Christine Ingleton
The University of Adelaide

Australia

Kerry O'Regan
University of South Australia

It seems paradoxical that mathematics is constructed as an objective, emotion-free discipline, yet mathematics engenders strong negative emotions among many learners. Emotions play a significant part in the learning process. In relation to the learning of mathematics, the focus in the literature is on the emotion of anxiety. This has largely been investigated from the perspective of psychology, with the use of psychometric tests, particularly the MARS (Mathematics Anxiety Rating Scale). The focus of this social-constructionist study is on the interactions that construct both emotion and learning in a social setting. In exploring the experiences of prospective and current teachers at primary, secondary and tertiary levels at two South Australian universities, strong emotions were found to pervade mathematics learning in experiences from early primary schooling to tertiary level. In some mathematics classrooms, public shaming for being wrong has been a common form of socialisation and control, diminishing confidence and arousing fear. We conclude that the interrelationships between pride and shame, and success and failure, in mathematics classrooms (and others) are associated with levels of threat and confidence that dispose students to act in certain ways towards mathematics.

Mathematics is a subject of some contradictions. It is commonly perceived as being impersonal, rational, fixed and rule-bound (Willis 1989; McKnight and & Cooney 1990; Leder 1986) but is associated with strong, typically negative, emotion reactions, perhaps more so than any other subject area (Blum-Anderson 1992; Willis 1989; Williams 1988; Joffe 1984). This aversion is reflected in the large-scale avoidance of mathematics by students when they have the opportunity to choose subjects other than mathematics. Avoidance has been a pattern for quite some time, both in this country and elsewhere (Barnes 1984; Meece 1990), and the latest figures for South Australia (SSABSA 1998) indicate that this is still the case. In 1997, only 8% of those entering secondary school four years previously proceeded to study Mathematics 1 and 2 at Year 12 level. This figure is slightly less than the corresponding 10% figure of ten years ago. In the United States, over half the students who initially select mathematics as a college major eventually change to another discipline area (Linn 1995). This attrition is particularly marked for female students. At beginning undergraduate level there are equal numbers of male and female students but, although females outperform their male counterparts in terms of achievement, they account for only 20% of PhD students in mathematics.

It appears that the choice to withdraw from the study of mathematics is related to emotional factors. However, the exploration of this has been largely through the perspective of cognitive psychology, the phenomenon being identified as Mathematics Anxiety (McCormick 1993; Bessant 1995; Barnes 1984; Norwood 1994; Hadfield 1992; Fairbanks 1992; Blum-Anderson, 1992). Mathematics anxiety is presented in the literature as a widely-occurring condition, the study by Norwood suggesting that as many as 70% of college students enrolled in mathematics classes experience high levels of anxiety associated with their study of mathematics. Although it has been recognised that mathematics anxiety is more complex than the label might signify, it has been studied as a psychological condition defined and measured through the use of psychometric tests, particularly the MARS (Mathematics Anxiety Rating Scale) or some variation of it, developed more than 25 years ago (Richardson 1972). The pervasive nature of mathematics anxiety is reflected in the strength of its presence on the Internet. A recent search by one of the authors revealed some 370 000 sites relating to mathematics anxiety, covering aspects such as literature, research and researchers, guidelines on how to identify and deal with mathematics anxiety, and programs and courses to help prevent and/or overcome it. The Sheila Tobias classic, Overcoming Math Anxiety of 25 years ago, has recently been revised (Tobias 1994). There still exists the educational/psychological/medical paradigm in which a large number of students suffer from a 'pathology' which needs to be dealt with either in terms of prevention or of remediation.

Feynman provocatively describes mathematics education as a 'self-propagating system in which people pass exams, and teach others to pass exams, but nobody knows anything', (cited in Higginson 1989 p 9). Reports over the years such as the Cockcroft Report in Britain (Cockcroft 1982) and the Gilding Report in Australia (Gilding 1988), have identified and attempted to address this perception of mathematics from an educational perspective. The Gilding Report acknowledged mathematics as being 'the one subject taken by a large number of students who have little interest or competence in the subject' (p 335). However, despite attempts to redress this, mathematics remains, to all but a select few (Burton 1994), a subject which often fails to inspire or excite, even some who will become teachers of mathematics.

Much of the research exploring attitudes towards mathematics among prospective teachers has focussed on the primary level. A study by Rech (1993) of elementary education majors in a US college found that they possessed more negative attitudes towards mathematics and scored lower in mathematics competency tests than the general college population. A program at LaSalle University, Pennsylvania (Freeman & Smith, 1997), attempting to reverse negative attitudes towards mathematics and science among pre-service elementary teachers, found those attitudes more resistant to change than had been anticipated. A current cohort of primary teacher education students at an Australian university has been found to have 'no student [with] a broad mathematics nor a strong mathematics background' (Chartres 1998). Of the 91 students in that cohort, about a third had studied Mathematics 1 or Mathematics Applied at Year 12, one third had studied Business Mathematics or Accounting and one third had taken no mathematics at Year 12. None had studied Mathematics 2.

This study explores connections between emotions and the learning of mathematics with a group of prospective primary teachers (undergraduates) as well as a group of prospective secondary teachers (postgraduates), and a group of tertiary teachers in a university mathematics faculty. Our interest in the research arises in part from our own teaching and learning experiences at secondary and tertiary level. O'Regan has been a secondary teacher of mathematics, and now provides learning support for students at tertiary level including those required to use mathematics in their tertiary courses. Ingleton chose to do subjects other than mathematics at secondary level and now works with staff to support student learning in a range of disciplines. We regarded the study as an opportunity for teachers and prospective teachers of mathematics to increase their awareness of their own learning of mathematics, their use of mathematics and the potential impact of their own teaching practices on their students.

Before describing the study, we introduce the theory of emotion on which our analysis is based. Emotions, particularly pride and shame (that is, those associated with acceptance and rejection), play a major role in the construction of self-identity (Kitayama 1995). It is through the experience of emotion that we judge the significance of an event and assess our 'readiness for and intention to act' (Scherer cited in Barbalet, 1998, p 114). Cognition and action are bound up with emotion, not separate from it. Barbalet defines an emotion as having three elements: 'a subjective component of feelings, a physiological component of arousal, and a motor component of expressive gesture' (Barbalet, 1998 p 86). He also characterises emotion as comprising cognitive and dispositional elements: emotion states include decision-making and disposition to act and so emotion is constitutive of reason and action as well as feeling. Barbalet defines confidence as an emotion on the grounds that it includes these three elements. Confidence functions in opposition to shame, shyness and modesty, which he describes as emotions of self-attention - 'thinking what others think of us' (p 86).

Confidence has its basis in particular experiences of social relationships - those situations in which a person receives acceptance and recognition. Conversely, anxiety or shame have their basis in situations in which a person is denied acceptance or recognition. Confidence is a critical emotion in the process of learning. Confidence depends on acceptance and recognition which are key components in establishing what Scheff calls the 'social bond' (Scheff 1991). His theory of the social bond describes the social relationships of solidarity and alienation as basic to the development of individuals' identity and self-esteem. Trust is fundamental in the building of self-esteem, through the strength of the social bond (Kitayama 1995). Trust is experienced in solidarity with others, while rejection is experienced as alienation. When the social bond and self-esteem are high, one may be disposed to act with the confidence of positive expectation, so important to self-efficacy. According to Scheff's theory of the social bond, pride accompanies solidarity while shame accompanies alienation. Self-esteem and confidence hinge on a cluster of 'self-conscious emotions, particularly pride' (Kitayama 1995 p 524).

Pride and shame are frequently experienced in learning, both in relation to one's own expectations of self, as well as those of parents, teachers and peers (Ingleton 1995). A study of teachers in a postgraduate teacher education course (Salzberger-Wittenberg et al. 1983) showed that hope and fear, love and rejection were emotions experienced by adult learners in a new learning situation. Those emotions were analysed in terms of transference onto teachers of early experiences of emotions in relation to acceptance and rejection by authority figures. This is a useful analysis in relation to the relationships of power and authority in the classroom. It also suggests why the strength of those emotions remains active over time, as will be shown later.

The role of emotion in learning may be summarised as follows:

social relationships disposition to learn

Figure 1 Emotion and learning (Ingleton 1998 forthcoming)

The model is based on the theory that pride and confidence emerge in social relationships of solidarity to create a positive disposition to learn. Shame and alienation emerge when threat is experienced, disposing action to protect self-esteem. Social relationships in the classroom give rise to emotions, conscious and unconscious, which interact with learning.

We undertook the study from a constructionist perspective in order to examine experiences in which confidence is built by learners and teachers in the classroom. Constructionists view 'all knowledge ... [as] contingent upon human practices, being constructed in and out of interaction between human beings and their world, and developed and transmitted within an essentially social context' (Crotty 1998 p 42). To explore the role of confidence in learning, we sought to identify social interactions in the classroom and beyond that create conditions which make for positive and negative dispositions towards learning. To understand some of those practices, we invited current and prospective mathematics teachers to examine their own learning experiences using the methodology of Memory-work.

Memory-work involves participants reflecting on, discussing and theorising about past events which are remembered for their significance (Haug 1987; Crawford 1992). This significance may be due in part to the emotions experienced in these events. The methodology assumes that what is remembered is significant, problematic, unfamiliar or in need of review (Crawford et al. 1992 p 38). The theory underlying the method is that 'memories are subjectively significant events which play an important part in the construction of self' (Crawford p 37). A memory may be seen as 'a construction of a real event in time: a construction that changes with reflection over time' (Crawford p 10). It is not the event which is important as much as the meaning that is negotiated in the remembering, the search for intelligibility in the construction and reconstruction of one's life story.

The Memory-work process includes:

The rules for writing (Crawford p 45) are as follows:

1 write a memory

2 of a particular episode, action or event

3 in the third person

4 in as much detail as possible

5 without importing interpretation, explanation or biography.

The groups comprised both university staff (faculty) and students who had attended briefing sessions on the purpose and method of the research and had then volunteered to participate. Each group met twice to discuss the general topic of the study (introduced below) and to decide on themes on which they would write. The written narratives were then read within the group and discussed according to the Memory-work process. The narratives we have selected for this paper illustrate the centrality of emotion in learning, particularly the experience of pride, shame and confidence in social interactions. The written narratives are titled and extracts from the group discussion preceding and following their reading are indented. The group discussions began with the following stimulus statement: 'Maths is constructed as an objective, emotion-free discipline, yet the study of maths generates strong negative emotions among many learners.' From the ensuing discussion, the students decided to write on the topics 'Encouragement in learning mathematics' and 'Discouragement in learning mathematics'.

In Memory-work, spoken and written narratives are used to explore the meanings of significant memories. The 'story or narrative provides the dominant frame for ... the organisation and patterning of lived experience' (Epston, 1992 p 80). ...[P]ersons' lives are shaped by the meaning they ascribe to their experience, by their situation in social structures, and by the language practices and cultural practices of the self and of relationship' (p 122). Of the thirty-two memories explored in this study, we have selected four to illustrate the particularity of situations in which students describe and give meaning to their experiences of confidence and threat in learning mathematics. The experiences are microcosms of the larger social structures and cultural practices in which identity, self-esteem and confidence are developed.

The narratives were analysed first by the groups who wrote them, then by the co-researchers paying attention to the situational context, what is going on, who is involved, the social and power relationships, and the role of language (Fairclough 1989 p 146). The narratives describe incidents experienced in years one and five of primary (elementary) schooling, and years eight and nine of secondary schooling. The writers are Sue who is in a university mathematics faculty, Margaret and Karen who are undergraduates training to be primary school teachers, and Colin who supports university students doing mathematics subjects. They wrote the narratives in the third person, according to the guidelines for Memory-work (above).

The Friday test was institutionalised in most primary schools, and in this school, the test results of each student are fed into the House system, magnifying and making public their achievements and losses. Learning is structured as a highly social activity, and the hierarchy of learners is apparent, from Sue to whom 'all of these things came fairly easily' to Paul, the stereotypical 'dunce'. Sue's narrative was explored further in the group discussion:

As House Captain, Sue takes on the task of bringing Paul's performance up to the House's expectations, and in doing so takes on the authority of the teacher. Sue coaches Paul out of his dunce status into an acceptable position, and he even comes to school early on a regular basis to learn. Not only does he experience success, he is contributing to his House. Sue's pride is strong:

Sue's encouragement comes from helping Paul, her success in that, the contribution to the House, and the teacher's affirmation of her maths ability - all contributing to positive feedback. The fact that the teacher is also the school principal heightens her confidence and pleasure.

From a social perspective, competition, co-operation and reward are part of the learning process. But let us look at what is happening to the students' perception of maths in this learning situation:

The extra time and help from a peer made an obvious difference in a short time. Not enough time to understand is often mentioned in the discussions - students needing more time, teachers having too little. Paul, Sue and the House were well rewarded for Paul's rote learning; mastering the rules, not understanding, was sufficient to do well. Understanding the logic requires elucidation and abstraction that is difficult for children to grasp and so procedures tend to be taught and tested rather than understanding. Implications of issues such as these for learning and teaching are expanded upon in O'Regan and Ingleton (forthcoming).

In the next narrative, confidence fails to develop within a context of threat and fear. Margaret is a mature age student who is currently studying for an Honours degree in Education with the intention of becoming a primary teacher. Her experiences of learning mathematics were far from happy. She is conscious of their significance for herself as a learner and practitioner of mathematics, and also as a teacher of mathematics at primary level. Margaret's narrative relates an incident that occurred very early in her school career and coloured much of her subsequent learning of mathematics.

Margaret describes her teacher as being 'really, really strict', the intensity of which is indicated by the repeated modifier. The power relationship between teacher and pupils is established forcefully with the teacher's physical and verbal intimidation of Margaret and her friend. Margaret's fear and sense of alienation are expressed in the absolute, 'completely scared', and her despair in having 'no idea what was going on' and 'didn't know...what she had to do'.

Margaret experiences humiliation and a devastating loss of control during this incident. She loses control of her bladder and wets her pants in front of the class. She also loses control over learning the mathematics. She experiences confusion and uncertainty. It is as if her learning is taking place in a space filled with the emotions of fear, shame and confusion. There is no place for engagement with the mathematics. The language gives a sense of the enormity of the event with the repeated use of the word 'just' in relation to the teacher's actions, as if this were the only thing that was happening. 'The teacher just yelled at her...just yelled at her...She was just yelling...the teacher just shook her'. Shame, fear, alienation and threat overwhelm her. 'Margaret didn't go back'. She eventually went back into that classroom (although we don't know the circumstances of that return) because she had to; she was not at an age when she could make other choices. From then on every maths lesson brought the same fear, uncertainty and alienation. In another sense she did not return: she did not take her place as a learner of mathematics.

The subsequent discussion explored Margaret's relationship with that teacher: 'She was a big, bad wolf'. The big, bad wolf metaphor brings with it connotations of childhood fears embodied in fairy tales. Wolves represent fear-inducing threat to the powerless and helpless. They represent extreme vulnerability and this is how Margaret saw herself in relation to the teacher and to mathematics: 'the whole way through school...I hated it'. Margaret talks of how she continued to carry that experience with her. Mathematics remained an unfathomable mystery: 'I don't know what I'm doing'. She dealt with this situation by assuming dependency and copying her friend which became a coping strategy in mathematics throughout her schooling. 'I didn't know what I was doing and everyone else did and I felt I had to copy...I didn't know what I was doing ever...I would just get help from the person next to me'. For Margaret, mathematics has one right way of being done. She knew that there was one way of doing it right and she did not know what that was. 'There's only one way to do it and this person knows how to do it so I'll just copy her'. In the third narrative, Karen shares Margaret's perceptions of mathematics as a subject which has a right answer and a right way of attaining that answer. Mathematics is not seen as an area of exploration or creativity.

Like Margaret, Karen is also a mature age student doing her Honours in Education with the intention of becoming a primary teacher. There are positive as well as negative emotions involved. For Karen the primary connection is between herself and mathematics. Family relationships and expectations, however, do seem to play an important part, and these fit into a wider socio-cultural context of expectations of girls in relation to mathematics.

Her first term at secondary school was very different for Karen, in terms of her mathematical experiences, from her primary school mathematics. In her account, Karen focuses on her engagement with the subject. For 'all of first term' (the 'all' suggesting a subjectively long period of time), her engagement with mathematics has been in terms of a struggle - something negative and difficult. The passive language used in the second sentence, with the absence of actor, suggests an absence of control in the process of being moved from one class to another, from one teacher to another. The fact that it was to a lower stream is stated in neutral language. The process of instruction is presented as a safe one, where each small step is guided and checked as it is made. Even so, Karen still has the expectation of failure. The safety net is removed and she has to take the risk of trying the 'list of long division problems' without the checking and reassuring that has previously accompanied each step. Karen 'attempted' the problems 'sure they would be wrong'. There are strong and positive emotions associated with the unexpected success - amazement and elation. The sense of struggle, of alienation, has gone and she acknowledges a return of her mathematical confidence.

The group discussion reveals the significance of the family setting for Karen. The message at home was that success in mathematics was gender-related:

This perception was challenged by her successful experience during that one year, by the encouragement of the teacher who said when she joined his class 'You'll be great; you'll work really well'. Although it did not last beyond that teacher and that year, the experience was a powerful one: 'I was so elated...I can still remember how I felt'. It is not clear from the narrative how Karen carried that experience with her as a learner, but the discussion indicates that she developed an awareness of the importance of the teacher in the learning process for she does not take on total blame for her lack of success in mathematics. 'I didn't think I could do mathematics...now I know actually I can, if I spend time with the right teacher to learn those skills'. She also sees implications for herself as a teacher, an area that will be discussed in a forthcoming paper.

In the fourth narrative we explore a different dimension of power, one in which the teacher gives more power to the students. He takes on the role of facilitator by creating a safe learning space in which students can test their understanding without his judgment.

Colin, aged about fourteen, is in the second year of secondary education at a government high school. Colin's focus is on the problem on the board; the unnamed teacher's is on the range of responses. The teacher proceeds to solve the problem using the students' suggestions, enabling the students to assess their understanding as the logic of their working unfolds. 'Now we'll vote ... ' he says, using the pronoun of inclusion, 'we', and keeping the focus on the board. Now the students have to commit themselves to the rightness or wrongness of their working. A lot of hands go up, followed by the undecided who opt for safety in numbers. No judgment is made by the teacher, who then invites students to respond to the second option. Only one hand goes up. Colin is surprised to find he is alone, but it is 'too late to recant'. Still the teacher's focus is on the board while he completes the solution to reveal Colin's answer to be correct.

In posing a fourth year problem to a second year class, the teacher implies his confidence in their ability to succeed, and throws them a challenge by which to judge themselves. He withholds judgment until the students themselves can see the logic of their argument as it unfolds through the working on the board. There is a freedom from personal judgment by the authority figure, and an emphasis on the students making their own judgments. This freedom creates what could be called a 'safe learning space' in which to problem-solve with confidence. Those students who are unsure of themselves join the weight of numbers by putting their hands up. While a 'collective groan' rises from the class, it is in response to their error, not to Colin. As a group, they are wrong, but their solidarity minimises their shame. Colin's reaction is a mixture of relief and satisfaction: release from the anxiety of being the only one wrong, and satisfaction in his singular cleverness. The teacher's or the class's response to Colin is not significant, although he is one of thirty students 'crammed into three long rows'. But we know that the teacher comments on Colin's enormous delight at a subsequent parent-teacher interview with his mother. Both his parent and his teacher share his success, a cause for pride.

In the analysis so far, we have commented on the uses of power by teachers, peers and individuals in particular classroom settings, and their impact on the learning environment. Sue, as House Captain, participates in institutionalised forms of control of House membership and competition and takes on the responsibility to teach Paul, in order to make better marks for the House. As his peer, and able to give time and attention to Paul, she is far more successful than the teacher. Margaret's teacher uses verbal and physical intimidation which become synonymous with her learning of maths for the rest of her schooling. Karen's teacher welcomes her into a new class saying 'You'll be great; you'll work really well,' trusting her, while Colin's teacher encourages students to work together to try out solutions to problems and discover the strength and weaknesses in their own logic.

The sense of power felt by students and peers in relation to mathematics and the teacher is significant. Among the university teachers, four of the five academics had a strong sense of personal power and remarkably little expression of negative emotion related to their own learning of mathematics. Louise reported recalling 'no negative experiences' in learning mathematics at school or in her first degree:

Although she described herself as lacking confidence as a person, she had strong confidence in her logic, just as Richard had confidence in his mathematics ability. Always ahead of the class, Richard was embarrassed in primary school when asked by the teacher not to participate in class chanting of tables because he knew them better than everyone else. On the other hand, when, at the age of fourteen, he was often sent out of the room for helping students, he defied the teacher because he knew that he was right to help his peers: 'They were not going to get the help they needed from the teacher'. Confidence resides in his habitual success with maths. Another of this group, Tom, recalls his success at the age of six or eight when his father was teaching him how to do fractions:

The confidence in manipulating the mathematics processes themselves was high in this group, and probably contributes to their agreement that it is not the construction of mathematics that is the problem, but the teaching of it. However, several of them agreed with the assertion that it is not the teacher that makes a difference, but how students react to the teaching:

This returns us to the opening thesis that the construction of relationships in the classroom and the learning of mathematics are intimately related through their impact on self-identity as a learner, whether in the grade one classroom or at university. Personality is only one dimension of the responses students make as they learn. The high levels of confidence of the university faculty are also reflected in the language used in the narratives, which, for the faculty group indicated little recall of negative memories and emotions. To give an overall picture of the emotions expressed in the 32 narratives, language from those narratives expressing and negative emotions is categorised in the tables below.

Positive emotions Negative emotions

Table 1 University Faculty

Positive emotions Negative emotions

Table 2 Graduate Diploma students

Positive emotions Negative emotions

Table 3 Primary and Junior Primary students

Altogether, only ten emotions are recorded in the narratives written by the university teachers in Table 1, while over thirty are recorded in Tables 2 and 3. For university teachers, only three of these are negative, while the graduate students record fourteen and the undergraduates nineteen, indicating a far higher incidence of negative emotions in the past experiences of learning mathematics for those who will be teaching the subject in primary and high schools. With the potential of past emotion to invade subsequent years of experience and the present through memory, we bring our emotion-laden experiences with us as learners, practitioners and teachers. In many instances in mathematics classrooms, where right and wrong are so strongly rewarded and punished from grade one, few students experience safety in learning.

Another significant dimension emerging from the study relates to perceptions of the nature of mathematics. These perceptions differed between school and university teachers, particularly relating to experiences where success through rote learning and the reproduction of process rather than understanding had been rewarded. Few students in this study were sufficiently confident of their control of the rules to play with numbers and know the satisfaction of alternative and elegant pathways in mathematics. Whether this sets up differences in the way the subject is taught at school and at university by the teachers and prospective teachers in this study is the subject of a forthcoming paper.

From this analysis, we argue that emotion is central in learning and in the disposition to learn. Affect is not just a product of learning, it is constitutive of learning. One model of learning in which emotion is given a place is Biggs' 3P Model of Learning (Biggs 1987). Biggs depicts a linear relationship between Presage, Process and Product - what is brought into the learning situation, what happens in it, and the outcomes. Affect is included only under Product, and is described as relating 'to how students feel about their learning, efficacy beliefs being one example' (pp 13, 14). 'Affective' is included with the categories 'quantitative, qualitative and institutional outcomes'. As the narratives show, emotions are not outputs; past emotions and memories are experienced powerfully in the present, and are ongoing in the meaning we make of our experiences, and in the maintenance of self-esteem. '[W]e always come up against earlier events in later ones, not as matter that has been fully formed and pushed aside, but absolutely present and alive'. They are not merely the product of individual personalities and experiences; they are constantly constructed in social settings that depend on interpersonal relationships of power and control in institutional settings. In the examples from memories of mathematics learning, we have illustrated that confidence is constructed through pride and self-esteem, but diminished through shame in the every day practices of teaching. Emotion is constitutive of learning, and as such, merits further consideration in the development of learning theory, and in the construction of learning and teaching practices in the classroom.

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