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,…

Mathematical definitions are central to learning and doing mathematics. Research has uncovered significant differences between how mathematicians and non-mathematicians construct, reason about, and refine mathematical definitions. Various strands of research provide insight into the development of definitional practices, yet an integrated approach…

Numbers are ubiquitous in life. At the same time, the symbols for numbers are highly abstract and their visual appearance does not carry any direct relation to their magnitude. This poses an important question for researchers interested in how numbers are mentally processed. Whereas research on numeric cognition suggests that culturally based…

Understanding the linear relationship of numbers is essential for doing practical and abstract mathematics throughout education and everyday life. There is evidence that number line activities increase learners' number sense, improving the linearity of mental number line representations (Siegler & Ramani, 2009). Mental representations of…

The vision of K-12 standards-based mathematics reform embraces a greater emphasis on students' ability to communicate their understandings of mathematics by utilizing adaptive reasoning (i.e., reflection, explanation, and justification of thinking) through mathematics discourse. However, recent studies suggest that many students lack the…