Connecting learning, memory, and representation in math education

In math education the goal is for children not only to master the materials and problems presented, but to understand underlying principles and properties that can be applied broadly to new problems and situations. Teachers in the classroom and policy-makers in Washington thus are both faced with what is essentially a cognitive question: What instructional regimes and practices will produce rapid learning, deep understanding, and broad transfer? This question has often been approached without connection to cognitive theories of learning, memory, and representation, but the gap has begun to narrow. On one hand, it is now known that domain-general learning mechanisms can acquire quite abstract and structured representations that go beyond the perceptual structure of the environment—a critical requirement for any theory of mathematical knowledge. Conversely studies in math cognition have revealed counter-intuitive behaviors that find ready explanations in cognitive models of learning in other domains. For instance, children, adults, and even math teachers reliably judge some three-sided figures to be better triangles than others, sometimes denying that irregular threesided figures are in fact triangles. Children transitioning from arithmetic to algebra often generate incorrect solutions to equations because they have learned to ignore the equal sign. Such examples suggest that math learning can be subject to the same factors that govern learning other domains. Yet it remains unclear whether such effects are epiphenomenal, or whether they hint at important common principles underlying concept acquisition across multiple domains. Our symposium investigates this question by bringing together scientists whose research spans the gap between cognitive and educational science in the domain of mathematical knowledge. Martha Alibali, Chuck Kalish and Tim Rogers consider how cognitive memory models from non-mathematical domains can shed light on the patterns of transfer shown by children and adults in arithmetic. Phil Kellman and Christine Massey will show that mathematical competency can improve when children learn to efficiently encode the perceptual structure of equations. Vladimir Sloutsky will consider interrelationships between learning of mathematical and object concepts in development. Jay McClelland and Kevin Mickey will discuss new research investigating the representational prerequisites that might underlie conceptual understanding of trigonometric functions. A short group question period will follow the four talks.