Reasoning and Proving in Algebra: Two Secondary Reform-Oriented Mathematics Textbook Units

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

Abstract

Due to the importance of reasoning and proof (RP) in mathematics (Schoenfeld, 2009) and the influence of written curricula on teachers’ instructional decisions (Grouws & Smith, 2000) researchers have shown an interest in analyzing RP elements in a middle school reform-oriented curriculum (Stylianides, 2009) and high school reform-oriented and conventional textbooks (Davis, 2011; Johnson, Thompson, & Senk, 2010). These studies, however, have focused on a single curriculum (Stylianides, 2009), made comparisons between reform-oriented and conventional textbook units (Davis, 2011), or investigated textbooks that have since been revised (Johnson et al., 2010). Therefore, the purpose of this study is to examine the dispersion of RP in exposition sections and student tasks in two recently created reform-oriented textbook units in the area of polynomial functions.

Theoretical Perspective
Identifying patterns, developing conjectures, and constructing arguments as defined by Stylianides (2009) comprise the main components of the theoretical framework. The testing of conjectures was also included in the framework (Johnson et al., 2010). Technology was considered an integral tool in reasoning and proof within the framework. Proof building blocks such as definitions and the development of theorems, corollaries, etc. were considered important components comprising RP.

Method
All three authors coded electronic versions of several sections from a chapter appearing within Algebra 2 (Education Development Center, 2009) (CME) and a chapter appearing in Advanced Algebra (Flanders et al., 2009) (UCSMP) using the framework. In the event of disagreements, discussions ensued until agreement was reached. When a significant level of agreement in our coding was achieved (UCSMP Kappa = 0.8205; CME Kappa = 0.8704) the second author coded the remaining sections in each chapter. Codes were applied using qualitative analysis software. The percentage of student tasks and exposition sentences that were coded as RP were calculated for each unit.

Results
The bulk of the RP instances in textbook expositions of both textbooks are devoted to the presentation of proof building blocks. The next most frequent RP category is the presentation of arguments and both textbook unit expositions have a nearly equal number of argument presentations. Over a third of the UCSMP exposition sentences involve RP while CME has a higher percentage of RP student tasks than UCSMP at almost 30%. Student tasks in CME contain over three times as many opportunities than UCSMP tasks to construct arguments. There is a greater frequency of conjecture testing in CME tasks, but more conjecture development in UCSMP. CME tasks encompass twice as many opportunities to identify definite and plausible patterns than UCSMP tasks. There are roughly equal numbers of technology RP uses in both units.

Significance of Findings
Background
Despite being considered reform-oriented textbooks these two units hold the potential to provide students with different RP experiences. The UCSMP unit provides students with a more passive RP experience as these instances are predominantly embedded in the textbook exposition while the CME unit is more active with the bulk of its RP opportunities appearing in tasks. This work lays the foundation for analyses of the enacted curriculum (Bieda, 2010).

Authors

• Jon D. Davis, Western Michigan University

• Dustin O. Smith, Western Michigan University

• Abhik R. Roy, Western Michigan University