Understanding mathematics anxiety: loss aversion and student engagement

Abstract

This paper applies the well-known cognitive bias of loss aversion from behavioural economics to student decisions over engagement with mathematically demanding coursework. This bias is shown to predict behaviour that is consistent with mathematics anxiety in a dynamic model of student engagement. It is shown that these forces can imply polarization in student outcomes with some students chronically disengaged in a low-attainment equilibrium, in the absence of any countervailing pedagogic interventions. However, the model illustrates that chronic disengagement is not necessarily equivalent to chronic apathy. Rather, students for whom the short-term cost of failure looms large are shown to be at heightened risk of disengagement. The model is used to understand and evaluate various elements of the mathematics anxiety literature including the role of formative assessment, the fixed and growth mindset models, the efficacy of task-oriented achievement goals, the cognitive interference and motivation enhancement models of test anxiety, the provision of remedial classes and technology-enhanced solutions to learning and assessment.

1 Introduction

Mathematics anxiety has rightly received the scholarly attention of pedagogues for almost 70 years (Aiken, 1974; Betz, 1978; Dutton, 1954; Gough, 1954; and Richardson & Suin, 1972 are some notable early contributions). García González & Sierra (2020) define mathematics anxiety as ‘a set of negative emotions, including fear, disappointment, panic, stress, shame and anger reflected in a visible state of discomfort in response to situations involving mathematical tasks’.¹ The phenomenon has been documented in students over a variety of subjects and across the spectrum of education levels (Hembree, 1990), and even among some groups of teachers (for example by García González & Sierra, 2020; and Harper & Daane, 1998). In the context of higher education, the issue of mathematics under-preparedness² has been a particular concern, especially as it relates to the transition from high school to university on a range of degrees (such as economics, finance, geography and engineering) that do not specialize in mathematics but are nevertheless mathematically demanding (Feudel & Dietz, 2019; McAlinden & Noyes, 2019; Hodgen et al., 2020).

Parallel to these developments, a hitherto unrelated literature has studied biases and inconsistencies in people’s responses to risky prospects following Kahneman & Tversky’s (1979) seminal contribution on prospect theory. The current paper uses loss aversion, a specific component of prospect theory, to deepen our understanding of mathematics anxiety. Put simply, loss aversion is the empirical observation that ‘Losses hurt about twice as much as gains make us feel good’ (Thaler, 2000, p. 137). This excess sensitivity to losses has been shown to induce important and pervasive biases in decision-making in areas as diverse as pension contributions, organ donation, tax payment, consumer decisions and disease prevention behaviour (O’Keefe & Jensen, 2007; Behavioural Insights Team, 2012; Cribb & Emmerson, 2016). The evidence base for loss aversion has been built largely on surveys of student populations from all over the world: Kahneman & Tversky’s (1979) paper was based on surveys of Israeli, Swedish and American university students and a single special issue of the Quarterly Journal of Economics in memory of Amos Tversky contained papers based on surveys of students from the Netherlands, the UK and the USA (Bateman et al., 1997; Gneezy & Potters, 1997; Thaler et al., 1997). For the interested reader, Rabin (2003), Rabin & Thaler (2001), and Thaler (2000), celebrate the rich contributions of prospect theory to our understanding of human decision-making, while Barberis (2013) provides a more recent assessment of this literature.

The main contribution of the present paper is to serve as a bridge between these literatures and to apply insights from loss aversion to further the pedagogic understanding of mathematics anxiety. To this end, the paper identifies the sharply dichotomous nature of success and failure as a key feature that distinguishes a typical mathematics exercise from other, more discursive, forms of university proficiency.³ A student typically knows within minutes of starting a mathematically intensive exercise in economics, finance, engineering or chemistry whether or not the attempt will be successful. In the absence of supportive pedagogies, students are likely to perceive the threat of failure to complete an exercise to be hedonically costly, and the promise of success to be a hedonic boon. As such, undertaking such an exercise implies a gamble over a student’s short-term perceived well-being.⁴ In these circumstances, prospect theory predicts that the decision-making processes of students may be subject to loss aversion.

The paper presents a simplified dynamic model of student engagement, which predicts that students who have initially low levels of mathematics preparedness are at heightened risk of mathematics anxiety and disengagement in highly numerate university degrees. In the model, anxiety and disengagement are consequences of negative framing of the threat of failure and low mathematics preparedness.⁵ In the absence of any countervailing pedagogic intervention, these forces are shown to imply polarization in student outcomes with some students sustaining engagement in a high-attainment equilibrium, whereas others are left persistently disengaged in a low-attainment equilibrium. These predictions are of particular concern given the substantial evidence base linking gender, ethnicity and class with heightened risk of disengagement in mathematics education (Mendick, 2005; Lim, 2008; Hottinger, 2016). Such concerns are amplified in the current landscape of higher education with larger and more heterogeneous student intakes (Department for Education, 2018; Commons Library Briefing, 2019), where a substantial proportion of students on mathematically demanding programmes hold relatively modest incoming mathematics qualifications (Hodgen, Atkins & Tomei, 2020).

The paper then turns to harnessing insights from the loss-aversion–based theory of mathematics anxiety developed in Section 2 to critically evaluate several remedial measures that have been proposed in the mathematics anxiety literature. In view of the theory developed here, the efficacy of potential interventions is likely to depend on the extent to which they are successful in reframing the perception of failure from a negative hedonic domain to a neutral or positive hedonic domain. Specifically, interventions that mitigate the perceived cost of making a mistake, such as efforts to cast formative assessments as a ‘safe space’ for students to learn how to improve their work (in the spirit of Sadler, 1989), to re-focus student ‘achievement goals’ toward building individual competence rather than peer group comparisons (as explained in Nicholls, 1984) and to foster a ‘growth mindset’ among mathematics learners (as proposed in Dweck, 2006) are found to be potentially effective responses because each intervention adopts at its heart a positive reframing of the threat of failure. The paper goes on to explore how a loss-aversion–based theory of mathematics anxiety can improve our understanding of the cognitive interference and motivation enhancement models of test anxiety, and the interrelationships between the two (Sarason, 1984; Hembree, 1988; Seipp, 1991; Cassady & Johnson, 2002  inter alia). A traditional response to heterogeneous ability cohorts is to offer remedial support to weaker students, although such support has been less than universally effective (Bahr, 2008; Lagerlöf & Seltzer, 2009; Di Pietro, 2011). Although a naïve interpretation of the model developed here provides some justification for remedial classroom support, a more careful interpretation reveals why this intervention may be of only limited practical benefit. Finally, the model is used to evaluate the effectiveness of technology-enhanced learning tools such as supplementary online resources and adaptive learning technologies that can be used to counteract some of these forces.

The remainder of the paper is organized as follows. Section 2 develops a simplified, dynamic model of mathematics anxiety and makes predictions about student engagement and attainment under loss aversion. Section 3 discusses the implications of these findings for a range of themes raised within the mathematics anxiety literature, while Section 4 concludes.

2 Mathematics anxiety under loss aversion

2.1 Ability, engagement and anxiety

The pedagogic function of formative assessment is to provide students with an opportunity to identify and rectify gaps in their learning without the potentially adverse consequences of summative assessments (Sadler, 1989). In theory then, failure to correctly complete a formative assessment should be costless, whereas the hedonic boon associated with ‘getting it right’ might yield a positive payoff, normalized here to equal 1. We may think of a student undertaking a mathematically intensive exercise as experiencing the following utilities:

Now suppose that for a given student and a given exercise there is a probability p that the attempt ends in success and (1-p) that it ends in failure. Then the expected utility of attempting the exercise is given by:

In principle, the decision of whether or not to engage with an exercise should fully internalize the long-term implications of engagement or disengagement on module and degree outcomes. However, a variety of studies have found that people, including the student samples on which the studies are based, suffer from ‘present bias’, where they systematically underweigh the long-term costs and benefits of their decisions in favour of the present (Gneezy & Potters, 1997; Takeuchi, 2011; Thaler et al., 1997 among many others). Although there are varieties of conceptualizations of present bias—including hyperbolic discounting (Loewenstein & Thaler, 1989 provide a useful summary) and models of willpower and personal rules (see for example Ainslie & Haslam, 1992; or Bénabou & Tirole, 2004)—and each of these is likely to interact with the forces studied here in interesting and nuanced ways, the present paper refrains from considering these in detail so as to maintain a sharp focus on the implications of loss aversion. Therefore, the paper starts by assuming that students are perfectly biased toward the present, or equivalently, that they fully ignore the long-term implications of their actions.⁶ Thus, the outside option or the utility of not attempting to engage with an assignment is 0:

Comparing equation (1) with equation (2) we observe that in theory, costless failure can incentivize students to engage with formative assessments, even when their decision-making is fully biased toward the present.

In practice, however, other factors may cause the hedonic effect of a failed attempt to complete an exercise to take on a more negative character. The assertion that an individual’s performance in a particular context can elicit an emotional response from that individual, with good performance eliciting positive emotions and poor performance eliciting negative ones, has strong foundations in the established psychology literature on appraisal theory (Arnold, 1960 and Lazarus, 1991 are notable early contributions, whereas Ellsworth, 2013 provides a more recent review). Achievement goal theories (Nicholls, 1984; Dweck, 1986; Elliot, 1999) draw a helpful distinction between ego-oriented goals where demonstrating superior performance in relation to others is emphasized, and task-oriented goals where working toward individual mastery of a given task is emphasized.⁷ For students with ego-oriented goals, failure is likely to imply a perceived loss of relative status. Failure may also be costly in a classroom setting, if exercises are conducted publicly, because it may cause students to lose standing among their peers. In addition, students may interpret failure to complete a formative exercise as a signal that they are likely to perform poorly in summative assessments, an association that may trigger test anxiety. These problems may be exacerbated if students treat some level of proficiency as a reference point and are sensitive to shortfalls thereof, as would be expected under prospect theory (Kahneman & Tversky, 1979). Thus, the implicit hedonic gamble that a student takes on when beginning a mathematical task may have payoffs defined over both the positive and negative domains so that the standard prospect theory payoffs, where ‘losses hurt about twice as much as gains make us feel good’ (Thaler, 2000, p. 137), applies as follows:

As above, chance chooses with probability p if a given attempt ends in success. I now focus on ability, a, and engagement, e, as the two key determinants of this probability of success, p.

There is mature literature in educational psychology on the determinants and correlates of mathematics ability. Some early contributions initially found that measures of mathematics ability were very closely correlated and potentially jointly determined with general intelligence (Hughes, 1928; Wilson, 1933). Subsequent studies identified a cluster of competences (arithmetic, numeracy and spatial reasoning) that are uniquely mathematical and so distinguish mathematics ability from general intelligence (Barakat, 1951; Wrigley, 1958 among others). In economic theory by contrast, ability has been defined in relatively broad terms, for example by Spence (1973) as ‘the productive capabilities of an individual’. Because the present paper builds a theory of mathematics anxiety in the economics tradition, it adopts a relatively broad definition of ability as the degree of skills, knowledge and understanding possessed by a student that facilitates mathematics proficiency at the university level. Here, this includes not only technical subject knowledge of arithmetic, geometry, algebra and statistics, but also attributes such as perseverance and resilience, which improve the probability of success in mathematical tasks.⁸

For simplicity, I model engagement as a dichotomous variable taking on a value of 1 if a student decides to attempt a problem, and 0 otherwise. The simplest functional form that allows for the probability of successfully completing an exercise to be increasing in ability and engagement is the unweighted average of these two arguments. Normalizing initial ability, to lie over the open interval between 0 and 1 ensures that the resulting probabilities are well behaved.⁹ Thus:

This simple structure is already sufficient to yield some preliminary insights, which are summarized in Fig. 1. If failure is costless then the expected utility or payoff to engaging with an exercise is increasing in ability but remains non-negative even for the lowest ability students. This is illustrated by the dashed line. This line is computed by substituting equation (5) and the payoffs in (1) into equation (2). Thus, costless failure and potential benefits from success in theory can incentivize engagement with formative assessments across the distribution of student abilities.

The solid line, by contrast, represents the expected utility of attempting a problem where failure is costly and students are loss-averse, that is, where the payoffs in (4) and equation (5) are substituted into (2). Here, there is a critical value of ability below which the expected utility of engagement is negative: when students perceive failure in mathematically demanding exercises to be costly—perhaps because they find it demotivating to fall behind their peers, or perhaps because they are not confident that the lecturer has offered a tractable problem at their level of mathematics knowledge—formative assessments may only be effective in eliciting engagement from students of relatively high ability because the threat of failure leads the remaining students to disengage. The excess sensitivity to losses predicted by prospect theory amplifies the negative payoff to failure and so biases student decisions toward disengagement.

The predicted behaviour of relatively low-ability students facing the prospect of costly failure above is entirely consistent with a range of conceptions of mathematics anxiety from the literature. In 1954, Sister Mary Gough, writing about what she calls ‘mathemaphobia’ among her students, observes that ‘without exception, their fear was coupled with the thought of failure’ (Gough, 1954). Measures of mathematics anxiety as implemented, for example, in the scales of attitude toward mathematics proposed by Aiken (1974) include statements such as ‘mathematics makes me feel uncomfortable and nervous’, and ‘it is my most dreaded subject’. More recently, García González & Sierra (2020) have defined mathematics anxiety as ‘a set of negative emotions, including fear, disappointment, panic, stress, shame and anger reflected in a visible state of discomfort in response to situations involving mathematical tasks’ (as quoted in the introduction). Interestingly, this last paper assumes a causal link from a lack of requisite mathematics knowledge to the experience of mathematics anxiety. Informed by these contributions, the present paper defines mathematics anxiety as the psychologically safeguarding impulse to disengage from mathematically demanding tasks that is brought about by fear of the threat of failure among students with low mathematics preparedness relative to their present needs. Thus, the model above demonstrates how adverse pedagogies that negatively frame the threat of failure can trigger loss aversion and so bias student decision-making toward disengagement, consistent with these definitions of mathematics anxiety.

To the extent that a key function of higher education is to serve as a signalling device for ability (as formalized by Spence, 1973), such patterns of engagement are not necessarily problematic: an educator could set exercises of a level of difficulty so as to allow students above some specified ability threshold to self-select into engaging with assessments. Clearly though, labour market signalling is not the only purpose of higher education. Education, particularly mathematics education, is a core input to human capital, to understanding the world around us and to graduate employment prospects. Universities seek to imbue their students with substantive academic development and growth in such abilities. It is to these more ambitious objectives of higher education that the anxiety and disengagement elicited by loss aversion pose a substantive threat. For students with low levels of incoming mathematics preparedness, the fear of failure, intensified by loss aversion, may prevent them from participating effectively in the learning process.

2.2 Dynamics: Polarization in engagement

The paper will now consider some aspects of the dynamics of engagement with loss-averse students. A key stylized fact that underpins much of the current delivery of university education is that sustained student engagement is an important input to academic proficiency (Lee, 2014 provides recent empirical evidence). This observation is formalized by assuming that ability in a particular period depends not only on the level of ability in the preceding period, but also on whether or not the student chose to engage with the academic material so that:

where the unweighted average and the single lag are again chosen to favour clarity and simplicity of exposition over complexity.¹⁰ The solid arrows in Fig. 2 present the dynamics that emerge from combining equation (6) with the behaviour predicted for loss-averse students by the solid line in Fig. 1.

In the model, a student who had an ability level greater than 1/3 in the preceding period would have chosen to engage with exercises in that period because the expected utility from doing so would have been positive and thus greater than that of the outside option, 0. Thus, their ability this period is given by |$\frac{a_{t-1}+1}{2}$|⁠, which is greater than |${a}_{t-1}$| so that over time, the student’s ability evolves toward the north-easterly corner of the diagram. By contrast, a student who had an ability level below 1/3 will not have engaged last period so that ability this period will be |$\frac{a_{t-1}}{2}$|⁠, which is less than ability last period. This depreciation in ability should be interpreted as occurring relative to the difficulty of academic content, which typically increases as students progress through a course of study. The predicted evolution of ability of students with low endowments of mathematics preparedness is represented in Fig. 2 by the solid arrow that evolves toward the south-westerly direction. Thus, in the absence of any shocks or countervailing pedagogic intervention, this model predicts that university-level mathematics preparedness and engagement can interact with one another to create virtuous and vicious cycles that if left unchecked can lead to polarization in classroom outcomes. This result is complementary to self-efficacy theory from psychology, which points to an individual’s previous performance outcomes in a particular task as the most powerful driver of that person’s belief in their own ability to succeed at that task going forward (Bandura, 1977). According to this theory, these beliefs feed into the level of effort expended, and how long that effort is sustained in the face of obstacles and so can significantly affect future outcomes.

2.3 Engagement transitions

Of course, a number of other forces may also affect whether or not a student chooses to engage in a particular period. Factors such as illness, family emergencies or relationship events can cause a historically engaged student to disengage for a particular period. Of particular concern here is the rapidly increasing number of starters in UK higher education declaring a mental health condition (UCAS, 2021). Conversely, a particularly inspirational teaching session, a helpful classmate or a temporary reprieve from some other form of life stress might cause a chronically disengaged student to temporarily engage.

To understand if short-term ‘shocks’ to engagement can have long-term effects on attainment, recall that in the model, students fall into a vicious cycle of persistent disengagement if ability dips below 1/3. Equation (6) assumes that next period’s ability is the unweighted average of this period’s ability and engagement. From this, a straightforward computation yields the result that that if current ability is below 2/3, disengagement today will imply that ability tomorrow is below 1/3. Therefore, all students with ability between 1/3 and 2/3 are ‘at risk’ of persistent disengagement if exposed to a one-off, adverse shock to engagement. The dotted arrow in Fig. 2 highlights these ‘at-risk’ students. Of course, these exact numbers are merely illustrative in nature and cannot be used as literal thresholds. Nevertheless, they serve to make the important point that at any given time, the set of students who are at risk of falling into the vicious cycle is in principle non-empty.

Conversely, the dashed arrows in Fig. 2 indicate the set of students who are ‘remediable’. These students are currently in a vicious cycle of persistent low engagement, but would transition to the high-ability, high-engagement trajectory if subject to a one-off, positive engagement shock. Inspection reveals that for the parameter values assumed here, a student of any ability level who engages for one period will have a subsequent level of ability greater than 1/3 and will therefore transition from the vicious cycle to the virtuous one. This admittedly optimistic feature of the current parameterization—that all it takes is one period of engagement for any low-ability student to transition to the virtuous cycle—is clearly a function of the assumed simplified lag structure and functional form. One can of course devise more complicated structures to generate a lower bound on the ability of students who are remediable with a one-off intervention, although it is not clear that the additional complexity would yield any further pedagogic insight.

3 Discussion

3.1 Positive framing: Formative assessment, achievement goals and the growth mindset

Formative assessments are designed to elicit student engagement by providing a safe space for essential academic trial and error that is necessary for students to identify how to improve their work (Sadler, 1989). The results above have shown that this function can be undermined for students with low levels of mathematics preparedness if other factors such as psychological, social or emotional forces intervene to make failure on mathematical tasks costly. In such cases, loss aversion can cause students to become overly sensitive to the costs of failure and so disengage from challenging mathematics content in a way that is consistent with mathematics anxiety. In the behavioural economics literature, framing has become a standard response to improving outcomes in areas subject to loss aversion as important and wide-ranging as pension contributions, tax payment, organ donation and disease prevention (O’Keefe & Jensen, 2007; Behavioural Insights Team, 2012; Cribb & Emmerson, 2016). To take a concrete example, Cribb & Emmerson (2016) show that when employees are asked to opt in to a pension, contributions become framed as losses from salary, and relatively few employees choose to contribute. On the other hand, if automatic enrolment is the default and employees are instead offered the option to opt out, enrolment increases by 37 percentage points. The literature has explained this apparently contradictory behaviour by observing that one frame contains an option that is defined over the negative domain (the deduction from salary) and so triggers loss aversion, whereas the other is defined over a non-negative domain (the already reduced salary is the status quo) and so does not. Thus, framing a situation in terms of gains does not remove the adverse prospect (those making pension contributions receive the same take-home pay regardless of what the default is), but rather it changes how the prospect is perceived. This established literature on gain-framed behavioural interventions has clear implications for good teaching practice.

Teachers who desire to ameliorate mathematics anxiety among their students should take steps to gain-frame the risk of failure and so rehabilitate student sensitivity to the threat of failure. If effective, such a strategy would transition students from the ‘costly failure’ to the ‘costless failure’ line in Fig. 1, implying an alleviation of mathematics anxiety, an increase in engagement and thus improved subsequent attainment. Positive pedagogies developed from both achievement goal theory (Nicholls, 1984; Dweck, 1986; Elliot, 1999) and the related growth mindset theory (Boaler, 2013, 2016; Dweck, 2006; Suh et al. 2011; and Sun, 2018) have the potential to gain-frame the threat of failure and so, from the perspective of the model developed above, are likely to find success in alleviating mathematics anxiety, as explained below.

Achievement goal theory distinguishes between ego-oriented goals, which prioritize demonstrating superior performance relative to others, and task-oriented goals, which prioritize attaining improvement and eventually, mastery over a given task. With ego-oriented goals, the threat of failure for an individual is likely to trigger a negative psychological frame because failure will cause a student’s appraisal of his or her own ability to decrease relative to the peer group. By contrast, task-oriented goals permit the threat of failure to be viewed through a non-negative frame as a step toward building competence, or even mastery. Thus, positive pedagogies that deliberately foster task-involved goals are likely to mitigate mathematics anxiety.

The theory of mathematics anxiety developed above is very much aligned with recent pedagogic innovations in ‘growth mindset’-oriented teaching (Dweck, 2006). Indeed, the present paper provides a complementary framework to understand why actively promoting a growth mindset in educational settings has had such a profound effect on learning (Boaler, 2013). Under the ‘fixed mindset’, which views mathematical ability as an unchanging endowment, failure to complete an exercise can only be viewed as a loss with a negative payoff. As argued above, such a frame is likely to trigger loss aversion, mathematics anxiety and student disengagement. This is particularly problematic because belief in innate ability is especially common for mathematics (Stevenson et al., 1993; Jonsson et al., 2012). However, under a growth mindset, the experience of failure is framed over a positive domain as an input to learning and a way to track growth. Thus, it is less likely to trigger loss aversion or mathematics anxiety, and more likely to elicit engagement. As such, the emerging literature on how teachers can best promote a growth mindset among mathematics learners (Boaler, 2016; Suh et al., 2011; and Sun, 2018) is likely to be an important and fruitful area of research.

An emphasis on inputs to learning such as time and effort spent by students on exercises (Lee, 2014 finds evidence that such inputs have a strong relationship with attainment) may also mitigate anxiety because it shifts the focus from risky outcomes such as success and failure, which drive loss-averse behaviour and anxiety in the model, to parameters over which students exercise direct control. Assigning in-class group presentations, as is common practice in business schools, can potentially play a role in shifting the narrative of failure in mathematical tasks from that of a personal loss to the creation of a community of learners. Given the results above, such a community may be effective in part because groups are less susceptible to being undermined by individual loss aversion. However, group work can also be subject to freeriding and so may itself create disengagement if not properly managed (as outlined for example by Hansen, 2006 and Maiden & Perry, 2011).

3.2 Salience of summative assessments: Motivation enhancement or cognitive interference?

An important implicit assumption in the narrative so far has been that students are present-biased, that is, their decision-making does not appropriately account for the considerable eventual costs that disengagement is likely to entail. This was implied in Fig. 1 earlier and is illustrated in Fig. 3 by modelling the expected utility of the outside option of disengagement as zero. The paper now considers the implications of relaxing this assumption. Persistent disengagement is associated with lower attainment, heightened risk of examination failure and resits, and in the worst case, heightened risk of degree non-completion (Singh et al., 2002; Salamonson et al., 2009; Lee, 2014 among others). Students who behave rationally should internalize these substantial costs of disengagement in their decision-making. It is possible, then, that undoing present bias by increasing the salience of contributory, summative assessments may increase engagement.

The assumption that students are fully present-biased is relaxed in Fig. 3, where the ‘negative outside option’ line illustrates the case where the payoff to disengaging is perceived to be substantial and negative. Comparing this with the ‘costly failure’ line shows that even when failure is costly, students across the ability distribution have an incentive to engage because the long-term consequences of disengaging are assumed to be even more costly. This is in sharp contrast to the behaviour of students in Fig. 1 who were assumed to be not only loss-averse, but also strongly present-biased. Thus, present biased students may be most at risk of disengagement. In theory then, reducing present bias by increasing the salience of examinations and the threat of degree non-completion can be a pedagogic intervention that incentivizes engagement among at-risk students. This result is consistent with ‘the motivation enhancement model’ (Seipp, 1991; Struthers et al., 2000) from the literature on test anxiety, which posits that examination stress can increase motivation and thus engagement and outcomes. However, the Yerkes–Dodson Law from psychology states that increased arousal (pressure or stress) only improves performance to a point. In fact, the point at which increased pressure begins to detract from performance occurs at lower levels of pressure for more complex tasks (Yerkes & Dodson, 1908). Thus, inducing further stress may not improve performance for a complex task like university mathematics, especially among students already at risk of mathematics anxiety.

This latter view is borne out in the test anxiety literature by the ‘cognitive interference model’ (Sarason, 1984; Hembree, 1988; Seipp, 1991; Cassady & Johnson, 2002; Zeidner, 2007), where stress impedes cognitive function more than it enhances motivation. The balance of the empirical evidence in the test anxiety literature seems to validate this latter view. Furthermore, inducing test anxiety may exacerbate disengagement if the two forms of anxiety are mutually reinforcing. Indeed, research indicates that these two phenomena may be more closely interlinked than previously thought, with many overlapping correlates (Kazelskis et al., 2000). Thus, priming students to interpret failure on formative assessments as a signal of future failure on summative assessments may induce test anxiety and so heighten mathematics anxiety. This is illustrated in Fig. 3 by way of the ‘induced anxiety’ line, where the expected utility of failure is even more negative than on the original ‘costly failure’ line. If the perceived cost of failure is sufficiently large as to make the induced anxiety line sufficiently steep, the model can generate the cognitive interference model where a greater fraction of students chooses to disengage after the threat of examination failure is made more salient than before.

Thus, Fig. 3 provides an important tool with which to understand both the motivation enhancement and cognitive interference models of attainment and the interplay between the two.

3.3 Polarization, diverse cohorts and layered curricula

Recent research (Hodgen, Atkins & Tomei, 2020) indicates that in the UK, mathematically demanding degrees such as economics, chemistry and computer science recruit substantial fractions of students both with and without an A-level or AS-level qualification in mathematics. It is also understood that the transition from high school mathematics education to university mathematics education is an especially complex developmental step for students in both the natural and social sciences (McAlinden & Noyes, 2019). The forces analysed above suggest that in cohorts with mixed levels of incoming mathematics experience, there is a real danger that students who start their degrees at a relative disadvantage may be at heightened risk of experiencing mathematics anxiety and disengagement if high proficiency is assumed or presented as the norm by their tutors.

In mathematically demanding subjects in the social and physical sciences, it is common for specific concepts and the accompanying mathematical tools to be presented to students in layers of increasing complexity over the weeks within a learning block and over the different learning blocks of a degree programme. For example, in an economics degree, it is typical for an introductory module to present profit maximization graphically, where an intermediate module might use univariate calculus, and an advanced module might require a bordered Hessian. The dynamics of engagement analysed above suggest that there may be unintended consequences to this type of layering that may disproportionately affect students who enter these degrees with low endowments of incoming mathematics proficiency. In these circumstances, the results of Section 3.1 above suggest that the convenors of first-year principles modules need to be especially careful to frame the risk of failure over the positive domain, to encourage task-oriented goals and to facilitate a growth mindset within the classroom.

3.4 Remedial support and technology enhanced tools

Understanding mathematics anxiety as rooted in loss aversion provides some justification for the provision of remedial mathematics seminars, but also raises some important questions about this approach. In Fig. 1, if students with negative expected utility of engaging are organized into a separate group and offered less complex exercises, the probability of failure would decrease and so the expected utility of engagement might increase. Thus, in theory, supplementary remedial classes might possibly improve engagement among students with low levels of mathematics preparedness who are at risk of mathematics anxiety without impeding the learning of others. Over time, this may enable students with initially low levels of mathematics preparedness to ‘catch up’ to their peers. Empirically however, such remedial mathematics support has proven to be less than universally effective (Bahr, 2008; Lagerlöf & Seltzer, 2009; Di Pietro, 2011). This may be because remedial support does not challenge the fundamental cause of mathematics anxiety identified here: that the threat of failure is negatively framed. Obliging students to self-select into a low-capability group in a way that is easily observable to their peers might exacerbate disengagement by magnifying perceived losses rather than ameliorating them. Consider as an example a student with low mathematics preparedness and ego-oriented achievement goals. The underlying prospect that the student is averse to having exposed—the cause for their panic and anxiety—is their own current lack of ability relative to their cohort. Allocation to a remedial group might in fact reinforce adverse motivational outcomes rather than helping the student to overcome them. Thus, the theory developed here suggests that remedial classes might be successful only if accompanied by careful supporting pedagogy that explicitly addresses the underlying causes of mathematics anxiety.

An alternative may be to leverage widely used virtual learning environments to address some of the shortcomings of remedial classes. When students engage with supplementary resources online, they may be less susceptible to the barriers that impede classroom participation (these barriers are carefully documented in Fassinger, 1995 and Fassinger, 2000). For example, computerized quizzes can provide an important tool for students to identify gaps in their knowledge without having to expose these gaps to their peers or faculty, a concern that may impede participation in interactive classroom sessions. Thus, the cost of failure may be substantially reduced in virtual learning environments in comparison with physical ones. Indeed, stigma may even be reversed as proactive, voluntary engagement with online resources might enhance self-image and motivational processes rather than detracting from them.

The developing field of gamification, i.e. the use of game characteristics in non-game contexts (Borges et al., 2014 provides a review of applications in education) has the potential to take these advances even further. ‘Dynamic difficulty adjustment’ (Hunicke, 2005) is a feature of video game design that matches game difficulty to contemporaneous estimates of player ability. This is done to maximize engagement as a player improves in skill with practice. If game difficulty rises too quickly, the player can get frustrated and disengage. On the other hand, if game difficulty rises too slowly, the player can get bored and disengage. Dynamic difficulty adjustment estimates current player ability from recent game play history and scales game difficulty accordingly in real time so as to maximize engagement. This has clear and exciting implications for the design of mathematical tasks on virtual learning environments. These can be programmed to dynamically match the difficulty level of questions with the revealed current ability level of the student (as estimated, for example, using the fraction of correct answers and question difficulty in the student’s recent history). The results of this paper help to understand how such algorithms can be effective: they allow formative assessment to be kept challenging enough to enable engagement and learning, but not so challenging as to induce anxiety and disengagement.

It does have to be said, however, that these positive effects on engagement might come at the cost of circumventing a potentially important learning experience: an overreliance on simple versions of such algorithms might remove opportunities for students to learn how to cope with the threat of failure, itself an important educational outcome. Such overreliance may also detract from the social nature of learning in and around the classroom, if not properly managed. But it is not difficult to imagine ways in which such technology-enhanced learning tools can be carefully deployed to overcome these possible shortcomings. Algorithms can be tweaked to encourage perseverance and resilience, while promising tools are already available on virtual learning environments to encourage and support group- and team-based learning. Thus, as part of a balanced portfolio of learning and assessment that spans the physical, social and virtual elements of the modern university, such technology-enhanced tools may well become an important component of a multifaceted approach to mitigating the effects of loss aversion and mathematics anxiety in heterogeneous mathematically intensive courses.

4 Conclusion

This paper has leveraged insights from behavioural economics (Kahneman & Tversky, 1979; Thaler, 2000; Rabin & Thaler, 2001) to develop an understanding of mathematics anxiety. The paper observed that the sharply dichotomous nature of success and failure that is usual in mathematically intensive university assessment can imply a gamble in students’ short-term perceived well-being. When sufficient care is not taken on the part of instructors to frame the outcomes of this gamble over the positive domain, loss aversion may bias study decisions toward disengagement in behaviour patterns that are consistent with mathematics anxiety. In a simplified, dynamic model of engagement and mathematics ability, it was shown that these forces can imply polarization in student outcomes.

The paper went on to use the loss-aversion–based theory of mathematics anxiety as a framework to understand and evaluate interventions that have been proposed in the literature to counteract mathematics anxiety. Interventions that are designed to positively frame the risk of failure in learning contexts were found to be potentially effective. These include efforts to cast formative assessment as a ‘safe space’ for students to learn how to improve their work (Sadler, 1989), to foster a growth mindset in the classroom (Dweck, 2006) and to encourage task-oriented achievement goals that are resilient to the threat of failure (Nicholls, 1984). Previous literature has also asked if the salience of summative assessments can be used to foster engagement in the motivation enhancement and cognitive interference models (Sarason, 1984; Hembree, 1988; Seipp, 1991; Cassady & Johnson, 2002). The dynamics of student engagement proposed here were applied to these models and shown to offer important insights on the interrelationships between them. The model also helped to understand why the provision of remedial classes, while potentially helpful in theory, is not always effective in practice (Bahr, 2008; Lagerlöf & Seltzer, 2009; Di Pietro, 2011). Technology-enhanced learning tools such as supplementary online resources and adaptive learning technologies (as described in Borges et al., 2014) that can dynamically match question difficulty to student ability were found to be potentially useful in sustaining engagement.

Mathematics struggles with diversity. Women, ethnic minorities and underprivileged groups are at heightened risk of persistent disengagement on mathematically intensive courses of study (Mendick, 2005; Lim, 2008; Hottinger, 2016). A theory that can help to understand persistent disengagement and identify measures that might help prevent it has the potential to make mathematics classrooms more diverse and inclusive spaces. Some educators have been tempted to conclude that chronically disengaged students are chronically apathetic to their studies. The theory developed here suggests otherwise. Rather, some students may experience anxiety and disengage because they are overly affected by the threat of failure. The positive pedagogies outlined above can help to equip such students with tools to cope with the threat of failure, thus alleviating mathematics anxiety and paving the way to persistent improvements in engagement and attainment.

Funding

No funding was received to assist with the preparation of this manuscript.

Conflict of Interest

The author has no relevant financial or non-financial interest to disclose.

Data availability

Data sharing is not applicable to this study because no datasets were generated or analysed during the current study.

Acknowledgements

I thank Frank Brower, Gabriella Cagliesi, Kelly Coate, Matthew Embry, Andrew Fiss, Zahira Jaser, Farai Jena, George MacKerron, Atonu Rabbani and Barry Reilly for their helpful feedback on this work. I thank the participants of the 2022 Winter Conference of the Economics Research Group in Dhaka, Bangladesh, for their helpful comments. I also thank two anonymous referees whose comments have greatly improved this paper. Any remaining errors are my own.

Footnotes

References

Dr. C. Rashaad Shabab is a Reader in Economics and the Director of Student Experience at the University of Sussex Business School. He is an award-winning teacher and holds a PhD from Sussex with research specialisms in household inequality, income risk, lifecycle theory and migration. He has published research papers in highly regarded economics journals and has co-edited or co-authored three books including Statistics for Economics, Accounting & Business Studies, 8th Edition (with Michael Barrow, forthcoming from Pearson).