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.

Time is fundamental to human experience: it plays a central role in our everyday lives; yet, we cannot feel it, touch it or hear it. How are we able to make sense of such

an abstract concept? Lakoff and Johnson (1980) proposed that our concrete experience of being embodied within and moving through space lays the foundation for how we think about abstract concepts, such as time. This idea has driven much research in the field of Cognitive Science, and dozens of studies have since demonstrated that how we think about time appears to be intimately linked with how we think about space. However, at any given moment, people have numerous spatial resources available to them, in what ways are these resources deployed to structure our thoughts about time?

The studies presented in this dissertation investigate the variety of ways that space is used to structure our thoughts about time (at least in the minds of English speakers). I start with the question: how are different types of temporal relationships associated with space? I then tease apart the use of different spatial axes by deictic and sequence time, using auditory stimuli and vocal responses. These effects are then replicated using visual presentation and manual responses. From there, I ask: what particular aspects of spatial experience are used to structure our thoughts about time? Here, I explore how spatial perspective and bodily motion through space influence how we spatialize time. Next, I examine the role that space plays in reasoning about time by employing a dual-task paradigm designed to interfere with people’s spatial resources during a temporal reasoning task. Finally, I look at what co-speech gesture reveals about how space is used in real time when thinking about time. Together, the psychological experiments and studies of co-speech gesture presented here reveal the flexibility, as well as the limits, of how we use space to structure our thoughts about time. This body of work hopes to draw attention to the richness with which space structures time and to illuminate some possible mechanisms that are responsible for our associations of time with space.