2 Literature Review
2.2 The study of affect in teaching and learning mathematics
2.2.2 The impact of emotion research
2.2.2.1 Concepts and structures of emotions
While the investigation of mathematical affect continues to date under the headings of belief, attitude, emotion and a fourth category of values that was added in 2006 by DeBellis and Goldin, from the 1990s onwards many mathematics education researchers chose to focus more narrowly on emotions and moods rather than on beliefs and attitudes as there appeared to be a high frequency of emotional expression in all types of classroom proceedings with a seemingly limitless capacity to influence student participation and attainment (Pekrun and Linnenbrink-Garcia, 2014, p.10). Definitions of emotions may be couched in either cognitive or Freudian terms but there is a general agreement that emotions are tripartite entities which comprise physiological processes that regulate the body, subjective experiences that regulate behaviour and expressive processes that regulate social coordination (Hannula, 2012; 2014, p.24). However, according to Shuman and Scherer (2014) and other theorists, emotions are even more complex and involve a cognitive ‘appraisal’ of a situation, motivational ‘action tendency’, physiological changes, a sensory motor component to effect expression and an affective component that gives rise subjective feelings. In a typical characterisation of test anxiety, for instance, five distinct emotional strands can be distinguished, including nervous, uneasy feelings (affective), worries about failing (cognitive), increased heart rate (physiological strand),
impulses to escape (motivational) and facial expressions associated with anxiety (expressive) (Pekrun and Linnenbrink-Garcia, 2014, p.2).
Emotion researchers generally view emotions as short-lived episodes which are instigated by a variety of actually-occurring, imaginary or remembered stimuli around an ‘event focus’. They are differentiated from moods, which are understood to possess similar properties to emotions, but are less intense and lack a specific referent (Shuman and Scherer, 2014, p.15). Individuals are not always aware of how they are processing emotions as cognitive ‘appraisals’ can be manifested consciously or unconsciously through diverse conceptual, schematic and sensory motor channels (Scherer, 2009). However, emotional responses cannot be regarded as simple stimulus-response reflexes as the action tendency component has been shown to exert control on the incremental steps which combine to produce an episode (Shuman and Scherer, 2014, p.17).
The collection of data in most studies of emotions is pursued through self-report questionnaires and descriptions of subjective feelings through interviews and computer- coded subjective feeling labels. However, the analysis of data can be approached through four distinct theoretical perspectives which involve differentiated conceptualisations of emotions, different measurement paradigms and different protocols for regulation (Shuman and Scherer, pp.18-24). A great deal of classroom research is predicated on basic emotion theories (Ekman, 1992; Izard, 1994, 2007) which derive from Darwin’s theories on emotions and restrict their focus to a limited number of emotions that have been shown to be necessary for survival and which are shared by humans and related species. In this perspective, an emotional episode is triggered by a stimulus and can be delineated as a sequence which consists of an emotional baseline, the stimulus, an emotional expression and a return to the baseline; the research objective is to discover the
function of an emotion. Observation-based coding systems such as the Facial Action Coding System created by Ekman et al (2002) can be used to identify expressions of happiness, anger, sadness, fear, disgust and surprise in social situations and emotional regulation is understood to be effected through changes in the stimuli to which an individual is exposed, the control of automatic action tendencies and/or the suppression of motor expressions.
Alternatively, appraisal theories identify cognitive appraisals as the triggers of an emotional episode; as it is understood that these triggers may be activated on a discrete or continuous basis, these theories avoid analyses of straightforward causes and effects in emotional sequences and postulate multiple possible pathways for emotional expression. For example, when an individual determines that an event is preventing her from reaching a highly important goal, she may appraise her potential to cope as low, feel utterly dejected and cry; but if she persuades herself that she can turn the situation around, she may become defiant and ready to act again. Researchers who adopt appraisal perspectives attempt to discover the events which lead to an emotion and assess the role of emotional expression in coping. The characterisation of subjective feelings is considered crucial as they are viewed as the integrated representation of all the changes which take place in an emotional sequence.
Psychological constructivist theories posit a concept of core affect that develops within an ever-present neurophysiological state that an individual can consciously access through an appreciation of the valence and arousal of feelings. In these frameworks, the emotional meta-experience, which can be defined as the conceptualisation of one’s perceived subjective state in terms of its discrete emotional components, is the focus of study and emotions are assessed typically in terms of valence, a measure of the positivity
or negativity of their qualitative states, or activation, a reference to its physiological impact, i.e. whether it activates in terms of excitement or deactivates in terms of relaxation. Wundt (1897) was an early exponent of this type of theory, which tends to contextualise emotional meta-experiences within cultural and linguistic influences.
Finally, non-linear dynamic system theories (Fogel et al 1992; Camras, 2011) describe emotions as ‘attractor states’ which result from the propagation of positive and negative emotional feedback loops which are instigated by any component of an emotional episode which may then draw other emotion components into an attractor state. For example, the act of smiling may stimulate the feelings, appraisals and physiological changes associated with happiness (Sherer and Shuman, 2014, p.23). Attractor states are understood to be very difficult to dissipate or dislodge and research is often concerned with describing distinctive attractor states in different social contexts.
2.2.2.2 Recent investigations of emotions linked to mathematics
In his review of the recent investigations of emotions linked to the teaching and learning of mathematics, Goldin (2014) confirms that research has continued to be carried out in the domains recognised by mathematics education researchers, including the affective factors surrounding obstacles in problem-solving (Goldin, 2000; Op’t Eynde et al, 2007; D’Mello and Graesser, 2014), emotions linked to the teaching and learning of mathematical concepts and procedures and the resulting cognitive restructuring and reinterpretation of existing knowledge (Lesh and Lamon, 1992; Firestone et al, 2004; Mora, 2011), mathematical anxiety (see below) and the mathematics-specific beliefs that can meet emotional needs and/or provide defences from pain (Handal, 2003; Leder et al 2002; Maasz and Schloglmann, 2009) (p.395). Now, however, studies tend to be either grouped according the educational function or the focus of the emotion(s) beingexamined, such as epistemic emotions, which emanate from the cognitive processes which ensue during the completion of a mathematical task and social emotions, which refer to the goals and outcomes associated with classroom-based teaching and learning. The other main distinction in educational research is to categorise emotions as either state emotions, which can be observed or inferred in the moment, and trait emotions, which are differentiated from moods but which are longer standing to their state counterparts, and are often used to describe how one feels at any given moment (Hannula, 2014).
2.2.2.3 Investigations of trait emotions
Goldin reports that, due to its prevalence in all areas of affect associated with mathematics, anxiety is by far the most widely studied trait emotion (p.398). Although Tobias (1993) uses interviews to investigate the phenomenon qualitatively, most investigations continue to be undertaken through large-scale studies which involve questionnaires and quantitative analysis of the information collected through the questionnaires. Beasley et al (2001) catalogue several anxiety measure scales in use, including the Mathematics Anxiety Questionnaire (MAQ) devised by Wigfield and Meece (1988), the Mathematics Anxiety Rating Scale (MARS), which exists in several formats, and the Mathematics Anxiety Scale for Children (MASC), which is essentially a shortened version of MARS. However, while researchers generally agree that mathematical anxiety increases with age and only becomes a significant factor in adolescence, at which point it appears to be more widely reported by females (Dowker, 2012), it has not yet been determined if mathematics anxiety is a single emotion or a combination of emotional components (p.400). While Beasley et al (2001) use the MASC with 278 pupils in Grade 6 and conclude that mathematics anxiety is unidimensional, Ho et al (2000), in a cross-national study of 671 Grade 6 students in China, Taiwan and the
nervousness, tension, dread and fear, and its cognitive components, which include the worry generated by negative expectations and self-deprecatory thoughts. Lee (2009) investigates the correlations of mathematics anxiety with test scores and conducts factor analyses of the self-constructs of mathematics self-concept, mathematics self-efficacy and mathematics anxiety using 2003 Program for International Student Assessment (PISA) data to support the hypothesis that mathematics anxiety is a separate, empirically distinguishable construct.
Researchers are also still divided as to the nature of the relationship between mathematics anxiety and mathematical achievement. A meta-analysis by Ma (1999) of 26 published and unpublished studies of students in Grades 5 to 12 finds that published studies indicate a statistically significant weaker relationship between mathematical anxiety and achievement than unpublished studies.
Investigations of positive trait emotions have been undertaken much less frequently. Data collected in the 2003 PISA indicate that in many countries students’ interest and enjoyment of mathematics are not correlated with mathematical performance. However, as in the investigations of mathematical anxiety, these studies are limited by their tendencies to present characteristic responses obtained through questionnaires and to neglect the emotional expressions that may be subsumed or overridden in the definitions of trait emotions.
2.2.2.4 Investigation of state emotions
Investigations of ‘in-the-moment’ state emotions tend to be embedded in comprehensive analyses of mathematical problem-solving, self-regulation and motivation and aim to formulate detailed descriptions of emotional episodes that can be substantiated by
theoretical conjectures engendered by theories of emotion (p.402). Most of this research eschews quantitative methods in favour of videotaped observation, open-ended tasks and retrospective interviews but the exact methodology used is dependent upon the nature of the research questions and the theoretical framework adopted by the researcher.
The largest category of state emotion studies uses case studies and observations of small classroom groups to analyse the emotions linked to mathematical problem solving. Walen and Williams (2002) observe two adult women and one Grade 3 pupil in the context of timed assessments and find that, while the assessments do not evoke mathematical anxiety in the strict sense of the term, they do instigate feelings of fear and distress. Lewis (2012) observes ‘Helen’, a college student highly disaffected by mathematics, experience hatred and anger interspersed with positive emotions engendered by her participation in group activities and her ability to help her colleagues and determines that a student’s relationship with mathematics is dependent upon complex motivational and emotional sequences. Nardi and Steward (2003) undertake a one-year investigation of three Year 9 middle ability classes in England using classroom observation, group interviews with pupils and coding procedures that reflect the frequency of pupil statements, many of which indicate a dissatisfaction with mathematics, a disconnect in conceptual procedural understanding, and an engagement with the subject that is maintained through obligation and pressure rather than enjoyment. DeBellis and Goldin (2006) focus on the interaction of emotion and cognition in pupils in Grades 4 to 6 as they performed videotaped tasks by inferring emotions from the children’s statements, interjections and tone of voice using Izard’s (1983) Maximally Discriminative Facial Movement Coding Scheme (MAX). Various affective pathways, i.e. sequences of emotions interspersed with mathematical cognition, are reported, including the transformation of a boy’s emotional expression as he encounters a mathematical insight which is inconsistent with his prior expectations.
Malmivuori (2006) combines a data analysis of the data sets generated by Finnish secondary students’ questionnaire responses and their mathematic achievements and the observation of ‘Frank’ as he solves a problem to highlight the role of affect and self- confidence as an essential aspect of students’ cognitive processes and self-regulatory patterns during mathematical activities. Op’t Eynde et al (2007) observe the emotional responses of 16 students from 14 junior high schools as they engage in problem-solving exercises and collect additional data through interviews, facial action coding and responses to the Mathematics-Related Beliefs Questionnaire to demonstrate that emotional expressions adjust themselves to continuous changes in the social context and at times appear to follow patterns. Heyd-Metzuyanim and Sfard (2012) study a small Year 7 group working on unfamiliar problems involving fractions and conclude, after coding participant statements, mapping moment-by-moment emotional sequences and capturing the ‘flows’ of emotional expression (ibid p.404) that pupils’ emotional experiences are highly intense and contextualised within identity struggles.
Goldin reports that various research teams at Rutgers University are studying the ‘engagement structures’ which constitute recurring patterns ‘behavioural/affective/social constellations’ in individuals by collecting a variety of qualitative data from small groups of middle school students and subjecting it to analysis through the four lenses of flow of mathematical ideas, the inference of strong emotion in key affective events, social interactions among students and significant teacher interventions (Epstein et al, 2007; Alston et al, 2007; Goldin et al, 2007, 2011; Schorr et al, 2010). As many as ten interacting strands have thus far been identified, including sequences of emotions linked to expressions of affect, patterns of behaviour for fulfilling designated motivations, meta- affect, meanings encoded in various emotions, inner-speech and self-talk associated with
emotion, and interactions of emotions with beliefs, attitudes and longer-term traits. Examples of engagement structures which have been delineated to date include “Look How Smart I Am”, “Get the Job Done”, “It’s Not Fair” and “Let Me Teach You”. However, the researchers are still struggling to understand which specific emotional components govern students’ overall engagement or disengagement with mathematics and Goldin notes that investigations of state emotions are limited by the unpredictability of emotional expression, the inability of researchers to replicate the classroom situations in which patterns of emotion are first observed to occur, the unreliability of the inference of emotion and issues regarding generalisability (pp.405-406).