Mathematical aptitude in adolescence

By the end of middle school, adolescents in Mexico should have developed a solid knowledge of the arithmetic of natural and rational numbers and should have started to develop their understanding of basic concepts of algebra and statistics [9]. However, first-year high school students still appear to lag behind these expectations [10,11]. Recently, Abreu-Mendoza, Chamorro, and Matute [10] reported that first-year Mexican high school students still have difficulties in solving arithmetic problems that involve fractions and decimals. Using a large sample (N = 1236), they performed an exploratory factor analysis on the 40 items of the Math Computation subtest of the Wide Range Achievement Test-4 (WRAT-4) [12]. A solution was identified with three factors: the first was the addition, subtraction, multiplication, and division of multi-digit numbers with and without decimals (arithmetic); the second was the solution of arithmetic problems of like fractions and simple equations (fractions and basic algebra); and the third was the conversion of decimals to percentages or fractions, and the solution of arithmetic problems involving fractions with different denominators (rational numbers). Although these abilities should have been consolidated at this stage of education, none of these factors showed a percentage of correct responses above 62%. Such difficulties are not exclusive to Mexican students. At roughly the same age, only a small percentage (27%) of U.S. eighth graders correctly estimate the sum of 12/13 + 7/8 [13]. Taken together, these studies suggest that individual differences in mathematical abilities during adolescence are found mostly in arithmetic problems that involve fractions and decimals.

Executive functions in adolescence

The executive functions are general purpose mechanisms that modulate the operation of various cognitive subprocesses [14] and allow an organized, purposeful, and goal-directed behavior [15]. Various models have described the abilities that fall under the executive function umbrella term. Miyake et al. [14] proposed "shifting," "inhibition," and "updating" as three basic, separated, yet related executive components that can be accurately operationalized and that are implicated in the performance of more complex executive behaviors. Shifting concerns the ability to switch from one mindset to another, often according to rules that would be incompatible with the other [16]. Updating involves monitoring and coding incoming information for relevance to a given task [17] and is closely related to the concept of working memory. Inhibition is the ability to deliberately suppress dominant, automatic, or prepotent responses when necessary [14].

Many studies have adopted this multifactorial framework. However, this model of executive functions is based on adult data, and it is not clear if it also describes developmental stages [18], especially when executive functions follow a protracted developmental path from childhood to adulthood [19]. In this regard, Lee et al. [18] reported that a two-factor model (combining inhibitory control and switching measures) provides a better fit during childhood (six to ten years of age) than the three-factor structure. The former is feasible until 11 years of age, and the three executive components became more differentiated around 15 years of age. A recent systematic review and re-analysis of confirmatory factor analysis-based models demonstrated a higher acceptance of parsimonious models (1 or 2 factors) in childhood and early adolescence [20]. Thus, late adolescence seems to be a good developmental stage to assess the three components proposed by Miyake and colleagues [14,21] in relation to academic achievement.

Mathematical aptitude and executive functions

Numerous studies have shown the importance of executive functions to the acquisition of mathematical skills. Performance in executive function tasks is associated with individual differences in numerical abilities from early childhood [7,8,22], and through childhood [23–25], adolescence, and adulthood [26–28].

Executive functions have also been linked to difficulties in acquiring such abilities. Individuals with "weak mathematical performance of developmental origin" [29] have impairments in verbal and visual working memory [30–33], inhibition [33], and shifting [24]. Following the traditional model of working memory [34], some studies have focused on comparing whether individuals with dyscalculia have impaired verbal or visual working memory that leads to different results. For instance, Ashkenazi, Rosenberg-Lee, Metcalfe, Swigart, and Menon [35] reported that children with mathematical disabilities have preserved verbal working memory skills, but also have deficits in visuospatial working memory that can be observed at both the cognitive and functional neuroanatomical levels. Moreover, some authors have suggested that visual working memory is a predictor for math abilities, but not for reading skills [36]. More recent studies have begun to investigate the full domain of executive functions. For instance, difficulties in performing arithmetic operations have been associated with impaired working memory and inhibition: working memory serves as a mental workspace for the manipulation of operators, operands, and numerical facts, and inhibition could suppress irrelevant information [33]. Working memory has also been linked with fact retrieval, another major impairment in children with MLD, which determines how much information is rehearsed and could also influence how much of this information is consolidated as long-term memory representation [37].

However, there is a need for further study of the link between executive functions and mathematical talent. For instance, indirect evidence suggests that high performance in math is related to an ability to alternate efficiently between strategies. By comparing the brain activity patterns of an expert calculator and six non-expert adults performing multi-digit multiplications, Pesenti and colleagues [38] found that experts switched between brain areas involved in computing and those involved in fact retrieval. More recently, Swanson [39] compared the performance of children scoring 1.5 SD above the mean on two standardized mathematical tests and those with average scores on a series of tests including measures of short-term memory, working memory, and inhibition. In his study, mathematical talent was associated with a latent variable representing working memory, which included the digit/sentence, backward digit, listening/sentence, and updating tasks. In a review of current literature on the cognitive correlates of mathematical talent, Myers, Carey, and Szucs [40] concluded that mathematical talent is associated with spatial processing, working memory, efficient strategy switching, and inhibition. Overall, working memory has been the component of executive function consistently found to contribute to both ends of the distribution of mathematical abilities [18,40].

The study

Are mathematical learning difficulties and mathematical talent related to the same underlying abilities? That is, are these two categories the lower and upper ends of the normal distribution of mathematical aptitude? Or are they discontinuous, qualitatively different cases? To study mathematical learning difficulties, researchers have traditionally taken the disability/ability approach, where difficulty is understood as the result of an impaired cognitive ability (e.g., number sense, working memory) or the malfunction of a specific brain region or regions [41–43]. Recent approaches have suggested that different levels of the same abilities whose deficits are responsible for learning disabilities may also explain the normal variation [30,44]. Here, we take a step further and suggest that a heightened degree of these same abilities could also be responsible for exceptional mathematical aptitude.

To obtain empirical support for this idea, we evaluated three groups of adolescents with distinct mathematical achievement levels: a mathematical learning difficulties (MLD) group, a typical performance (TP) group, and a mathematical talent (MT) group. Participants in the three groups had similar reading speeds, verbal and non-verbal intelligence, and processing speeds. They were evaluated with a suite of executive function tasks (verbal and visuospatial working memory, inhibition, and shifting) and with verbal and visuospatial short-term memory tasks. Given the associations found between mathematical aptitude or learning difficulties and individual differences in verbal and visual working memory, we predicted that working memory would contribute to both ends of the distribution: that is, that adolescents in the MT group would outperform those in the TP group, who would in turn outperform those in the MLD group. Fig 1 shows the hypothetical patterns for the contributions of different executive functions to mathematical performance. Fig 1C provides a visual representation of our predicted pattern for the measures of working memory: effect sizes for both pairwise comparisons (MLD vs. TP and TP vs. MT) will be significant.

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Fig 1. Patterns of hypothetical contributions of executive functions (EF) to the whole continuum of the mathematical abilities.

(A) Contribution to Mathematical Talent (MT), (B) contribution to Mathematical Learning Difficulties (MLD), (C) contributions to both ends of math performance, and (D) no contributions.

https://doi.org/10.1371/journal.pone.0209267.g001

Participants

A total of 2,682 first year high school students aged 14 years to 16 years 11 months, from three public schools in Guadalajara, Mexico, were screened for mathematical abilities using the WRAT Math Computation subtest [12]. A subsample of 74 students and their parents gave written informed consent for an expanded assessment. This evaluation included the Vocabulary (verbal intelligence), Matrix Reasoning (non-verbal intelligence), and Arithmetic subtests from the Mexican version of the WISC-IV [45]; the Written Math and the Reading a Text Aloud subtests from the Evaluación Neuropsicológica Infantil (ENI) [Neuropsychological Assessment for Children] [46]; and a suite of executive function and short-term memory tasks. Forty-eight adolescents met criteria for one of three groups of interest that formed the final sample: MLD (n = 16, mean age = 16.38 [SD = 0.69], 56% girls), TP (n = 16, mean age = 16.28 [SD = 0.99], 56% girls), and MT (n = 16, mean age = 16.28 [SD = 0.92], 50% girls). The groups were similar in their mean age: F(2, 45) = 0.75, p = .92, η² = .003.

For the MLD group, the criteria included: (a) a score at least 1.5 SD below the mean for Mexican first-year high school students [10] on the WRAT Math Computation subtest, and (b) a score below the 10th percentile on the ENI Written Math subtest. For the TP group, the criteria included (a) a score within the range of ± 1 SD of the Mexican standard mean on the WRAT Math Computation subtest, and (b) a score above the 10th percentile on the ENI Written Math subtest. Finally, for the MT group, the criteria included (a) a score ≥ 1.5 SD above the Mexican standard mean on the WRAT Math Computation subtest, and (b) a scaled score above the 90th percentile on the ENI Written Math subtest. Participants from all groups were required to have (c) a scaled score ≥ 7 on the WISC Vocabulary subtest, (d) a scaled score above the 10th percentile on the ENI Reading a Text Aloud subtest, and they were required (e) not to meet the diagnostic criteria for ADHD based on a parent questionnaire [47].

The TP group was used as the comparison group: adolescents from MLD and MT groups had similar age, reading skills, and verbal and non-verbal intelligence as those in this group. Fig 2 summarizes performance of each group for the standardized subtests. To further describe their math abilities, Table 1 shows the mean correct responses of each group on the WRAT Math Computation subtest and the mean proportion of correct responses for each of the three factors of this subtest described in Abreu-Mendoza et al. [10]. Across these factors, the largest significant difference between the MLD and TP groups was in the Arithmetic factor, while the largest difference between the MT and the TP groups was in the Rational Numbers factor.

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Fig 2. Mean scaled scores for each group of criterion subtests.

Group means and 95% confidence intervals are shown. Below the X-axis, effect sizes (Hedges' g) are given for the difference between the pairwise comparisons across groups. MLD = Mathematical learning difficulties, TP = Typical performance, MT = Mathematical talent. * p < .05, *** p < .001.

https://doi.org/10.1371/journal.pone.0209267.g002

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Table 1. Mean proportion (standard deviation) of correct responses on the WRAT math computation subtests by group, and effect size (Hedges' g) for the differences between groups.

https://doi.org/10.1371/journal.pone.0209267.t001

The University of Guadalajara ethics committee approved this research study (approval number ET112015-205), which included verification that the consent form, procedures, and methods were in compliance with the Helsinki principles.

Experimental tasks

To cover the whole range of executive functions proposed by Miyake and colleagues [14,48] and both components of working memory (updating and capacity) in its two modalities (verbal and visual), we used established tasks as outcome measures of these functions.

Procedure

Criterion subtests were administered in the following fixed order: the Vocabulary, Matrix Reasoning, and Arithmetic subtests of the WISC-IV, followed by the Reading a Text Aloud and Written Math subtests of the ENI; they were administered either in participants' schools or in our lab. Those taking the criterion subtests at their schools performed the experimental tasks in a second session in our lab; those taking the tests in the lab were given a five-minute break before the experimental tasks. The tasks were presented in the following order: digit span, RT, Corsi-block tapping tasks; the letter n-back, go/no-go, visual n-back, and local/global tasks were then presented in a counterbalanced order without presenting the n-back tasks consecutively. The RT task was programmed and presented using E-prime 2.0, and the last four tasks were programmed using Psychtoolbox running under MATLAB. All computerized tasks were presented on a 23-inch LED monitor and participants were seated approximately 65 cm from the monitor. The administration of the criterion subtests lasted approximately 40 minutes, and the experimental tasks lasted about an hour. At the end of the session, families received a travel reimbursement of approximately $10 USD.

Statistical analyses

After confirming the absence of multivariate outliers using the Mahalanobis distance and verifying homogeneity with Levene's test (Levene's p > .001), we performed statistical analyses for each of the evaluated cognitive constructs: reaction time, short-term memory, and executive functions. For the simple reaction time task, we performed a one-way ANOVA with Group (MLD, TP, and MT) as between-subjects factor.

To compare performance across the three groups on the two short-term memory and six executive function tasks, we performed two independent, one-way MANOVAs with Group (MLD, TP, and MT) as between-subjects factor. In a first step, we tested the MANOVA assumption of correlation between the dependent variables by performing a series of Pearson correlations with the scores of each of the two types of tasks. The second step was to introduce these scores as the dependent variables of the one-way MANOVAs with Group as the between-subjects factor. In the third step, we conducted a series of follow-up ANOVAs to determine the subtest scores in which the main effect was present. Finally, for ANOVAs that were significant after correcting for multiple comparisons using Holm correction, we performed least significant difference (LSD) post-hoc tests between the MLD and TD groups, and between the MT and TD groups.

Mindful of recent criticism of the use of p-values [53], we also calculated standardized effect sizes and bootstrapped confidence intervals (CIs). Confidence intervals not only indicate whether a result is statistically significant, they also communicate the precision of the effect size and can be considered significant if they do not include zero. Parametric methods for calculating CIs make several assumptions about the shape of the data distribution. To avoid these assumptions, bootstrap methods approximate the distribution from the data itself by repeatedly drawing samples, with replacement, from the original data sample. Here we calculated bootstrapped, bias-corrected and accelerated (BCa) confidence intervals using the bootES package [54] of the statistical programming language R, utilizing 100,000 permutations with replacement.

We used Hedges' g as a measure of effect size, as it is more appropriate for the estimation of unbiased effect sizes in studies with small sample sizes (n < 20) than Cohen's d [53,55]. As Holm correction cannot be used to determine adjusted confidence intervals, we used a Bonferroni correction, which resulted in 97.5% and 99.375% CIs for the short-term memory and executive function tasks, respectively. Depending on the task, effect sizes (Hedges' g) were interpreted as being significant if the 97.5% or the 99.375% bootstrap CIs excluded zero. Using similar guidelines as those for Cohen’s d [56], Hedges’ g values of 0.2, 0.5, and 0.8 can be considered small, medium, and large effects, respectively.

A post hoc sensitivity power analysis showed that our sample had sufficient power (β = .80) at a significance level of ɑ = .05 to detect medium to large effect sizes (f ≥ .463) in a one-way ANOVA, and to detect large effect sizes (Cohen's d ≥ 1.02) in an independent sample t-test. Power analysis was done with G-Power [57]; all statistical analyses were performed using R [58].

Simple reaction time task

The ANOVA showed no significant effect of Group on reaction times: F(2, 45) = 1.46, p = 0.24, η² = 0.06.

Short-term memory tasks

Table 2 shows the descriptive statistics for these tasks. The MANOVA did not show a main effect of group: Pillai's Trace = 0.17, F(4, 90) = 2.12, p = .08, η² = .10.

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Table 2. Mean (standard deviation) for the simple RT task and the two short-term memory tasks.

https://doi.org/10.1371/journal.pone.0209267.t002

Executive function tasks

The MANOVA yielded a main effect of Group: Pillai's Trace = 0.74, F(16, 78) = 2.87, p = .001, η_p² = 0.45. Univariate ANOVAs were used to examine individual dependent variable contributions to main effects. After controlling for multiple comparisons, only the performance of the go/no-go task, F(2, 45) = 5.34, p = .0499, η_p² = 0.19, the visual 2-back condition, F(2, 45) = 15.55, p < .001, η_p² = 0.41, and the local global task, F(2, 45) = 5.61, p = .047, η_p² = 0.20, showed a main effect of group. Detailed statistics for each of the ANOVAs are shown in Table 3.

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Table 3. Mean Matthews correlation coefficient and mean accuracy (%) with 97.5% confidence intervals for executive function tasks.

https://doi.org/10.1371/journal.pone.0209267.t003

In the go/no-go task, least significant difference (LSD) post hoc analyses for the planned pairwise group comparisons showed that the TP group (M = 0.86, SD = 0.07) outperformed the MLD group (M = 0.79, SD = 0.11, p = 0.03). In the visual 2-back task, the TP group (M = 0.32, SD = 0.11) also outperformed the MLD group (M = 0.13, SD = 0.16, p < .001). Finally, in the local/global task, the MT group (M = 0.89, SD = 0.05) outperformed the TP group (M = 0.94, SD = 0.03, p = .04).

Effect sizes and bootstrapped confidence intervals

Figs 5 and 6 summarize the effect sizes (Hedges' g) and their bootstrapped confidence intervals of the group differences (TP minus MLD and MT minus TP) in each of the short-term memory and executive function tasks. Consistent with the parametric tests, confidence intervals did not include zero for the effect sizes for the differences between the TP and MT groups in the local/global task or the differences between the MLD and TP groups in the visual 2-back task. According to Cohen's criteria [56], these effect sizes can be considered large. These effect sizes remained significant when outliers (> 2 SD from the groups’ mean) were removed from the analyses: visual 2-back task, MLD (n = 15) vs. TP (n = 15) Hedges' g = 1.22 [0.16, 2.18]; local/global task, TP (n = 15) vs. MT (n = 15) Hedges’ g = 1.17 [0.23, 2.05]. Furthermore, regression line analyses, using the Math Computation scores as a continuous variable, showed a similar pattern: the slope for the visual 2-back task scores was significant only for the Math Computation scores within the range of the MLD and TP groups (see Fig A, upper panel, in S1 Appendix), while the slope for the local/global task scores was significant only for the Math Computation scores within the range of the TP and MT groups (see Fig A, lower panel, in S1 Appendix).

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Fig 5. Group differences (expressed in Hedges' g) for the two short-term memory subtests: Digit subtest (forward condition) and corsi block tapping test (forward condition).

Error bars represent the adjusted 97.5% bootstrapped CI (only the lower level is shown). Effect sizes are significant if error bars do not include zero. MLD = Mathematical learning difficulties, TP = Typical performance, MT = Mathematical talent.

https://doi.org/10.1371/journal.pone.0209267.g005

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Fig 6. Group differences (expressed in Hedges' g) for the six executive functions tasks.

Error bars represent the adjusted the 99.375% bootstrapped CI (only the lower level is shown). Effect sizes are significant if error bars do not include zero. MLD = Mathematical learning difficulties, TP = Typical performance, MT = Mathematical talent. * p < .05.

https://doi.org/10.1371/journal.pone.0209267.g006

It is noteworthy that even though the p-value for the differences between the TP and MT groups in the local global task was relatively large (p = .04), the effect size for this difference can be considered large (Hedges' g = 1.17). As p-values are more vulnerable to sample size than effect sizes, we believe the group differences are best interpreted based on the latter, as well as on confidence intervals. Finally, even though the effect size for the outperformance of the TP group relative to the MLD group in the go/no-go task was medium (Hedges' g = 0.661 [-0.32, 1.54]), its confidence interval included zero, suggesting that this result has low precision.