surrounding the new

curriculum; and (4) show appreciation for both the scientific and indigenous ways of

knowing and interpreting experience with nature. Although the process of how people

(including these subjects) change their minds on a given subject are not fully known, it is

evident from the findings of this study that many factors are probably involved and that

argumentation could contribute to our understanding of that process.

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A case study on the influence of environmental factors on the

implementation of science inquiry-based learning at a township school in

South Africa

Umesh Ramnarain

Department of Science and Technology Education, University of Johannesburg, South Africa

[email protected]

This mixed-methods case study investigated teachers‟ perceptions of environmental factors

influencing the implementation of inquiry-based learning at township school. Quantitative

data were collected by means of an adapted version of the Science Curriculum

Implementation Questionnaire (SCIQ) (Lewthwaite, 2001). The adapted version was

renamed the Scientific Inquiry Implementation Questionnaire (SIIQ) and was administered to

6 teachers at a township school in Gauteng. The teachers were interviewed in order to solicit

in-depth information on the findings that emerged from the questionnaire analysis. The

findings highlight factors such as resources adequacy, time, professional support and school

ethos as constraining the implementation of inquiry-based education at the school. The data

collected from SIIQ provides a foundation for understanding at a school level how factors

influence the delivery of a curriculum underpinned by inquiry.

Introduction

One of the key imperatives in the transformation of education in South Africa is the need to

provide quality education for all (Department of Education, 2001). In response to this

imperative the South African government developed policies that sought to enhance the

quality of education. The Department of National Education‟s White Paper 1 on Education

and Training (1994) provided a framework for the transformation of the education system.

The main thrust for science education in this document is the improvement in the quality of

school science for black students towards equity. A strong force driving change in sciences

education was the assertion that the previous curriculum was both inaccessible and irrelevant

to Black students. Curriculum reform initiatives in this country reflect a paradigm shift from

a teacher-dominated to a learner-centred approach. In this regard, scientific „inquiry‟ has been

advocated as a common curriculum goal in school science education in South Africa, and

also throughout the world. Inquiry science is viewed as a means to advance students‟

understanding of scientific concepts, the processes of scientific investigation, and the nature

of science (Abd-El-Khalick et al. 2004). In South Africa this imperative is expressed in the

new Curriculum and Assessment Policy Statement (CAPS) document where Specific Aim

One states that „the purpose of Physical Sciences is to make learners aware of their

environment and to equip learners with investigating skills relating to physical and chemical

phenomena‟ (Department of Basic Education, 2011, p. 8). A similar curriculum goal is

expressed in the Natural and Life Sciences CAPS documents.

There is empirical evidence to suggest that the priority in curriculum reform given to inquiry

is warranted. Studies have reported that inquiry-based learning stimulates interest in science

(Gibson & Chase, 2002), improves understanding of concepts (Gott & Duggan, 2002), leads


188

to an understanding of the nature of scientific knowledge (Quintana, Zhang & Krajcik, 2005),

facilitates collaboration between learners (Hofstein & Lunetta, 2003) and helps to develop

experimental skills (Drayton & Falk, 2001).

Factors affecting curriculum change

Fullan (1992) affirms that the success of curriculum implementation and improvement efforts

are influenced by several factors, and that no one single factor can be targeted alone to effect

curriculum reform. Lewthwaite (2006) refers to these factors as intrinsic and extrinsic or

environmental. He refers to science teaching self-efficacy, professional science knowledge,

science teaching interest and motivation as intrinsic factors of teachers that are critical

dimensions to science curriculum reform. However, although the teacher lies at the centre of

effective science curriculum delivery, the environment in which a teacher works is also

significant. Extrinsic or environmental factors are identified equally as critical elements to the

effective delivery of science programs in schools (Lewthwaite, Stableford, & Fisher, 2001).

A commonly cited list of environmental factors includes resource adequacy, time, school

ethos, and professional support (Lewthwaite, 2001). Often the success of curriculum reform

such as inquiry-based education is fostered or impeded by the availability of instructional

materials, as well as equipment (Author, 2008; Lewthwaite, 2001; Rogan & Grayson, 2003).

Widespread change towards inquiry is not possible without appropriate and high quality

resources (Anderson, 2007). Time is a factor known to influence the success of curriculum

reform efforts. The availability of time is critical in the teaching of inquiry because “inquiry

takes more time, and the teacher wanting to give more emphasis to inquiry faces a dilemma

of significant proportions” (Anderson, 2007, p. 816).

The success of any science programme is strongly influence by the school culture or ethos.

Although there is no single universally accepted definition of school culture, there is general

agreement according to Deal and Peterson (1990) that it involves “deep patterns of values,

beliefs, and traditions that have formed over the course of [the school‟s] history” (p. 218).

Many studies (e.g. Feiman-Nemser, 2003; Kriek, 2005; Loucks-Horsley, Hewson, Love, &

Stiles, 1998) have shown that the availability of professional support is a major factor in the

implementation of curriculum reform. This professional support includes support from within

the school, as well as support from outside agencies. Within the school, teachers must

experience the active, concerned support of their colleagues and be given the opportunity to

negotiate their involvement in curriculum innovation (Stewart & Prebble, 1985). Rogan and

Grayson (2003) refer to outside agencies as “organisations outside the school, including

departments of education that interact with a school in order to facilitate innovation” (p.

1191).

In South Africa, the differences in these environmental factors can be quite varied across the

educational landscape. The previous Apartheid education system was segregated into

departments for Blacks, Whites, Coloureds, and Indians. The funding of these departments

was unequal, with the per capita expenditure for a White student five times that for a Black

student (Foundation for Research Development, 1993). Most teachers at high schools for

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black learners were under-qualified to teach science (Murphy, 1992). Learners at such

schools produced dismal results in high stakes national examinations in the subject (Naidoo

& Lewin, 1998).

Understanding the context in which innovation is to occur is at the heart of school

development (Stewart & Prebble, 1993). This understanding is established through the

gathering of high quality information that provides insight into the forces that are impeding or

contributing to curriculum implementation at a school. This information then becomes the

foundation from which discussion and reflection takes place, so that deliberate focused

change can begin (Stewart & Prebble, 1993). Accordingly, this study evaluates the

environmental factors affecting the implementation of inquiry-based teaching and learning at

a township school.

Against this background, the following research question was formulated:

What are the perceptions of science teachers at a township school on environmental factors

influencing the implementation of inquiry-based learning?

Method

The methodology used in this research inquiry is the case study. This study adopted a

„sequential explanatory mixed methods‟ design (Creswell, 2002). This design enabled me to

“collect both quantitative and qualitative data, merge the data, and use the results to best

understand a research problem” (Creswell, 2002, p. 564).

The school

Progress High School (pseudonym) is situated in a densely populated township in the north-

eastern province of Gauteng. The school is similar to other schools that are situated in

disadvantaged communities, in terms of the availability of resources, the social, economic

and cultural background of learners. The location of the school was convenient as it was

accessible to in terms of travelling distance. The township residents belong mainly to a low

income group, and there is a high rate of unemployment. The school has 1200 Black learners.

The pass rate for the Grade 12 national exit examination in the previous year was 45%. The

school fee was R1000, with a 60% collection rate. The average class size is 41. The school

has six science teachers who teach Natural Sciences, Life Sciences or Physical Sciences, or a

combination of these subjects. All six teachers formed the focus of this research. Table 1

provides a demographic description of the teachers in this sample.

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Table 1. Demographic description of teachers

Teacher Male/Female Age Diploma/degree Science teaching

experience (years)

Teaching subjects

Teacher 1 Female 46 T 18 Physical Sciences and

Natural Sciences

Teacher 2 Female 44 T 14 Life Sciences

Teacher 3 Female 29 E 4 Natural Sciences

Teacher 4 Male 37 T & E 15 Life Sciences

Teacher 5 Male 45 E 16 Life Science and

Natural Sciences

Teacher 6 Male 34 E 8 Physical Science and

Natural Sciences

T = teaching diploma, E = education degree

Data collection and analysis

Quantitative data were collected by means of an adapted version of the Science Curriculum

Implementation Questionnaire (SCIQ) (Lewthwaite, 2001). The adapted version of SCIQ is

now referred to as the Scientific Inquiry Implementation Questionnaire (SIIQ). The SCIQ

was used in the evaluation of factors influencing science program delivery at schools in New

Zealand, Canada and Australia, and has been the foundation for data collection in numerous

research publications (for example Lewthwaite 2004, 2005). SCIQ is a forty-nine-item

questionnaire that provides accurate information concerning environmental and intrinsic

(teacher attribute) factors influencing science program delivery at the classroom and school

level. The items are statements to which teachers respond on a 5-point Likert scale that

ranges from 1 (strongly disagree) to 5 (strongly agree). In adapting SCIQ, only items

pertaining to the four environmental scales identified in this questionnaire: Resource

Adequacy; Time; School Ethos; and Professional Support were considered. Each item was

studied with a view to adapting the item to measure the influence of a factor on the

implementation of inquiry-based teaching. For example the item “The school is well

resourced for the teaching of science” in the scale Resource Adequacy was changed to “The

school is well resourced for inquiry-based education”. Similarly, all other items in SCIQ

related to an environmental factor were adapted. The content validity of this questionnaire in

terms of which items related to each of the four environmental factors was established by

having it reviewed by three researchers in science education at three South African

universities. The instrument was then field-tested with a group of 25 sciences teachers. They

were asked to identify and comment on items which were considered unclear or not readable.

As a result of this feedback I reworded four items in the questionnaire. A description of each

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of the scales in the new SIIQ is provided in Table 2. SIIQ (Appendix A) was administered to

the 6 science teachers at Progress High School.

Table 2 . Scales and Sample Items from the Scientific Inquiry Implementation Questionnaire

Scale Description of scale Items per scale Sample item

Resource

adequacy

Teacher perceptions of the

adequacy of equipment,

facilities and general

resources

required for teaching of

inquiry.

3;5;15;19;23 Teachers at this school have

ready access to resources

and materials for inquiry-

based education

Time Teacher perceptions of time

availability for preparing and

delivering the inquiry-based

requirements of science

curriculum.

2;6;10;13;16;21;24;26 There is not enough time in

the school program to teach

inquiry

Professional

support

Teacher perceptions of the

support available for teachers

in inquiry-based teaching

from both in school and

external sources.

4;8;12;14;18;22;27 Teachers at this school have

the opportunity to receive

ongoing science curriculum

professional support in

inquiry

School ethos The status of inquiry-based

education as

acknowledged by staff, school

administration and

community

1;7;11;17;20;25 The school management

recognises the importance of

inquiry as a science

curriculum goal in the

overall school curriculum

The questionnaire data were analyzed by computing scores on the above constructs (scales).

Mean (average) calculations were performed to identify general trends in perceptions for each

of the scales and items, and standard deviations were calculated to determine the degree of

consistency amongst respondents for each scale and again each item. After the data collected

through the questionnaire had been analysed, I arranged individual interviews with the

science teachers. Through these interviews, I solicited in-depth explanations of some of the

findings which emerged from the quantitative survey. The interviews were initiated through

the question, "What is your view of the status of inquiry learning at this school"? Based on

the manner in which teacher responded to this question, I asked follow-up questions to seek

clarity when necessary and also to probe teachers on the views they were expressing. The

teachers were also asked to describe the influence of the environmental factors upon their

practice of inquiry teaching. The interviews were transcribed and analysed using computer-

aided qualitative data software, Atlas.ti. Data were then coded and classified (Mouton, 2001)

through a process guided by the trends and patterns which had emerged from the analysis of

the questionnaire data in relation to the four environmental factors on inquiry implementation

being investigated.

The findings from the analysis of the questionnaire survey were integrated with the findings

from the teacher interviews into a coherent whole. The interview data explained some of the

findings which emerged from the questionnaire analysis. This integration of quantitative and

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qualitative data supported the production of assertions (Gallagher & Tobin, 1991) on the

teacher perceptions of the influence of environmental factors in the implementation of

inquiry-based education at their school. These assertions are presented in the results section.

Results

The statistical results from the SIIQ questionnaire are presented in Table 3 below. This

statistical data together with the interview data was invoked in generating the assertions that

are presented below.

Table 3. Scale statistics for SIIQ

Scale name Cronbach alpha Scale mean Scale standard

deviation

Resource

adequacy

0.78 1.5 0.3

Time 0.81 1.6 0.4

Professional

support

0.86 1.7 0.5

School

ethos

0.84 1.9 0.6

Assertion 1: The school has inadequate physical resources and learner support materials

for inquiry-based teaching and learning, and teachers are now starting to use

improvised resources

The relatively low mean score of 1.5 on the resource adequacy scale of SIIQ shows that the

science teaching staff of Progess High believes that their school is inadequately resourced for

inquiry-based teaching and learning. The standard deviation of 0.3 suggests that the teachers

are quite consistent in this perception. In the interviews, teachers elaborated upon the lack of

resources for inquiry teaching. The following excerpts from the interviews are presented in

this regard:

At this school we do not have much resources for learners to do inquiry. The

cupboards are empty. We have asked many times for apparatus and chemicals, but we

have been ignored. (Teacher 3)

We are being frustrated by not having the correct stuff (resources) all the time. I spoke

to my facilitator on this and he always promises we will get some equipment. (Teacher

4)

I can see why the learners need to do inquiries, but the materials are not there. It is a

big challenge for us here. (Teacher 4)

The interviews also revealed that against the lack of traditional resources for practical work,

teachers were using improvised low cost resources that were being sourced from the home. In

the interview, teacher 2 explained how she used improvised resources in supporting inquiry-

based learning in their classrooms. An excerpt from this interview is presented below:

193

I do not want to deny the learners an opportunity to do inquiry. I started to try out, as a

substitute, things that learners can bring from their homes. When on acids and

teaching about testing for acids, I know that learners can do inquiry by using the

household acids and we can also make our own indicators. They were really excited

when I asked them to bring things like lemon juice and vinegar. I bought a red

cabbage and made some indicator from this. It worked excellently when they had to

investigate what things were acid and what was alkaline. (Teacher 2)

In addition to the lack of physical resources, teachers also remarked that the textbooks that

they were using did not facilitate the inquiry-based approach to learning. They found the

textbooks to be too didactic and less learner-centred, and this impeded inquiry learning. They

indicated the need for more supplementary materials such as activity worksheets.

Assertion 2: There is insufficient time for planning inquiry lessons and inquiry teaching,

and these results in the inquiry-based approach being underplayed

The results from the SIIQ survey revealed that time availability was perceived to be a factor

that impeded the implementation of inquiry-based education at this school ( = 1.6). There

was a high degree of consistency in the manner in which teachers perceived time to be a

factor inhibiting the implementation of in which inquiry-based learning (SD = 0.4). Time

features twofold as a factor in inquiry-based education. Firstly, teachers maintained that there

was not sufficient time for them to implement the inquiry-based approach in their class. This

is evidenced in their response to item 13 “There is not enough time in the school program to

fit inquiry teaching in properly” where the mean score was 4.6. In the interviews, the teachers

elaborated on this dimension of time availability as follows:

I struggle to include inquiry in my lessons. It takes time because the learners need to

do so much on their own. It is no longer me telling them about something, but they

must investigate on their own. (Teacher 1)

Inquiry is demanding on time. The curriculum is loaded with content. I cannot get

through if learners are going to learn everything by inquiry. I will be in trouble with

my HOD if I do not cover all topics. (Teacher 5)

When teachers were asked how this lack of inquiry teaching time could be addressed, they

stated the need for more science teaching time, and suggested this could be achieved by

increasing the length of the academic school day.

Secondly, teachers indicated that inquiry required much lesson preparation, and they felt they

did not have enough time for this. The teacher either disagreed or strongly disagreed when

responding to item 10 “Teachers have the time to prepare for inquiry teaching requirements

of the national science curriculum.” This is reflected in the mean score of 1.5 for this item.

The interview data re-affirmed this perception. The teachers commented as follows on the

lack of preparation time for inquiry lessons. These comments referred to the great demands of

planning an inquiry lesson compared to other lessons.

Inquiry takes a lot of planning. I have got to think about what they will be inquiring

on. There is the question of materials. If I do not have this, then I must go to another

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school and make an arrangement to get it here. I must think about how the class will

be organised for it. Who is going to be in what group and so on? (Teacher 2)

The inquiry teaching is not the same like teaching lesson on the chalkboard. You must

plan a lot of things for the class. It is about preparing well for a successful lesson. I

start by asking what do I want them to learn? Can they learn it by inquiry and what

experience must they have? So you plan not for what you will do but what the learner

does. The most time-consuming is to design a worksheet for them to fill in while they

do it. (Teacher 4)

Assertion 3: The teachers express the need for more professional support in inquiry

teaching, especially from outside agencies such as tertiary institutions

The SIIQ results suggest that teachers consider the professional support received in inquiry

teaching to be weak ( = 1.7). There was consistency in this perception of teachers (SD =

0.5). Responses to item 18 “The curriculum leadership in science foster capabilities in those

who require support in teaching inquiry” and item 27 “Teachers at this school have the

opportunity to undertake professional development in inquiry from outside agencies” (items

means of 1.3 and 1.2, respectively) indicate that teachers are not satisfied with the

professional support both from within the school and outside agencies.

The interviews with the teachers confirmed this finding. The excerpts below illustrate the

frustration teachers feel in not receiving the necessary professional support.

We were not ready for the many curriculum changes. Inquiry is something quite

foreign to me. I know the requirements from CAPS, but the problem is that we do not

have a guideline on how this must happen. I wish the education department could plan

some development for us on it. (Teacher 2)

We sometimes get the examples of activities from the education department, but we

then have to figure it out on our own. I can say that the question of variables and

hypothesis still confuses me. Where is the support we are always told about? (Teacher

6)

In the interviews, the teachers also explained that they had not directly experienced inquiry in

their teacher education programmes, and this contributes to their lack of confidence in

teaching inquiry. This is evident below:

When I was trained more than 10 years ago we did not even hear about inquiry.

Sometimes we handled apparatus, but what I did not learn is about how do you get the

learners to be investigating something. (Teacher 1)

I went to a college many years ago, and there was hardly anything on practicals let

alone inquiry. Now we are told you must teach it this way and I know it is problem for

me. (Teacher 5)

From the teacher responses it also became clear that teachers at this school felt the need for

support that recognised the context in which they taught. In this regard they also condemned

the so called „one-shot‟ workshops that were offered by the Department of Basic Education,

and considered this training to be irrelevant and removed from their classroom realities. This

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is encapsulated well in the response by teacher 4 who stated that “I wanted something that

was going to look at my own situation and then move me forward”.

When teachers were asked who should provide this support, they indicated that outside

agencies such as universities should play a greater role in their professional development

through short learning programmes. In this regard they also believed that their subject

advisors did not have much expertise in inquiry and hence were not in a position to support

them in inquiry. The following interview excepts attest to this.

I have spoken to the SES (senior education specialist) about practical work and about

how he can help us. It was disappointing to learn this person may know less than me.

You know the universities have lots of knowledge and they must offer some

programmes on it. I heard about one university that does short courses. If I can find

the funds I will go for it. (Teacher 2)

I can now only see the professors from the universities like you to be helping us. You

must be support people like us who are now teaching, and not just the young ones

starting out. (Teacher 6)

Assertion 4: The school management does not recognise the importance of inquiry as a

science curriculum goal in the overall school curriculum

The low mean ( = 1.9) for the school ethos factor suggests that teachers do not perceive

their school to recognise the status of inquiry in the science curriculum. The low standard

deviation (SD = 0.6) showed that there was a high consistency in this perception. In

responding to item 1 (The school management recognises the importance of inquiry as a

science curriculum goal in the overall school curriculum), a mean score of 1.3 was achieved.

This reflects quite decisively, based on teacher perception that the school management does

not as yet view inquiry to be of much importance in the teaching of science. When teachers

were asked to explain this finding, they referred to the management being fixated with the

summative results of learners and not too concerned with quality of learning experience and

the pedagogical approach adopted by teachers. This is evidenced in the interview excerpts

below:

I sometimes speak to the principal, Mr Dhlomo (pseudonym) about what I want to

achieve in science and how we must give the learners a quality experience. I will also

bring up inquiry learning because the advisor spoke about it. He just says make them

pass and this will make everybody happy.

You know this inquiry is an unknown to everybody. I wish we can all sit together and

talk about how we are teaching. My school management see this as a luxury. I asked

my deputy principal once about us buying some batteries for electricity practical. He

looked at me funny. This tells he is not concerned about the quality of learning and

how they must learn, but only on the final result.

Discussion

This study used an adapted version of the Science Curriculum Implementation Questionnaire

(SCIQ), now referred to as Scientific Inquiry Implementation Questionnaire (SIIQ) to elicit

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data on extrinsic factors influencing the implementation of scientific inquiry at a township

school in South Africa. The questionnaire results revealed that resource adequacy, time to

inquiry; professional support and school ethos were factors that impede the implementation

of inquiry at this school. The interview data reaffirmed this perception of teachers and also

provided an in-depth understanding of teachers‟ experiences in contending with these factors.

The findings of this study resonate well with other studies both locally and internationally.

The factor of resource adequacy in constraining science curriculum reform has been evident

in other studies in this country. Recent South African studies point to the implementation of

inquiry being constrained by classroom realities such as the lack of resources (Author, 2011;

Dudu & Vhurumuku, 2012; Muwange-Zake, 2004). Despite attempts by the post-Apartheid

government to redress the historical imbalances, these township schools remain poorly

resourced (Magopeni & Tshiwula, 2010). It is quite probable that this scenario of schools in

disadvantaged communities will persist, and therefore other alternatives need to be sought.

An option that should be exploited more extensively is the use low-cost improvised materials

(Author, 2011). This is supported by Poppe, Markic and Eilks (2010) who maintain that

locally available resources can be used for creative inquiry activities. It is further argued that

by learners using the resources common to them and from their homes, they will operate

within their zone of comfort, and thereby overcome some of the abstractness often associated

with science learning (The Commonwealth for Learning, 2001).

The issue of time has been flagged in other studies as a dilemma for teachers who are

attempting to move towards to an inquiry-oriented science education encounter (Anderson,

2007; Author, 2008). The pressure to cover topics in the syllabus and the lack of school time

for scientific inquiry impinge upon the implementation of this approach (Author, 2008).

Research, however shows that the time spent developing inquiry investigations can lead to

“more in-depth student comprehension of science principles” (Schmidt, 2003, p. 30). The

National Research Council (NRC) of the United States (2005) offers block-scheduling as a

means by which more time can be made available in the school time-table for inquiry. In this

approach classes meet every other day for longer blocks of about 90-100 minutes, instead of

every day for 40 or 45 minutes.

General literature on education reform reports that educational change will be stifled without

professional support for teachers (Fullan, 2001). The findings of this study confirm this

assertion. The teachers strongly made the point that the anticipated support in inquiry-based

teaching was not forthcoming, and that the support from the Department of Basic Education

was inadequate. In planning professional support for teachers, two guiding principles need to

be adhered to. Firstly, it needs to be contextual, and secondly it needs to be sustainable.

Anderson (2007) decisively makes the point that “There is no gold-standard, all-purpose way

of providing systemic support for changing towards inquiry-oriented education” and that it

“must be designed for a given situation and for the people and place at hand” (p.827).

The finding in relation to school ethos does suggest the need at a systemic level for

deliberations on the vision of science education held by not only science teachers, but also

those entrusted with decision-making powers. Lewthwaite (2004) identifies the instructional

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leadership provided by a principal as a major factor influencing the effective delivery of the

science curriculum. Fullan (2002) asserts that principals are central agents in sustaining

innovations and achieving turnarounds, because it is they that carry the message as to

whether some curriculum innovation is to be taken seriously (Hopkins, Ainscow, & West,

1994).

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Appendix. Scientific Inquiry Implementation Questionnaire

There are 27 items in this questionnaire. The statements are to be considered in the context in

which you teach.

Mark your response by placing a cross in the appropriate block.

Strongly

disagree

Disagree Uncertain Agree Strongly

agree

1 The school management recognises the importance of

inquiry as a science curriculum goal in the overall school

curriculum

2 There is not enough time in the school program to teach

inquiry

3 Teachers at this school have ready access to resources

and materials for inquiry-based education

4 Teachers at this school have the opportunity to undertake

professional development in inquiry

5 The resources at this school are well organised for

inquiry-based education

6 Lack of time is a major factor inhibiting the

implementation of inquiry at this school

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7 Inquiry-based education has a high profile as a science

curriculum goal at this school

8 Collegial support is a positive factor in fostering the

implementation of inquiry-based education at this school

9 The school-based system for managing of science

resources for inquiry-based education is well maintained

10 Teachers have the time to prepare for inquiry teaching

requirements of the national science curriculum

11 The school places a strong emphasis on inquiry-based

education in the science curriculum

12 Teachers at this school are supported in their efforts to

teach inquiry

13 There is not enough time in the school program to fit

inquiry teaching in properly

14 The senior management actively supports inquiry as goal

of the science curriculum

15 The school is well resourced for inquiry-based education

16 The science curriculum is crowded. Inquiry-based

education suffers because of this.

17 The school‟s ethos positively influences the teaching of

inquiry

18 The curriculum leadership in science fosters capabilities

in those who require support in teaching inquiry

19 The equipment that is necessary to teach inquiry is

readily available

20 Inquiry-based education is valued at this school

21 Teachers have the time to prepare for inquiry teaching

requirements of the national science curriculum

22 Collegial support evident in this school is important in

fostering capabilities in teachers who find inquiry

difficult to teach

23 The facilities at this school facilitate inquiry-based

education

24 Teachers believe there is adequate time in the overall

school program to teach inquiry

25 Inquiry-based education has a high status as a science

curriculum goal at this school

26 There is not enough time in the school week to do an

adequate job of meeting the requirements of the national

science curriculum for inquiry-based education

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27 Teachers at this school have the opportunity to undertake

professional development in inquiry from outside

agencies

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