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This chapter analyses mathematical thinking as a whole. In particular, it scrutinizes the common appeal to a distinction between algebraic and geometric thinking. By looking at a number of examples, it is shown that what underlies this appeal is not a division but two poles of something more like a spectrum; and that any division of mathematical thinking into just two kinds is bound to be misleading.

Computational thinking is a way of solving problems, designing systems, and understanding human behavior that draws on concepts fundamental to computer science. The advanced placement course, AP Computer Science Principles, introduces students to basic concepts and challenges them to explore how computing and technology impact the world. Computational thinking across the K–12 curriculum compliments, rather than competes with, efforts to expand computer science education. Computer science courses include algorithmic thinking, logic, abstraction, decomposition, and debugging. Computational and mathematical thinking have much in common. The book In Pursuit of the Unknown: 17 Equations that Changed the World is an excellent introduction to mathematical thinking by describing the impact of equations.

Language does not necessarily support all human cognition. Comparative research continues to uncover the links between the cognitive abilities of humans and other animal species that lack language. Representing and manipulating numbers are important for most aspects of human life, and much of this involves abstract symbolic representations. Given this perspective, the authors illustrate how the comparative approach has been used in the domain of numbers to identify the origins of mathematical thinking. The authors in this chapter found out that a system for representing numbers nonverbally is shared by both humans and many nonhuman animal species. A host of behavioral parallels between human and animal numerical cognition such as ratio dependence, semantic congruity, cross-modal matching, and nonverbal arithmetic are revealed in this chapter. The homologous brain structures appear to support numerical representations in humans and macaque monkeys.

In the foundations of mathematics there has been an ongoing debate about whether categorical foundations can replace set-theoretical foundations. The primary goal of this chapter is to provide a condensed summary of that debate. It addresses the two primary points of contention: technical adequacy and autonomy. Finally, it calls attention to a neglected feature of the debate, the claim that categorical foundations are more natural and readily useable, and how deeper investigation of that claim could prove fruitful for our understanding of mathematical thinking and mathematical practice.