Mathematics anxiety what is known and what is still missing 1st edition irene c mammarella editor

Mathematics Anxiety What Is Known and What is Still Missing 1st Edition

Irene C. Mammarella (Editor)

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MATHEMATICS ANXIETY

Feelings of apprehension and fear brought on by mathematical performance can affect correct mathematical application and can influence the achievement and future paths of individuals affected by it. In recent years, math anxiety has become a subject of increasing interest in both educational and clinical settings. This ground-breaking collection presents theoretical, educational and psychophysiological perspectives on the widespread phenomenon of mathematics anxiety.

Featuring contributions from leading international researchers, Mathematics Anxiety challenges preconceptions and clarifies several crucial areas of research, such as the distinction between math anxiety from other forms of anxiety (i.e., general or test anxiety); the ways in which math anxiety has been assessed (e.g. throughout self-report questionnaires or psychophysiological measures); the need to clarify the direction of the relationship between math anxiety and mathematics achievement (which causes which).

Offering a re-evaluation of the negative connotations usually associated with math anxiety and prompting avenues for future research, this book will be invaluable to academics and students in the psychological and educational sciences, as well as teachers working with students who are struggling with math anxiety.

Irene C. Mammarella is Associate Professor at the University of Padua, Italy. Her research interests include the role of working memory and emotional aspects in specific learning disorders, and other neurodevelopmental disorders. She is the cofounder of a clinical university centre for neurodevelopmental disorders (LabDA srl) and coauthored the book Nonverbal learning disabilities (Guilford Press, 2016).

Sara Caviola is a Lecturer in Developmental Psychology, at the School of Psychology, University of Leeds, UK. She won a Marie Skłodowska-Curie fellowship and spent two years at the Centre for Neuroscience in Education, University of Cambridge. Her main interests include analyses of cognitive and emotional underpinnings of mathematical cognition, in both children and adult populations.

Ann Dowker is University Research Lecturer at the Department of Experimental Psychology, Oxford University, UK. She has edited and coedited several books, and is the author of Individual Differences in Arithmetic: Implications for Psychology, Neuroscience and Education (Psychology Press, 2005; second edition to be published in 2019). She is the lead researcher on the Catch Up Numeracy intervention project.

MATHEMATICS ANXIETY

What is Known and What is Still to be Understood

Edited by Irene C. Mammarella, Sara Caviola and Ann Dowker

First published 2019 by Routledge

2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN and by Routledge 52 Vanderbilt Avenue, New York, NY 10017

Routledge is an imprint of the Taylor & Francis Group, an informa business © 2019 selection and editorial matter, Irene C. Mammarella, Sara Caviola and Ann Dowker; individual chapters, the contributors

The right of Irene C. Mammarella, Sara Caviola and Ann Dowker to be identified as the authors of the editorial material, and of the authors for their individual chapters, has been asserted in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988.

All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers.

Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe.

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A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data

A catalog record for this book has been requested

ISBN: 978-0-367-19033-0 (hbk)

ISBN: 978-0-367-19039-2 (pbk)

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Mark H. Ashcraft 2 Different ways to measure math anxiety

Krzysztof Cipora, Christina Artemenko and Hans-Christoph Nuerk

Chiara Avancini and Dénes Szűcs

Ann Dowker

Dominic Petronzi, Paul Staples, David Sheffield and Thomas Hunt

Maria Chiara Passolunghi, Marija Živković and Sandra Pellizzoni

7 The different involvement of working memory in math and test anxiety

Ee Lynn Ng and Kerry Lee

8 Math anxiety in children with and without mathematical difficulties: the role of gender and genetic factors

Sara Caviola, Irene C. Mammarella and Yulia Kovas

9 Probing the nature of deficits in math anxiety: drawing connections between attention and numerical cognition 156

Orly Rubinsten, Hili Eidlin Levy and Lital Daches Cohen

10 Gender stereotypes, anxiety, and math outcomes in adults and children

Carlo Tomasetto

11 The role of parents’ and teachers’ math anxiety in children’s math learning and attitudes

Julianne B. Herts, Sian L. Beilock and Susan C. Levine

Irene C. Mammarella, Sara Caviola and Ann Dowker

PREFACE

During a cold December afternoon, we were discussing a research project related to mathematical learning and related emotional difficulties, drinking cups of tea and coffee, when we realized the absence of a handbook entirely dedicated to this topic. Almost simultaneously we looked at each other (Irene and Sara) and had the same insight: why not try to edit a book on this topic?

Once we realized that we had decided to follow through on the idea, we found ourselves lost (deep) in conversation, trying to list, organize and select all the fundamental topics we thought were worth including in the book. Thus, we wrote to Ann, who possesses tremendous expertise in this field, and asked her to join us. Happily, she immediately embraced the project. The final structure of this book followed on from a symposium that we organized for the first international conference on Mathematical Cognition and Learning Society held in Oxford (Sara and Ann) in April 2018, and includes chapters from leading researchers in psychology, neuroscience and education from all over the world. There is a shared understanding that learning mathematics involves a complex interplay of cognitive, motivational and emotional processes (Carey, Hill, Devine, & Szücs, 2016; Dowker, Sarkar, & Looi, 2016; Hill et al., 2016; Mammarella, Hill, Devine, Caviola, & Szűcs, 2015). Indeed, mathematical difficulties may be associated with not only specific mathematical learning disorders but also domain-general cognitive weaknesses (e.g. phonological memory, working memory, executive functions) and negative emotions (Maloney & Beilock, 2012; Vukovic, Kieffer, Bailey, & Harari, 2013). Interest in the interference of these negative affective factors, usually defined in the literature by the term ‘mathematics anxiety’, has grown from the students’ mathematics outcomes observations: these feelings of apprehension and fear aroused during a mathematical performance can hamper or even impede its correct execution (Ashcraft & Kirk, 2001; Dowker et al., 2016).

For the domain of mathematics as a whole, long-standing quantitative research concerning the relationship between students’ math anxiety and their general mathematical achievement has been carried out, and the literature has mostly revealed a substantial negative relationship between the two (Ashcraft, Krause, & Hopko, 2007; Ashcraft & Moore, 2009; Carey et al., 2017; Hembree, 1990; Ma, 1999). In his meta-analysis, Hembree (1990) pointed out that cognitive-behavioural interventions developed for the treatment of test anxiety or general anxiety were effective in reducing or eliminating the effects of math anxiety on mathematics performance. Interventions merely focused on changes in classroom curricula, relaxation therapy or group counselling were less effective. Later studies have investigated the effects of more transitory disruptions involving ‘choking’ in response to threat, and have focused on relieving the cognitive symptoms of anxiety, and particularly their impact on working memory resources, with some promising results (Ramirez & Beilock, 2011; Supekar, Iuculano, Chen, & Menon, 2015).

Since the overall aim of the book is to gain a greater understanding of math anxiety and, consequently, of ways to prevent or ameliorate the phenomenon, it is important to obtain converging evidence from as many fields as possible. Developmental psychologists, educationists, neuroscientists, educational and clinical psychologists, teachers and policymakers often tend to proceed independently, sometimes neglecting relevant findings from the research and practice outside their own disciplines. Just as with mathematical cognition in general, math anxiety research intersects with a wide array of sub-fields, such as cognitive and educational psychology, neuroscience and developmental psychology. In this book, we broadened the perspective, bringing together converging international researchers working on different areas with the aim to shed light on a) the theoretical background of math anxiety, b) the development of this phenomenon in both typical and atypical populations, c) the main cognitive processes involved and d) the importance and role of different social contexts. The result is a collection of eleven essays and constitutes a comprehensive survey of state-of-the-art studies on important facets of math anxiety.

The book begins with two chapters about the theoretical backgrounds and the psychophysiological consequences of math anxiety. The first chapter, by Mark H. Ashcraft, provides a comprehensive summary of four major approaches to understanding the phenomenon and highlights the strengths and weaknesses of each of them. The chapter can be viewed as an overview and introduction to the in-depth chapters that comprise the entire volume. It is followed by Cipora, Artemenko and Nuerk’s chapter, which presents an exhaustive review of math anxiety measurement techniques, and by the chapter of Avancini and Szűcs, who discuss how math anxiety induces physiological reactions within individuals, and how psychophysiological measures may offer new ways to assess this phenomenon without relying on self-report questionnaires.

The subsequent group of chapters discuss the development features of math anxiety. In particular, Ann Dowker points out the relationships and differences

between math anxiety and other related constructs (e.g. general- and test-anxiety, and attitudes to mathematics), and discusses reasons for the well-established negative relationship between math anxiety and maths performance. Dominic Petronzi and colleagues, after discussing issues regarding math anxiety measurement, propose a review of the literature surrounding the onset of math anxiety, focusing on several factors that may influence the development of such feelings of apprehension: e.g. negative evaluation from peers and teachers, pessimistic attitudes, low self-efficacy and reduced motivation.

Another important aspect involves the cognitive processes, especially working memory processes often associated with math anxiety. Two chapters specifically address this topic, providing a detailed summary of the current research. Passolunghi and colleagues discuss the relationship between math anxiety and mathematical performance by reviewing studies of both children and adult populations. The relationship between math anxiety and mathematics achievement is further analysed by Ng and Lee. In their chapter, they focused on not only math anxiety but also test anxiety, and discuss the overlap between these two constructs. A different insight is offered by the subsequent chapter, where Rubinsten and colleagues, after defining the meaning of attentional bias, report the most recent studies aimed at investigating whether math anxiety is characterized by an attentional bias toward math-related stimuli.

Caviola, Mammarella and Kovas give an overview of the literature in not only typical but also atypical populations, and in particular on children with mathematics difficulties or developmental dyscalculia. Their chapter tries to provide a fresh framework of the individual differences (considering both genetic and environmental factors) involved in young children’s low mathematics performance. In the following chapter, Tommasetto presents the most recent research on how gender differences in math anxiety are moderated by gender stereotypes (and self-concept beliefs). Finally, the last chapter, by Herts, Beilock and Levine, provides a detailed description about the social determinants of children’s and adolescents’ math anxiety by examining their parents’ and teachers’ math achievements and attitudes. Parents’ and teachers’ math anxiety is conceptualized as a moderator, determining the strength and direction of the relationships between children’s math anxiety and math education outcomes.

Thus, the purpose of this book is to stimulate theoretical reflection on the ways in which math anxiety can influence the achievement and consequently the future paths of individuals. Findings with regard to gender differences, cognitive networks, types of assessment and psychophysiological correlates may help generate a better definition of math anxiety and clarify what is known about it. In this respect, a novel contribution of this book is to bring together different research fields into one single volume. In doing so, we also hope to challenge preconceptions about math anxiety and offer a re-evaluation of the negative connotations usually associated with the term.

September 2018

References

Ashcraft, M. H., & Kirk, E. P. (2001). The relationships among working memory, math anxiety, and performance. Journal of Experimental Psychology. General , 130 (2), 224–237. https://doi.org/10.1037/0096-3445.130.2.224

Ashcraft, M. H., Krause, J. A., & Hopko, D. R. (2007). Is math anxiety a mathematical learning disability? In D. B. Berch & M. M. M. Mazzocco (Eds.), Why is math so hard for some children? The nature and origins of mathematical learning difficulties and disabilities (pp. 329–348). Baltimore, MD: Paul H Brookes Publishing.

Ashcraft, M. H., & Moore, A. M. (2009). Mathematics Anxiety and the Affective Drop in Performance. Journal of Psychoeducational Assessment , 27(3), 197–205. https://doi. org/10.1177/0734282908330580

Carey, E., Devine, A., Hill, F., Szűcs, D., Ng, P., & Chan, M. (2017). Differentiating anxiety forms and their role in academic performance from primary to secondary school. PLoS One, 12(3), e0174418. https://doi.org/10.1371/journal.pone.0174418

Carey, E., Hill, F., Devine, A., & Szücs, D. (2016). The chicken or the egg? The direction of the relationship between mathematics anxiety and mathematics performance. Frontiers in Psychology, 6 (JAN), 1–6. https://doi.org/10.3389/fpsyg.2015.01987

Dowker, A., Sarkar, A., & Looi, C. Y. (2016). Mathematics Anxiety: What Have We Learned in 60 Years? Frontiers in Psychology, 7, 508. https://doi.org/10.3389/fpsyg.2016. 00508

Hembree, R. (1990 ). The Nature, Effects, and Relief of Mathematics Anxiety. Journal for Research in Mathematics Education , 21(1), 33–46. https://doi.org/10.2307/749455

Hill, F., Mammarella, I. C., Devine, A., Caviola, S., Passolunghi, M. C., & Szűcs, D. (2016). Maths anxiety in primary and secondary school students: Gender differences, developmental changes and anxiety specificity. Learning and Individual Differences, 48, 45–53. https://doi.org/10.1016/j.lindif.2016.02.006

Ma, X. (1999). A Meta -Analysis of the Relationship between Anxiety toward Mathematics and Achievement in Mathematics. Journal for Research in Mathematics Education , 30 (5), 520–540. https://doi.org/10.2307/749772

Maloney, E. A., & Beilock, S. L. (2012). Math anxiety: Who has it, why it develops, and how to guard against it. Trends in Cognitive Sciences, 16 (8), 404–406. https://doi.org/ 10.1016/j.tics.2012.06.008

Mammarella, I. C., Hill, F., Devine, A., Caviola, S., & Szűcs, D. (2015). Math anxiet y and developmental dyscalculia: A study on working memory processes. Journal of Clinical and Experimental Neuropsychology, 37(8), 878–887. https://doi.org/10.1080/13 803395.2015.1066759

Ramirez, G., & Beilock, S. L. (2011). Writing about testing worries boosts exam performance in the classroom. Science (New York, N.Y.), 331(6014), 211–213. https://doi. org/10.1126/science.1199427

Supekar, K., Iuculano, T., Chen, L., & Menon, V. (2015). Remediation of Childhood Math Anxiety and Associated Neural Circuits through Cognitive Tutoring. The Journal of Neuroscience : The Official Journal of the Society for Neuroscience , 35(36), 12574–12583. https://doi.org/10.1523/JNEUROSCI.0786-15.2015

Vukovic, R. K., Kieffer, M. J., Bailey, S. P., & Harari, R. R. (2013). Mathematics anxiety in young children: Concurrent and longitudinal associations with mathematical performance. Contemporary Educational Psychology, 38 (1), 1–10. https://doi.org/10.1016/j. cedpsych.2012.09.001

1 MODELS OF MATH ANXIETY

Mark H. Ashcraft

Mathematics anxiety: “a feeling of tension and anxiety that interferes with the manipulation of numbers and the solving of mathematical problems in a wide variety of ordinary life and academic situations.”

(Richardson & Suinn, 1972, p. 551)

There is good reason to begin this volume with an introductory chapter on models of math anxiety – discussing the several models that have guided investigations of math anxiety almost necessarily involves a general review of the research on math anxiety, or at least the highlights of that research. As such, this chapter can serve as an introduction to the more specific, in-depth chapters that follow. As this introduction will show, the models that have been proposed for understanding math anxiety reflect researchers’ varying viewpoints about expected consequences of math anxiety as well as factors suspected of influencing it, researchers’ own theoretical orientations, and, to a degree, developments in the field that have made new kinds of research possible. Thus, the models show how our thinking about math anxiety has evolved and how different research orientations have enriched our understanding of math anxiety.

In the title of their recent review, Dowker, Sarkar, and Looi (2016) asked rhetorically “What have we learned in 60 years?” in our research on math anxiety. The 60 years in question date from the first modern research paper on the topic by Dreger and Aiken (1957 ) in which those authors tentatively advanced the term “number anxiety” as a label for the emotional reaction to numbers and mathematics. To be sure, there were precursors to this 1957 article; for example, Browne’s (1906) report on performance on the four arithmetic operations made passing reference to emotional reactions to math, and Gough (1954) contributed anecdotal evidence about students’ “mathemaphobia” (along with advice

for other teachers). But the precursors were largely anecdotal or clinical; indeed, several were psychoanalytic writers who suggested that “failure in arithmetic may be related to maternal overprotection” ( Dreger & Aiken, 1957, p. 344).

But the Dreger and Aiken paper was the first clear example of an empirical research approach. In their study, Dreger and Aiken added three math-focused questions to the Taylor Scale of Manifest Anxiety (and dropped three questions from the Taylor scale that had low validity), and also collected scores on an intelligence test, and final grades in a university math course; a subsample (n = 40) of the 704 participants were also given an arithmetic test while Galvanic Skin Response (GSR) deflections were recorded. All measures were inter-correlated, and the three math-focused questions were factor analyzed. The results showed that “number anxiety” appeared to be a separate construct from more general anxiety, that it was unrelated to general intelligence, and that it correlated negatively with math grades. All of these results have been replicated many times since this original report.

Math anxiety as a personality construct

The Dreger and Aiken (1957) paper began a tradition of research on math anxiety that treated math anxiety as a personality construct, that is, as a factor or dimension of the individual that needed to be explored in relation to other personality characteristics, factors, traits, or differences. An early effort in the research, not surprisingly, involved assessment. What test or survey was to be used to measure math anxiety? Although several different tests were devised, the Mathematics Anxiety Rating Scale (MARS), by Richardson and Suinn (1972), became the most widespread assessment tool for determining an individual’s level of math anxiety; it, along with its various revisions and versions for younger individuals, was the underlying test used in over half of the studies that appeared in Hembree’s (1990) and Ma’s (1999) influential meta-analyses on math anxiety.

The original MARS was a 98-item test, with the 98 items describing scenes or situations that might invoke math anxiety (having to reconcile a checkbook, checking a restaurant bill that you think has overcharged you, getting ready to take a math quiz). The test asked for a self-report of how anxious each situation would make the respondent feel, on a Likert scale of 1 (not anxious) to 5 (very anxious). Later versions of the test, including the sMARS (s for shortened), by Alexander and Martray (1989), a 25 item test extracted from the original MARS, and Hopko, Mahadevan, Bare, and Hunt’s (2003) Abbreviated Math Anxiety Scale (AMAS), a 9-item test, are also in wide use. All have good to excellent reliability and show substantial inter-correlations, ranging from .50 to .85 ( Dew, Galassi, & Galassi, 1983; Hopko et al., 2003; see Chapter 2 for a full discussion).

Beyond the basic work on assessing math anxiety, considerable effort was devoted across the ensuing two decades to determine whether math anxiety was in fact a separate construct from general anxiety or the more specific construct of test anxiety. The well-known meta-analysis by Hembree (1990) devoted appreciable effort to this question and argued that math anxiety is indeed a separate

construct, although one that overlaps with test and general anxiety to a degree (for further detail, see Chapter 7 in this volume). Repeatedly, the correlations between math anxiety and test anxiety were reasonably high, but not as high as those between alternate tests of math anxiety – for instance, various math anxiety tests tend to inter-correlate in the range of .50 to .85, whereas the overall math-to-test anxiety correlation (Hembree, 1990) is .52 (see also Dew, Galassi, & Galassi, 1984). The correlation between math anxiety and general anxiety is usually smaller; in Hembree’s (1990) meta-analysis, the value was .35.

Of more interest, research summarized in Hembree’s (1990) and Ma’s (1999) meta-analyses covered the relationships investigated since the advent of the MARS (and its successors) and a variety of personality and achievement factors. This work revealed an extensive list of worrisome correlations with math anxiety (see Table 1.1 for a list of factors and correlations). On the educational side, math anxiety correlates negatively with math achievement at both the precollege and college levels, and also negatively with high school and college math grades. Math anxiety also correlates negatively with the extent of math taken in

TABLE 1.1 Selected correlations with math anxiety (MARS) summarized in Hembree’s (1990) and Ma’s (1999)

(Adapted from Hembree, 1990; Ma, 1999)

high school (elective coursework), and individuals’ intent to enroll in elective math courses in college; these correlations are routinely interpreted as indicators of avoidance. Note, however, that math anxiety has a fairly low correlation with overall intelligence (−.17), and is uncorrelated with IQ when only verbal aptitude or achievement is considered (−.06).

The correlations between math anxiety and attitudes concerning math are more strongly negative, and are also considered as supportive evidence for an overall pattern of avoidance. Math anxiety correlates negatively with enjoyment of math, self-confidence in math, motivation to learn math, and views about the usefulness of math.

Hembree’s (1990) paper considered the theoretical models for test anxiety as a guide for initial models of math anxiety and focused on two models in particular, the interference model and the deficits model. According to the interference model, test anxiety was thought to disrupt recall of prior learning. The model also claimed that interference included an individual’s worry during test taking, which would divert attention away from the test itself. The deficits approach, in contrast, claimed that an individual’s lower scores on a test were due to poor study habits and deficient test-taking skills. The individual in a test-taking situation, accordingly, would remember previous poor test performance, and this would cause test anxiety in the present moment. Because his earlier work on test anxiety had supported the interference account, Hembree proposed that math anxiety too might be better approached from the standpoint of the interference model.

Interestingly, very little of the research leading up to the time of Hembree’s meta-analysis appeared to advance theoretical proposals or models of math anxiety. Instead, the research focused on two general topics. First, researchers explored other personality characteristics and factors with which math anxiety was associated (for a related perspective on the lack of such theoretical work, see McLeod, 1989). This is the work just discussed, such as studies of the associations between math anxiety and factors like self-confidence in math, enjoyment of math, selfefficacy, and so forth. The other focus during this period was research relating math anxiety to educational outcomes, that is, math achievement. A variety of studies examined the negative association between math anxiety and grades, and between math anxiety and math achievement, with the overall correlations (in Hembree, 1990) found to be −.30 (pre-college) and −.27 (college) for grades, and −.27 (pre-college) and −.31 (college) for math achievement. Relationships of similar magnitude continue to be obtained, although the relationship is now believed to be more nuanced than a simple overall negative relationship (see, e.g., Ramirez, Chang, Maloney, Levine, & Beilock, 2016, and Chapter 4, this volume).

Math anxiety as a cognitive construct

Early on, researchers and theorists acknowledged that math anxiety, along with other forms of anxiety, involved a cognitive component (e.g., Dew et al., 1983).

Early writings routinely noted that test anxiety involved both an affective component, emotionality, and a cognitive component, conscious worry. The theoretical model that brought this thinking into the realm of cognitive psychology was the important processing efficiency theory by Eysenck (1992; Eysenck & Calvo, 1992). According to this theory, worry is an internal process that occupies consciousness during an anxiety reaction. Critically, this preoccupation was predicted to consume the resources of the limited Working Memory system. Thus, Eysenck predicted quite specifically that an anxious individual should show disruption on a cognitive task to the extent that the task relies on working memory resources.

Interestingly, just before Hembree’s (1990) important meta-analysis on math anxiety, and likewise just before Eysenck’s (1992) theory, Ashcraft and Faust (1988) presented a conference report on an initial study concerning the cognitive consequences of math anxiety. Their study examined the underlying cognitive processes of doing mental arithmetic by individuals varying in their math anxiety; as noted when the study was subsequently published (Ashcraft & Faust, 1994), it appeared to be the first to pose the question whether math anxiety actually influenced the mental processing involved in doing arithmetic.

In this exploratory work, Ashcraft and Faust presented simple addition and multiplication problems (e.g., 4 + 3 = 7, 8 × 4 = 38), two-digit addition problems with and without a carry (e.g., 24 + 17 = 43), and a set of complex problems containing all four arithmetic operations (e.g., 18 + 16 = 34, 47 – 18 = 19, 12  × 14 = 168, 156 ÷ 12 = 13), all for true/false judgments. Participants were given the MARS assessment and were divided into four math anxiety groups.

For the most part, the simple addition and multiplication problems revealed no math anxiety effects, these problems showing only the standard effects found in regular tests of addition and multiplication, namely, that latencies and errors increased as the problems grew larger. But the two-column addition problems revealed two particularly interesting math anxiety effects. First, the higher anxiety groups were considerably slower to these problems than the lowest anxiety group. Second, the higher anxiety groups seemed particularly slowed down by the presence of a carry problem; that is, when a problem involved a carry, only the low anxiety group demonstrated efficient performance, whereas performance in groups 2, 3, and 4 was particularly disrupted.

In a second set of studies, Faust, Ashcraft, and Fleck (1996) replicated and extended the exploratory studies, and found several additional effects of math anxiety on cognitive performance. Three results deserve particular mention. First, in the initial experiment, we again studied performance of simple addition problems in the true/false task. But we expanded the range of values (termed “split”) for the incorrect answers; here, incorrect answers could be wrong by 1, 5, 9, or 23 (e.g., 7 + 5 = 35). Unlike the typical result, which shows performance improving as the split grows larger, we found the higher anxiety groups actually made more errors (and had more extreme scores) when the split grew larger. This suggested to us a deficiency in “number sense,” in that we expected

an unreasonable answer like 35 for the problem 7 + 5 to be rejected immediately based on a plausibility judgment (see Suarez-Pellicioni, Nunez-Pena, & Colome, 2013, for a replication using event-related potentials methodology). Second, we again found that the carry operation in two-column addition led to slower and more error prone performance for the higher levels of anxiety. And finally, to rule out a simple math competence as a confound in explanations of the math anxiety effects, we tested all of the experimental stimuli in an untimed, paperand-pencil format in a separate study, thus removing the time pressure of the laboratory methods. In this study, there was no relationship between performance and measured level of math anxiety at all, either in correlations or in analysis of variance with the four-level math anxiety groups.

Mainstream research in mathematical cognition had firmly established the important role of working memory in procedural aspects of performance, that is, carrying in two-column addition (e.g., Ashcraft & Stazyk, 1981; Widaman, Geary, Cormier, & Little, 1989). Thus, we were confronted with evidence that this working memory-intensive aspect of performance was especially troublesome for those higher in math anxiety. Accordingly, we extended Eysenck’s (1992) processing efficiency theory to the realm of math anxiety and conducted a direct test of the hypothesis that math anxiety causes a disruption in working memory processing while doing math, visible only when the math task relies on the resources of working memory. To perform this test, Ashcraft and Kirk (2001) placed participants in a dual-task setting, asking them to perform addition problems of increasing difficulty: basic facts (single digit operands), medium size problems (a two-digit plus a one-digit operand), and large problems (two two-digit operands). Half of the problems required a carry operation. In a letteronly control condition, either two or six random letters were shown prior to the addition problem, and then had to be recalled after the addition problem was removed from the screen. Because we showed the answer to the addition problem for the participant to read aloud, no actual math processing was required in this condition. But in the dual-task condition, the letters had to be held in working memory while solving the addition problem, and then recalled from memory after giving the answer to the problem.

Our prediction was that the larger addition problems, especially those involving the carry operation, would be the most taxing of working memory resources, and thus would interfere the most when six letters had to be held in working memory for recall. This is the classic dual-task interference effect, of course. But our prediction went further. The prediction we derived from Eysenck (1992) was that the dual-task interference effect should be especially pronounced for the high math anxious group, which would essentially be performing in a triple-task situation –difficult carry problems being solved under a heavy working memory load while their working memory was being drained by their anxiety-induced worries.

In short, this is exactly what the results demonstrated. Higher working memory loads led to more errors, especially when the addition problem involved carrying, but this of course was only found in the dual-task condition; letter

recall was nearly perfect in the letter-only control condition. But the critical prediction involving math anxiety was confirmed; the high math anxious group scored almost 40% errors on the working memory task when working memory load was high and the problem involved a carry, compared to 18% for the low anxious group. And these two groups had identical error rates in the six-letter load condition when the problem did not involve carrying. Thus, the critical role of working memory in the procedure of carrying was confirmed, and the crippling of working memory on the part of highly math anxious participants was also demonstrated (see Chapters 6 and 7 for additional information on working memory; for related work showing working memory’s involvement in algebra problem solving, see Trezise & Reeve, 2014).

Ramirez, Shaw, and Maloney (2018) have termed this model of math anxiety the disruption account ; math anxiety disrupts math performance because it reduces the working memory resources necessary for successful performance. Of course, working memory is the attentional and executive control system that manipulates and maintains the limited amount of information immediately relevant for a task; it keeps important information available while inhibiting attention to irrelevant information (see, e.g., Baddeley, 1986; Engle, 2002; Miyake & Shah, 1999). In this disruption account, it is the worry and ruminations about one’s math anxiety that cause the disruption in working memory, either that working memory’s resources are in some sense consumed by those worries and ruminations, or perhaps equivalently that one fails to inhibit attention to those worries and ruminations.

Recall that in Faust et al. (1996), the final study gave participants an untimed, pencil-and-paper test on the math problems and found no differences in performance (accuracy) at the different levels of math anxiety. This is consonant with the implication of the disruption account, that the math anxiety effect is one of a transient disruption of performance due to interference in working memory, rather than a global disruption due to overall lower competence at math. As Ashcraft and colleagues have been careful to note, however, this assumption about competence cannot be made at higher levels of math difficulty, given the wealth of evidence that people higher in math anxiety avoid taking higherlevel (elective) math coursework. There surely are math competence differences between levels of math anxiety when truly high-level math is involved; however, to date, only relatively lower-level math problems have been studied in experimental work, where competence, in terms of educational attainment, seems to be equivalent across the levels of math anxiety (though see Trezise & Reeve, 2014). But when higher levels of math are concerned, given the clear evidence that individuals with higher levels of math anxiety take fewer math courses (e.g., Hembree, 1990), we would expect a lower level of competence as a function of math anxiety. This has been described as an education-based reduced competency account of math anxiety by Ramirez et al. (2018).

In an offshoot of this work, several studies have examined what particular aspect of working memory might be disrupted by math anxiety. A likely

candidate, based on reported studies, is the inhibition function, that is, the aspect of the central executive system within working memory that inhibits one’s attention to task-irrelevant stimuli. In this account, information contained in the math problem along with one’s own memory concerning math are relevant to the math task, while worries and ruminations are clearly irrelevant to the problem solving task. Thus, attention to those worries should be inhibited in order for math processing to proceed in an uncompromised fashion. Accordingly, several investigators have asked whether high math anxious individuals have difficulty in inhibiting their attentional processes to threat-related stimuli when their math anxiety is aroused.

In one study, Hopko, Ashcraft, Gute, Ruggiero, and Lewis (1998) had participants read passages of italicized text into which irrelevant distractor words had been inserted in regular font (strings of x’s were used in the control condition); participants were instructed to read the italicized text aloud. Notably, in many of the tested paragraphs, the topic was generally about math, and mathrelated words were often used as distractor words (e.g., a paragraph about balancing a checkbook, distractor words “negative exponent” and “add ten formula”). Although reading times increased for all groups when distractor words were present, the medium and high math anxiety groups took considerably longer to read the texts when either unrelated or related distractor words were present, compared to the low math anxious group. And on the follow-up test on content accuracy, the medium and high math anxious groups had considerably higher error rates when distractor words had been present, again compared to the low anxious group. In other words, during processing, the higher math anxious participants were far less able to inhibit their attention to the irrelevant distractor words, and then, despite their slower reading, still answered fewer of the questions correctly on the final test for content. Related work has shown comparable evidence of attentional bias toward math-related stimuli (i.e., emotionally negative for a math anxious individual) using a Stroop task (Suarez-Pellicioni, Nunez-Pena, & Colome, 2015) and the dot probe task (Rubensten, Eidlin, Wohl, & Akibli, 2015). Again, high math anxious participants preferentially process highly math anxious stimuli, or the location where such stimuli were presented, as if unable to inhibit attention to those anxiety-related stimuli. (For similar evidence concerning weaknesses in inhibitory control on the part of high math anxious secondary school students, see Passolunghi, Caviola, De Agostini, Perin, & Mammarella, 2016, and Chapter 8 in this volume.)

On the other hand, a more recent line of research questions whether there might be a more central competency deficiency in those with high math anxiety. This possibility, also referred to as a reduced competency account (Ramirez et al., 2018), suggests that high math anxious individuals simply do not have the math skills of their lower math anxious peers, that their lower competency leads to poorer learning and performance, thus yielding math anxiety. Interestingly, this version of reduced competency does not appeal to education-based reasons, but

suggests that individuals are simply lower in math skills, particularly in basic numerical and spatial skills. Evidence for this view comes especially from a series of studies by Maloney and colleagues ( Ferguson, Maloney, Fugelsang, & Risko, 2015; Maloney, Ansari, & Fugelsang, 2011; Maloney, Risko, Ansari, & Fugelsang, 2010). In one, these researchers had high and low math anxiety groups of participants perform a straightforward enumeration task, counting the number of squares displayed on the screen. The groups were equivalent in performance in the subitizing range (1–3 or 4 squares), but the high math anxious group was significantly slower in the counting range (more than 4) of the task. In another, participants completed the number comparison task, either deciding which of two presented digits was the larger, or, in a second study, comparing the single presented digit to an unchanging standard (5). In both cases, high math anxious participants showed a steeper numerical distance effect than low anxious participants, that is, they had more difficulty in choosing the larger of two digits when the digits were closer in value. In the third series, participants were tested on a spatial orientation task. As in the other studies, high math anxious participants were far less accurate than their low math anxious peers.

The simplicity of these tasks, their presumed reliance on rather primitive number representations (as opposed to formally taught arithmetic) or elementary procedures (counting), led Maloney and colleagues to suggest that math anxiety may well be due to basic skill and numeracy deficiencies, deficiencies that then become more visible later in schooling when math content becomes more demanding.

Note that none of the effects investigated in these cognitive approaches to math anxiety challenge or dispute the findings discussed in the personality approach to math anxiety. The cognitive theorists have not rejected claims about math anxiety’s associations with self-confidence, motivation, and test anxiety and have not argued that the personality approach should be overthrown in order to promote a cognitive approach. Instead, in most respects the cognitive approach has sought to both advance plausible explanations that stem from the personality approach and derive testable hypotheses from that approach. For example, the demonstrated avoidance characteristics of math anxiety (e.g., Liew, Lench, Kao, Yeh, & Kwok, 2014), including the avoidance of elective coursework, should lead to deficits in knowledge, as proposed by the reduced competency account of math anxiety. Noting that worry and ruminations routinely accompany an anxiety reaction led to specific predictions about cognitive processing requiring working memory resources.

And as we discuss two final models of math anxiety, the same will be true there. These two approaches to understanding math anxiety do not overthrow what has come before in terms of offering alternative explanations of the phenomenon. Instead, they flesh out or provide further insights into possible reasons for the onset of math anxiety in the first place, potential risk factors in other words, leading to additional ideas to be explored for possible interventions.

Math anxiety as a sociocultural construct

It seems true, almost by definition, that it will be difficult to determine the causal factors that lead to math anxiety. As we all know, a truly experimental study, with random assignment, is needed to determine causality, which is impossible with a variable like math anxiety, and the correlational designs that are largely available to us to study math anxiety are simply not suitable for studying causal relationships. Researchers have devised clever manipulations, however, that border on cause-effect results, or at least results that strike many as far more persuasive than correlational studies by themselves. Some such manipulations have already been discussed, for example the dual-task study by Ashcraft and Kirk (2001), testing a specific prediction about math anxiety (reduced working memory resources) in a specific math setting (two-column addition with carrying). But now we turn to studies of possible influences on the individual’s math anxiety due to social and cultural factors.

It has long been suspected, or even asserted based on anecdotal evidence, that teachers play a role in the development of their students’ math anxiety. Factors such as pedagogical practice, teacher affect, and the teacher’s own math anxiety have all been considered as possible negative influences on students’ attitudes and math anxiety. As is commonly known, Hembree’s (1990) meta-analysis reported that the highest mean math anxiety score, by college major, was found for students majoring in elementary education. It is thus plausible that elementary school teachers, or teachers more generally, affect their students’ attitudes toward math by their own behaviors and attitudes; and if the teachers are highly math anxious, then the influence being transmitted would likely be negative.

In what is now a classic demonstration of this, Beilock, Gunderson, Ramirez, and Levine (2010) assessed first- and second-grade teachers’ math anxiety, and then examined their students’ math achievement and stereotype endorsement about math (e.g., “boys are good at math, and girls are good at reading”). Although there were no differences in children’s attitudes at the beginning of the school year, by the end of the year girls exhibited much more gender-stereotyped beliefs and lower math achievement when their teachers were higher in math anxiety; boys, however, were unaffected, perhaps not a surprise since children tend to imitate same-gender models (all teachers were female). Thus, teacher math anxiety was directly related to students’ ability beliefs and math achievement. Particularly because gender stereotypes concerning women and math are so prevalent, this result regarding stereotyped beliefs is a particularly worrisome result with respect to women’s life-long pursuit and success in math-related careers.

In related work, Maloney, Ramirez, Gunderson, Levine, and Beilock (2015) studied a different possible source of children’s math anxiety: their parents. These researchers reasoned that parents might spend time at home helping their children with their math homework. For parents with low math anxiety, this could easily be beneficial. But for parents with high math anxiety, one might easily

expect that the parents themselves could be transmitting anxious, negative attitudes about math to their children, to the extent that the parents displayed frustration, lack of understanding, and anxiety themselves during the help sessions. In a large field study spanning a school year, parents’ math anxiety was assessed, and parents kept careful logs of the frequency and duration of their homework help sessions with their children. The results confirmed the research hypothesis; higher math anxious parents who helped frequently with math homework had children who, at the end of the year, were higher in math anxiety themselves, compared to children whose high math anxious parents seldom helped with homework, or to children who were helped by non-anxious parents (for a review of parent effects, see Batchelor, Gilmore, & Inglis, 2017; Chapter 11 in this volume discusses both parent and teacher effects on math anxiety).

Finally, it seems clear that peers – and society at large – play a role in shaping children’s and adults’ attitudes about math, and likely their math anxiety as well. As many authors have noted, poor attitudes about math abound in Western societies. Ashcraft (2002) noted that math is frequently viewed as inherently difficult (as Barbie dolls used to say, “Math class is hard”), aptitude is considered as more important than effort (e.g., Geary, 1994, chap. 7), and mastering math is often viewed as either unimportant or even optional by many (see the correlations in Table 1.1 concerning motivation and intent to enroll, for example). Unfortunately, there appears to be no clear evidence concerning peer influence on math anxiety. At the global level, however, there is now evidence that economically developed and more gender equal countries have lower overall levels of math anxiety than less developed, less equal countries (Stoet, Bailey, Moore, & Geary, 2016) – but even so, those more favored countries show a larger national gender difference in math anxiety compared to less developed countries.

Math anxiety and the role of gender

The role of gender in math anxiety is perplexing. The research routinely finds that females score higher on standard assessments of math anxiety than do males; see, for example, Betz (1978), Wigfield and Meece (1988), and Hopko et al. (2003), and the full meta-analysis account in Hembree, 1990. In the classic figure from Hembree (1990), showing mean math anxiety levels from grades 6 through college, the curve representing female participants shows that females score approximately 20 points higher than males at every grade level. But despite the negative connection between math anxiety and math performance, there are generally no overall differences between the genders in actual mathematical performance (e.g., Spelke, 2005); or, when a gender difference is found, it is usually rather small (e.g., Kovas et al., 2007 ). Interestingly, however, females tend to rate themselves as having lower math performance than males rate themselves (e.g., Dowker, Bennett, & Smith, 2012; Goetz, Bieg, Ludtke, Pekrun, & Hall, 2013; Hill et al., 2016).

It may be that the tendency for females to show higher levels of math anxiety is related to the general tendency for females to show generally higher levels of anxiety, and for females to have a higher prevalence of clinical anxiety disorders (e.g., Beidel & Alfano, 2011; Feingold, 1994; McLean, Asnaani, Litz, & Hofmann, 2011). If so, then this would conceivably be due to the overlap between math anxiety and more general anxiety reactions. Another possibility, as articulated in Ashcraft (2002), is that women may feel more comfortable reporting anxiety than men do. This could explain the puzzling reason why the gender difference in math anxiety is not accompanied by a gender difference in math performance. Yet another alternative, explored more fully in a later section (see also Chapter 10 in this volume), relates to the notion of stereotype threat, the phenomenon that one’s performance will dwindle if a negative stereotype about one’s group, say gender, is brought to mind at the time of a test. It has been shown ( Beilock, Rydell, & McConnell, 2007 ) that women do more poorly on a math-based test if a negative stereotype about women and math is aroused prior to testing (“we are trying to understand why men do better than women on this task”). Interestingly, in this study it was only on problems requiring the resources of working memory that the effects of math anxiety were apparent; thus the cognitive interpretation of the effect was parallel to that of the Ashcraft and Kirk (2001) study, that anxiety caused a drain on working memory resources.

A final possibility, of course, is that women have internalized to some degree the social and cultural attitudes involving the stereotypical belief that math is a male domain. When inevitable challenges arose during education, they may have interpreted their difficulties as evidence of a lack of ability, and developed a degree of math anxiety as a result. According to Dweck’s (1999) social-cognitive theory of intelligence, this would demonstrate an entity theory of intelligence on the part of such females, a belief that their abilities in math are largely fixed and unable to change. When confronted with negative feedback, say, poor math test performance, such individuals often withdraw their effort from the task, that is, they adopt an avoidance pattern of behavior. Of course, as noted earlier, avoidance is a prime characteristic of math anxious individuals. And as demonstrated by Burkley, Parker, Stermer, and Burkley (2010), female students who endorse this entity or fixed abilities belief system were less likely to enjoy math, to pursue a college major requiring math, or to consider career paths involving math. These characteristics, of course are prominent in the Table 1.1 correlations involving math anxiety. They may be especially prominent, or exacerbated, among females. (The opposite set of beliefs in Dweck is the incremental theory, the belief that intelligence and abilities are malleable, and can be improved with practice.)

Math anxiety as a neuro-biological construct

Very little research has been conducted on the possible biological, that is, genetic, bases of math anxiety, although one study that has been reported is rather

compelling. Wang et al. (2014) tested 216 monozygotic and 298 same-sex dizygotic twins, from an ongoing longitudinal project; at the time of testing the twins’ average age was 12.25 years. The tests that were administered included a math anxiety test, a test of general anxiety, a math problem solving test, and a reading comprehension test. Overall, the results showed that approximately 40% of the variation in math anxiety was accounted for by genetic factors, with the remaining 60% accounted for by individually specific environmental factors. Math anxiety was influenced by the genetic and environmental risk factors associated with general anxiety. There was also evidence for an independent genetic component related to math problem solving. In their conclusion, the authors note that their results suggest a rather worrisome possibility, the possibility of a dynamic, downward spiraling process. In particular, they suggest that the genetic influence related to poor math problem solving, along with the genetic and individual-specific environmental influences for high general anxiety, serve as risk factors in the development of math anxiety. When these risk factors combine, there may be further impairment in math performance, with attendant negative consequences (low grades, emotionality, decreases in math self-concept). Such reactions can in turn exacerbate the individual’s math anxiety (e.g., avoidance, negative attitudes), setting up another cycle of declining performance and increasing anxiety.

Three other well-known studies also deserve mention here, not because they explored possible genetic determinants of math anxiety, but because they examined brain mechanisms and regions underlying the anxiety reaction. And, for those who might have doubted that math anxiety was truly a documentable phenomenon, they provided brain-based evidence of its existence and operation.

In the first study, Young, Wu, and Menon (2012) tested children ages 7–9 in an fMRI environment on addition and subtraction problems; problems were shown with answers, and children made true/false judgments. Simple problems for both operations had an operand of 1 (e.g., 5 + 1 = 6; 7 – 1 = 5). Beyond obtaining conventional results on behavioral measures (errors, latencies), Young et al. also found rather definitive outcomes in terms of neural activations. Math problem-solving was associated with greater activation in the right amygdala in the high math anxious group, compared to the low anxious group. The right amygdala also showed greater effective connectivity with brain areas associated with social and general anxiety. On the other hand, the results showed lower activation for the high anxious group, compared to low anxious participants, in many cortical areas associated with typical math processing, including the intraparietal sulcus (IPS), superior parietal lobule, and right dorsolateral prefrontal cortex. Likewise, the right amygdala in high math anxious participants showed less effective connectivity to the typical “math” regions, namely, the posterior parietal cortex, including the IPS and angular gyrus. Thus, Young et al. determined that math anxiety is associated with hyperactivity in the amygdala, a region of the brain typically associated with processing fear and negative emotions. Furthermore, their evidence revealed that high math anxious children also

showed reduced brain activations in regions that have repeatedly been found to be associated with mathematical and numerical reasoning in children and adults (see also Klados, Simos, Micheloyannis, Margulies, & Bamidis, 2015).

In the second two studies, Lyons and Beilock (2012a , 2012b) tested low and high math anxious adults in an fMRI setting. In both studies, participants were given a cue several seconds before receiving a block of either math or word trials, with the cue alerting them to which kind of trial would follow. Math stimuli had to be judged as true or false; easy problems had operands less than 5, and hard problems had operands of 5 or greater. The word trials showed either 4 or 7 letter strings; participants responded true if the letters spelled an English word when reversed (e.g., yrestym would yield mytsery, and participants should respond “false”). In one study ( Lyons & Beilock, 2012b), the focus was on the different regions and networks that become activated during the cue period, that is when the participant is anticipating either having to do math or having to do the word task. The compelling result was that high math anxious participants showed increased bilateral dorso-posterior insula and mid-cingulate cortex activity during the anticipation period, that is, during the cue period prior to actually processing the math problem. The critical point here is that these regions are associated with visceral threat detection, and even with the experience of pain itself. Thus, anticipating an upcoming math problem registers in the brain as an experience of visceral pain, in regions of the brain associated with threat to the physical body (see also Pletzer, Kronbichler, Nuerk, & Kerschbaum, 2015).

In a paired study, Lyons and Beilock (2012a) noted a corollary result; while the high math anxiety group averaged nearly twice as many errors on difficult problems as the low math anxious group, some of the high math anxious participants seemed relatively unimpaired in their math performance. Lyons and Beilock wondered if the key to this result might be found during the anticipation phase, after participants had been given the cue telling them which kind of trial would follow. Perhaps these participants were engaging in some form of preparation, some kind of cognitive restructuring that enabled them to then process the math problem unhindered by the interference of anxiety.

Indeed, when the fMRI results were examined in terms of behavioral differences, those high math anxious participants who showed better math performance (a smaller discrepancy between their math and word performance) showed increased neural activity in frontoparietal regions during the anticipation period, that is, during the cue phase when they were preparing for the upcoming trial. Frontoparietal regions would be just the regions one would expect to be more active during math processing, suggesting that indeed these participants were ramping up their cognitive resources in preparation for the upcoming math problem. Lyons and Beilock suggested, accordingly, that it is perhaps not necessarily one’s level of math anxiety but, instead, one’s ability to engage cognitive control processes before problem solving has even begun that predicts math performance. Such control might include reappraising one’s approach to problem solving, increasing one’s motivation, or even marshaling deliberate control

to inhibit attention to distracting worries (for additional evidence concerning anticipatory anxiety, see Klados, Pandria, Micheloyannis, Margulies, & Bamidis, 2017 ).

Math anxiety in the future – the interpretation account

Ramirez et al. (2018) have recently proposed an overarching account of math anxiety that departs in several significant ways from current thinking, in that it emphasizes the individual’s interpretation of previous math experiences and outcomes, rather than the individual’s avoidance, reduced competency, or worries per se, as determiners of math anxiety. The approach stems from existing appraisal theory (e.g., Lazarus, 1991), in its view that emotional outcomes and attitudes are based on one’s interpretations of events, internal states, physiological cues, and the like. Consider, for example, studies showing that earlier achievement has a stronger effect on subsequent math anxiety than earlier math anxiety has on subsequent math achievement ( Meece, Wigfield, & Eccles, 1990). The reasoning here is that students may believe that their math outcomes, that is, their math performance on standardized tests, are indicators of their abilities. As such, students form expectations about their future success – or lack of success – based on those performance measures. Thus, those who attribute their poor performance to lower ability may be at greater risk for developing math anxiety than are those who attribute their poor performance to lower effort, or those who acknowledge that mistakes are routine when learning math. Note the similarities here to Dweck’s (1999) distinction between entity versus incremental theories of intelligence, described earlier; someone with an entity theory does poorly, attributes that to fixed (and low) abilities, and then fails to engage in the practice and effort that would improve performance, thus falling further behind and likely developing worse attitudes.

While the Ramirez et al. approach is a novel approach to explaining math anxiety, it is nonetheless far too new to evaluate how well it explains all of the extant evidence that has now accumulated about math anxiety. It does, however, make some clear claims about designing interventions for math anxiety. They all generally have to do with reinterpretation or reappraisal of the experiences and outcomes that lead an individual to adopt math anxious attitudes, or that perpetuate the anxiety and attitudes. And in general, although there are only a handful of studies that have tested such manipulations, they do seem to show promise. For example, Jamieson, Peters, Greenwood, and Altose (2016) tested community college students who were enrolled in remedial mathematics. Two exams were given to the classes, one prior to the manipulation to serve as a pre-test (which showed group equivalence). The reappraisal intervention was given prior to Exam 2. In it, students were told about the adaptive functions of stress; they were told that increased arousal was not harmful, that stress responses evolved to help address acute demands, and that increased arousal aids performance. The control group was simply told that the best way to improve outcomes during

stressful test situations was to ignore the stress, and to ignore negative thoughts associated with stress during the test ( Jamieson et al., 2016, p. 581). Students in the reappraisal group performed significantly better on Exam 2 than did students in the control group, and they also reported lower math evaluation anxiety during Exam 2 compared to controls. Thus reappraisal of the stress experienced by high math anxious students during the second math test led to improved performance. (For similar effects observed after a short period of expressive writing, see Park, Ramirez, & Beilock, 2014.)

Whatever the ultimate fate of the Ramirez et al. (2018) interpretation account of math anxiety, its different approach, along with the other models of math anxiety has made it clear that math anxiety is a multifaceted phenomenon, which has profited from the multiple approaches researchers have taken in attempts to understand it. In a very real sense, the several models that have been summarized here are all mutually compatible – no single model or construct contradicts another model in major ways. Instead, the models and different approaches have been largely supportive, and have enriched the study of math anxiety. As an example, determining that the anxiety response consumes essential working memory resources, thus compromising the cognitive effort that can be devoted to math problem solving, led directly to brain imaging searches for evidence both of the emotional, anxiety-based processing and the standard cognitive-based math processing. And the personality-based research revealed clear evidence of avoidance and lower math achievement, important warnings for researchers in terms of the potential confounding factors between math anxiety and math competence at higher levels of math difficulty.

This chapter was intended to be an overview, a brief introduction to the various topics that are relevant to a full understanding of math anxiety. Having been introduced to those topics, the reader is now invited to full-length, in-depth treatments of these topics, from the leading researchers in the field.

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