Reasoning and Proving in Textbooks: The Case of High School Geometry

• In Event: Reasoning and Proving in Mathematics Textbooks Across the Grades

Abstract

Summary
Although reasoning-and-proving is central to mathematics, it is not typically central to students’ school mathematics experiences. Instead, reasoning-and-proving is largely confined to high school geometry (Hanna & de Bruyn, 1999; Herbst, 2002) By closely examining the opportunities for reasoning-and-proving that exist in geometry, the field of mathematics education may move thoughtfully toward answering the calls for reasoning to be pervasive in school mathematics (NCTM, 2009; NGA & CCSSO, 2010). Toward this end, the present study analyzes the reasoning-and-proving opportunities in six U.S. high school geometry textbooks, adding to similar work that has been done in middle school (Stylianides, 2009), high school algebra and pre-calculus (Johnson, Thompson, & Senk, 2010), and a high school integrated series (Davis, 2010).

Theoretical Perspective
Past analyses of reasoning-and-proving opportunities have focused on types of student activity—conjecturing, proving, finding a counterexample, and so forth. In this study, we used the necessity principle (Harel & Tall, 1989) to consider not only reasoning-and-proving activities but also the mathematical contexts of those activities. Specifically, we posit that activities involving general mathematical claims are more likely to allow students to see the intellectual necessity of deductive reasoning-and-proving than are activities around particular mathematical diagrams or situations. Thus, we ask not only, “How often are students expected to prove a mathematical claim?” but also, “What sorts of mathematical claims do students have opportunities to prove?”

Method
In each of the six geometry textbooks, analysis was conducted on a 40% stratified random sample of expository sections as well as a review section for each chapter. Within expository text, mathematical statements (general or particular) and their justifications (deductive, empirical, left-to-student, etc.) were coded. Within student exercises, mathematical statements (general, particular, or general with particular instantiation provided) and expected activities (prove, explain, conjecture, fill-in-the-blanks of a proof, etc.) were coded. Additionally, statements or exercises explicitly about the practice of reasoning-and-proving were coded. Reliability surpassed 90% on each dimension.

Findings
The six geometry textbooks were more similar in their treatment of reasoning-and-proving than they were distinct. This presentation will focus on general trends across the six geometry textbooks and a comparison of these results to the findings of studies in other domains (e.g., Davis, 2010; Johnson et al., 2010; Stylianides, 2009).

Within geometry textbooks, approximately one-quarter of the student exercises involved reasoning-and-proving and less than one-fifth of these involved developing a mathematical proof. Textbook exposition typically presented general mathematical statements, whereas the vast majority of reasoning-and-proving exercises involved particular mathematical statements. This prevalence of particular statements in the exercises has implications for the extent to which such opportunities help students see the necessity of reasoning-and-proving and provide them with authentic experiences with this central mathematical practice. Additionally, even in geometry, reasoning-and-proving was rarely made an explicit object of reflection.

Significance of Findings
This study contributes to the research on reasoning-and-proving by mapping the opportunities in U.S. geometry textbooks, where reasoning-and-proving is most prevalent, and by expanding analysis beyond the activities expected of students to also consider the mathematical context of those activities.

Authors

• Lorraine Marie Males, University of Nebraska - Lincoln

• Samuel Otten, University of Missouri - Columbia