Reasoning-and-Proving in School Mathematics Textbooks

Abstract

Despite widespread agreement that the activity of reasoning-and-proving should be central to all students' mathematical experiences, many students face serious difficulties with this activity. Mathematics textbooks can play an important role in students' opportunities to engage in reasoning-and-proving: research suggests that many decisions that teachers make about what tasks to implement in their classrooms and when and how to implement them are mediated by the textbooks they use. Yet, little is known about how reasoning-and-proving is promoted in school mathematics textbooks. In this article, I present an analytic/methodological approach for the examination of the opportunities designed in mathematics textbooks for students to engage in reasoning-and-proving. In addition, I exemplify the utility of the approach in an examination of a strategically selected American mathematics textbook series. I use the findings from this examination as a context to discuss issues of textbook design in the domain of reasoning-and-proving that pertain to any textbook series.

This article is based in part on the author's doctoral thesis, which was completed at the University of Michigan under the supervision of Edward Silver. The author wishes to thank Deborah Ball, Hyman Bass, and Priti Shah for comments that influenced the development of the ideas in the thesis. The author wishes to also thank Lyn English and the anonymous reviewers for useful feedback on earlier versions of the article.

Notes

¹I acknowledge that the extent of textbook use by teachers is lower in some other countries such as England.

²I talk about opportunities designed for (as opposed to opportunities offered to) students in mathematics textbooks in order to (1) indicate my focus on the written curriculum, i.e., what is included in the students' textbooks and the teachers' editions of a textbook series; and (2) emphasize that there may be differences between the written and implemented curriculum, whereby implemented curriculum I mean the way in which the written curriculum is enacted in the classroom.

³CMP is currently undergoing revision. Parts of the new version of CMP are now available. All the analysis and discussion in this paper refers to the first iteration of CMP. Also, I use the phrase “reform-based textbook series” to refer to textbook series that were designed to embody the recommendations of the NCTM (1989, 2000) Standards. I use the phrase “conventional textbook series” to refer to textbook series that do not associate their existence with the Standards. Although many of the conventional textbook series have been updated to address the NCTM Standards, their updating has been primarily one of retrofitting as opposed to complete redesign.

⁴The notion of “pattern” in this framework includes also statements that describe covariation between structures, properties, or variables, i.e., it includes patterns that describe relationships between different patterns. A more refined approach in the treatment of pattern in the framework would be to distinguish between patterns that relate different patterns and patterns that do not. The former category can be more challenging for students because it requires coordination of multiple structures, properties, or variables and, thus, may require special attention in instruction.

⁵See in the method section for how I use the term “task.”

⁶In the cover page of each CMP unit, the textbook authors denote which of the four strands they consider to be primarily promoted in that unit.

⁷It was not necessary to calculate separately the inter-rater agreement for the purposes of patterns and conjectures because this inter-rater agreement was embedded in the second reliability value.

⁸With this statement I do not suggest that students' capabilities in deductive reasoning and proof develop only with age. The teacher has an important role to play in this development. Mathematics education research offers evidence that, in supportive classroom environments, even elementary students are able to successfully engage in deductive reasoning and other forms of reasoning involved in the development of proofs (see, e.g., Ball & Bass, 2003; Lampert, 1992; Reid, 2002; Stylianides, 2007a; see also earlier research into young children's abilities to reason, e.g., Donaldson, 1978; Gattegno, 1987).