Reasoning and Proving Opportunities in Elementary Mathematics Textbooks

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

Abstract

Background
Initiatives such as the NCTM Principles and Standards and the more recent Common Core State Standards for mathematics (CCSSI, 2010) call for mathematics instruction at all grade levels to emphasize students’ reasoning and abilities to generate and critique mathematical arguments in a meaningful and consistent way. Although existing work shows that limited reasoning and proving opportunities exist in middle school curriculum (Stylianides, 2009), as well as high school (non-geometry) texts (Johnson, Thompson and Senk, 2010), no research to date has determined the extent to which elementary textbooks engage students in reasoning and proving tasks.

Theoretical Framework
We developed the following definition to operationalize our understanding of reasoning and proving tasks at the elementary grade levels: reasoning involves engaging in processes to generalize mathematical phenomena and/or conjecturing about mathematical relationships, whereas proving involves justifying a mathematical claim to be true for the domain to which the claim applies, using logically valid arguments presented using representational forms accepted by the classroom community.

Method
We conducted analyses of seven grade 5 mathematics textbooks, with four National Science Foundation-funded and/or NCTM Principles and Standards-based textbooks and three commercially-prepared texts. We chose to analyze grade 5 books on the assumption that there is a greater likelihood in upper elementary texts that students would be engaged in tasks of high cognitive demand such as reasoning and proving.
Potential reasoning and proving tasks were first identified by looking for keywords and phrases (explain, show, tell why, tell how, justify, prove), then we reviewed student and teacher materials to interpret both the purpose of the problem and the type of justification warranted. Tasks from every other lesson in all units in each textbook were considered, where lettered parts of a numbered problem were considered as separate tasks. The total number of possible tasks in the sample was 28, 210.

Results
The average percentage of reasoning and proving tasks across the elementary texts analyzed was 4.0%, with no significant differences in the proportions of total reasoning and proving tasks to total possible tasks between the seven textbooks. However, analyses of the purpose of the tasks and the type of justification warranted revealed distinctions between the text materials in terms of the 1) cognitive demand of the tasks and 2) placement of tasks in the lesson sections. One commercially-prepared text contained a total of 286 tasks eliciting students to provide justifications, however 48% of such tasks were found in enrichment sections of the lessons. In contrast, one National Science Foundation-funded, NCTM Principles and Standards-based text contained only 161 tasks written to elicit justifications but only 8% of the tasks were found in enrichment sections. Placement of reasoning and proving tasks is crucial as enrichment activities are written to be beyond the scope of the lesson.

Significance of Findings
The findings show reasoning and proving tasks are scarce in elementary textbooks. However, our research suggests that important differences between textbooks exist that can potentially influence the extent to which elementary students have opportunities to engage in mathematical reasoning and proving.

Authors

• Kristen Bieda, Michigan State University

• Justin Drwencke, Michigan State University

• Andrew William Picard, Michigan State University