These projects are investigating the core foundations which build up conceptual reasoning in science, technology, engineering, and math (STEM) disciplines.
From Brains to Classrooms: Negative Numbers
This interdisciplinary project explores questions of how people acquire fundamentally new mathematical concepts, particularly looking at the role that perceptual-motor functionalities play in abstract mathematics. We focus on the domain of the integers, which builds on the counting numbers to include negative numbers and 0. Drawing on laboratory, classroom, and neuroscience studies, we propose a theory by which learning mathematical concepts occurs by bringing together independent perceptual systems in new ways. For example, for the positive numbers, research shows that our brain's magnitude perception system (which is evolutionarily very old) has been recruited to help us understand the culturally constructed symbolic number system. The integers introduce new mathematical structure on top of the positive numbers - the additive inverse property (x + -x = 0).
Our behavioral and fMRI studies suggest that the perceptual function of symmetry may be recruited to support this new structure and bound with the magnitude system in adult understandings of integers. For example, when adults are asked to find the midpoint of two integers, such as -3 and 7, they get faster as the numbers get closer to symmetric around 0, and areas of the brain related to visual symmetry also increase in activity. Based on these findings, we developed a curriculum for teaching integers that uses hands on materials to focus students attention on symmetry around 0. In a classroom study with fourth graders, we found our symmetry-focused materials helped students to develop mental models that are more effective in integer problem solving and generative in unfamiliar mathematical problem solving situations, compared to more traditional forms of integer instruction.
- Blair, K. P., Tsang, J. M., & Schwartz, D. L. (in press). The bundling hypothesis: How perception and culture give rise to abstract mathematical concepts. To appear in S. Vosniadou (Ed.), International Handbook of Research on Conceptual Change II. New York: Taylor & Francis.
- Schwartz, D. L., Blair, K. P., & Tsang, J. (2012). How to build educational neuroscience: two approaches with concrete instances. British Journal of Educational Psychology Monograph Series, 8.
- Blair, K. P., Rosenberg-Lee, M., Tsang, J., Schwartz, D. L., and Menon, V. (2012). Beyond natural numbers: Representation of negative numbers in the intraparietal sulcus. Frontiers in Human Neuroscience, 6 (7).
- Varma, S., & Schwartz, D. L. (2011). The mental representation of integers: An abstract-to-concrete shift in the understanding of mathematical concepts. Cognition, 121, 363-385.
- Tsang, J.M., & Schwartz, D.L. (2009). Symmetry in the semantic representation of integers. In N. Taatgen, & H. van Rijn (Eds.), Proceedings of the 31st Annual Conference of the Cognitive Science Society (pp. 323-328). Austin, TX: Cognitive Science Society.