Mathematical Cognition 1 A Parallel-Distributed Processing Approach to Mathematical Cognition1

How should we think about the nature of our knowledge of mathematical concepts, and the mechanisms we use to learn and use these concepts when we do mathematics? Here we describe a perspective on the answers to these questions and a future research program to address them that is grounded, in part, in the parallel distributed processing [PDP] approach to cognition and learning [1,2] implemented in artificial neural networks. We begin with a more basic question, namely the nature of mathematics and mathematical reasoning, and proceed from there to consider mathematical knowledge, learning, and thinking, stressing the roles of culture and experience in the creation and learning of mathematics. We then review the PDP perspective on the nature of knowledge and learning, and consider how it can address many findings in the mathematical cognition literature, and how it provides alternative ways of understanding what nature may provide and what nurture may create. Next we discuss the exciting challenges facing the approach and how they might be addressed, organizing the discussion around the question: How might a neural network-based approach meet the challenge of learning to achieve a level of competence in algebra and geometry sufficient to pass a highschool proficiency exam in geometry, in a human-like way, from experiences similar to those of human learners? We conclude with a discussion of the implications of the approach for learners and teachers of mathematics and for the processes of teaching and learning.