Title: What it means to understand mathematics, and why some people are better at it than others
Abstract: The question of how humans understand mathematics is central to cognitive science. Most research to date has been restricted to the mental representation and processing of the natural numbers, including arithmetic operations defined on this concrete number class. This talk will describe how my research program is pushing beyond the current boundary. The first line investigates how people use symbol systems to understand more abstract number classes: the integers, rationals, and irrationals. It reveals how these symbol systems bring structure to evolutionarily ancient and intuitive semantics for number and magnitude. The second line explores forms of mathematical thinking that are critical for success in STEM fields. It offers definitions of mathematical intuition and mathematical insight, exhibits paradigms for studying these forms of thinking in the lab, and provides evidence that individual differences in intuition and insight predict individual differences in mathematical achievement (e.g., ACT-Math scores). The third line of research is building a bridge between mathematical thinking and algorithmic thinking. These ongoing studies are investigating how people reason about polynomial and exponential expressions, and also how spatial referents and motor actions can support understanding of inductive proofs. Along the way, I offer theoretical proposals and formal models to explain the empirical findings, and also draw the implications of these findings for the learning sciences.
Bio: Dr. Varma earned his BS in Mathematics and Cognitive Science from Carnegie Mellon University, his PhD in Cognitive Psychology (minor in Computer Science) from Vanderbilt University, and completed a post-doctoral fellowship in the Center for Innovations in Learning at Stanford University. He is currently an Associate Professor of Learning and Cognition in the Department of Educational Psychology at the University of Minnesota. His research investigates those complex forms of cognition that are uniquely human – and indeed make us human – from multiple disciplinary perspectives. His primary line of research is in mathematical cognition, where he investigates how people use symbols systems to understand abstract mathematical concepts, how they develop intuitions about and insights into mathematics, and the mental mechanisms shared between mathematical reasoning and algorithmic thinking. His secondary line of research focuses on the development of cognitive architectures. These are unified theories of the mind – of its basic representations, processes, and control structures – expressed as programming languages. He co-developed the CAPS family of architectures and, within these architectures, models of language comprehension, problem solving, visuospatial reasoning, and multitasking that explain behavioral and brain-imaging data collected from typical adults and neuropsychological patients. Finally, he has written extensively on the application of neuroscience findings to education, with a focus on dyscalculia.