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How ‘Natural’ are the Natural Numbers? Inconsistencies Between Formal Axioms and Undergraduates’ Conceptualizations of Number


It is widely assumed within developmental psychology 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., Carey, 2004; Leslie et al., 2008; Rips et al., 2008). Specifically, developmental researchers assume 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. This formally-consistent knowledge is thought to emerge spontaneously as children learn how to count, without the need for formal instruction in the axioms of natural number. In developmental psychology, the assumption that spontaneously-arising conceptualizations of natural number match the formal characterization has been 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 given in the Dedekind-Peano Axioms? In the four studies that comprise the dissertation I use an open-ended problem-solving task, a computer-based judgment task, a number-line estimation task, and semi-structured interviews to explore undergraduates’ conceptualizations of natural number. I find evidence that undergraduates who have not received explicit instruction in the axioms of natural number often recruit conceptualizations that are at odds with the formal characterization given in the Dedekind-Peano Axioms, and that many students demonstrate formally-consistent reasoning in only the simplest and most foundational mathematical contexts. Taken together, these results call into question the assumption that formally-consistent notions of natural number arise spontaneously from mastery of counting. Based on these findings, I make recommendations for future research into number concept development and discuss practical implications for classroom instruction for concepts that build on the natural number system (for instance, mathematical induction), as well as mathematics teacher development.

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