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## Proofs without Words: A Visual Application of Reasoning and Proof

Reasoning and Proof is one of the process standards set forth in NCTM's principles and standards for school mathematics (2000).

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## Opportunities to Learn Reasoning and Proof in High School Mathematics Textbooks

This article addresses the nature and extent of reasoning and proof in the written (i.e., intended) curriculum of 20 contemporary high school mathematics textbooks. Both the narrative and exercise sets in lessons dealing with the topics of exponents, logarithms, and polynomials were examined. The extent of proof-related reasoning varied by topic and textbook. Overall, about 50% of the identified properties in the 3 topic areas were justified, with about 30% of the addressed properties justified with a general argument and about 20% justified with an argument about a specific case. However, less than 6% of the exercises in the homework sets involved proof-related reasoning, with developing an argument and investigating a conjecture as the most frequently occurring types of proof-related reasoning.

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## Classify & Capture: Using Venn Diagrams and Tangrams to Develop Abilities in Mathematical Reasoning and Proof

Article describes a game of classifying objects according to their attributes using a three-set Venn diagram, then analyzing the classification of the objects to determine their accuracy. Students attempt to capture opponents' game pieces (tangrams) by making mathematical arguments that identify misplaced pieces.

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## Transforming Perceptions of Proof: A Four-Part Instructional Sequence

Mathematics teachers are expected to engage their students in critiquing and constructing viable arguments. These classroom expectations are necessary, given that proof is a central mathematical activity. However, mathematics teachers have been provided limited opportunities as learners to construct arguments and critique the reasoning of others, and hence have developed perceptions of proof as an object that must follow a strict format. In this article, we describe a four-part instructional sequence designed to broaden and deepen teachers' perception of the nature of proof. We analyzed participants' reflections on the instructional sequence in order to gain insight into (a) the differences between this instructional sequence and participants' previous proof learning opportunities and (b) the ways this activity was influential in transforming participants' perceptions of proof. Participants' previous learning experiences were focused on memorizing and reproducing textbook or instructor proofs, and our sequence was different because it actively and collaboratively engaged participants in constructing their own arguments, critiquing others' reasoning, and creating criteria for what counts as proof. Participants found these activities transformative as they became more clear about what counts as proof, began to view proof as socially negotiated, and expanded their conception of proof beyond a rigid structure or format.

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## Proofs without Words in Geometry

Pictures and diagrams help high school geometry students develop reasoning and proof-writing skills.

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## Interactive Geometry Software in the B.C. (Before Computers) Era

The use of 3×5 cards to explore geometric relationships through the first three van Hiele levels of geometric reasoning. Students engage in reasoning and proof as they explore concepts related to parallel lines and quadrilaterals.

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## Improving Preservice Secondary Mathematics Teachers' Capability With Generic Example Proofs

Preservice mathematics teachers are entrusted with developing their future students' interest in and ability to do mathematics effectively. Various policy documents place an importance on being able to reason about and prove mathematical claims. However, it is not enough for these preservice teachers, and their future students, to have a narrow focus on only one type of proof (demonstration proof), as opposed to other forms of proof, such as generic example proofs or pictorial proofs. This article examines the effectiveness of a course on reasoning and proving on preservice teachers' awareness of and abilities to recognize and construct generic example proofs. The findings support assertions that such a course can and does change preservice teachers' capability with generic example proofs.

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## Mathematical Explorations: Reasoning with Algebraic Contexts

According to Principles and Standards for School Mathematics, two standards that should be addressed in all grades are (1) Algebra and (2) Reasoning and Proof (NCTM 2000). Both involve generalization and are, therefore, naturally related (Lannin 2003). There is often much discussion about the need to engage students with tasks related to algebra; however, it is unclear if as much attention is given to engaging students with reasoning and proof, although this topic is also fundamental to mathematics. As Hanna (2000) noted, “Students cannot be said to have learned mathematics, or even about mathematics, unless they have learned what a proof is” (p. 24).

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## An Odd Sum

Principles and Standards for School Mathematics (NCTM 2000, p. 342) states, “Students should develop an appreciation of mathematical justification in the study of all mathematical content. In high school, their standards for accepting explanations should become more stringent, and they should develop a repertoire of increasingly sophisticated methods of reasoning and proof.” This article helps teachers and students further develop their understanding of proof and, at the same time, gain insight into some of the many interesting properties of the natural numbers.

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## Families Ask: Why Not Just Tell Students How to Solve the Problem?

If you have been attempting to implement the NCTM's Process Standards (Problem Solving, Reasoning and Proof, Communication, Representation, and Connections), then you likely have been posing rich problems for students to solve. As students encounter difficulties, you try to ask questions to guide their thinking, without taking away the challenge of the task. This major shift away from telling students how to solve the problem and having them practice a standard way, however, can result in students reporting to their families that you will not help them in mathematics class.