Exploring new dimensions of mathematics-related affect: embodied and social theories

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Exploring new dimensions ofmathematics-related affect: embodiedand social theoriesMarkku S. Hannula aa Department of Teacher Education, University of Helsinki, Finland

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Exploring new dimensions of mathematics-related affect: embodied andsocial theories

Markku S. Hannula*

Department of Teacher Education, University of Helsinki, Finland

This paper will review theoretical approaches for research on mathematics-relatedaffect from the 1990s until today. In order to organise this field, a metatheory ofthe affective domain is developed, based on distinctions along three dimensions:1) cognitive, motivational and emotional aspects of affect; 2) rapidly changingaffective states versus relatively stable affective traits; and 3) the social,psychological and physiological nature of affect. Using ideas from enactivismand other system theories, the third dimension is elaborated. The embodiedperspective brings forth on the one hand the evolutionary basis of human affect,and on the other the individual developmental perspective. Classroom micro-culture and cross-cutting social variables (e.g., gender and ethnicity) are identifiedas two different ways of theorising the social dimension of mathematics-relatedaffect.

Keywords: metatheory; affect; social; beliefs; attitudes; embodiment

Introduction

This article develops further my paper in the proceedings of the Seventh Congress of

the European Society for Research in Mathematics Education (CERME7), which

suggested a new metatheoretical foundation for relating different branches of

research on mathematics-related affect to each other (Hannula 2011). The most

important dimensions to consider, when relating different branches of research, are:

1. which aspects of affect (emotional, cognitive or motivational) are being

studied;

2. the perception of affect as a state or as a trait; and

3. whether affect is being studied as a biological, psychological or social

phenomenon.

The affective domain is typically considered as the non-cognitive aspects of

human thought. Also, in this approach, mathematical knowledge of facts and

routines is not considered to be part of affect. However, most research on beliefs,

motivation and values includes some cognitive aspects as part of the definition. For

example, beliefs are considered to be ‘‘largely cognitive in nature’’ (McLeod 1992,

579). Therefore, it is important to consider the cognitive dimension when building

this metatheory. The distinction between the continuously fluctuating emotional

states and the relatively stable affective dispositions is usually addressed in theories of

*Email: markku.hannula@helsinki.fi

Research in Mathematics Education

Vol. 14, No. 2, July 2012, 137�161

ISSN 1479-4802 print/ISSN 1754-0178 online

# 2012 British Society for Research into Learning Mathematics

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affect, although not always explicitly. More importantly, the distinction between

state and trait aspects is considered to be important for all different aspects of affect,

be it cognitive, emotional or motivational. Furthermore, we acknowledge that

although neurophysiological and social theories may address the affective issues,

their conceptualisations of the phenomenon are different from the psychological

research tradition and, therefore, it is important to recognise these three different

levels of theorising.In this article, some of the details that were sufficiently elaborated then will be

discussed only briefly, in order to use the opportunity to say more about some other

aspects. More specifically, this paper will provide an abbreviated review and criticism

of McLeod’s (1992) seminal paper, in which he reviewed and reconceptualised

research on mathematics-related affect. Elaborations regarding the state-trait

distinction and emotion/cognition/motivation dimension are brief. Instead, there

will be a more thorough explication on the enactivist perspective as a foundation to

conceptualise affect, and more extensive discussion of the social dimension of

mathematics-related affect1.

An overview of research approaches in mathematics-related affect

This first section of this article will sketch the research area in mathematics-related

affect and its development from the 1970s to today. The overview will be divided into

a) research that aligns nicely with McLeod’s framework; b) critique of different

aspects of that framework; and c) new venues of research that seem not to be

compatible with McLeod’s framework. In this article, the word affect is used as an

umbrella concept for those aspects of human thought which are other than cold

cognition, such as emotions, beliefs, attitudes, motivation, values, moods, norms,

feelings and goals.

Research on attitudes, beliefs and emotions

Early research on mathematics-related affect mainly consists of surveys about

mathematics anxiety or attitudes towards mathematics (Zan et al. 2006). These

studies identified differences between countries, and an overall tendency for students’

relations with mathematics to become more negative over the school years (McLeod

1992). Research on mathematics-related emotions was less active. Buxton’s (1981)study Do you panic about maths?, and Mason, Burton and Stacey’s (1982) studies on

the Aha!-experience were examples of early studies that highlighted the role of

emotions in problem solving.

By the end of the 1980s, a number of local theories for mathematics-related affect

had been developed, such as self-efficacy in mathematics (Bandura and Schunk

1981), affect in mathematical problem solving (Schoenfeld 1985; McLeod and

Adams 1989), and mathematics anxiety (Hembree 1990), each with their own usage

of terminology. An important step in theorising research on mathematics-related

affect was McLeod’s (1992) reconceptualisation of the research domain. His aim was

to build an overall framework of mathematics-related affect that would be consistent

with research that is cognitively oriented. His review provides an excellent overview

of the state of the art in the early 1990s, and his conceptualisation of the research

138 M.S. Hannula

area, as well as his suggestions for future research, have had a major influence on

research on mathematics-related affect for the last two decades.2

McLeod’s (1992) framework identifies three major categories in the research of

mathematics-related affect: beliefs, attitudes and emotions. Firstly, these concepts are

characterised by their varying degree of stability, intensity and cognitive involvement.

Beliefs represents the cool, cognitive, and stable end of these dimensions, emotionsthe other extreme, and attitudes covers the middle ground. Secondly, the relation-

ships between these categories were identified in a theoretical framework. McLeod

considered repeated emotional reactions to be the origin of attitudes, while social

context (culture) and individual experience were seen to contribute to the formation

of beliefs. Beliefs were seen to play an important role in students’ emotional reactions

in mathematical situations. In the framework, mathematics as a discipline, self,

mathematics teaching (and learning), and social context were identified as objects

of mathematical beliefs.

McLeod’s framework became the norm in the field, and even today it provides

a structure that can be used to synthesise much of research that has been completed

around mathematics-related affect. One significant branch of research has confirmed

the positive correlation between mathematical affect and achievement. However,

it has been more problematic to establish the direction of causality (see Ma and

Kishor 1997a, b; Ma 1999, for meta-analyses of studies). More specifically, there has

been a lack of studies that use a longitudinal design (see Ma and Xu 2004). Minatoand Kamada (1996) reviewed eight longitudinal studies on the relationship between

mathematics achievement and attitude towards mathematics, and found out that

most studies indicated no predominance of either attitude or achievement. In a few

instances a causal direction from attitude to achievement was found. However, a

larger and more representative sample found a contrasting dominant causal

relationship from achievement to attitude (Ma and Xu 2004). Taken together, these

studies suggest a reciprocal rather than unidirectional causality between achievement

and affect. Such a reciprocal relationship has been identified between self-efficacy

and achievement in mathematics across countries (Williams and Williams 2010).

Another specific field of research on mathematics-related affect that has

accumulated strong evidence is the role of gender. When mathematics-related affect

has been constructed as a single variable, studies have generally found boys to have

more positive affect towards mathematics (Hyde et al. 1990). However, when more

refined analysis of the different dimensions of attitude have been made, interesting

variations have been found. For example, studies in Finland have consistently found

no gender difference in how much students like mathematics or how useful theyperceive mathematics to be (Mattila 2005; Niemi 2010; Hannula 2010). On the other

hand, gender differences have been quite robust in relation to students’ self-

confidence in mathematics (Mattila 2005; Niemi and Metsamuuronen 2010; Hannula

et al. 2005; Leder 1995; Hannula 2010). Lower self-confidence has been found

among female students, even at the level of individual tasks, in the case of both

correct and incorrect answers (Hannula et al. 2002).

Studies of students’ emotions have typically been made in the context of problem

solving (e.g., Schoenfeld 1985; DeBellis and Goldin 1997; Goldin 2000). Many of

these studies have confirmed that, although all individuals tend to experience similar

emotions in the process of problem solving, experts control their emotions better

than novices (e.g., Allen and Carifio 2007; Schoenfeld 1985).

Research in Mathematics Education 139

Critique of McLeod’s framework

Although McLeod’s (1992) conceptual framework has become important in the field,

it has not accomplished what it aimed for. McLeod attempted to provide a general

framework for research on mathematics-related affect, and even explicated the

relationship between his framework and several other approaches (e.g., confidence,

causal attributions, motivation, and intuition). Yet, terminological ambiguity has

continued to be a problem. For example, leading researchers in the field could not

agree on any of the definitions for attitudes, beliefs or conceptions provided in the

literature (Furinghetti and Pehkonen 2002). Such persistent and continuing diversity

in definitions suggests that there may be more theoretical concepts in the research

area than there are names for them. Later in this article, I will suggest a framework

that allows a more detailed and more systematic language for describing these

concepts.

Probably the most problematic concept in McLeod’s framework is attitudes.

He defines attitudes as ‘‘affective responses that involve positive or negative feelings

of moderate intensity and reasonable stability’’ (p. 581). Other researchers have

provided a variety of definitions of the concept ‘attitude’. Often, the definition is only

implicitly identifiable through the instrument they have chosen to use. Some define

attitude as positive or negative degree of affect, others identify emotions and beliefs

as two components of attitude, while yet others define attitude as consisting

of cognitive (beliefs), affective (emotions), and conative (behaviour) dimensions (for

a detailed discussion, see Di Martino and Zan 2010).Within mathematics attitude research, the tripartite framework (e.g., Hart 1989)

has been the most popular. If we try to combine this view of different aspects of

attitude with McLeod’s framework of different components of affect, we see that

attitude is at the same time a parent and a sibling to emotions and beliefs (Figure 1).

This apparent mismatch highlights the different usages of terminology in the field,

and the consequent problems to a synthesis of research.

Attitude is also problematic in a more profound way. Knowing a student’s

attitude does not really help a teacher, as it is seen from the viewpoint of an observer,

and may stem from different psychological reasons (Ruffel, Mason and Allen 1998;

Hannula 2002). Di Martino and Zan (2010) have noticed that teachers often use the

concept ‘attitude’ as an excuse, ‘a claim of surrender’, when they are unable to help a

student.

Emotion

Emotions Beliefs Behaviour

Attitude Belief

AffectMcLeod

Figure 1. An unsuccessful attempt to combine McLeod’s (1992) framework for affect with

Hart’s (1989) tripartite framework for attitude.

140 M.S. Hannula

Moreover, McLeod (1992) suggested that more attention should be paid to the

characteristics of different attitudes. For example, a student with a negative attitude

may feel anxiously afraid of failure, utterly bored, or absolutely hate mathematics,

all of which have a different impact on behaviour. In a similar way, feeling excited orserenely confident about mathematics are very different positive attitudes. Yet,

almost all work on attitudes misses important distinctions regarding its quality,

simply focusing on the direction and magnitude of attitude. Recently, however, the

wide spectrum of emotional traits has been addressed in research about academic

emotions (Pekrun, Elliot, and Maier 2006; Pekrun and Stephens 2010).

In McLeod’s (1992) framework, emotions are considered to be unstable, or at

least less stable than beliefs and attitudes. However, people can have very stable

patterns for emotional arousal across similar situations, which is the foundation ofthe whole concept of attitude. In fact, different types of anxiety are defined as

emotional states of fear relating to discrete situations, e.g., mathematical ones

(Hembree 1990). Is this any different from beliefs that appear only in an appropriate

context (e.g., self-efficacy beliefs about word problems)? We shall return to this issue

when we discuss the state and trait aspects of affect.

Beyond beliefs, attitudes and emotions

More recent research in mathematics education has highlighted affective concepts

that are not included in McLeod’s (1992) framework, such as values (DeBellis and

Goldin 1997; Bishop 2001; Law, Wong and Lee 2011), identity (Beijaard, Meijer, and

Verloop 2004; Frade, Roesken, and Hannula 2010; Sfard and Prusak 2005),

motivation (Hannula 2006; Middleton and Spanias 1999; Yates 2000), and norms

(Yackel and Cobb 1996; Partanen 2011). Especially influential has been the research

on motivation, where numerous quantitative studies have produced results quite

similar to research on attitudes and beliefs: motivation and achievement arepositively correlated, motivation is malleable in the early school years but crystallises

later, and male students are more motivated to study mathematics than female

students (for a review, see Middleton and Spanias 1999).

Much of the original theorising around these concepts has been done outside

mathematics education before they have been introduced in the context of

mathematics. Another important field of research that has advanced significantly

is human neuropsychology (e.g., Damasio 1994; LeDoux 1998), and this approach

is gradually becoming applicable also in the field of education. For example, Brownand Reid (2006) have elaborated the role of ‘somatic markers’ (Damasio 1994) in the

preconscious phase of narrowing down the possibilities when making decisions.

The ambiguity of terminology is a problem for many of these approaches, too.

For example, there are a multitude of theoretical approaches to the study of

motivation that use overlapping terminology (Murphy and Alexander 2000). Also,

when Wigfield and Eccles (2002, 94�5) reviewed some definitions given for values,

an obvious problem was that those definitions did not differentiate values from other

affective variables.A more general shift in mathematics education research has been the ‘social turn’

in mathematics education (Lerman 2000). He identified two different levels of

acknowledging the social in theories of learning. Recognising the influence of social

conditions for learning is generally accepted in this field. However, Lerman (2006)

identified three theoretical fields, cultural psychology, anthropology and sociology,

that take the notion of ‘social’ much further, calling these the ‘strong’ social turn.

Along with these theoretical fields, he argued that all but the most primitive aspects

of human behaviour are essentially social, including learning and knowing. This

strong social perspective sees all meanings as socially produced, with experiences

being interpreted through local cultural practices.

Much of this new complexity was captured in the graphic representation that

Peter Op ‘t Eynde drew for the final presentation session of the affect group at

CERME5 (Figure 2). The figure identifies three main conceptual categories:

cognition, motivation, and affect, and their partial overlapping. The figure also

positions several of the frequently-used concepts in relation to these categories and

each other. Moreover, the figure identifies the local classroom context and the socio-

historical context, where the individual student’s or teacher’s affect is being formed

and is developing. At that time we did not even attempt to define exact borders

between domains or concepts. Hence, each concept in the diagram is located where

its core is perceived to be, and the most significant information regards the relative

positioning of concepts.

I will return to the social perspective later in this paper, but here I want to

highlight the fact that McLeod’s (1992) framework observes the social context

through the individual’s perception of it and, therefore, is not able to respond to the

challenges posed by the strong social position.

A new metatheoretical foundation

McLeod’s review and reconceptualisation was able to summarise most of the

contemporary research done on mathematics-related affect. However, more recent

Socio-historical context

Classroom context

Student/teacherCognition

AffectMotivation

Meta-emotion/affect

Math knowledgeand strategies forlearning/teaching

Metacognition

Metamotivation

Belief system

Emotion Attitude

Figure 2. Peter Op ‘t Eynde’s representation of the affective domain at CERME5 (Hannula

et al. 2007).

142 M.S. Hannula

critique has shown some shortcomings in the framework, and new venues that have

been opened since require a new approach to make a synthesis of the field.

A variety of perspectives was brought together in the special issue on affect in

Educational Studies in Mathematics (Zan et al. 2006). Although just one journal

issue cannot summarise all approaches comprehensively, it provides an illustration of

the richness of the research area. The theoretical frameworks applied to research

on affect are so numerous, complex and diverse, that it may be impossible to make a

synthesis that does justice to each of them. Yet, there is a need for coherence tofacilitate discussion across the different frameworks.

What is presented in this paper could be seen as a metatheory (Wagner

and Berger 1985). Metatheories are ‘‘overarching frameworks that link, separate,

and contextualise other theories’’ (Edwards 2008, 63). As McLeod’s framework has

proven important in our field, the new framework should seek to incorporate its

basic elements: emotions, attitudes, and beliefs; and its dimensions: intensity,

stability, and cognitive-affective. This was carried out successfully, except for the

dimension ‘intensity’, which did not seem to separate theories of affect from one

another.

Instead of attributing stability to beliefs and instability to emotions, the present

framework identifies stability as an independent dimension. In this way, emotional

disposition (the narrow definition of attitude, Di Martino and Zan 2010) forms the

trait aspect of the emotional dimension of affect. Moreover, while McLeod suggested

dealing with motivation through beliefs and emotions, motivation is considered

as a third type of affect in the theory being presented here. Motivation has a

distinctive influence on choices, which cannot be exhaustively analysed through

cognitive and emotional processes. Moreover, motivation research is a very extensive

field of educational research, and for that reason alone it deserves special attention.Thirdly, there is general agreement that emotions consist of three processes:

physiological processes that regulate the body, subjective experience that regulates

behaviour, and expressive processes that regulate social coordination (Buck 1999;

Power and Dalgleish 1997). Emotions are a fundamental aspect of affect, and these

three levels are also identifiable in cognition and motivation. Moreover, there are

distinct research traditions that focus on biological, psychological and social

conceptualisations of the human mind. Therefore, the metatheory needs to identify

the distinct ways of conceptualising affect in three different theories: physiological �psychological � social.

These three dimensions were found to provide a satisfactory way to analyse the

different frameworks used in affect-related research. Hence, the metatheory for

mathematics-related affect was based on three distinct dimensions:

1. cognitive, motivational and emotional aspects of the affect;

2. rapidly changing affective states versus relatively stable affective traits; and3. the physiological (or embodied), psychological and social nature of affect.

These distinctions are between categories, and the dimensions are unrelated, thus

producing a matrix with 18 cells (Figure 3).

In the next sections of the article, we will elaborate these dimensions. The first

two dimensions will only be touched on briefly here, as a more thorough discussion

of them can be read in Hannula (2011).

Cognition, emotion, and motivation

The terms cognition, motivation and emotion have been in frequent use among

psychologists and educators as explanatory factors of behaviour and learning. The

essence of each of these categories can be seen in their function in learning and other

behaviour. Cognition deals with information (self and the environment), while

motivation directs behaviour (goals and choices). Success or failure in goal-directed

behaviour is reflected in emotions (e.g., shame). These emotions, in turn, act as a

feedback system to cognitive and motivational processes.

Theoretical frameworks have typically foregrounded just one of these categories,

while the other two categories are positioned in subordinate roles. For example,

motivation theories may handle emotion as part of motivation, or emotion theories

may handle cognition as part of emotion (Meyer and Turner 2006).

There are some examples of theories within mathematics education that

incorporate all three. DeBellis and Goldin (1997, 2006) have suggested a framework

of emotions, attitudes, beliefs, and values to conceptualise mathematics-related

affect. They define values as ‘‘deep, ‘personal truths’ or commitments cherished by

individuals. They help motivate long-term choices and shorter-term priorities’’

(2006, 135). Clearly, this is closely related to motivation. Also Schoenfeld’s

(1998) framework for teachers’ decision making, and more recently for decision

making in general (Schoenfeld 2010), lists knowledge, goals and beliefs as the

three key components. In his terminology, ‘beliefs’ is a broad category which in-

cludes also emotional aspects. Goals, on the other hand, is clearly a motivational

concept.

Malmivuori (2001, 2006) provides a thorough elaboration of student self-

regulation in the context of learning mathematics, emphasising their self-beliefs

Figure 3. Identifying the three dimensions for a metatheory of mathematics-related affect.

144 M.S. Hannula

and self-motivation. Op ’t Eynde, De Corte and Verschaffel (2006, 98) also

emphasise the need to ‘‘stay aware of the close interactions between affective,

motivational, and cognitive processes within emotional processes and mathematics

learning’’ in their socioconstructivist approach.

State and trait

The distinction between trait and state-type psychological constructs seems to date

early in the history of psychological research, and Bergmann (1955) considered the

terms ‘state’ and ‘trait’ to be idiomatic. In anxiety research, the distinction between

trait and state (Spielberger 1966) was an important step forward.In mathematics education research, this dimension has been identified, but

different levels of stability have been attributed categorically to different concepts.

McLeod (1992) attributed the highest stability to beliefs and the lowest to emotions.

Goldin (2002) made a more explicit distinction between local and global (more

stable) affect. He also attributed stability to beliefs and values, while emotions were

seen to change more rapidly. More recently, Goldin, Epstein, Schorr and Warner

(2011) made a detailed analysis of the interaction between trait and state type affect

and motivation. Also Malmivuori (2001) explicitly addresses both trait and state

aspects of self-related emotions, motivations, and beliefs in self-regulated mathe-

matics learning. In motivation research, a similar important distinction is made

between personal interest (a trait) and situational interest (a state) (e.g., Mitchell

1993).

The present perspective also attributes state and trait aspects to emotions,

motivations and beliefs. For example, in the case of mathematics anxiety, an anxious

student experiences fear (emotional state) when faced with a mathematical task

and, on the other hand, has a tendency to experience this fear (emotional trait).

In a similar way, many beliefs and motives can be analysed as general traits, or

the focus can be on the activation of specific beliefs and motives in a specific

situation.

Enactivism and the system-theoretical perspective

This article explains more thoroughly the physiological, psychological and social

theorisings of mathematics-related affect, the third of the metatheory’s dimensions,

than was possible in the conference proceedings (Hannula 2011). Dealing with thisdimension requires explication of the fundamental ideas behind the framework,

namely enactivism and system theory, before we can elaborate the physiological and

social dimensions of mathematics-related affect.

Within the educational context, affective phenomena are typically considered

either as psychological or as social, i.e., they are perceived as individual mental

processes or as aspects of social interaction and structure. Here, affect is seen in a

broader perspective, which suggests additional ways to view it. In my own thinking,

enactivism (Maturana and Varela 1992; Reid 1996; Hannula 1998) provides a

background for these perspectives, but similar perspectives are provided also by other

system theories. System theories originate in the natural sciences, but they have been

applied also in the social sciences (Chen and Stroup 1993).

The enactivist or system-theoretical perspective suggests a paradigmatic shift in

the way we see the human mind:

Within this paradigm, the knower and the known are codetermined, as are the learnerand what is learned. Thus, cognition is about enacting or bringing forth adaptive andeffective behaviour, not about acquiring information or representing objects in anexternal world. (Nunez, Edwards and Matos 1999, 49)

This view emphasises not only the situatedness of learning, but also refutes the

‘content’ view of knowledge, perceiving cognition in terms of adaptive situated

behaviour.

The key concepts of enactivism are autopoiesis, structure determinism, structural

coupling, emergence and coemergence (Maturana and Varela 1992). Autopoiesis is

the spontaneous self-organisation of complex, dynamic systems, such as cells,

animals, beehives, ecosystems and institutions. Autopoietic entities are resilient, i.e.,

relationships within a system can absorb change and the structure will persist

(Holling 1973, 17).

Structure determinism refers to the idea that the system’s structure determines

action. Autopoietic entities tend to organise themselves into interactive networks

and thus form new autopoietic systems. This is referred to as structural coupling.

Living organisms form ecosystems; individuals form institutions and cultures.

Emergence relates to properties that appear through this structural coupling.

Emergent phenomena are generated from underlying processes, and yet they are

somehow autonomous (Bedeau 1997, 2002). For example, humans have eyes and

can speak, yet neither eyes nor an ability to speak exist at a cellular level. The

emergent approach rejects both types of reductionism: the tendency to seek the key

explanation at ever smaller units, and the tendency to reduce to a holism that ignores

the basic elements (Walby 2007).

Coemergence refers to interactions within a system, where micro-level elements

influence each other so that their interaction produces emergent properties. For

example, the cells of an embryo interact in ways that produce organs as the emergent

property of an individual. In a similar fashion, social organisations and organisa-

tional climates coemerge in a group as members of the group interact with each

other.

To summarise the enactivist perspective, learning takes place on different levels of

life, from cells to culture. The units of analysis are the different self-organising

autopoietic systems that are structurally coupled to form more complex, higher level

autopoietic systems. Cognition, motivation and emotion have their origins in the

cellular level, but as emergent properties they cannot be reduced to biology alone.

On the other hand, human cognition, emotion and motivation underlie many social

level phenomena, but again, cannot exhaustively explain socially emergent phenom-

ena. Learning is adaptation to the environment, and it takes place on all levels, from

cells to social relations. For example, when a social norm is established in a classroom

(social level of learning), it requires that at least some individual students change

their behaviour (psychological level), which is reflected in the neural connections

of the brain (physiological level).

It is problematic to define borders for autopoietic entities (Clark and Chalmers

1998; Wilson and Clark 2009; Froese and Stewart 2010; Virgo, Egbert and

146 M.S. Hannula

Froese 2011). In the sea, using flippers allows us to move more efficiently and having

a mask allows us to see more clearly. It would not make sense to view the flippers and

mask as a part of the environment. Yet, they are not really part of the individual

either. Tools that are used frequently become an essential part of us, as in Merleau-

Ponty’s (1962) famous example of the blind man’s walking stick (see also Bateson

1972). However, humans are extremely well adapted to using a variety of tools and

even making improvised tools for the need at hand.

One way to view such tools is to consider them as affordances (opportunities for

action that objects, events or places in the environment provide: Hirose 2002). More

integrated tools, such as contact lenses and prosthetic limbs, are affordances

provided by technology. However, we have also learned to control our affect

chemically. Coffee and alcohol are everyman’s choices, while legal and illegal drugs

exemplify more radical methods. Could these possibilities to manipulate our own

affective functioning also be considered affordances?

Virgo, Egbert and Froese suggest that:

much of the conflict in this field comes from the conflation of two concepts that shouldbe kept distinct: the physical boundary of an autopoietic system, which is produced bythe system and makes an important contribution to the working of the system; and whatwe call its operational limits, which determine which processes are part of the system.(Virgo, Egbert, and Froese 2011, 240)

For example, any information on mathematics posters on the classroom wall is

within students’ operational limits, even though it is outside their physical

boundaries.

The fuzziness of boundaries is even more pronounced when we consider social

structures. School is a relatively well-defined social structure, yet, it is difficult to

define its boundaries. For example, it depends on the issue in hand whether one

considers the maintenance personnel or students’ families as part of the school. Here,

even the distinction between physical boundaries and operational limits seems to

provide little help.

As any structure needs to interact with its environment in order to be ‘alive’, there

is always a temporary structural coupling between the structure and the elements

of its environment, causing the boundary between environment and ‘invironment’ to

move back and forth. This movement is the metaphorical ‘inhaling and exhaling’ of

the autopoietic system. On a physiological level this may be very literal. Inhaling

stale classroom air with too much carbon dioxide will make students drowsy

(structural determinism of the body). On a psychological level, a student using paper

and pencil as cognitive tools, has ‘inhaled’ these tools as part of their cognitive

system. On a social level, a school inhales and exhales students and teachers. The

school is empty at night, in hibernation. In the morning the school wakes up and

becomes dynamic, while inhaling teachers and students. From a student perspective,

it makes more sense to consider the class of students to be the structure, which

inhales and exhales a number of different learning environments and teachers

throughout the school day. Instead of trying to define the boundary of the

autopoietic entity, we acknowledge its dynamic expanding and contracting nature.

This is extremely important with respect to social systems, as will become apparent

later in this article.

Embodiment

What has been discussed above about autopoietic entities, structure determinism,

structural coupling, and emergence in quite general terms will now be elaborated in

the context of affect and mathematical thinking and learning. In particular, the ideas

will be used to specify the third dimension of the suggested meta-theoretical

framework, the embodied, psychological and social nature of affect. This discussion

will not deal with the psychological perspective separately. Rather, it will explore the

new points of view provided by the embodied perspective and, in the following

chapter, the social perspectives.

Nunez, Edwards and Matos (1999, 49) summarised four different ways of using

the term’embodiment’:

For some, embodiment refers to the phenomenological aspects of the human bodilyexperience [. . .], and the resulting psychological manifestations [. . .]. Certain theoristsstress the unconscious aspects of bodily experience that underlie cognitive activity andlinguistic expression [. . .]. Others focus on the organization of bodily action underprinciples of non-linear dynamics [. . .]. Yet others emphasize the biologicostructuralcodefinition that exist between organisms and the medium in which they exist, fromwhich cognition results as an enactive process [. . .].

An embodied view perceives the human individual as the autopoietic system, and

denies the mind-body dualism. So-called mental phenomena, (e.g., consciousness,

emotions, motivation) are emergent properties of the physiological structure of

humans. These emergent phenomena are of a different nature than the underlying

basic physiology, and therefore they need to be analysed using different theoretical

frameworks.

We can perceive the embodied foundations of affect from two perspectives. First,

there is the biological system of the human species, the genetic level of hard-wired

patterns that have proven their power in evolution. These produce the universal

similarities between individuals across different cultures and individuals, such as the

universally recognisable facial expressions of basic emotions (Ekman 1972). From

this perspective, the human species in its environment is perceived as an autopoietic

system. Feedback takes place through survival, reproduction and promotion of one’s

own gene pool.

This perspective highlights the Darwinian, evolutionary role of emotions.

Emotions serve survival, play an essential role in human reproductive behav-

iour, and are key ingredients in social coordination. Emotions have also more subtle

functions in thinking and learning, e.g., through biasing cognition (Forgas

2008). Some of this evolutionary heritage is useless, and some even problematic in

modern society. For example, while anger might have given primitive man the

necessary boost to break a hard nut, it will not help with a hard mathematics

problem.

The second perspective of the embodiment looks at the individual as autopoietic

entity. Genetic heritage provides only the starting point and the basic rules. It is

through our interaction with our environment that our cognition, motivation and

emotions emerge and develop. The similarity of everyone’s basic physical experiences

in the world explains the similarity of our cognitive structures (Lakoff and Nunes

2000). This perspective coincides with the traditional psychological perspective

148 M.S. Hannula

emphasising the importance of early childhood in cognitive and affective develop-

ment. This early development is the basis for our most fundamental preferences,

behavioural patterns and often automatic emotional reactions. However, through

structural couplings with our physical and social environment, we continue todevelop throughout our lives.

With regard to research on affect, we may draw some conclusions from such

evolutionary and developmental perspectives. Firstly, neuropsychological studies

signify the role of affect in learning. For example, it is now well established that

emotions direct attention and bias cognitive processing: fear (anxiety), for instance,

directs attention towards threatening information and sadness (depression) biases

memory towards a less optimistic view of the past (Power and Dalgleish 1997;

Linnenbrink and Pintrich 2004). A recent study has identified that activity inthe amygdala (a part of the limbic system of the brain which performs a primary

role in the processing and memory of emotional reactions) during an Aha! experi-

ence is a strong predictor of which solutions will remain in long-term memory

(Ludmer, Dudai and Rubin 2011). Such studies indicate that in addition to cognitive

learning processes, the emotional processes of the learner also make a difference

to the learning outcome. Moreover, the recording of physiological measures

such as eye pupil size (e.g., Lindstrom and Gulz 2008) or facial expressions

(e.g., D’Mello, Lehman and Person 2010) provides an opportunity to record theemotional dynamics related to mental activities without interrupting the cognitive

process.

Another lesson from embodiment is the understanding of automatic emotional

reactions (Power and Dalgleish 1997; LeDoux 1998). Emotional reactions to certain

key stimuli are innate (e.g., fear of heights), and other emotional responses to certain

stimuli may become automatic after traumatic or repeated experiences. Such

automatisation has offered an evolutionarily advance by shortening reaction times

to possible threats. On the downside, automatic reactions lack flexibility and aredifficult to change once formed (Power and Dalgleish 1997). Unlearning such

automated emotional reactions is a slow process, which may partially explain

Hembree’s (1990) findings that the most efficient treatment for mathematics anxiety

was systematic desensitisation (a slow therapy). Malmivuori (2001) analysed in detail

the interplay of automatic affective regulation that functions on preconscious levels

and the consciously regulated active regulation of affective responses.

Thirdly, we know from ethnographic (e.g., Lutz and White 1986) and

psychological (e.g., Harlow 1958) research that affect is intricately linked to socialinteraction. Hence, there is a strong link between the embodied perspective to affect

and the social perspective that we will focus on in the next chapter. In Hannula

(2006), need for social belonging was suggested as one important origin of the goals

that students choose, thus explaining some behaviour that from a learning

perspective seemed counterproductive and even irrational. Of course, these three

are just examples of the many lessons that can be learned from the embodied

perspective of affect.

How about the social turn?

Research on embodied cognition confirms our knowledge that social interaction is

an essential feature of human behaviour agnd that it is strongly linked to our affect.

In the present discussion we are taking a strong social position (Lerman 2006), and

we are not considering here the individual’s perceptions of, preferences for, or

feelings about, the social setting for their learning. It is in human nature to self-

organise into autopoietic social systems, and here we take a look at the emergent

properties of affect at the level of social systems. In this chapter we will elaborate the

rich variety of social systems.

A social mosaic

To be social is an essential characteristic of human nature. We are structurally

coupled with other individuals, forming social, autopoietic entities. This coupling

between individuals is largely mediated by interpersonal affective relationships: love

and hate, friendship, loyalty, rivalry, etc. Through this coupling, social phenomena

such as social order, discourse, and division of labour, emerge.

If we think of the prehistoric human, that was perhaps enough. The individual

was born into one social system (clan/tribe), and the core of that social system

typically continued the same throughout one’s life. This social environment formed

an autopoietic structure that preserved, or rather regenerated, its structure through

the structural coupling of individuals. This structural coupling took place through

language (shared meanings), norms (shared values), and discourse.

In (post)modern society, the structural coupling of the individual becomes more

complex. Stryker and Burke (2000, 285) describe modern society as:

a mosaic of relatively durable patterned interactions and relationships, differentiated yetorganised, embedded in an array of groups, organisations, communities, and institu-tions, and intersected by crosscutting boundaries of class, ethnicity, age, gender, religion,and other variables.

Crosscutting social variables such as gender determine what kind of positions or

roles are available for individuals of that type. Some of these positions are

determined by the actual resources members of that category have. For example,

in many countries the wealth of the family largely determines how long, and of how

good a quality, the child’s formal education will be. Some other positions are

determined by custom, e.g., the gendered roles in the classroom.

We can identify at least three dimensions that are relevant for analysing different

social structures. First, resilience is often reflected in the temporal duration of the

structure. Some social organisations may last for hundreds of years (e.g., nations and

universities), while others may form and disappear within one day (e.g., teacher in-

service training workshop). The second dimension relates to the magnitude of the

group, the number of people involved. Nations have millions of citizens, while a

family can have as few as two members. The third dimension is the group members’

level of belonging, or commitment, to the group. For example, a student may be

much more committed to the aims and accomplishments of a street gang than to

those of a school.

These social structures are dynamic; they ‘inhale’ and ‘exhale’ individuals into

their networks of interaction. How can we identify them, and define their boundaries

and operational limits? Luhmann (2008) suggested communication as the particular

150 M.S. Hannula

mode of autopoietic reproduction of social systems. He identified two different types

of closure for this communication:

Societies are encompassing systems in the sense that they include all events which, forthem, have the quality of communication. . . . Interactions, on the other hand, form theirboundaries by the presence of people who are well aware that communication goes onaround them without having contact with their own actual interaction. [. . .] interactionsalso are closed systems, in the sense that their own communication can be motivated andunderstood only in the context of the system. (Luhmann 2008, 87)

Indeed, citizens of a nation-state often share language, culture and an under-

standing of history. The same is true to a lesser extent for other long-lasting social

entities; they generate their own shared understandings and discourse. This is also the

case for mathematics classrooms (e.g., Sherin 2002).Communication may be essential in the formation of social structures, but it is

also important to recognise the material and symbolic resources that a social

autopoietic entity possesses. A social institution may own tools that influence

communication and artefacts that have embedded meanings, for example, introdu-

cing a dynamic geometry software in class may radically change the way of learning

geometry. Such material and symbolic resources are an essential aspect of the

autopoietic entity’s structure. Sometimes, these material and symbolic resources seem

to become independent of people, the social system becoming an unstoppable

machine. For example, changing all employees of a well-established company would

probably not cause more than a temporary disturbance to its functioning. It is the

materials and symbolic resources of the company that keep it going, rather than

its employees. In particular, social entities that endure for a long time have different

people involved at different times. Moreover, the alternation of active and

hibernating states is typical for most modern social structures.

From the point of view of individuals, they are navigating between different social

groups and taking different roles in them. As the school closes for the day, members

of the school community become active in other social structures, taking their roles

as children, friends, partners, team members, coaches, scouts etc. Each group and

each role requires building interpersonal relations and negotiating about shared

norms, values and understandings, i.e., learning in the community of practice

(Wenger 1998). For this negotiation, it is not necessary to explicate values and norms.

Rather, norms and values become established as participants enact them. In this

process of negotiation, both the individual and the social system change (Bandura

1978). Even a passive adaptation to existing rules and norms influences the system,

validating the status quo.

Sometimes the discourse, values, and norms learned in one social system turn out

to be useful resources in other social structures; sometimes different roles require

contextual adaptation. The disparity of appropriate norms, values and meanings in

different contexts might be related to the organisation of beliefs in clusters around

specific situations and contexts, more or less isolated from each other (Green 1971).

However, it is likely that attempts to hold contradictory roles separate from each

other will not be fully successful. Research on role-playing recognises how the

player’s thoughts and feelings are often influenced by those of their fictional

character, or vice versa (Montola 2010). Why would this not happen between

different social roles that one has? Discourse analysis acknowledges this

interdiscursivity, people using concepts and values of other discourses (Evans,

Morgan and Tsatsaroni 2006). Evans (2002) gives an example of an adult learner,

who positions herself with respect to mathematics similarly to how her father used to

position her: as not understanding her father’s work, and thus excluding her from

knowing, from his work, perhaps from his love. Schorr et al. (2010) report another

study where such interdiscursivity seems to be at play. They recognised ‘‘archetypal

affective structures’’ describing patterns in urban inner city students’ engagement

in mathematical investigations, and some affective reactions (e.g., in the ‘‘Don’t

disrespect me’’-style) seem to be influenced by the street code.

Mathematics education research about social systems indicates that the

characteristics of learning communities are powerful predictors for students’

academic success. Research within the achievement goal theory (i.e., motivational

traits) has acknowledged and studied the role of the classroom goal structure

(e.g., Kumar, Gheen, and Kaplan 2002). A recent study has found that mastery goal

structures have a positive effect on class interpersonal relations, whereas perform-

ance goal structure have negative or non-significant effects on different types of

interpersonal relations and, moreover, these effects are not mediated by personal

goals (Polychroni, Hatzichristou and Sideridis 2011).

Another typical research area has been the classroom microculture of teacher-

student interactions. With respect to mathematical affect, an important approach has

been to analyse the social and sociomathematical norms (i.e., motivational traits)

and how those are being established (i.e., the process of states) in the classroom.

However, researchers often realise that the more institutionalised school culture

and broader socio-cultural situation where schooling takes place, penetrate to the

microculture (Cobb and Yackel 1996; Partanen 2011). However, the microculture of

the classroom may also build resilience against overall educational policy. For

example, classroom culture (community, autonomy and mastery goal orientation)

has been found to mitigate the influence of the prevalent performance pressure in the

U.S. educational system to students’ motivational orientation (Ciani et al. 2010).

Such studies on classroom affect and motivation often emphasise the teacher role in

the establishment of classroom discourse and motivational orientation (e.g., Turner,

Meyer, and Schweinle 2003), paying less attention to the students’ role in the

establishment of the classroom climate.

Another approach is to focus on the broadest level of social systems and to

conduct cross-cultural comparative studies. Such studies indicate, for instance, that

high performing Asian countries such as Korea and Japan demonstrate compara-

tively low mathematics self-concept and mathematics self-efficacy and high mathe-

matics anxiety when contrasted with some lower performing countries. On the other

hand, some of the Western European countries such as Finland, the Netherlands,

Liechtenstein, and Switzerland show ‘balanced’ outcomes, with high mathematics

performance and low levels of mathematics anxiety (Lee 2009). The assumption is

that some cultural differences explain the differences in student affect. These social

systems can be said to form a hierarchical, or nested, set of systems: the classroom

is part of the social system of the school, which is part of the educational system,

which is embedded in society at large. Such a perspective makes it tempting to

analyse different top-down and bottom-up mechanisms. However, students and

152 M.S. Hannula

teachers also belong to other social systems (e.g., sports clubs and on-line gaming

communities) which do not fit into these hierarchical categories.

The third typical approach focuses on the influence of social variables, such as

gender and ethnicity, that cross-cut other social systems, or at least most of them.

The cultural gender norms are perhaps reflected in all social systems, also through all

levels of the educational system, from the society to the classroom microculture.

One interesting field of research is to analyse the interaction between group and

cross-cutting variables. For example, research results regarding gender and ethnic

differences are sometimes taken for granted in new cultural contexts. On the

contrary, the validity of such results ought to be questioned before accepting the

results, not only for each new nation, but also for different sub-cultures.

Identities

What was elaborated in the previous section was primarily viewed from the

perspective of the social. What about the individuals; how do people navigate

through this mosaic of social systems? Identity is the conceptualisation of the

individual’s relationship with their social environment and it, too, has a variety of

definitions.

Stryker and Burke (2000, 293) suggested that two different research traditions

identify different bases of identity: ‘‘Social identity theory has focused on category-

based identities (e.g., black or white, Christian or Jew); identity theory has focused

primarily on role-based identities (e.g., parent or child, teacher or student)’’.

Brewer distinguishes three fundamental self-representations: the individual self,

the relational self, and the collective self. Stated otherwise, people seek to achieve

identity in three fundamental ways: (a) in terms of their unique traits; (b) in terms

of dyadic relationships; and (c) in terms of group memberships (Brewer and Gardner

1996; Brewer 2001).

The enactivist or system-theoretical view of social organisation emphasises the

relational view of identity. All the other identities in Brewer’s classification can also

be perceived as relational. No group or organisation can survive if its members do

not partake in reconstructing it. Even the most individual identities are still

generated in relation to the society they live in. This is the standpoint of identity

theory:

Identity theory thus adopts James’ (1890) vision of persons possessing as many selvesas groups of persons with which they interact. To refer to each group-based self, thetheorists chose the term identity, asserting that persons have as many identities asdistinct networks of relationships in which they occupy positions and play roles.In identity theory usage, social roles are expectations attached to positions occupied innetworks of relationships; identities are internalized role expectations. (Stryker andBurke 2000, 286)

There are different levels of agency as a person chooses social identities. Many of

the identities one grows into, not having a choice regarding nationality or gender, for

example. Some choices, such as participation in compulsory education and possible

streaming and setting, are made for the individual. Sometimes, one is allowed to

choose, for example the level of mathematics one studies. Moreover, some of the

identities are public (e.g., gender) and they influence the negotiations for roles in

other social systems (e.g., in school).

Behaviour can be seen to be directed towards the goal of fulfilling ones identity

(Stryker and Burke 2000; Krzywacki-Vainio and Hannula 2010) and, morespecifically, students’ learning is an effort to actualise their identity through

participation in classroom activities (Op ’t Eynde, De Corte and Verschaffel 2006).

Emotions rise due to an increasing or a decreasing discrepancy between identity

standards and a perceived situation (Stryker and Burke 2000; Op ’t Eynde, De Corte

and Verschaffel 2006).

Using the metatheory to analyse and compare three theoretical frameworks

The purpose of this article has been to provide a metatheory of research on

mathematics-related affect that would enable dialogue across different theories. More

specifically, there has been elaboration of a system-theoretical approach that might

be able to relate research on mathematics-related affect based on a basic neuroscien-

tific framework, a traditional psychological framework, and different social frame-

works. Here, I will use this metatheoretical framework to compare McLeod’s (1992)

theory of mathematics-related affect, achievement goal theory, and a local ‘theory’ of

mathematics anxiety and working memory, as examples of its power.McLeod’s (1992) theory suggests that student beliefs have their origin in

individual experiences and the social context. Beliefs influence the interpretation

of mathematics-related events, and thereby the student’s emotional experiences.

Repeated emotional experiences are the origin of attitudes towards mathematics.

Achievement goal theory is a motivation theory where students’ definitions

of success as either mastery of content or as better performance in comparison with

peers are predictive of their behaviour in class. Research has confirmed that

individual mastery goal orientation is related to better learning strategies andemotional well-being, yet performance orientation seems to be a better predictor

of achievement. The theory also incorporates the classroom goal structure, which is

seen to influence individual students’ goal orientations. In addition, both personal

goals and classroom goal structures are seen as predictors of a number of social,

affective and cognitive consequences on an individual and at group level (e.g., Urdan

and Schenfelder 2006; Ciani et al. 2010; Polycroni et al. 2012).

The negative relationship between mathematics anxiety and achievement is

well recorded in numerous studies. One theory for the mechanism is the overloadingof working memory, as the subject is preoccupied with mathematics fears and

anxieties (Ashcraft and Krause 2007; Rubinstein and Tannock 2010).

These three theories use very different language and different conceptualisations

of affect. On the surface, there is little connection between them. However,

researchers in the area can recognise that they are partially overlapping, and finding

connections between them is important for mathematics education research. The

metatheoretical framework presented in this paper will make the connections

apparent.Both achievement goal theory and McLeod look at social level traits and their

influences on individual traits. As achievement goal theory focuses specifically on

motivational traits, it could be seen as a special case of McLeod’s theorising for the

social influence on individual beliefs. Moreover, both theories indicate that

154 M.S. Hannula

individual affective traits (motivation/beliefs) influence the individual’s emotions

(well-being/emotions and attitudes). In a way, these two theories view emotional trait

as the outcome of social influence and individual experience. On the other hand,

anxiety theory takes a specific emotional trait as the starting point, and elaborateshow the related emotional state influences cognitive processing in the brain,

consuming working memory and thus compromising performance. Moreover,

research on mathematics anxiety suggests that, in addition to social context, lower

than average working memory capacity (a physiological cognitive trait) also makes

students prone to mathematics anxiety (Ashcraft and Moore 2009).

Conclusions

The review of research on mathematics-related affect indicates that the dominant

framework of beliefs, emotions and attitudes is not sufficiently broad to incorporate

all research in this area. More specifically, embodied perspectives and strong social

theories go beyond that frame. Emotions are, by their very nature, linked closely

both to the biological human body and to social systems. The system-theoretical

perspective was found to be a feasible framework to analyse these new approaches

to research on mathematics-related affect. The analysis of social systems in relation

to affect highlights the continuous renegotiation of the role (position) that theindividual has in different social systems. This negotiation is the basis for the

reciprocal coemergence of beliefs, norms and values adopted by the individual and

in the group. However, the balance of this codetermination varies, a single individual

not having much influence on large and resilient systems, such as a nation.

The metatheoretical framework (Figure 3) provides a way to relate different

research approaches used for mathematics-related affect. However, seldom does

research fit nicely within just one of the cells of the framework. For example, typical

survey studies on mathematics-related beliefs or attitudes tend to mix self-beliefs(‘‘I am good at maths’’) and emotional traits (‘‘I like maths’’). In fact, many

frameworks aim to theorise the interaction between different aspects. For example in

Schoenfeld’s (1998, 2010) theory, the explanatory factors for decision making

(‘‘knowledge, goals and beliefs’’) refer to different cognitive, motivational and

emotional traits in our framework. Yackel and Cobb (1996) are explicit in their

framework about relating the social norms to individual beliefs.

The metatheoretical framework makes some historical trends visible. Research in

mathematics education has originated in the testing of mathematical knowledge,which is located in the psychological cognitive traits cell of the framework. Thence,

research has expanded in all directions, important phases of transition being the

social turn (Lerman 2000) and the advancement of brain research (e.g., Damasio,

1994).

One rationale that Edwards (2008) suggests for making metatheoretical analysis

is to identify potential research areas that have not received sufficient attention.

There is a clear imbalance in favour of studies that focus on traits over studies that

focus on states, and a similar imbalance favouring a psychological approach overothers. In particular, studies that focus on the dynamics of emotional or motivational

states in a classroom or other learning community are still rare (as examples, see

Barsade 2002; Jarvela, Jarvenoja, and Veermans 2008). Mathematics educators seem

not yet to have ventured into these areas.

1. I wish to acknowledge the importance of discussions in the Working Groups on affectthrough the ERME conferences CERME3 to CERME6 for the development of the ideasbeing presented below.

2. According to Harzing’s (2011) ‘Publish or Perish’ software, it has received over 700citations so far.

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