Exploring the naturalness of natural numbers: Do undergraduates' everyday conceptualizations of natural number match the formal characterization?

Josephine Relaford-Doyle

It is widely assumed that spontaneously-arising conceptualizations of natural number ? those that develop without explicit mathematics instruction ? match the formal characterization of natural number given in the Dedekind-Peano Axioms (e.g. Leslie, Gelman, & Gallistel, 2008; Carey, 2004; Rips, Bloomfield, & Asmuth, 2008). Specifically, it is assumed that fully-developed conceptualizations of natural number are characterized by knowledge of a starting value ?one' and understanding of the successor principle: that for any natural number n, the next natural number is given by n+1. In developmental psychology the assumption that spontaneously-arising conceptualizations of natural number match the formal characterization is taken as unproblematic and has not been subject to empirical investigation. In this dissertation I seek to provide a more rigorous, thorough, and empirically-grounded characterization of spontaneously-arising natural number concepts. What do we know about natural number, without being explicitly taught? And to what extent are these conceptualizations consistent with the formal mathematical definition? In the four studies that comprise the dissertation I use an open-ended problem-solving task, number categorization task, number-line estimation task, and semi-structured interviews to explore undergraduates' conceptualizations of natural number. By identifying specific ways in which spontaneously-arising conceptualizations of natural number may actually deviate from the formal mathematical definition, this dissertation will inform classroom instruction for concepts that build on the natural number system (for instance, mathematical induction). This project will also help teachers to make more accurate assumptions about students' prior knowledge and design instructional experiences that effectively bridge between students' everyday conceptualizations and formal mathematical knowledge.

About Josephine Relaford-Doyle

Josephine Relaford-Doyle is a doctoral candidate in Cognitive Science at the University of California, San Diego. She is interested in mathematical certainty, justification, and the relationship between everyday experience and mathematical knowledge. Prior to her doctoral studies Josephine spent five years teaching high school mathematics as a Math for America Los Angeles fellow. In her current work she uses a variety of qualitative and quantitative methods to address questions about the nature of mathematics as a human conceptual system. Specifically, her research explores the extent to which mathematical concepts learned through everyday experience may deviate from formal mathematical definitions. By characterizing the nature of students' pre-existing mathematical ideas, she hopes to help teachers build bridges to formally consistent mathematical conceptualizations in their students. She holds a BA in Cognitive Science from the University of California, Berkeley, where she was a Regents' and Chancellor's Scholar, and an MA in Education from Claremont Graduate University.