Reasoning and Proving in Textbooks for Future Elementary Teachers

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

Abstract

Summary
This study aims to understand how reasoning and proving are presented in textbooks for undergraduate mathematics classes for future elementary teachers. Studies of textbooks have focused on K-12 texts, developing topical lists against which to compare texts. Studying reasoning and proving in texts for future teachers is important because research shows that they face significant difficulties in understanding logical principles and distinguishing between empirical and deductive forms of argument. It seems unlikely that prospective K-8 teachers will develop adequate knowledge of reasoning and proving unless the mathematics courses they take offer them opportunities to develop this knowledge.

Theoretical Perspective
We examined whether textbooks for teachers made reasoning and proof an explicit topic, and if not, where and how reasoning and proof appear in these texts. We adopted the three-part definition of proof presented by Stylianides (2007), and used it to interrogate the 13 textbooks in print specifically written for undergraduate mathematics classes for future teachers.

Method
We identified major mathematical elements or components of reasoning and proving and created a list of related concepts and topics; then we looked for them in the 13 textbooks in print for mathematics classes for elementary teachers in three ways: 1) Looking in the table of contents to identify specific chapters or sections where these topics were included; 2) Using indexes to find pages where these topics were mentioned;3) Analyzing selected individual topics in depth, noting occurrences of and attention to reasoning and proving. Textbooks included the following approaches:
1. A chapter with reasoning or proof in the title
2. A chapter or section on logic
3. A chapter on problem solving
4. Other chapter or chapters, or throughout the book
5. Not explicitly covered
Most often, references in the index pointed to the chapters in which reasoning and proof were taught, with few references to any of these terms outside of the sections specifically aimed at teaching reasoning and proof.

Findings
Although more findings will be shared in the session, a sample of our findings include: 1) We found the terms “mathematical induction”, “explanation”, “definition” and “assumption” in almost none of the books; 2) The books noticeably differed in how they approached reasoning and proof, from a definition-theorem-proof book in the style of a classic mathematics text to books in which no proofs were found and mathematical reasoning was vague; 3) Several books made careful mathematical arguments and sometimes offered proofs, but predominately on an informal level.

Significance of Findings
Our results suggest that most of these textbooks do not give adequate, explicit attention to reasoning and proving. Since these books are widely used for required mathematics courses for teachers, we conclude that, unless instructors are supplementing content through their instruction or other materials, students in courses that use most of these books have very little opportunity to build a strong understanding of mathematical reasoning and proving, and would have a difficult time using these books to build such knowledge.

Authors

• Raven S. McCrory, Michigan State University

• Andreas J. Stylianides, University of Cambridge