Learning to see less than nothing [electronic resource] : symmetry in the mental representation of integers
Keywords: Education.

Four studies were conducted on the representation and learning of the integers, which are positive and negative whole numbers and zero. Broadly, the studies explored the relationship between perceptual capacities and abstract mathematical knowledge. Specifically, we asked whether it is possible that a sense of perceptual symmetry contributes to people's mental representations of integers. Symmetry is a quality that people readily identify in their surroundings, and it can describe an important relational property of the integers, the additive inverse, X + -X = 0. However, it is questionable whether people can use perceptual symmetry to support integer reasoning because negative integers stand for quantities that are, in some sense, not perceivable (one cannot see negative three flowers). The first three studies investigated perceptual symmetry in integer representations using a mental integer bisection task. Adult participants reported the midpoint between two integers, and their response times and functional Magnetic Resonance Imaging (fMRI) brain responses were measured. The bisection pairs varied in proximity to symmetric across zero. Results indicated that people were fast to bisect integer pairs when the pairs were perfectly symmetric across zero (e.g., [-7, 7]) and nearly symmetric (e.g., [-6, 8]), and slower to solve problems that were far from symmetric (e.g., [-3, 11]). Furthermore, a network of brain regions increased response level as participants sped up with proximity to symmetry, suggesting that these regions "came online" to support integer bisection when problems were nearly symmetric. One of the regions found is located close to areas previously reported to be sensitive to visual symmetry. Collectively, the results suggested that adults incorporate the symmetry of the integers into their mental representations, and that this facilitates integer operations when symmetry is salient in the problem. Thus, people may be able to harness their perceptual capacities in service of reasoning with even abstract numbers. The fourth study tested the implications of the initial experiments for integer learning and instruction. Symmetry is not commonly taught in integers curricula in schools. If adults show evidence of symmetry in their integer representations, does this imply that symmetry emphasis in integers instruction could benefit students' learning? Fourth grade students (9-10 years old) who had received no prior instruction in the integers were assigned to one of three instructional treatments. Two were based on common models of integer instruction and did not include mention of symmetry. The third was also based on common models of integer instruction, but additionally incorporated an explicit symmetry framing for the integers. The results showed that students in all three treatment conditions learned to execute basic symbolic calculations with integers. However, students in the symmetry-framed condition (1) chose symmetry-oriented strategies for integer problems more than the other conditions, (2) performed better on problems that required them to generalize their integer knowledge to new contexts, and (3) showed an interference effect on a response-time task that suggested symmetry had become a salient feature in their integer mental models. Thus, the symmetry-framed curriculum helped students to incorporate symmetry structure into their mental models of integers, and this helped the students to interpret and approach familiar and unfamiliar problem solving situations. The four studies show that symmetry is incorporated in the mental representation of integers, and that students benefit from instruction specifically targeting the symmetric structure of the numbers. More broadly, the results suggest that perception can contribute to thinking and learning about higher levels of mathematics than the natural numbers.