The Mathematical Nature of Reasoning-and-Proving Opportunities in Geometry Textbooks

Abstract

International calls have been made for reasoning-and-proving to permeate school mathematics. It is important that efforts to heed this call are grounded in an understanding of the opportunities to reason-and-prove that already exist, especially in secondary-level geometry where reasoning-and-proving opportunities are prevalent but not thoroughly studied. This analysis of six secondary-level geometry textbooks, like studies of other textbooks, characterizes the justifications given in the exposition and the reasoning-and-proving activities expected of students in the exercises. Furthermore, this study considers whether the mathematical statements included in the reasoning-and-proving opportunities are general or particular in nature. Findings include the fact that the majority of expository mathematical statements were general, whereas reasoning-and-proving exercises tended to involve particular mathematical statements. Although reasoning-and-proving opportunities were relatively numerous, it remained rare for the reasoning-and-proving process itself to be an explicit object of reflection. Relationships between these findings and the necessity principle of pedagogy are discussed.

ACKNOWLEDGMENTS

A portion of the results contained herein were presented at the 33rd annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. The authors would like to thank Kristen Bieda and Sharon Senk for their helpful comments and conversations during the course of this study.

Notes

¹We join Stylianides (2009) in using the term “reasoning-and-proving” to refer broadly to processes of conjecturing and justification that are meant to articulate a mathematical statement and convince oneself or others of its truth-value, wherein the justification or reasons given are acceptable to a particular classroom community if not the formal mathematical community. We use the term “mathematical proof” or “proof” to mean an explicit and deductive argument that would be accepted as valid by mathematicians knowledgeable in the particular content area.

²We note that the empirical problem being described here has to do with the inappropriate use of inductive arguments to “prove” a general statement that requires a deductive justification. For example, it would be inappropriate to use 1 + 3 = 4 and 3 + 5 = 8 to “prove” that the sum of any two odd numbers is even. We do not mean to imply that there is no role for empirical reasoning in the reasoning-and-proving process. In fact, we agree with Polya (1959, 1981) that empirical reasoning is central to the formulation of conjectures and with Lakatos (1976) that empirical reasoning is central to the refinement of definitions and theorems.

³It is important to note that our analytic framework excludes patterning exercises (e.g., determine the hundredth entry in a given pattern) from reasoning-and-proving, although Stylianides (2009) and Davis (2010) included such exercises in their studies of reform-oriented curricula. If, however, a geometry textbook asked students to conjecture or prove a statement based on a pattern, we included the item as reasoning-and-proving.

⁴With respect to the prominence of rationale exercises, one might contend that an “explain” prompt provides students with an opportunity to construct a proof because a key function of mathematical proofs is explanation (de Villiers, 1995; Hanna, 1995). This leads to the question, however, of whether students realize that a proof would be an effective response to a rationale exercise. Answering this question would take us beyond the realm of textbook analysis, although we can note that Herbst and Brach (2006) found that geometry students viewed proof tasks to be distinct from explanation tasks.

⁵These proof opportunities around particular statements, although they may achieve many of the state standards for proof in geometry, do not seem to align with the new proof standards as outline in the Common Core State Standards (NGA & CCSSO, 2010). As Kosko and Herbst (2011) noted, the Common Core proof standards often relate to important pieces of content in such a way that these standards are not likely to be met with “token proof exercises whose conclusions are not memorable” (p. 7).