The relationship between executive functions (EF) and academic achievement is wellestablished, but leveraging this insight to improve educational outcomes remains elusive. Here, we propose a framework for relating the role of specific EF on specific precursor skills that support later academic learning. Starting from the premise that executive functions contribute to general math skills both directly -supporting the execution of problem solving strategies -and indirectly -supporting the acquisition of precursor mathematical content, we hypothesize that the contribution of domain-general EF capacities to precursor skills that support later learning can help explain relations between EF and overall math skills. We test this hypothesis by examining whether the contribution of inhibitory control on general math knowledge can be explained by inhibition's contribution to processing rational number pairs that conflict with individual's prior whole number knowledge. In 97 college students (79 female, age = 20.58 years), we collected three measures of EF: working memory (backwards spatial span), inhibition (color-word Stroop) and cognitive flexibility (task switching), and timed and untimed standardized measures of math achievement. Our target precursor skill was a decimals comparison task where correct responses were inconsistent with prior whole number knowledge (e.g., 0.27 vs. 0.9). Participants performed worse on these trials relative to the consistent decimals pairs (e.g., 0.2 vs. 0.87). Individual differences in the Stroop task predicted performance on inconsistent decimal comparisons, which in turn predicted general math achievement. With respect to relating inhibitory control to math achievement, Stroop performance was an independent predictor of achievement after accounting for age, working memory and cognitive flexibility, but decimal performance mediated this relationship. Finally, we found inconsistent decimals performance mediated the relationship of inhibition with rational number performance, but not other advanced mathematical concepts. These results pinpoint the specific contribution of inhibitory control to rational number understanding, and more broadly are consistent with the hypothesis that acquisition of foundational mathematical content can explain the relationships between executive functions and academic outcomes, making them promising targets for intervention.

Current research in the number development field has focused in individual differences regarding the acuity of children's approximate number system (ANS). The most common task to evaluate children's acuity is through non-symbolic numerical comparison. Efforts have been made to prevent children from using perceptual cues by controlling the visual properties of the stimuli (e.g., density, contour length, and area); nevertheless, researchers have used these visual controls interchangeably. Studies have also tried to understand the relation between children's cardinality knowledge and their performance in a number comparison task; divergent results may in fact be rooted in the use of different visual controls. The main goal of the present study is to explore how the usage of different visual controls (density, total filled area, and correlated and anti-correlated area) affects children's performance in a number comparison task, and its relationship to children's cardinality knowledge. For that purpose, 77 preschoolers participated in three tasks: (1) counting list elicitation to test whether children could recite the counting list up to ten, (2) give a number to evaluate children's cardinality knowledge, and (3) number comparison to evaluate their ability to compare two quantities. During this last task, children were asked to point at the set with more geometric figures when two sets were displayed on a screen. Children were exposed only to one of the three visual controls. Results showed that overall, children performed above chance in the number comparison task; nonetheless, density was the easiest control, while correlated and anti-correlated area was the most difficult in most cases. Only total filled area was sensitive to discriminate cardinal principal knowers from non-cardinal principal knowers. How this finding helps to explain conflicting evidence from previous research, and how the present outcome relates to children's number word knowledge is discussed.

Children can successfully compare continuous proportions as early as age 4, yet struggle to compare discrete proportions least to age 10, especially when the discrete information is misleading. This study examined whether inhibitory control contributes to individual differences in discrete proportional reasoning and whether reasoning could be enhanced by priming continuous information. Forty-nine second-graders completed two tasks. In the Hearts and Flowers (H&F) task, a measure of inhibition, children pressed either on the corresponding or opposite side depending on the identity of the displayed figure. In the Spinners task, a measure of proportional reasoning, children chose the spinner with the proportionally larger red area, across continuous and two discrete formats. In the discrete adjacent format, the continuous stimuli were segmented into sections, which could be compatible with the proportional information or misleading; the discrete mixed format interspersed the colored sections from the discrete adjacent conditions. Finally, two priming groups were formed. Children who saw the continuous immediately before the discrete adjacent format formed the Continuous-priming group (n =26). Children who saw discrete mixed immediately before the discrete adjacent format formed the Discrete-priming group (n =23). Our results showed that children who performed better in the H&F task also had better performance on the discrete counting misleading trials. Furthermore, children in the Continuous-priming group outperformed children in the Discrete-priming group, specifically in contexts where discrete information was misleading. These results suggest that children's proportional reasoning may be improved by fostering continuous representations of discrete stimuli and by enhancing inhibitory control skills.

Are mathematical learning difficulties caused by impairment of the abilities that underlie mathematical talent? Or are mathematical difficulties and talent qualitatively different? The main goal of this study was to determine whether mathematical learning difficulties are explained by the same executive functions as mathematical talent. We screened a pool of 2,682 first-year high school students and selected 48 for evaluation, dividing them into three groups: those with mathematical learning difficulties (n = 16), those with typical performance (n = 16), and those with mathematical talent (n = 16). Adolescents from the learning difficulties and talented groups had age, reading skills, and verbal and non-verbal intelligence that were similar to those of the typical performance group. Participants were administered a suite of tasks to evaluate verbal and visual short-term memory and executive functions of inhibition, shifting, and updating. Different executive functions showed different contributions at the two ends of the math ability continuum: lower levels of performance in updating visual information were related to mathematical learning difficulties, while greater shifting abilities were related to mathematical talent. Effect sizes for the differences in performance between groups were large (Hedges' g > 0.8). These results suggest that different executive functions are associated with mathematical learning difficulties and mathematical talent. We discuss how these differences in executive functions could be related to the different types of mathematical abilities that distinguish the three groups.