This chapter discusses ways in which conceptual analyses of mathematical ideas from a radical constructivist perspective complement Realistic Mathematics Education’s attention to emergent models, symbolization, and participation in classroom practices. The discussion draws on examples from research in quantitative reasoning, in which radical constructivism serves as a background theory. The function of a background theory is to constrain ways in which issues are conceived and types of explanations one gives, and to frame one’s descriptions of what needs explaining. The central claim of the chapter is that quantitative reasoning and realistic mathematics education provide complementary foci in both design of instruction and evaluation of it. A theory of quantitative reasoning enables one to describe mathematical understandings one hopes students will have, and ways students might express their understandings in action or communication. It is argued that conceptual analyses of mathematical ideas cannot be carried out abstractly. In contrast, it is found to be highly useful to imagine students thinking about something in discussions of it. In relation to this, the focus is on what one imagines to be the “something” teachers and students discuss, and on the nature of the discussions surrounding it. This type of conceptual analyses overlaps considerably with the Realistic Mathematics Education notion of emergent models in instructional design. There is, however, a difference in one respect; Realistic Mathematics Education attends to tools which will influence students’ activity, while from a quantitative-reasoning perspective the focus is more on things students might re-perceive and things about which a teacher might hold fruitful discussions.
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