Theses Doctoral

# Proof and Reasoning in Secondary School Algebra Textbooks

Dituri, Philip Charles

The purpose of this study was to determine the extent to which the modeling of deductive reasoning and proof-type thinking occurs in a mathematics course in which students are not explicitly preparing to write formal mathematical proofs. Algebra was chosen because it is the course that typically directly precedes a student's first formal introduction to proof in geometry in the United States. The lens through which this study aimed to examine the intended curriculum was by identifying and reviewing the modeling of proof and deductive reasoning in the most popular and widely circulated algebra textbooks throughout the United States. Textbooks have a major impact on mathematics classrooms, playing a significant role in determining a teacher's classroom practices as well as student activities. A rubric was developed to analyze the presence of reasoning and proof in algebra textbooks, and an analysis of the coverage of various topics was performed. The findings indicate that, roughly speaking, students are only exposed to justification of mathematical claims and proof-type thinking in 38% of all sections analyzed. Furthermore, only 6% of coded sections contained an actual proof or justification that offered the same ideas or reasoning as a proof. It was found that when there was some justification or proof present, the most prevalent means of convincing the reader of the truth of a concept, theorem, or procedure was through the use of specific examples. Textbooks attempting to give a series of examples to justify or convince the reader of the truth of a concept, theorem, or procedure often fell short of offering a mathematical proof because they lacked generality and/or, in some cases, the inductive step. While many textbooks stated a general rule at some point, most only used deductive reasoning within a specific example if at all. Textbooks rarely expose students to the kinds of reasoning required by mathematical proof in that they rarely expose students to reasoning about mathematics with generality. This study found a lack of sufficient evidence of instruction or modeling of proof and reasoning in secondary school algebra textbooks. This could indicate that, overall, algebra textbooks may not fulfill the proof and reasoning guidelines set forth by the NCTM Principles and Standards and the Common Core State Standards. Thus, the enacted curriculum in mathematics classrooms may also fail to address the recommendations of these influential and policy defining organizations.

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• Dituri_columbia_0054D_11165.pdf application/pdf 16 MB Download File