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A yearlong professional development project investigated types of discourse and argumentation that students engage in, participation structures and routines that teachers can include to support students, and types of tasks that promote mathematical argumentation.

This article contributes to the research literature concerning prospective elementary teachers' mathematical thinking and learning with a focus on flexibility. We present a case study of a prospective elementary teachers' development of flexibility in mental addition and subtraction during a Number and Operations course. Building upon the construct of strategy ranges, we introduce scaffolded strategy ranges, which describe the sets of strategies that people use given the opportunity to solve a task in multiple ways. Like many prospective elementary teachers, Brandy initially appeared inflexible in mental addition and subtraction. In fact, her unscaffolded strategy ranges were limited to just the mental analogs of the standard algorithms. However, Brandy's scaffolded strategy ranges revealed greater potential for flexibility. Furthermore, the way of reasoning that appeared in Brandy's scaffolded strategy ranges (a) influenced her interpretations of nonstandard strategies that she encountered in the Number and Operations course and (b) foreshadowed the direction in which her flexibility would develop.

The Standards for Mathematical Practice (SMP) in the Common Core State Standards for Mathematics (CCSSM) (CCSSI 2010) highlight the mathematical habits that educators should be fostering in mathematics classrooms throughout K-grade 12 education. That argumentation and discourse are important components of developing mathematically proficient students has been well established, and this fits well with SMP 3, which states that students will \"construct viable arguments and critique the reasoning of others\" (CCSSI 2010, p. 6). Given that this practice is essential, how do teachers effectively incorporate mathematical argumentation into their upper elementary-level lessons? What does this practice look like, and what can teachers expect from students who have had minimal experience with this form of instruction? How do teachers strategically embed argumentation into the appropriate mathematical content? The authors of this article address these questions as they present evidence-based instructional strategies for promoting argumentation. Although these general instructional strategies would apply to mathematical argumentation within various mathematics topics, the specific examples they use to illustrate the strategies are all within the context of exploring the arithmetic properties. The strategies include: (1) Provide language supports; (2) Discuss rich, familiar content; (3) Specify conditions; (4) Introduce false claims; and (5) Manipulate familiar content to be unfamiliar.

In this article the authors describe a project during which they unpacked fraction standards, created rigorous tasks and lesson plans, and developed formative and summative assessments to analyze students' thinking about fraction multiplication. The purpose of this article is to (1) illustrate a process that can be replicated by educators interested in using rigorous mathematical tasks and assessments to support and advance their students' mathematical thinking; and (2) share the artifacts and instructional products that educators can use to improve mathematics assessment practices.

We examined ways of improving students’ unit concepts across spatial measurement situations. We report data from our teaching experiment during a six-semester longitudinal study from grade 2 through grade 5. Data include instructional task sequences designed to help children (a) integrate multiple representations of unit, (b) coordinate and group units into higher-order units, and (c) recognize the arbitrary nature of unit in comparison contexts and student’s responses to tasks. Our results suggest reflection on multiplicative relations among quantities prompted a more fully-developed unit concept. This research extends prior work addressing the growth of unit concepts in the contexts of length, area, and volume by demonstrating the viability of level-specific instructional actions as a means for promoting an informal theory of measurement.

There are different ways to view the role of the teacher, from someone who imparts knowledge to their students, to someone who monitors learning of students, to someone who facilitates the learning of their students, to a more knowledgeable other (Munter, 2014), see green parts of Figure 1. In this article we want to provide a classroom example to show how argumentation can enhance conceptual understanding of students and provide a vision for what teachers as learners in community with their students can look like. Background During a professional development project we worked with a school district to explore how to engage students in noticing and wondering, conjecturing, justifying, and extending ideas. With these previous activities and student knowledge in mind, we planned the next lesson to include opportunities for students to notice and wonder, make conjectures, justify, and extend so that we could continue to make their thinking, including possible misconceptions, visible to the community.

This Teaching Tip describes the use of children's literature to help second‐grade students meet Common Core State Standards for English Language Arts and for Mathematics. During a shared reading experience, students used manipulatives to represent plot and characters while demonstrating mathematical reasoning. The article offers instructional ideas, guiding questions, authentic student work samples, and a rationale for using children's literature and mathematics manipulatives to help students make connections across disciplines.

The goals for this study were to investigate how fourth-grade students developed an understanding of the arithmetic properties when instruction promoted mathematical argumentation and to identify the characteristics of students' arguments. Using the emergent perspective as an overarching theoretical perspective helped distinguish between two components of classroom learning: the process of argumentation as the social perspective and the content of arithmetic properties as the psychological perspective. I administered pre- and post-assessment interviews individually to students. I modified eight lessons from the traditional textbook and introduced sociomathematical norms that supported mathematical argumentation into the whole-class teaching experiment. I identified and analyzed 67 arguments that emerged from the lesson transcripts and used an empirically-modified model adapted from Toulmin's (1958) model of argumentation and Stylianides' (2007) elements of argumentation to characterize the students' arguments. With regard to the characteristics of the students' arguments, I partitioned all of the arguments into six model templates and all of the claims into two categories. The justifications that students relied on seemed related to many factors, including the origin of the claim, the mathematical context of the claim, and the type of claim. The study suggests that arithmetic properties should be integrated into new and familiar contexts rather than placed in one section of the fourth-grade curriculum and that argumentation is a beneficial process for teaching the properties to children.