Introducing Young Children to the Role of Assumptions in Proving

Abstract

The notion of assumptions permeates school mathematics, but instruction tends to highlight this notion only in the advanced grades. In this article, I argue that it is important for even young children to develop a sense of the role of assumptions in proving, and I investigate what it might mean and look like for instruction to promote this goal. Toward this end, I study an episode from third grade that describes the first time that the students in the class were introduced in a deliberate and explicit way to the role of assumptions in proving. The central role of the mathematical task in the episode is identified, and features of mathematical tasks that can generate rich mathematical activity in the intersection of assumptions and proving are discussed. In addition, issues of the role of teachers in fostering productive interactions between students and mathematical tasks that have those features are considered.

Notes

¹I use the term proving to describe the activity associated with the search for a proof. In turn, I use the term proof to describe—in the context of a classroom community at a given time—a mathematical argument that fulfills three criteria: (1) it builds on true statements that are accepted by the community and that can be used without further justification, (2) it uses valid modes of argumentation that are known or conceptually accessible to the community, and (3) it employs appropriate modes of representation that are known or conceptually accessible to the community (see Stylianides, 2007, for elaboration). The terms “true,” “valid,” and “appropriate” that are used in the descriptions of the three criteria for a proof should be understood in the context of what is typically agreed on in the field of mathematics within the domain of particular mathematical theories. In addition, it should be noted that the notion of assumptions corresponds to the first criterion regarding the statements on which a given argument or a proof is based.

²These arguments can be considered as proofs in the given context.

³For example, the person in the problem can make his trips according to the following process. For the first trip, the person goes up and down once between the third floor and the fourth floor, and then ends up at the second floor. For each new trip, the person goes up and down one time more than in the previous trip between the third floor and the fourth floor, and then ends up at the second floor. This procedure contains no terminating condition and enters a process of infinite recursion.

⁴Evidence from Deborah Ball's third-grade class shows that young children can produce, after appropriate teacher scaffolding, such general arguments. In particular, a student in Ball's class developed the following generic proof (Balacheff, 1988; Harel & Sowder, 1998; Rowland, 2002) for the claim that there are infinitely many number sentences for 10: “We would take any number, it wouldn't matter what number, say 200. And then we would minus 200, then we would plus 10, and it would always equal 10. So you could go on for a long, long time, just keep on doing that. […] So, since numbers they never stop, you could go on and on and on and on and on and on and on […]” With Ball's help, the students represented this argument algebraically, using the sentence x − x + 10 = 10, and noting that x can stand for any number. For elaboration on this classroom episode (including an analysis of the students' proof and of the teacher's actions that supported the reformulation of the proof in algebraic form), see Stylianides (2007).