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From numerical concepts to concepts of number

Lance J. Ripsa1, Amber Bloomfielda2 and Jennifer Asmutha3

a1 Psychology Department, Northwestern University, Evanston, IL 60208 rips@northwestern.edu http://mental.psych.northwestern.edu

a2 Department of Psychology, DePaul University, Chicago, IL 60614 abloomfi@depaul.edu

a3 Psychology Department, Northwestern University, Evanston, IL 60208 asmuth@northwestern.edu

Abstract

Many experiments with infants suggest that they possess quantitative abilities, and many experimentalists believe that these abilities set the stage for later mathematics: natural numbers and arithmetic. However, the connection between these early and later skills is far from obvious. We evaluate two possible routes to mathematics and argue that neither is sufficient: (1) We first sketch what we think is the most likely model for infant abilities in this domain, and we examine proposals for extrapolating the natural number concept from these beginnings. Proposals for arriving at natural number by (empirical) induction presuppose the mathematical concepts they seek to explain. Moreover, standard experimental tests for children's understanding of number terms do not necessarily tap these concepts. (2) True concepts of number do appear, however, when children are able to understand generalizations over all numbers; for example, the principle of additive commutativity (a+b=b+a). Theories of how children learn such principles usually rely on a process of mapping from physical object groupings. But both experimental results and theoretical considerations imply that direct mapping is insufficient for acquiring these principles. We suggest instead that children may arrive at natural numbers and arithmetic in a more top-down way, by constructing mathematical schemas.

Lance Rips is Professor of Psychology at Northwestern University. He is the author of The Psychology of Proof and The Psychology of Survey Response (with Roger Tourangeau and Kenneth Rasinski). He is also co-editor with Jonathan Adler of the anthology Reasoning: Studies of Human Inference and Its Foundations. His research is currently directed at concepts (especially concepts of individual objects and events as they change in time), reasoning (especially reasoning about new mathematical systems), and autobiographical memory.

Amber Bloomfield is a Postdoctoral Fellow at DePaul University. As a doctoral student at Northwestern University, she studied decision making under risk, including the effects of outcome type and aspirations on the framing effect, and the relationship between intertemporal discounting and risk. She also studied the effects of topic content on conditional reasoning performance. Her current research interests include the effects of comparison on risk interpretation, social comparison effects on anticipated and perceived performance, and age differences in loss aversion.

Jennifer Asmuth is a doctoral student in the Cognitive Psychology Program at Northwestern University. Her research interests include conceptual change, mathematical reasoning, and the representation of relational information. Her dissertation work explores the role of structure in learning novel mathematical information.

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