How can children's natural perceptuo-motor skills be harnessed for teaching and learning mathematical structure? We address this question in the case of the integers. Existing research suggests that adult mental representations of integers recruit perceptuo-motor functionalities involving symmetry. Building on these findings, we designed a hands-on curriculum that emphasizes symmetry to teach integer concepts to fourth graders. Compared to two control conditions, children who went through the experimental curriculum showed evidence of incorporating symmetry into their mental representations of integers and performed higher on problems beyond the scope of instruction, including negative fractions and algebra-readiness problems. Gains did not come at the expense of basic integer computation skill. This study has direct practical implications, as current integers curricula generally omit symmetry. The research demonstrates an approach to designing instruction that involves identifying perceptuo-motor functionalities underlying numerical cognition and creating learning activities to recruit them.

Unlike natural numbers, negative numbers do not have natural physical referents. How does the brain represent such abstract mathematical concepts? Two competing hypotheses regarding representational systems for negative numbers are a rule-based model, in which symbolic rules are applied to negative numbers to translate them into positive numbers when assessing magnitudes, and an expanded magnitude model, in which negative numbers have a distinct magnitude representation. Using an event-related functional magnetic resonance imaging design, we examined brain responses in 22 adults while they performed magnitude comparisons of negative and positive numbers that were quantitatively near (difference <4) or far apart (difference >6). Reaction times (RTs) for negative numbers were slower than positive numbers, and both showed a distance effect whereby near pairs took longer to compare. A network of parietal, frontal, and occipital regions were differentially engaged by negative numbers. Specifically, compared to positive numbers, negative number processing resulted in greater activation bilaterally in intraparietal sulcus (IPS), middle frontal gyrus, and inferior lateral occipital cortex. Representational similarity analysis revealed that neural responses in the IPS were more differentiated among positive numbers than among negative numbers, and greater differentiation among negative numbers was associated with faster RTs. Our findings indicate that despite negative numbers engaging the IPS more strongly, the underlying neural representation are less distinct than that of positive numbers. We discuss our findings in the context of the two theoretical models of negative number processing and demonstrate how multivariate approaches can provide novel insights into abstract number representation.

Unlike natural numbers, negative numbers do not have natural physical referents. How does the brain represent such abstract mathematical concepts? Two competing hypotheses regarding representational systems for negative numbers are a rule-based model, in which symbolic rules are applied to negative numbers to translate them into positive numbers when assessing magnitudes, and an expanded magnitude model, in which negative numbers have a distinct magnitude representation. Using an event-related fMRI design, we examined brain responses in 22 adults while they performed magnitude comparisons of negative and positive numbers that were quantitatively near (difference < 4) or far apart (difference > 6). Reaction times for negative numbers were slower than positive numbers, and both showed a distance effect whereby near pairs took longer to compare. A network of parietal, frontal, and occipital regions were differentially engaged by negative numbers. Specifically, compared to positive numbers, negative number processing resulted in greater activation bilaterally in intraparietal sulcus (IPS), middle frontal gyrus, and inferior lateral occipital cortex. Representational similarity analysis revealed that neural responses in the IPS were more differentiated among positive numbers than among negative numbers, and greater differentiation among negative numbers was associated with faster reaction times. Our findings indicate that despite negative numbers engaging the IPS more strongly, the underlying neural representation are less distinct than that of positive numbers. We discuss our findings in the context of the two theoretical models of negative number processing and demonstrate how multivariate approaches can provide novel insights into abstract number representation.

Educational neuroscience is an emerging discipline, but it is not a uniform endeavor. There are different ways for it to make progress. We describe two broad approaches, which we agnostically label Culture A and Culture B. Culture A is currently the more frequent approach. It relies on individual differences to advance the science with a special emphasis on solving the challenges faced by learners with special needs. Culture B is less common. It examines the effects of contextual variables on typical learners to make headway at solving theoretical problems in education and improving general instruction. Both are valuable and both seek to improve education. By describing their differences, along with concrete examples of their logic, findings, and cultures of work, we hope to help both neuroscientists and educators answer a key question about one another’s work, “Why do they find that worth doing?”

Symmetry in the Semantic Representation of Integers Jessica M. Tsang (jmtsang@stanford.edu) Stanford University School of Education, 485 Lasuen Mall Stanford, CA 94305 USA Daniel L. Schwartz (danls@stanford.edu) Stanford University School of Education, 485 Lasuen Mall Stanford, CA 94305 USA One possibility is that the new structure is carried symbolically by a set of manipulation rules and categories (e.g., if two digits are the same absolute amount and one has a negative sign, then they are symmetric). A recent study by Varma and Schwartz (2009) suggests that this is not the case for adults. The authors compared seventh graders to adults on a number comparison task. Seventh graders showed no symbolic distance effect when comparing a positive and negative number. The absence of this robust marker of analog representation suggests the children used a rule like “a positive is larger than a negative” to compare the numbers. On the other hand, adults showed an „inverse‟ distance effect on these problems – they answered far comparisons slower than near comparisons – indicating that they had developed a semantic representation of integers. Given that adults use a semantic representation to operate Abstract The integers include more structure than the natural numbers; for example, they exhibit symmetry about zero. Do adult mental representations directly encode this increased structure? In two studies, adults completed a numerical bi- section task in which they were presented with two symbolic integers and were asked to report the digit at the midpoint of the interval. The reaction times demonstrated a “tuning curve” such that people were faster when the midpoint or end point of an interval was close to zero. The results suggest that the mental representation of negative numbers and zero has incorporated analog properties to represent the increased structure of the integers. Keywords: analog representation; mathematics; bisection; symmetry; integers Introduction The positive integers, or natural numbers, have ready perceptual referents; for example, six divided by three can be materialized in the world as six cookies shared between three people. The structural properties of natural numbers (e.g., ordinality, cardinality, magnitude) can be gleaned from applying numbers to physical situations (Griffin, Case, & Siegler, 1994). This may help explain why people have an analog representation of natural numbers, as indicated by the symbolic distance effect (Moyer & Landauer, 1967) and more recent brain evidence (Piazza, Izard, Pinel, Le Bihan, & Dehaene, 2004). In contrast, negative integers are abstract entities that do not map readily to tangible things; negative three cookies are hard to imagine, and a negative times a negative is typically handled symbolically in school curricula. Historically, negative numbers were considered fictitious even when effectively used in calculations (Schwarz, Kohn, & Resnick, 1993/1994). Schwarz, et al. (1993/1994) say, “It was only in the 19 th century that negative numbers emerged as directed magnitudes (e.g., in the domain of electricity) and that the set of integers was axiomatically defined in such a way as to give negatives a symmetric status to that of positives.” The full set of integers includes greater quantitative structure than the natural numbers; for example, it includes zero as an identity; and it is possible for quantities to exhibit symmetry. How do people represent the increased structure of integers? Figure 1: Protocol for bisection task. A) Study 1. B) Study 2.

We concur with Cohen Kadosh & Walsh (CK&W) that representation of numbers in the parietal cortex is format dependent. In addition, we suggest that all formats do not automatically, and equally, access analog magnitude representation in the intraparietal sulcus (IPS). Understanding how development, learning, and context lead to differential access of analog magnitude representation is a key question for future research.

Functional MRI studies of mental arithmetic consistently report blood oxygen level–dependent signals in the parietal and frontal regions. We tested whether white matter pathways connecting these regions are related to mental arithmetic ability by using diffusion tensor imaging (DTI) to measure these pathways in 28 children (age 10–15 years, 14 girls) and assessing their mental arithmetic skills. For each child, we identified anatomically the anterior portion of the superior longitudinal fasciculus (aSLF), a pathway connecting parietal and frontal cortex. We measured fractional anisotropy in a core region centered along the length of the aSLF. Fractional anisotropy in the left aSLF positively correlates with arithmetic approximation skill, as measured by a mental addition task with approximate answer choices. The correlation is stable in adjacent core aSLF regions but lower toward the pathway endpoints. The correlation is not explained by shared variance with other cognitive abilities and did not pass significance in the right aSLF. These measurements used DTI, a structural method, to test a specific functional model of mental arithmetic.

The integers include more structure than the natural numbers; for example, they exhibit symmetry about zero. Do adult mental representations directly encode this increased structure? In two studies, adults completed a numerical bi- section task in which they were presented with two symbolic integers and were asked to report the digit at the midpoint of the interval. The reaction times demonstrated a "tuning curve" such that people were faster when the midpoint or end point of an interval was close to zero. The results suggest that the mental representation of negative numbers and zero has incorporated analog properties to represent the increased structure of the integers.