Predicting maths anxiety from mathematical achievement across the transition from primary to secondary education

The primary- to secondary-education transition is a substantive life event for many children. The transition can be associated with changes in the developmental trajectories of both emotional health and academic achievement. The current study looked at whether the trajectory of mathematical attainment and emotional health (anxiety) across this transition predicted later maths anxiety. A secondary analysis of data from the Twin Early Development Study (TEDS) was performed. The statistical models were fit on the 753 participants (one from each twin pair) for which there were measures of mathematical performance across the primary- to secondary-education transition and maths anxiety at age 18. Two multi-level growth models were fit predicting mathematical attainment and anxiety over the primary- to secondary-education transition. The intercepts and slopes for each child were extracted from these models and used as predictors of subsequent maths anxiety at age 18. These effects were adjusted for biological sex, socio-economic status, verbal cognitive ability and general anxiety. Maths anxiety at age 18 was significantly predicted by both pre-transition levels of anxiety and mathematical attainment and their rate of change across the primary- to secondary-education transition. However, the effects were small, suggesting that theories of maths anxiety may have overplayed the role of prior mathematical attainment and general anxiety.


Comments to the Author(s)
Comments to the manuscript: Predicting Maths Anxiety From Mathematical Achievement Across the Transition From Primaryto Secondary-Education Summary: The present manuscript analyzed a sample of 1104 participants in which math attainment and emotional difficulties were measured at ages 9 and 12, while math and general anxiety were measured at age 18. The main aim of the study was to understand whether math and emotional changes predicted math anxiety in older participants, by also controlling for verbal intelligence, SES, gender and general anxiety.
General remarks: The manuscript is interesting and clearly written. The strength of the study is the high number of participants and the presence of some variables tested longitudinally. However, I have some concerns which are summarized here below.
Major points: 1. The introduction is well structured and clear. However, the following papers should be considered. At p. 2 when the Authors reported the estimates of math anxiety I suggest considering the paper of Devine, Hill, Carey, & Szucs, 2018 who tested a large sample of children in UK looking for differences between math anxiety levels and math difficulties. Moreover, on the same page, the most recent meta-analysis of Namkung and Peng, 2019 on school-aged students should be mentioned. 2. In the results section, I suggest to delete Table 2, or to move it in the supplementary materials. Descriptive statics are reported, however, the correlations among variables should be added to have an overall view of the data. Moreover, the linear models are difficult to read and interpret and the suggestion to transform the b values in terms of changes in math anxiety is correct but convoluted. I suggest to report standardized measures in the tables in order to simplify the interpretation of the results. Although many participants were included in the model, math change had a t-value of 1.96, with a p=.05 so it seems to me that the relation with math anxiety was not very high, as the Authors admitted. Anyway, with standardized measures, the interpretation could be easier. Finally, in the models reported in Tables 4 and 5, it is not clear why degrees of freedom changed so much. It can depend on the number of participants, of course, who changed across measures, but it is not clear whether measures were added one after the other or all together. 3. The Authors should clarify why also Math pre-transition and SDQ pre-transition were entered into the models. In my view changes in these measures could contribute to explain math anxiety at age 18, but I'm not sure to have well understood the hypothesis related to those measures at age 9. 4. In the limits of the study, the Authors should add that their math attainment derived from teachers' ratings but not from objective measures.

17-Oct-2019
Dear Professor Field On behalf of the Editors, I am pleased to inform you that your Manuscript RSOS-191459 entitled "Predicting Maths Anxiety From Mathematical Achievement Across the Transition From Primary-to Secondary-Education" has been accepted for publication in Royal Society Open Science subject to minor revision in accordance with the referee suggestions. Please find the referees' comments at the end of this email.
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Once again, thank you for submitting your manuscript to Royal Society Open Science and I look forward to receiving your revision. If you have any questions at all, please do not hesitate to get in touch. Both reviewers and I are in agreement that this paper addresses an important and current topic, is clearly written, and has several notable methodological strengths, specifically the longitudinal design, the large sample size, and the analytical approach. There are nonetheless some relatively minor changes which would further improve the paper, and Reviewer 2 makes several constructive suggestions which will need to be addressed. In particular, please include the correlations among predictors and with outcome variables. Also, although I appreciate your efforts to interpret the magnitude of the effects, this could be made clearer -please consider Reviewer 2's suggestion of including standard scores. Finally, please clarify the concluding statements on p. 10 (lines 12-15), which currently come across as contradictory statements about whether or not the results support the reciprocal model of maths anxiety.
Reviewer comments to Author: Reviewer: 1 Comments to the Author(s) A handful of recent papers have examined the issue of which construct, math anxiety or math achievement, leads to or influences the other, in an attempt to gain some evidence on causality on this knotty question. As the authors note, three theories have been articulated -the Debilitating Anxiety Model, the Deficit Model, and the Reciprocal Model -but all suffer from the same limitations, that the data brought to bear on the issue necessarily come from correlations, hence cannot support causal interpretations. The recent papers, on the other hand, have attempted to gain some leverage on the issue by examining patterns of influence (again, largely the magnitudes of correlations) in longitudinal designs, asking for example if poorer performance at time 1 leads to worse anxiety at time 2 or, instead whether worse anxiety at time 1 leads to worse performance at time 2. The current manuscript examines the same kind of evidence, but I believe is considerably stronger than the recent papers I've seen, for a variety of reasons. First, it presents data from a far larger sample, well over 1000 cases, in contrast to the 100 or 200 cases normally tested. Second, it spans a considerably longer time range to examine the longitudinal effect; here the range of interest is prior to the transition from primary to secondary education, around ages 9-12, with the final assessment of math anxiety at age 18 (thus the effective age span tested was age 9 through 18). The authors seem somewhat concerned that their results show rather modest effect after analyzing the data, possibly because other, similar studies have sometimes shown stronger effects. I would suggest that with their more adequate sample size and more thorough age sampling range, they are probably providing a more accurate estimate of the true effect size under consideration. I found the overall coverage of the manuscript to be quite adequate, including the thoroughness of the literature review, the explication of the analysis strategy, and the interpretation of results. The results are entirely sensible, the limitations of the study are portrayed clearly, and need not be expanded in my view. It is a strong paper, one that will contribute to our understanding of math anxiety.

Reviewer: 2
Comments to the Author(s) Comments to the manuscript: Predicting Maths Anxiety From Mathematical Achievement Across the Transition From Primaryto Secondary-Education Summary: The present manuscript analyzed a sample of 1104 participants in which math attainment and emotional difficulties were measured at ages 9 and 12, while math and general anxiety were measured at age 18. The main aim of the study was to understand whether math and emotional changes predicted math anxiety in older participants, by also controlling for verbal intelligence, SES, gender and general anxiety.

General remarks:
The manuscript is interesting and clearly written. The strength of the study is the high number of participants and the presence of some variables tested longitudinally. However, I have some concerns which are summarized here below.
Major points: 1. The introduction is well structured and clear. However, the following papers should be considered. At p. 2 when the Authors reported the estimates of math anxiety I suggest considering the paper of Devine, Hill, Carey, & Szucs, 2018 who tested a large sample of children in UK looking for differences between math anxiety levels and math difficulties. Moreover, on the same page, the most recent meta-analysis of Namkung and Peng, 2019 on school-aged students should be mentioned. 2. In the results section, I suggest to delete Table 2, or to move it in the supplementary materials. Descriptive statics are reported, however, the correlations among variables should be added to have an overall view of the data. Moreover, the linear models are difficult to read and interpret and the suggestion to transform the b values in terms of changes in math anxiety is correct but convoluted. I suggest to report standardized measures in the tables in order to simplify the interpretation of the results. Although many participants were included in the model, math change had a t-value of 1.96, with a p=.05 so it seems to me that the relation with math anxiety was not very high, as the Authors admitted. Anyway, with standardized measures, the interpretation could be easier. Finally, in the models reported in Tables 4 and 5, it is not clear why degrees of freedom changed so much. It can depend on the number of participants, of course, who changed across measures, but it is not clear whether measures were added one after the other or all together. 3. The Authors should clarify why also Math pre-transition and SDQ pre-transition were entered into the models. In my view changes in these measures could contribute to explain math anxiety at age 18, but I'm not sure to have well understood the hypothesis related to those measures at age 9. 4. In the limits of the study, the Authors should add that their math attainment derived from teachers' ratings but not from objective measures.

31-Oct-2019
Dear Professor Field, I am pleased to inform you that your manuscript entitled "Predicting Maths Anxiety From Mathematical Achievement Across the Transition From Primary-to Secondary-Education" is now accepted for publication in Royal Society Open Science.
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Editor
RC: Finally, please clarify the concluding statements on p. 10 (lines 12-15), which currently come across as contradictory statements about whether or not the results support the reciprocal model of maths anxiety.
AR: We have rephrased these sentences as: Nonetheless, small though the effect is, it is consistent with the wider literature (Maloney, Ansari, and Fugelsang 2011;Maloney et al. 2010;Núñez-Peña and Suárez-Pellicioni 2014;Dowker, Sarkar, and Looi 2016;Hembree 1990;Ma and Xu 2004;Wu et al. 2012). In this sense, the results support the hypothesis from the reciprocal and deficit models of maths anxiety that poor performance leads to increased maths anxiety, but also imply that this relationship is very weak.

Reviewer #1
RC: The current manuscript examines the same kind of evidence, but I believe is considerably stronger than the recent papers I've seen, for a variety of reasons. First, it presents data from a far larger sample, well over 1000 cases, in contrast to the 100 or 200 cases normally tested. Second, it spans a considerably longer time range to examine the longitudinal effect; here the range of interest is prior to the transition from primary to secondary education, around ages 9-12, with the final assessment of math anxiety at age 18 (thus the effective age span tested was age 9 through 18). The authors seem somewhat concerned that their results show rather modest effect after analyzing the data, possibly because other, similar studies have sometimes shown stronger effects. I would suggest that with their more adequate sample size and more thorough age sampling range, they are probably providing a The Table 2 in this revision is a table of correlations for the variables in the models as requested.
RC: Moreover, the linear models are difficult to read and interpret and the suggestion to transform the b values in terms of changes in math anxiety is correct but convoluted. I suggest to report standardized measures in the tables in order to simplify the interpretation of the results. Although many participants were included in the model, math change had a t-value of 1.96, with a p=.05 so it seems to me that the relation with math anxiety was not very high, as the Authors admitted. Anyway, with standardized measures, the interpretation could be easier. Finally, in the models reported in Tables 4 and  5, it is not clear why degrees of freedom changed so much. It can depend on the number of participants, of course, who changed across measures, but it is not clear whether measures were added one after the other or all together.} AR: I respectfully disagree with the suggestion that standardized parameter estimates simplify the interpretation. I think this is largely a matter of opinion rather than anyone being correct. Raw coefficients have the benefit of retaining the original scale of measurement. In the current paper, I believe this makes it easier to see the 'real world' effect of the predictors. There is a clear mapping between a unit change on a predictor and the corresponding change in maths anxiety. What you lose, is the ability to compare predictors (i.e. see the relative effect of one predictor vs another). Baguley (2009) has convincingly (in my opinion) argued that raw effect sizes should always be reported and rarely is the standardized effect size more helpful than the unstandardised one (https://pdfs.semanticscholar.org/86b6/ bef80331f6afbbcb7371bd23ab3abc3ba0b2.pdf_). However, I also want to be responsive to the reviewer. My compromise has been to reproduce Tables 4 and 5 from the main paper in the Supplementary materials but with values deriving from models in which predictors and outcomes were standardized (in other words, the parameter estimates are standardized). This information gives readers the best of both worlds -they can follow the arguments in the paper, but refer to standardized coefficients in the additional materials if they find that helpful to aid their understanding. I hope this is a reasonable compromise.
The MS text has been amended to refer to these tables: Table 4 shows the model parameters for predictors of maths anxiety at age 18 (an equivalent table reporting the same model fitted to standardized scores can be found in the supplementary information). Table 5 shows an unplanned, post hoc, model that quantifies the degree to which biological sex moderated the effects on maths anxiety of maths attainment and SDQ emotional symptoms prior-to and across the school transition. Biological sex did not significantly moderate any of the effects of maths attainment of emotional symptoms on subsequent maths anxiety. An equivalent table reporting the same model fitted to standardized scores can be found in the supplementary information.
The degrees of freedom in the models are a function of the multiple imputation procedure, which is why the values seem strange. I have clarified that all predictors were entered simultaneously in the Data Analysis Plan section of the paper: In phase two, the main model was simply a linear model predicting maths anxiety at age 18 from SES, biological sex, verbal attainment, general anxiety at age 18 and the four variables from the previous models: SDQ emotional symptoms (pre-transition), SDQ emotional symptoms (change), Maths attainment (pre-transition), and Maths attainment (change). All predictors were entered simultaneously. Table 2 shows the correlations between these variables.
RC: The Authors should clarify why also Math pre-transition and SDQ pre-transition were entered into the models. In my view changes in these measures could contribute to explain math anxiety at age 18, but I'm not sure to have well understood the hypothesis related to those measures at age 9.
AR: We have clarified their inclusion at the end of the introduction: With respect to the school environment the transition from primary-to-secondary education represents a significant change in the environment that has a negative impact on many children. In addition, the impact that it has is likely to be worse for children high on trait anxiety. Specifically, it might heighten or maintain high levels of trait anxiety. Therefore, trait anxiety before and after the transition from primary-to-secondary education should also predict future maths anxiety.
RC: In the limits of the study, the Authors should add that their math attainment derived from teachers' ratings but not from objective measures.