Neural representations of transitive relations predict current and future math calculation skills in children
Introduction
The ability to understand and manipulate logical relations has long been thought to be associated with the acquisition of math skills in children. For instance, the pioneering developmental psychologist Jean Piaget famously proposed that the development of math cognition relies on the emergence of logical skills from childhood to adolescence (Piaget, 1952). Educational policies have also often been influenced by the idea that learning math would contribute to the development of logical thinking in children, a claim whose premises can be found in the writings of philosophers such as Plato, Locke, and Bacon (Inglis and Attridge, 2017). To date, however, evidence that math learning is related to logical reasoning remains scarce, most likely because research has largely focused on investigating to what extent math development relies on evolutionally old mechanisms for representing quantities (Feigenson et al., 2004).
Yet, mathematical development is about more than simply processing quantitative information. It is also about understanding relations between those quantities and making inferences based on these relations. For example, a type of logical relation that is prevalent in many math domains is that of transitivity. Transitivity is a property that arises from a set of items that can be ordered along a single continuum (Wright, 2001). A relation is said “transitive” when it allows reasoners to infer a relationship between two items (e.g., A > C) from two other overlapping pairs (e.g., A > B; B > C). The ability to recognize transitive relations and make associated inferences may contribute to the acquisition of many mathematical concepts. For example, transitive reasoning is fundamental to the acquisition of measurement skills in children (Inhelder and Piaget, 1958; Bryant and Kopytynska, 1976; Rabinowitz and Howe, 1994). It also allows for the processing of ordinal and categorical information, which in turn contributes to domains such as arithmetic, algebra and geometry (Bryant and Kopytynska, 1976; Rabinowitz and Howe, 1994; Wright, 2001).
Consistent with a role for transitive reasoning in math learning, a few studies have shown that the ability to understand transitive relations relate to math skills in children. For example, Morsanyi et al. (2013) found that 10-year-olds with math learning disability (i.e., dyscalculia) performed significantly worse than typically developing controls in a transitive reasoning task. In contrast, children with high math ability performed significantly better than typically developing controls on that same task. This is in line with a prior report showing that math performance is generally related to transitive (as well as conditional) reasoning ability in elementary school children (Handley et al., 2004).
What neural mechanisms may underlie the relationship between transitive reasoning and math learning in children? Two lines of evidence suggest that this relationship may be mediated by brain mechanisms involved in relational processing in the posterior parietal cortex (Wendelken, 2015). First, neuroimaging studies indicate that the processing of transitive relations consistently activates the intraparietal sulcus (IPS) (for a review, see Prado et al., 2011a), a region that is also typically involved in tasks that require the manipulation of numbers (Nieder and Dehaene, 2009). Arguably, both transitive reasoning and numerical cognition require individuals to process ordinal information. Therefore, this common reliance on IPS mechanisms may stem from the fact that this region is central to ordinal processing more generally, in line with studies showing that the IPS is also activated when participants represent learned series such as letters, days or months (Fias et al., 2007).
However, it is also possible that transitive relations and numbers are encoded in similar regions of the IPS because they both rely on representations that are inherently spatial and for which the posterior parietal cortex is key (Prado et al., 2010a, 2010b). For example, numbers are often thought to be represented on a mental number line (Hubbard et al., 2005) in adults and children. This is notably suggested by the behavioral distance effect observed in number comparison tasks (i.e., reaction times decrease with the distance between numbers; Moyer and Landauer, 1967), indicating that numbers that are close (e.g., 4 vs 5) are more difficult to distinguish than numbers that are far (e.g., 2 vs 8). This phenomenon has also been reported in transitive reasoning (i.e., reaction times decrease with the distance between items in a transitive ordering task; Prado et al., 2010a, 2010b), suggesting that transitive items are also arranged on a mental representation of space (Alfred et al., 2018). Furthermore, activity in the same region of the IPS has been found to decrease with the distance between numbers (Pinel et al., 2001; Pinel et al., 2004; Mussolin et al., 2010) and transitive items (Prado et al., 2010a), suggesting that transitive relations and numbers engage common spatial representations in the IPS.
A second line of evidence suggesting that the relationship between transitive reasoning and math learning may be mediated by the posterior parietal cortex is provided by a recent neuroimaging study. In that study, we contrasted brain activity of typically developing children to brain activity of children with math learning difficulties during a transitive reasoning task. While transitive relations were associated with IPS activity in typically developing children, that was not the case in children with math learning difficulty (Schwartz et al., 2018). Children with math learning difficulty also showed significantly less activity in the IPS than typically-developing children during the transitive reasoning task. Thus, the extent to which IPS mechanisms process transitive relations may be related to levels of math competence in children.
It is important to note that our previous study only provides indirect correlational evidence that math abilities may be associated with reasoning-related activity in the IPS. However, it raises an intriguing possibility that may strengthen the claim that transitive reasoning plays an important role in math learning: current and future math skills in children might be predicted (at least to some extent) by the neural processing of transitive relations in the IPS. The goal of the present study was to formally test this hypothesis. FMRI activity of 31 typically-developing children from 9 to 13 was measured while they passively listened to transitive (e.g., A > B, B > C) and non-transitive (e.g., A > B, C > D) relations that were embedded in a coherent story that was designed to be as interactive (it took the form of a “choose your own adventure story”) and as engaging as possible (Schwartz et al., 2018). This task allowed us to measure activity associated with transitive relation in a relatively ecological context (i.e., discourse comprehension). Arithmetic and math problem-solving abilities were measured for each child at the time of the fMRI session (T1), as well as 1.5 years later (T2). The predictive power of the neural representations of transitive relations in the IPS on current and future math skills was assessed using a machine learning approach with cross-validation (Gabrieli et al., 2015). Specifically, we evaluated whether multivariate patterns of IPS activity associated with the processing of transitive relations (as compared to non-transitive relations) at T1 could accurately predict math scores of children at both T1 and T2.
Section snippets
Sample size justification
To our knowledge, this is the first study to examine the link between individual differences in math skills and the neural processing of transitive relations in typically developing children. For this reason, and because decoding accuracy in multivariate analyses may not reflect effect sizes (Hebart and Baker, 2018), it is difficult to determine the optimal sample size for the multivariate analyses. However, we note that previous studies that have used multivariate analyses to predict math
ROIs definition
ROIs involved in transitive reasoning were identified by contrasting activity associated with transitive relations to activity associated with non-transitive relations. As shown in Table 3 and Fig. 2, this contrast revealed significant activity in two clusters of the left IPS and one cluster of the right IPS. These clusters defined the voxels that were used in the subsequent univariate and multivariate analyses (see Methods). Specifically, both left IPS clusters constituted the left IPS ROI
Discussion
Over the past decades, developmental research on math learning has largely focused on exploring the role of mechanisms for representing numerical quantities (Ansari, 2008). Yet, acquiring math skills also involves understanding relations between quantities and drawing inferences based on these relations (Singley and Bunge, 2014). In keeping with this idea, recent behavioral studies suggest that understanding relations of transitivity may significantly contribute to math learning in children (
CRediT authorship contribution statement
Flora Schwartz: Methodology, Investigation, Data curation, Formal analysis, Writing - review & editing. Justine Epinat-Duclos: Investigation, Data curation. Jessica Léone: Investigation, Data curation. Alice Poisson: Supervision. Jérôme Prado: Funding acquisition, Conceptualization, Methodology, Supervision, Formal analysis, Writing - review & editing.
Declaration of competing interest
The authors declare no competing financial interests.
Acknowledgments:
This research was supported by a grant from the Agence Nationale de la Recherche (ANR-14-CE30-0002) to J.P. We thank the Hospices Civils de Lyon for sponsoring the research, as well as Romain Mathieu, Auriane Couderc, Inès Daguet, and the MRI engineers (Franck Lamberton and Danielle Ibarrola) at the CERMEP-Lyon platform for their assistance in collecting the fMRI data.
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