Analysis of the revisions that pre-service teachers of mathematicsmake of their own project included in their learning portfolio

M.J. Cáceres a,1, J.M. Chamoso a,*, P. Azcárate b,2

aDept. of Educational Mathematics and Experimental Sciences, Faculty of Education, University of Salamanca, Paseo de Canalejas 169, 37008 Salamanca, SpainbDept. of Didactic, Faculty of Education, University of Cádiz, Avda. República Saharaui s/n, 11519 Puerto Real, Cádiz, Spain

a r t i c l e i n f o

Article history:

Received 24 April 2009

Received in revised form

7 December 2009

Accepted 4 January 2010

Keywords:

Mathematics education

Pre-service teacher

Learning portfolio

Learning project

Reflection

a b s t r a c t

Learning portfolios are increasingly being used in university teacher-education programs as assessment

instrument. With formative assessment in mind, this study provides a method to assess the modifications

made by each pre-service teacher in his/her project included in his/her learning portfolio. The project

consisted of designing a lesson plan for teaching mathematical knowledge taking into account Content,

Activities, Methodology and Reflection. The outcomes showed significant differences in the revisions

carried out in all categories except Activities. Although the use of portfolio promoted the successful

development of each pre-service teacher's initial ideas, the training received during the teacher-education

program had limited influence.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

During the last few years, many changes have taken place not

only regarding learning but also teaching (De Rijdt, Tiquet, Dochy, &

Devolder, 2006). Many countries are now moving towards a more

constructivist view of learning and towards developing students'

skills in an active, self-regulated and collaborative way (Van der

Shaaf, Stokking, & Verloop, 2008).

To foster this change, the training of mathematics teachers

should involve pre-service teachers in a variety of interesting

activities that allow them, for example, to work in a collaborative

way, to ask and answer questions, to take part in substantial

discussions regarding aspects of Mathematics Education, to reflect

on what is being learned and on the nature of mathematics. It is

necessary to propose projects and open tasks so that pre-service

teachers can learn to reason critically, carry out complex activities

and apply their knowledge to real situations. Reflection about how

to solve situations and problems linked to their future educational

practice is what allows ideas and forms of understanding Mathe-

matics Education to be put into practice (Llinares & Krainer, 2006).

This means that evaluation should not be subjected to the

limitations that have been described as technical, mechanistic or

technological (Craig, 2003) but rather involve new methods of

analysis and interpretation to allow pre-service teachers to

construct their own personal and collective knowing and learning

(Silver & Kenney, 1995). In essence, assessment needs to take into

account classroom discourse and activities, work done outside the

classroom and the overall learning process. This involves using

a selection of instruments that provide evidence about learning,

such as control lists, observations, interviews, reflective diaries,

cases, exhibitions, problem-based inquiries or tests (Darling-

Hammond & Synder, 2000).

The portfolio can include several of these aspects. It is a poten-

tially valuable method in teacher-education programs that are

based on the construction of pre-service teachers' learning from

experience, creating their own meaning and developing both

expertise and commitment to the process of reflection. Further-

more, the portfolio is a medium for capturing future teachers'

developing understanding of the teaching-learning process carried

out in the teacher-training classroom (Wade & Yarbrough, 1996).

The same portfolio can be used for appraisal of learning for both

formative and summative assessment purposes in order to examine

the product of learning at the end of the process (Tillema & Smith,

2007). In the present study, we focus on the former aspect.

Pre-service teachers should therefore have clear knowledge of

the criteria being used to assess them. Indeed, one indicator of

excellence in their progress would be for them to be able to assess

* Corresponding author. Tel.: þ34 923294400x3469; fax: þ34 923294703.

E-mail addresses: majocac@usal.es (M.J. Cáceres), jchamoso@usal.es (J.M. Cha-

moso), pilar.azcarate@uca.es (P. Azcárate).1 Tel.: þ34 923294400x3469; fax: þ34 923294703.2 Tel.: þ34 956016200; fax: þ34 956016031.

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Teaching and Teacher Education

journal homepage: www.elsevier .com/locate/ tate

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doi:10.1016/j.tate.2010.01.003

Teaching and Teacher Education 26 (2010) 1186e1195

their own work in the same way their teachers do. Having knowl-

edge of assessment criteria not only allows them to satisfy their

sense of justice but also, and more importantly, it helps them to

understand the objectives being sought, improve their learning and

develop the metacognitive knowledge necessary to monitor their

own work. An understanding of what the criteria mean, the ability

to apply them to their own work and, even change the rules in

certain cases are important aspects in the training of future

teachers. Furthermore, self-assessment has a social and motiva-

tional function improving knowledge acquisition by allowing pre-

service teachers and their lecturers to collaborate (Shepard, 2001).

Several kinds of teaching portfolios have been identified in the

literature depending on their purpose, what goes into the portfolio,

thenature andqualityof social interactions that pre-service teachers

experience in the process of their construction or what happens

with the portfolio once it is completed (Zeichner & Wray, 2001).

Usually researchers distinguish among learning portfolio, assess-

ment portfolio, and employment portfolio. The learning portfolio,

often used throughout a pre-service teacher-education program, is

generally defined as a personalized collection of work to promote

reflection, has the purpose of engaging pre-service teachers in

inquiry about their teaching and documenting their development

(Dinham & Scott, 2003; Xu, 2003; Zeichner & Wray, 2001).

In this study we focus on the use of the learning portfolio. The

learning portfolio has been advocated and tested in recent years,

but there has been little systematic attention given to the ways in

which it is being used in teacher-training classrooms. Fully inte-

grating teaching into the curricular structure through the use of the

learning portfolio will require a better appreciation of how it can be

used as an instrument for reflection and how it indicates the

monitoring of pre-service teachers learning as they carry out

activities within their training program. Our work has focused

specifically on how pre-service teachers reconstruct the initial

ideas they set out in a project that was included in their learning

portfolio at the beginning of their teacher-education program and

based on the instruction received during this program.

2. Theoretical framework

Reflection on teaching practice is one of the main objectives of

teachers' professional work. Schön (1983, 1987) emphasized the

importance of the context and the moment in which reflection

takes place by distinguishing between “reflection in action” which

occurs while one is teaching and “reflection on action”which refers

to what one thinks about before and after a classroom session.

Schön considered that the challenge lies in helping teachers to be

more competent so that they are aware of what happens and

instantly change their behaviour in situations of uncertainty or

surprise that arise in the classroom. This idea corroborates Dewey's

(1910) recognition of the importance of reflection in the teaching

and learning process. Subsequently, many interpretations have

beenmade of its application to teaching and it has fosteredmuch of

the research carried out in the field of education (Valli, 1997).

Although reflection is a private act, we consider what pre-service

teachers write, both in response to a task carried out in the teacher-

education classroom and to their ownwork related to the teaching-

learning process in Primary level Mathematics, to be examples of

reflection (Chamoso & Cáceres, 2009).

While much has beenwritten about the importance of reflective

practice, scarce attention has been given to the reflective process

which future teachers have to experience in making a learning

portfolio (Tillema, 1998). To be more precise, there are hardly any

empirical studies about the use of reflective thinking with pre-

service teacherswhen they try to improve their ownwork as a result

of the training theyhave received in the teacher-educationprogram.

Portfolios have historically been based on the “best practice”

concept and can give an idea of what pre-service teachers under-

stand as good teaching and how they learn (Wray, 2007). A teacher-

education program is only the beginning of one's education for

teaching, but it does provide the foundation for future investiga-

tion. The portfolio may help to do just that (Farr Darling, 2001).

Portfolios are seen as a way to enhance the authenticity of future

teacher appraisal and as a useful learning and professional

knowledge tool (Shulman, 1998; Wade & Yarbrough, 1996). To keep

a portfolio requires reflexive writing skills (Lee, 2005), among

which may be counted a general sensitivity for critical events, the

ability to reflect on one's own role in a conflict, and the willingness

to accept errors and mistakes as learning opportunities.

In the last few years, the use of portfolios has become

commonplace internationally in teacher education for different

purposes (e.g. Farr Darling, 2001; Van Tartwijk, Van Rijswijk,

Tuithof, & Driessen, 2008; Zeichner & Wray, 2001). For example,

they stimulate pre-service teachers to reflect on specific themes in

their context (Mansvelder-Longayroux, Beijaard, & Verloop, 2007)

or document learning and growth in a specific project (Wade &

Yarbrough, 1996). Although great expectations have been placed

on the use of the portfolio, research studies on the efficiency of the

instrument have been few and far between (Imhof & Picard, 2009).

We need to learn more about the nature and quality of reflection

that emerges under different conditions of portfolio use (Zeichner

& Wray, 2001). Empirical research is still needed to determine

what effect portfolios have made on pre-service teachers' indi-

vidual thinking (Wade & Yarbrough, 1996). By reason of the value

of portfolios as a mechanism for promoting reflection and

professional learning, Xu (2003) recommended extending the use

of portfolios beyond the pre-service years to promote teachers'

future professional development.

In this sense, currently, four aspects seem to define the context

for a learning portfolio (Imhof & Picard, 2009): (1) Constructivist

theories of self-regulated learning describe learning as a series of

cycles from planning through assessment, progressing over time

and across different learning environments, (2) Recognition of

individualized learning in a self-reflexive and self-regulated mode,

(3) Competence and standard orientation in teacher education, (4)

Documentation and continuous reflection about professional

experience.

Along with Dinham and Scott (2003), we consider the learning

portfolio to be essentially formative, capable of alteration by

growth in experience and development in understanding. In this

study, the concept of the learning portfolio is defined as the

collection, selection and organization of a pre-service teacher's

work over time that shows evidence of reflection and learning. In

this sense, the creation of a learning portfolio by pre-service

teachers entails some generally accepted fundamental notions:

- It is strongly focused on the process of reflection that pre-

service teachers have to undergo (De Rijdt et al., 2006).

- It involves the cultivation of two important habits of mind: the

view of teaching as an ongoing inquiry and of collaboration as

a valuable way of coming to know teaching (Grant & Huebner,

1998).

- It involves pre-service teachers having to make important

decisions about what ideas to include, what to omit and to

communicate their reasons (Loughran & Corrigan, 1995).

- It involves real “student work” that tests cannot reveal

regarding all facets of a person's growth and potential (Wade &

Yarbrough, 1996).

- It is not an activity carried out at one particular moment but

a process that needs to be realised over a certain span of time

(Meyer & Tusin, 1999).

M.J. Cáceres et al. / Teaching and Teacher Education 26 (2010) 1186e1195 1187

- It serves the purposes of both teachers and students (Wade &

Yarbrough, 1996).

A portfolio for the documentation of professional competence is

increasingly being required as a regular condition for certification

in pre-service teacher education as well as for the advanced

teaching certificate. But there is a current lack of explicitness in the

criteria used for determining the quality of documentation of

portfolios due to heavy reliance on the appraisal process of context

and circumstance in portfolio construction (Tillema & Smith, 2007).

In fact, readers who engage in assessments of portfolios often limit

their reading and interpretation to recognizing superficial features

of performance and have great difficulty providing explanations to

substantiate their judgments (Delandshere & Arens, 2003). This

means that there is a lack of transparency that hinders pre-service

teachers as they try to determine what is required of them. Some

researchers feel that the evaluation of the portfolio process ought to

concentrate on the quality of the reflection captured in the indi-

vidual texts (e.g., Ward & McCotter, 2004).

Raising standards in the appraisal of portfolios is an issue of

considerable debate (Shepard, 2000). The actual practice of

assessing and grading a portfolio seems to take on many forms and

is conducted through a diversity of methods (Tillema & Smith,

2007). One of the forms most used to assess portfolios is that of the

Vermont Department of Education. This consists of the following

categories (Wilson,1995): (1) Understanding of the task, (2) Quality

of approaches and procedures, (3) Decisions along the way, (4)

Outcomes of activities, (5) Language of mathematics, (6) Mathe-

matical representations, (7) Clarity of presentation. The category,

“Decisions along the way”, includes the reconsideration of one's

ownwork. In addition to this are itemized changes in approach, oral

or written explanations, validation of the final solution and sources

of evidence (Wilson, 1995). Here is one of the fundamental aspects

on which learning portfolios are based (Campbell, Melenyzer,

Nettles, &Wyman, 1999; Hartmann, 2004). This was adapted to the

training of teachers of mathematics at the primary level.

This study was set within the context of a proposal for innova-

tion that linked a teaching-learning process with an assessment

process that included the construction of a learning portfolio. In

this context, the main specific research question was: What kind of

changes did pre-service teachers of primary mathematics make in

one of the projects included in their learning portfolios after

receiving instruction in the university teacher-education program?

We investigated the nature of the reflection that emerged

from the learning portfolios when “reflection” was processed in

a specific way. That is, the reflection that each student made upon

modifying his/her own project (one of the papers to be included

in their learning portfolios and consisting of designing a lesson plan

for mathematical knowledge at the primary level) according to

what they were learning in the teacher-education classroom. To do

so, we constructed an assessment tool that would allow us to

compare the versions of the initial project (hereafter IP) with those

of the final project (hereafter FP).

3. Method

3.1. Context

The experiment was carried out during the academic year

2007e08 with a regular group of pre-service teachers in a univer-

sity mathematics teacher-education program at the Faculty of

Education of the University of Salamanca, Spain. The 45-h module

was conducted in 2-h sessions during the pre-service teachers'

second year. This was before they had begun their traineeship in

the local primary schools. The objective was to develop capabilities

related to specific competencies in the training of professionals to

teach mathematics at the primary level using contents relating to

plane and spatial geometry and measurement. At this time, the

Spanish curriculum for Primary-school teachers was based on three

years of study.

3.2. Participants

Of the 37 students enrolled, thirty pre-service teachers (10 men,

33%, and 20 women, 67%) completed all the activities of the

module. Their average age was 22. None of them had previous

experience in the construction of portfolios. The experiment was

carried out by their regular lecturer whowas one of themembers of

the research team and had 23 years experience in the profession.

For the past 4 years, this member had worked with pre-service

teachers on applying learning portfolios in their initial training.

3.3. Development of the experience

3.3.1. Introduction

It was considered that in their initial training, besides theoret-

ical knowledge (knowing), future mathematics teachers at the

primary level needed to possess sufficient skills to teach the

contents (know-how). This was organized in terms of competencies

from two perspectives, each of which in turn is divided into two

different sub-aspects:

1. Mathematical competency in two senses:

a. content, understood as the knowledge of mathematical

concepts, properties and activities at the primary level and

b. in-depth knowledge, understood, for example, as the ability

to experiment with mathematical contents, carry out open

activities or establish relations with other contents or areas

of knowledge.

2. Professional competency, to teach primary level mathematics, in

two senses:

a. content, understood as knowledge of the contributions of

mathematics education in facilitating teaching and learning

as regards materials and resources, and the peculiarities of

primary-school students when they are faced with learning

or methodological aspects, and

b. in-depth knowledge, understood as the ability to apply one's

knowledge to the teaching and learning of primary level

mathematics contents, as for example, in the design of how

to apply a lesson plan in the primary classroom; reflection

on one's own actions, those of others and other elements

that characterize an educational situation or the design of

creative activities for this stage.

With this approach in mind, we designed and developed

classroom sessions focused on the work of the pre-service teacher

in which knowledge and in-depth knowledge of mathematical and

professional competencies were combined with the development

of work proposals for the pre-service teachers linked to the four

sub-aspects described above (For more details, see Chamoso &

Cáceres, 2009). According to the number of sub-aspects involved,

theywere called: Exercises, if they involved a single sub-aspect (For

example, Exercise 2 was: “Build the Euler's straight line using only

a piece of paper”); Activities, involving more than one sub-aspect

(For example, Activity 5 was aimed at identifying how each pre-

service teacher related and applied his/her understanding of

examples of mathematics taken from other cultural contexts to his/

her own practice. Because of this, a classroom session was devel-

oped in order to use examples of mathematical practice from

different cultures relating to geometry, arithmetic and problem

M.J. Cáceres et al. / Teaching and Teacher Education 26 (2010) 1186e11951188

solving (See examples in Planas, Chamoso, & Rodríguez, 2007).

Later, each pre-service teacher had two weeks to answer this

question: “Prepare your own definition of intercultural attitude and

look for examples of cultural differences relating to mathematics

that support your definition”. The pre-service teachers presented

their activities for discussion during a normal classroom session

a few days later); and Projects, which included all four sub-aspects

(Project 1 is presented below). In all, 6 exercises, 8 activities and 4

projects were proposed.

A typical class session began with a presentation by the lecturer,

followed by work in small groups and then a final sharing session.

Some work proposals for the pre-service teachers were carried out

in small groups, but each pre-service teacher had to make the final

presentation individually in his/her portfolio. Some of the pre-

service teachers' work proposals could be completed outside the

classroom while others had to be undertaken entirely outside. The

lecturer acted as mediator and informed the pre-service teachers

about the teaching-learning objectives, methodology, criteria and

forms of assessment. Guidance was given in carrying out and the

revision of their work proposals.

As support, we used the University of Salamanca virtual plat-

form called Studium, based on Moodle, in order to provide the pre-

service teachers with prior information regarding the objective,

development and documentation of each session. Studium was also

used to handle the overall development of the course, to manage

the collection of the pre-service teachers' work (although this was

sometimes handed in) and to increase the possibilities of pre-

service teacher interaction with the lecturer.

3.3.2. The learning portfolios

In regard to the assessment system, each pre-service teacher

had to prepare a learning portfolio. The learning portfolio was to

include a selection of his/her work that had been studied in the

classroom, the 4 projects and the activities done entirely outside

the classroom, some voluntary papers concerning aspects that

showed their knowledge and a weekly critical reflection about the

teaching-learning process identified in the teacher-training class-

room. Each pre-service teacher assessed his/her tasks in his/her

reflections and had the opportunity to make any revision and

improvement. All of this was presented at the beginning of the

module and agreed on by the pre-service teachers and the lecturer.

3.3.3. The project

Of the four projects that the pre-service teachers had to include

in their portfolios, the following was selected for this research: at

the beginning of the course the contents of the syllabus were

distributed among the pre-service teachers and they were given

a week to design individually how to teach them at the primary

level, taking into account: (a) Content at the teaching level chosen,

(b) Activities of different types relating to the contents involved and

adapted to the level selected, (c) Methodology for teaching, and (d)

Reflection on the work carried out. Their capacity for achievement

in each case would allow us to assume their capabilities in any

other case. At the end of the module each pre-service teacher

revised his/her own work based on the training received during the

module and included it in their learning portfolio.

The project presented by each pre-service teacher in each case

followed, in general, the elements suggested. Most of the pre-service

teachers included images, photographs and diagrams of their own

taken from other sources. Their length ranged from 6 to 47 pages.

3.3.4. Classroom training in relation to the pre-service

teacher project

Some training sessions were carried out in which the project in

question was mentioned directly, while in other sessions it was not

mentioned at all, although all the sessions were expected to have an

impact on the development of each pre-service teacher's final

presentation:

1. Directly, by referring to the project in question (22% of

the module):

- After the pre-service teachers had handed in their IP the

lecturer gave them an example of what could be done. He

made an elementary plane geometry construction using no

material other than paper. The pre-service teachers analysed

how they could incorporate this presentation in a written

paper such as they were being asked to do. In other sessions,

the lecturer presented different types of activities, for

example, tasks based on photographs of the immediate

surroundings, comics, materials such as the tangram or traffic

lights and origami. These were made available for the pre-

service teachers. Subsequently, the pre-service teachers

analysed how these activities could be classified according to

their application in the primary classroom, both from the

point of view of the teacher and the students (for more

details, see Chamoso & Rawson, 2004).

- Some pre-service teachers presented their work in the

classroom to be evaluated by their peers.

2. Indirectly, without mentioning the project in question, and

organized according to the four aspects identified in the project

(78% of the module):

a. Content: No specific preparation was given in this sense

because the pre-service teachers were considered to have

sufficient grounding in it according to official guidelines.

The mathematical content was only considered as a means

for developing the different training sessions.

b. Activities (18% of the module): As with the Content, the pre-

service teachers were expected to have sufficient basic

knowledge of activities. Nevertheless, we worked on

different ways in which the activities could be introduced

into the classroom according to the objective of the

teaching-learning process in question. We considered

whether they could be used, for example, to introduce

a class session, explain a concept, to go more in-depth,

explain formulas, demonstrate theorems, discover

geometric forms in our surroundings, pose activities and set

out routine and not-so-routine problems. Further inquiry

covered how the students could approach these activities as

exercises, problems, research objects, or other types of open

activities (Chamoso, 2007; Chamoso & Rawson, 2001).

c. Methodology (42% of the module): This was the main

objective of a good part of the classroom sessions and was

used in different ways. For this reason, most of the sessions,

including those with other main objectives, were taught

using different methodologies, such as the importance of

group work, classroom dialogue, the way in which open

activities can be used, the importance of taking into account

multicultural factors, or the ability to be creative in unfore-

seen situations. Others focused on evaluating the use of

materials of all types in the classroom, such as video, tech-

nology, and so forth. A third group of sessions was devoted

to relating mathematics to other areas, such as aspects of

daily life or the environment, and for this the pre-service

teachers created stories or developed maths trails (Chamoso,

2003, 2004; Chamoso, Fernández, & Reyes, 2009).

d. Reflection (18% of the module): The pre-service teachers had

no previous experience or training in making reflections

within the framework of a training programme.

- Joint reflection in the classroom in each of the first 3

sessions. This was then followed by the pre-service

M.J. Cáceres et al. / Teaching and Teacher Education 26 (2010) 1186e1195 1189

teachers writing down and handing in their reflections

individually. Once they had been revised by the members

of the research team, they were returned with comments,

usually in the form of questions in the margins, with

a view to clarifying the objective pursued in them.

- Classroom sessions covering the following: a) A personal

reflection by the lecturer on the development of the

course during the first few weeks of the module, b) some

presentations on the importance of reflection and how to

apply it, c) a final appraisal was made of the development

of the module by both the pre-service teachers and the

lecturer.

3.4. Data collection and analysis

The instruments used for the collection of the data were the

papers that the pre-service teachers included in their learning

portfolio. These referred to the project, described earlier, and

detailed the pre-service teachers' thinking at the beginning and the

end of the course. The objective was to analyse the progress that

each pre-service teacher showed in his/her work as a result of the

training received in the module. To evaluate the process, we

considered the following:

1. The differences in the assessment of each pre-service teacher's

IP and FP overall and in each of the following categories:

Content, Activities, Methodology and Reflection.

2. The modifications made in the FP with respect to the IP of each

pre-service teacher, analysing both the level of depth and the

aspects to which they refer in their work.

To evaluate the work carried out, we established three general

levels for classifying each pre-service teacher's degree of learning,

based on their work in the IP and the FP, in the following way:

Level 1: The pre-service teacher participated in the process as

a spectator, describing the knowledge without

becoming involved in how to teach it at the primary

level.

Level 2: The pre-service teacher participated actively in the

process by trying to understand the meaning of the

knowledge involvedandhowtodealwith it in theprimary

level classroom.

Level 3: The pre-service teacher, besides participating, became

involved in the process and improved his/her own

knowledge by taking his/her own decisions as to its

application in the primary level classroom.

These general criteria were adapted to evaluate each of the

elements identified in the project in question in the four categories:

Content, Activities, Methodology and Reflection (Table 1).

An assessment template was drawn up for each of the four

aspects, based on the analysis of each pre-service teacher's work

which had previously been transcribed. This template was

modified throughout the process both in the description of indi-

cators for each category and in the specific and overall aspects to

be applied. Such modification was based on the specific special

features to be considered in each case until we achieved an

assessment guide for each of them. Below, we give an example of

the definitive template for assessing the proposal of activities in

each project (Table 2). The first column shows the level, the

second the indicators for each level and the third gives concrete

examples (the template for evaluating Reflection is based on

Chamoso & Cáceres, 2009).

The first step in analysing the work of each pre-service teacher

was to identify complete units of information understood as

a single idea, thought or knowledge about a particular topic or

event. Among all the pre-service teachers there were, in total, 1231

units, of which-referring to the IP and the FP, respectively, in each

case- 137 and 190 had to do with Content, 178 and 403 with

Activities, 59 and 82 with Methodology and 78 and 104 with

Reflection. The units ranged from one short sentence to longer

pieces of several hundred words. Next, the corresponding assess-

ment was applied to the work of each pre-service teacher. This was

organized in tables in which the first column included the pre-

service teachers' work divided into complete information units, the

second column, the category to which it pertains according to the

corresponding guidelines and the third column, the public contents

that summarise what the pre-service teachers had reflected. All of

this was done independently by the members of the research team

and then together they agreed upon the information units to code.

This work was subsequently revised by an independent party,

a researcher in educational psychology, who for the most part was

in agreement (88% in Content, 95% in Activities, 86% inMethodology

and 97% in Reflection). Discussion among the researchers and the

independent party helped to resolve disagreements and endorse

agreement.

After categorizing the work of each pre-service teacher, the

results for each aspectwere collected in tables inwhich each section

1, 2 or 3 reflected the amount of complete information units of each

pre-service teacher corresponding to Content, Activities, Method-

ology and Reflection both in absolute values and in percentages.

The sum of the outcomes of each pre-service teacher multiplied,

respectively, by the factor of the corresponding category and, the

final outcome, divided by the sum of all the complete information

units, was considered the score in each of the aspects considered.

For example, the outcomes of the pre-service teacher MIE in Activ-

ities were: 4 units of information in the first level, 12 in the second

level, and 14 in the third level, so the final score for MIE in Activities

was ð4 1þ 12 2þ 14 3Þ=ð4þ 12þ 14Þ ¼ 70=30 ¼ 2:33.

Secondly, to determine the nature of the revisions, we analysed

both the depth of the modifications made by each pre-service

Table 1

Levels for assessing the pre-service teacher projects in the categories: Content, Activities, Methodology and Reflection.

Content Activities Methodology Reflection

(1) Presents the contents in a linear

form without relating the different

parts to each other

Presents the activities in an isolated

way, without perceiving any

relationship among them

Reproduces what is contained in

the official guidelines or other

sources

Describes aspects related to his/her

own teaching and learning process

(2) Relates the different parts of the

contents and develops them

following a logical sequence

Relates the activities to each other

or to the content to which they refer

Refers to aspects relating to the

teachingelearning process

although in an isolated manner

Argues, justifies or draws

conclusions about his/her own

teaching and learning process

(3) Explains the contents with his/her

own arguments to clarify its

meaning and create original models

Designs activities to involve the

student actively in the construction

of his/her own knowledge

Draws up a teaching proposal

which gives a view of what should

be accomplished in the classroom

Makes suggestions in order to

improve his/her own teaching and

learning process

M.J. Cáceres et al. / Teaching and Teacher Education 26 (2010) 1186e11951190

teacher in the FP with respect to the IP in each of the four categories

and the frequency of references to aspects relating to the teaching-

learning process in the primary classroom.

To study the depth of the modifications made in the FP, we

established the following levels according to the reflection process

each pre-service teacher made in each case in order to improve

their IP:

Level 4: The IP was completely revised.

Level 3: The IP was completely reorganized and perhaps some

things were added.

Level 2: Only some parts of the IP were reorganized or modified.

Level 1: New knowledge was added without modifying or reor-

ganizing the IP.

Level 0: The IP was not modified in any way.

For the analysis of the quantitative data, related samples t-test

and Pearson's chi-square test were used.

4. Results

The IP and FP presented by each pre-service teacher were ana-

lysed in two ways: i) assessment of the work done according to the

templates designed for each category (Content, Activities, Method-

ology and Reflection) and ii) the nature of the modifications made in

each FP with respect to the IP, with attention to both the depth of

the revisions and the aspects of the teaching-learning process

referred to.

In the first place, regarding the assessment of the work pre-

sented by the pre-service teachers in both the IP and the FP, the

mean scores were compared (Fig. 1), as were the percentages in

each evaluation level in the different categories (Fig. 2).

A related samples t-test showed that the overall assessment of

each pre-service teacher considering the 4 categories increased

significantly in the FP after revision of the IP as a result of the

reflection carried out owing to the training received [t(29) ¼ 5.820,

p < 001]. Specifically, in a comparison of the scores in each of the

Table 2

Levels, indicators and examples for assessing the activities proposed by the pre-service teachers in their projects.

Level Indicators Examples

(1) Presents the activities in an

isolated way, without

perceiving any relationship

among them

e The activities demand the direct application of routines,

such as applying formulas or identifying something that

has been presented previously.

e They are taken directly from other sources.

e Possibilities are posed without clearly specifying what the

primary level student should do.

e “Given triangle a, b, c. If side a measures 3 and the

hypotenuse 7, what is the length of side b? Solve this

problem using Pythagoras' theorem.”

e “A list of sentences is given and the student must decide if

the situation they describe is one of inverse

proportionality.”

(2) Relates the activities to each

other or to the content to

which they refer

e A concern can be sensed for doing something out of the

routine that demands reasoning, construction or manipu-

lation on the part of the primary level student.

e Mathematics are related to other areas or to daily life

although not entirely accurately.

e Various activities are organized for the same learning

objective.

e “Cut out the different angles, write down their names

(acute, obtuse, etc.) and their degrees (90 , 180 , etc.). They

will then be pasted on a mural grouped according to type.”

e “Bring to school objects with different geometric shapes so

that we can name them, measure them, calculate areas and

volumes and classify them.”

(3) Designs activities to involve

the student actively in the

construction of his/her own

knowledge

e Activities are designed and adapted to the level chosen in

order to involve the students actively, and demand their

participation and collaboration in a coherent way, thus

constituting a teaching project.

e Activities are related to other areas of knowledge or to

daily life in a reasoned way.

e Activities are grouped according to a clear criterion,

sequenced by level of difficulty or their usefulness is

demonstrated in the teachingelearning process.

e The pre-service teacher justifies why an activity has been

proposed and how it is related to the content or to other

aspects.

e “All the students will create a giant mosaic together, and

each one must bring a piece for it. First they must decide

which pieces each one should make by discussing what

regular and irregular figures can be used, and then they

will paste them together, deciding whether they are valid

or not and how unsuitable pieces can be modified. In the

end, each student writes a summary in his/her notebook.”

e “Invent a story in which the main characters are geometric

figures and persons who made mathematical discoveries

(e.g. Pythagoras or Thales of Miletus), including the

discovery and, where applicable, the formulation of the

corresponding theorems (related to Language and

Literature).”

Fig. 1. Mean obtained by the pre-service teachers in the categories Content, Activities,

Methodology and Reflection in IP and FP.

Fig. 2. Percentage of pre-service teachers in each assessment level in the categories

Content, Activities, Methodology and Reflection, in IP and FP (for example, in Content,

20% of pre-service teachers got a score between 2 and 3 in IP, while 40% of pre-service

teachers got a score between 2 and 3 in FP).

M.J. Cáceres et al. / Teaching and Teacher Education 26 (2010) 1186e1195 1191

categories Content, Activities, Methodology and Reflection, related

samples t-test showed significant differences between the scores

obtained by the pre-service teachers in the IP and FP in all the

categories except Activities (Table 3).

We also analysed the homogeneity of the scores obtained by the

pre-service teachers among the different categories, both in the IP

and in the FP. In the former, there were significant differences

between the levels reached in Activities and Methodology and

Reflection [t(29) ¼ 2.587, p < 0.05 and t(29) ¼ 2.067, p < 0.05,

respectively], whereas in the FP there were significant differences

between Content and Methodology [t(29) ¼ 2.365, p < 0.05] and

Activities and Methodology [t(29) ¼ 2.687, p < 0.05].

Secondly, the scores obtained in the study of the depth of the

revisions carried out by the pre-service teachers in each category

according to the 5 levels considered, both in absolute value and in

overall percentage are presented in a Table (Table 4). In this table,

each score is broken down, in the row immediately below, into the

number of pre-service teachers whose score obtained in the FP was,

from left to right, higher than, equal to or lower than that of the IP,

respectively, both in absolute terms and in partial percentages in

each case. For example, 10 pre-service teachers out of 30 (33%)

completely modified the Content in the presentation of the FP with

respect to the IP, and 6 of the 10 improved their score whereas 4

had a poorer outcome.

The number of pre-service teachers who made a completely

new FP in all the categories was greater than the numberwhomade

another type of modification except in the case of Activities, where

the number was the same as that of those who only added new

knowledge to the IP without making other modifications. Most of

the pre-service teachers improved their score in the FP with respect

to the IP in all the categories except Content and Activities, where 4

(13% of the total) and 6 (20% of the total) pre-service teachers

obtained a lower score in the FP, 3 of whom coincided in both cases.

As to the level of the modifications, all the pre-service teachers

made some type of revision in at least 2 of the 4 categories; 6 (20%)

of them completely redid the work in all the cases whereas 3 (10%)

redid the work in 3 of the 4 categories. That is, 30% of the pre-

service teachers redid their work in all or most of the categories. On

the other hand, 8 (24%) pre-service teachers did not reach this level

in any of the categories.

Based on the analysis of the pre-service teachers' work, we

compared the aspects of the knowledge towhich they referred both

in the IP and the FP. For example, in the IP, 7 (23%) pre-service

teachers referred to proposals of group work and 8 (27%) proposed

open problems or activities. Despite the fact that most of the

classroom sessions were carried out in groups and used open

Table 3

Comparative results of the mean scores obtained in the categories Content, Activities,

Methodology and Reflection in the IP and FP of each pre-service teacher.

Difference of means t p

Content 0.247 4.034 0.000**

Activities 0.148 1.723 0.096

Methodology 0.884 5.646 0.000**

Reflection 0.525 3.682 0.001**

*n.s. < 0.05 **n.s. < 0.01.

Table 4

Number of pre-service teachers in each category, in absolute values and in percentages, according to the depth of the modifications made in the FP with respect to the IP, and

a breakdown of how their initial scores improved, remained the same or decreased, respectively. (In some cases, the sum of the percentages is not exactly 100 due to rounding).

M.J. Cáceres et al. / Teaching and Teacher Education 26 (2010) 1186e11951192

activities, in the FP only 15 (50%) pre-service teachers referred to

proposals of group work and 19 (63%) proposed open problems or

activities, in almost all cases in an isolated fashion. In spite of this

scarce transference, the chi-square statistic with predetermined

alpha level of significance 0.01, and 1 degree of freedom allows us

to affirm, with a probability lower than 1%, that there were

significant differences in both cases higher in the FP with respect to

the IP (c2 ¼ 9.130, p < 0.01 y c2 ¼ 12,139, p < 0.01, respectively). On

the other hand, none of the pre-service teachers took multicultural

aspects into account either in the IP or the FP, even though

a training session was carried out to this end and the pre-service

teachers carried out a task with that objective. As regards the

Activities, 4 (13%) pre-service teachers classified them with 3

different criteria in the IP, whereas in the FP, despite the training

received in this sense, only 15(50%) did so (besides the previous

classifications they proposed 4 more possibilities).

We also compared whether the proposals of the pre-service

teachers referred to aspects of the teaching-learning process at the

primary level; that is, whether they took into account aspects such

as the type of task posed, its nature, the role it plays in the process,

the type of knowledge involved, consideration of motivations and

obstacles, attention to diversity, the methodological sequence,

teaching resources, socio-personal relationships, classroom orga-

nization and evaluation. The number of pre-service teachers who

considered these aspects in the FP with respect to the IP went from

6 to 9 in Content, from 9 to 11 in Activities, from 14 to 22 in Meth-

odology and from 10 to 17 in Reflection, so that the least differences

were again observed in Content and Activities (Table 5). Despite

these small differences, the chi-square statistic with predetermined

alpha level of significance 0.01 and 1 degree of freedom established

significant differences in all the cases, with a probability lower than

1% (c2 ¼ 10.159, p < 0.01 in Content; c2 ¼ 22.208, p < 0.01 in

Activities; c2 ¼ 9.545, p < 0.01 in Methodology and c2¼ 11.471,

p < 0.01 in Reflection).

5. Discussion and conclusions

A teaching-learning system integrating an assessment process

with learning portfolios for educating pre-service teachers, based

on activities developed to enable them to construct their own

learning and reflect on their work, was designed and investigated.

The pre-service teachers were expected to engage with different

activities such as written papers, oral presentations, problem

solving, projects and open tasks, shared reflections, discussion and

assessments of their work. The pre-service teachers completed the

project after receiving the syllabus for the course then, as we had

expected, there was evidence of improvement in the final product.

In addition, the devised assessment system allowed us to analyse

the modifications made by the pre-service teachers in their own

work after reflecting on the training received in a university

teacher-education module.

Little research has been done on the revisions pre-service

teachers make in their ownwork to determine the consequences of

the outcomes obtained. Therefore, the present study should

contribute to the empirical investigation of pre-service teacher

thought processes and how they make sense of the training they

have received. This is especially relevant when it is noted that each

pre-service teacher is given opportunity to reflect on his/her

learning portfolios. We acknowledge that more research is needed

on this matter. Consequently, this study provides a method to

assess the work of pre-service teachers and the modifications they

make in their written reports that could be used in future research.

This method can also be helpful to teacher educators working both

in the context of teacher training and in lifelong professional

education.

The pre-service teachers made more modifications to Method-

ology and Reflection, putting less emphasis on Content and Activities.

This could be due to the lower emphasis placed on these two

aspects during pre-service-teacher training while, on the contrary,

different methodologies were used in the classroom sessions along

with individual and group reflection carried out explicitly as well as

being dealt with implicitly in other ways. Another possible reason is

that the pre-service teachers had less knowledge of these aspects

and thus were readily disposed to adapt to these new experiences.

The portfolio was considered as a developmental piece of work

(Dinham & Scott, 2003). For this, it contained a large number of

activities that the pre-service teachers could use in their modifi-

cations. This led some of them to make a considerable revision of

their IP. We suggest this indicates that the learning process taking

place in the teacher-education classroom was responsible for

enabling some students to engage at a high level of reflection. It

would appear that their initial ideas, usually closer to a more

directed style of teaching, had adapted to the proposals embedded

in the course methodology. This group of pre-service teachers

showed that they understood knowledge as an ongoing sense of

reconstruction, and this was reflected in their portfolio.

What invites our attention is, although we devoted specific

sessions and work proposals to certain aspects that the pre-service

teachers could incorporate into their learning portfolio, such as the

consideration of multicultural circumstances, none of the pre-

service teachers took them into account in their project revisions.

The importance to prepare pre-service teachers to work in diverse

contexts is recognized. However, there is much evidence in the

literature on multicultural teacher education that pre-service

teachers often bring deficit perspectives in this respect and teacher-

education programs often fail to help them re-examine these views

and develop more culturally relevant teaching practices (Ladson-

Billings,1999). Perhaps this is an aspect that needs greater research.

It may also be the case that the pre-service teachers are viewing

this knowledge through the lens of their own classroom experi-

ences. Nonetheless, some pre-service teachers considered other

practices to which they are also usually unaccustomed such as

group work or open task proposals. This small step may be very

important as one may infer that this reveals an understanding that

learning results from interaction in groups and engagement in

purposeful activity. However, making these connections between

the broad ideas of teaching and learning are difficult to formulate in

practice (Delandshere & Arens, 2003). On the other hand, it must be

Table 5

Results of the comparison between the IP and the FP, by category, taking into account whether the pre-service teachers referred to aspects of the teaching process at the

primary level.

FP

Content Activities Methodology Reflection

Yes No Yes No Yes No Yes No

IP Yes 5 1 9 0 14 0 10 0

No 4 20 2 19 8 8 7 13

M.J. Cáceres et al. / Teaching and Teacher Education 26 (2010) 1186e1195 1193

highlighted that a small number of pre-service teachers did incor-

porate proposals made in the teacher-education classroom to their

FP, such as inventing a story (2, 7%) or providing opportunities for

reflection after performing an activity (2, 7%).

Flores, López, Gallegos, and Barojas (2000) point to the need to

moderate expectations of change in teacher-education activities.

They suggest progressive transitions would lead to consolidated

progress in understanding of future teachers instead of taking

great leaps in the dark that have little chance of being maintained

over time. Reflection on the design of teaching practice, one of

the predominant strategies in teacher-education programs, has

a clear influence on the ideas of future teachers as we have seen

in this study. Nevertheless, they need to experiment with

their new ideas and reflect on that experimentation in order to

consolidate the changes in their way of thinking (Watts & Jofili,

1998).

This cannot be compared with general theoretical ideas but

rather with other, alternative, ideas of a practical nature. That is

why teacher education should adopt progressive and constructivist

approaches, such as we propose in this study. This will enable

future teachers to create practical referents that each will have to

adapt to his/her training and beliefs. Future teachers need to

contrast their own views with alternative practices and not only

with theoretical information (Duit & Treagust, 2003).

Although use of the learning portfolio promoted the successful

development of pre-service teachers' initial ideas, we must also

highlight the lack of consistency in both their IP and FP with which

every pre-service teacher in our study applied to the four cate-

gories. The absence of connections between the different sections

of the portfolio is, at times, incoherent and inconsistent (Zeichner &

Wray, 2001). This suggests that, even after the development of the

learning portfolio, our group of pre-service teachers continued to

perceive teaching as managing a list of things to know, to do and to

believe in but with few justifications or rationales (Delandshere &

Arens, 2003). Furthermore, many pre-service teachers enrol in

a pre-service teacher-education programwith the expectation that

they will be told how to teach, and this perception is frequently

accompanied by a well-developed but transmissive view of

learning (Clark, 1988). Even though they may want to be more

independent learners, it is a developing process, not change that

can be achieved immediately (Loughran & Corrigan, 1995).

Perhaps this study should have been complemented by means

of a formal oral presentation, interviews or surveys of at least

a sample of respondents, thus providing a student appraisal of

the process (Kagan, 1990). Dialogue enhances the learning of pre-

service teachers and reveals a measure of course satisfaction not

inherent in the response format. Opportunities for appraisals of

this nature should provide the potential for a more collaborative,

apprenticeship model of teacher education, although this

involves creating an environment which nurtures a free flow of

ideas (Roe & Stallman, 1994). Nevertheless, we must keep in

mind that not all will be revealed during these interview sessions

because, as Polanyi (1967) points out, we know much more than

we can tell.

Postholm (2008) understood that Schön's (1983) “reflect in and

on actions” fell too short in describing reflections that could pave

the way for development and suggested that reflections on prior

experiences before taking action would prepare teachers for future

actions. As practitioner researchers, we think that another kind of

reflection is needed in addition to this. We should also take into

account reflection outside action, that is, what is done by pre-

service teachers in their university training before their practical

experience in schools. It is in that environment they are supposed

to become aware of the teaching-learning process that goes on in

the Primary classroom.

The research presented here could be extended by further study

of a similar nature. Future research could be applied to repeating

the experience with another group of pre-service teachers in order

to compare results, to repeating the experience with the same

group of pre-service teachers over several modules to analyse how

the experience affects both the pre-service teachers and the

lecturer, to comparing the results obtained in the revisions of other

work included in the learning portfolio or to comparing the

outcomes of each pre-service teacher in the project analysed with

the outcomes they obtained in other tasks as suggested by

Chamoso & Cáceres, (2009). We could also analyse the reasons for

the reduced influence of certain issues treated in the training

classroom, such as the multicultural perspective, in the FP of the

pre-service teachers. Furthermore, the same portfolios can be used

for appraisal of learning and summative assessment purposes.

Therefore the outcomes obtained in this study could be compared

to the lecturer's evaluation as the summative product of learning of

pre-service teachers at the completion of the process. On the other

hand, there is very little literature available that discusses the

nature and consequences of using electronic portfolios in teacher-

education programs as opposed to traditional paper and pencil

portfolios with pre-service teachers. Further research could also

take up this challenge (Zeichner & Wray, 2001).

Acknowledgements

This work has been partially supported by the Ministry of

Education and Science (Spain) under grant SEJ2006/11249 and the

Regional Ministry of Culture of Castilla-León (Spain) under grant

SA032A08.

References

Campbell, D. M., Melenyzer, B. J., Nettles, D. H., & Wyman, R. M. (1999). Portfolio andperformance assessment in teacher education. Needham Heights, MA: Allyn andBacon.

Chamoso, J. (2003). Considering dialogue as a social instrument in the Mathematicsclass. For the Learning of Mathematics, 23(1), 30e40.

Chamoso, J. (2004). In pursuit of patterns: a dialogued enquiry. MathematicsTeaching, 188, 22e26.

Chamoso, J. (2007). A mathematics vision from the usual environment. ScientificResearch and Essays, 2(7), 222e231.

Chamoso, J., & Rawson, W. (2001). En la búsqueda de lo importante en el aula deMatemáticas. Suma, 36, 33e41.

Chamoso, J., & Rawson, W. (2004). Contando la Geometría. Colección Diálogos deMatemáticas. Madrid: Nivola.

Chamoso, J. M., & Cáceres, M. J. (2009). Analysis of the reflections of student-teachers of Mathematics when working with learning portfolios in Spanishuniversity classrooms. Teaching and Teacher Education, 25, 198e206.

Chamoso, J., Fernández, I., & Reyes, E. (2009). Burbujas de arte y matemáticas.Colección Diálogos de Matemáticas. Madrid: Nivola.

Clark, C. (1988). Asking the right questions about teacher preparation: contribu-tions of research on teaching thinking. Educational Researcher, 17, 5e12.

Craig, C. J. (2003). What teachers come to know through school portfolio devel-opment. Teaching and Teacher Education, 19, 815e827.

Darling-Hammond, L., & Synder, J. (2000). Authentic assessment of teaching incontext. Teaching and Teacher Education, 16, 523e545.

De Rijdt, C., Tiquet, E., Dochy, F., & Devolder, M. (2006). Teaching portfolios in highereducation and their effects: an explorative study. Teaching and Teacher Educa-tion, 22, 1084e1093.

Delandshere, G., & Arens, S. A. (2003). Examining the quality of the evidence inpreservice teacher portfolios. Journal of Teacher Education, 54, 57e73.

Dewey, J. (1910). How we think: A restatement of the relation of reflective thinking tothe educative process. Boston, MA: D.C. Heath and Company.

Dinham, S., & Scott, C. (2003). Benefits to teachers of the professional learningportfolio: a case study. Teacher Development, 7, 229e244.

Duit, R., & Treagust, D. (2003). Conceptual change: a powerful framework forimproving science teaching and learning. International Journal of ScienceEducation, 25, 671e688.

Farr Darling, L. (2001). Portfolio as practice: the narratives of emerging teachers.Teaching and Teacher Education, 17, 107e121.

Flores, F., López, A., Gallegos, L., & Barojas, J. (2000). Transforming science andlearning concepts of physics teachers. International Journal of Science Education,22, 197e208.

M.J. Cáceres et al. / Teaching and Teacher Education 26 (2010) 1186e11951194

Grant, G., & Huebner, T. A. (1998). The portfolio question: the power of self-directedinquiry. In N. Lyons (Ed.), With portfolio in hand: Validating the new teacherprofessionalism (pp. 156e171). New York: Teacher College Press.

Hartmann, C. (2004). Using teacher portfolios to enrich the methods courseexperiences of prospective mathematics teachers. School Science and Mathe-matics, 104, 392e407.

Imhof, M., & Picard, C. (2009). Views on using portfolio in teacher education.Teaching and Teacher Education, 25, 149e154.

Kagan, D. M. (1990). Ways of evaluating teachers cognition: inferencesconcerning the goldilocks principle. Review of Educational Research, 60,419e469.

Ladson-Billings, G. (1999). Preparing teachers for diversity: historicalcontexts. In L. Darling-Hammond, & G. Sykes (Eds.), Teaching as a learningprofession: Handbook of policy and practice (pp. 86e124). San Francisco:Jossey Bass.

Lee, H. J. (2005). Understanding and assessing preservice teachers' reflectivethinking. Teaching and Teacher Education, 21, 699e715.

Llinares, S., & Krainer, K. (2006). Mathematics (student) teachers and teachereducators as learners. In A. Gutierrez, & P. Boero (Eds.), Handbook of research onthe psychology of mathematics education. Past, present and future (pp. 429e459).Rotterdam, The Netherlands: Sense Publishers B.V.

Loughran, J., & Corrigan, D. (1995). Teaching portfolios: a strategy for developinglearning and teaching in preservice education. Teaching and Teacher Education,11, 565e577.

Mansvelder-Longayroux, D. D., Beijaard, D., & Verloop, N. (2007). The portfolio asa tool for stimulating reflection by student teachers. Teaching and TeacherEducation, 23, 47e62.

Meyer, D. K., & Tusin, L. F. (1999). Preservice teachers' perceptions of portfolios:process versus product. Journal of Teacher Education, 50, 131e139.

Planas, N., Chamoso, J. M., & Rodríguez, M. (2007). Retazos interculturales para laformación del profesorado de matemáticas. XIII Jornadas para la Enseñanza y elAprendizaje de las Matemáticas (XIII JAEM). Granada, julio 07.

Polanyi, M. (1967). The tacit dimension. London: Routledge and Kegan Paul.Postholm, M. B. (2008). Teachers developing practice: reflection as key activity.

Teaching and Teacher Education, 24, 1717e1728.Roe, M. F., & Stallman, A. C. (1994). A comparative study of dialogue and response

journals. Teaching and Teacher Education, 10, 579e588.Schön, D. A. (1983). The reflective practitioner: How professionals think in action. New

York: Basic Books, Inc.Schön, D. A. (1987). Educating the reflective practitioner: Toward a new design for

teaching and learning in the profession. San Francisco: Jossey-Bass.Shepard, L. A. (2000). The role of assessment in a learning culture. Educational

Researcher, 29, 4e15.Shepard, L. A. (2001). The role of classroom assessment in teaching and learning. In

V. Richardson (Ed.), Handbook of research on teaching (pp. 1066e1101). Wash-ington: American Educational Research Association.

Shulman, L. S. (1998). Teacher portfolios: a theoretical activity. In N. Lyons (Ed.),With portfolio in hand: Validating the new teacher professionalism (pp. 23e37).New York: Teachers College Press.

Silver, E. A., & Kenney, P. A. (1995). Sources of assessment information forinstructional guidance in mathematics. In T. A. Romberg (Ed.), Reform in schoolmathematics and authentic assessment (pp. 38e86). Albany: State University ofNew York Press.

Tillema, H. H. (1998). Design and validity of a portfolio instrument for professionaltraining. Studies in Educational Evaluation, 24, 263e278.

Tillema, H., & Smith, K. (2007). Portfolio appraisal: in search of criteria. Teaching andTeacher Education, 23, 442e456.

Valli, L. (1997). Listening to other voices: a description of teacher reflection in theUnited States. Peabody Journal of Education, 72, 67e88.

Van der Shaaf, M. F., Stokking, K. M., & Verloop, N. (2008). Teacher beliefs andteacher behaviour in portfolio assessment. Teaching and Teacher Education, 24,1691e1704.

Van Tartwijk, J., Van Rijswijk, M., Tuithof, H., & Driessen, E. W. (2008). Using ananalogy in the introduction of a portfolio. Teaching and Teacher Education, 24,927e938.

Wade, R. C., & Yarbrough, D. B. (1996). Portfolios: a tool for reflective thinking inteacher education? Teaching and Teacher Education, 12, 63e79.

Ward, J. R., & McCotter, S. S. (2004). Reflection as a visible outcome for preserviceteachers. Teaching and Teacher Education, 20, 243e257.

Watts, M., & Jofili, Z. (1998). Towards critical constructivist teaching. InternationalJournal of Science Education, 20, 173e185.

Wilson, M. (1995). Assessment nets: an alternative approach to assessment in mathe-maticsachievement.InT.A.Romberg(Ed.),Reforminschoolmathematicsandauthenticassessment(pp.236e259).Albany:StateUniversityofNewYorkPress.

Wray, S. (2007). Teaching portfolios, community, and pre-service teachers'professional development. Teaching and Teacher Education, 23, 1139e1152.

Xu, J. (2003). Promoting school-centered professional development throughteaching portfolios: a case study. Journal of Teacher Education, 54, 347e361.

Zeichner, K. M., & Wray, S. (2001). The teaching portfolio in US teacher educationprograms: what we know and what we need to know. Teaching and TeacherEducation, 17, 613e621.

María José Cáceres is preparing his PH.D in Mathematics Education in the Universityof Salamanca, Spain. She is Mathematics secondary teacher. His current researchinterests include teacher education and assessment in Mathematics Education.

José María Chamoso obtained his PH.D in Mathematics Education from the Universityof Salamanca, Spain, in 2000. He is professor and researcher in this University. Hiscurrent research interests include technology, maths trail, teacher education, andassessment in Mathematics Education.

Pilar Azcárate obtained his PH.D in Philosophy and Education Sciences from theUniversity of Cádiz, Spain, in 1995. She is professor and researcher in this University.His current research interests include teacher education, statistic education, assess-ment and methodology in Mathematics Education.

M.J. Cáceres et al. / Teaching and Teacher Education 26 (2010) 1186e1195 1195