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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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## Spotlight on the Principles/Standards: Problem Solving in the Middle Grades

IN WHAT WAYS WOULD YOU LIKE YOUR middle-grades students to experience problem solving in the mathematics curriculum? Do you want the curriculum to capture the excitement of geometry and measurement, algebra, statistics, and number relationships? Do you want it to help students understand and build new mathematical knowledge and explore new mathematical relationships? Do you want the curriculum to be filled with opportunities for students to ponder, create, and critique arguments about mathematics? If this is your vision for your students, then you should be pleased with, and excited by, the Problem Solving Standard in Principles and Standards for School Mathematics (NCTM 2000).

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## Authentic Argumentation With Prospective Secondary Teachers: The Case of 0.999…

Engaging prospective secondary teachers in mathematical argumentation is important, so that they can begin to learn to engage their own students in creating and critiquing arguments. Often, when we attempt to engage prospective secondary teachers in argumentation around topics from secondary mathematics classes, the argumentation is not authentic, as they believe they already know the answers. I suggest that there are problems related to the secondary curriculum around which we can engage students in authentic argumentation, and I propose one of them is whether 0.999… = 1. Purposefully engaging and supporting students in discussing this problem, and others like it, can lead to productive discussions that go beyond the answer to the question, including, for instance, what counts as evidence in mathematics.

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## Using Toulmin's Model to Develop Prospective Secondary Mathematics Teachers' Conceptions of Collective Argumentation

As creating and critiquing arguments becomes more of a focus in mathematics classes, teachers need to develop their abilities to facilitate collective arguments. Many mathematics education researchers find Toulmin's (1958/2003) model of argumentation to be useful in analyzing arguments, raising the question of whether mathematics teachers would find it useful as well. We introduced the model to prospective secondary mathematics teachers and asked them to analyze arguments using it. We found that the prospective teachers developed an appropriate understanding of what collective argumentation looks like in the classroom, and the model provided them a lens for analyzing teaching practice. This suggests the use of Toulmin's model is a promising step in helping prospective teachers develop their conceptions of collective argumentation.

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## From Rules That Expire to Language That Inspires

opportunity to construct and critique arguments. They are deprived of this when the teacher always assumes the role of verifying correctness of solutions. Rather than build skills in justifying their own solutions, students will wait for the teacher's approval