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Amy F. Hillen and Tad Watanabe

Conjecturing is central to the work of reasoning and proving. This task gives fourth and fifth graders a chance to make conjectures and prove (or disprove) them.

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Amy F. Hillen and LuAnn Malik

A card-sorting task can help students extend their understanding of functions and functional relationships.

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Amy F. Hillen and Margaret S. Smith

Communication is an important part of classrooms in which students are engaged in challenging mathematics (NCTM 2000). Students communicate by “participating in social interaction, sharing thoughts with others, and listening to others share their ideas” (Hiebert et al. 1997). As students take part in sharing their mathematical thinking, they have opportunities to make and justify conjectures, consider the validity of others' ideas, engage in mathematical argumentation, and reflect on their current understandings (NCTM 2000). Thus, the mathematical discourse in classrooms provides important opportunities for students to learn mathematics (Hiebert et al. 1997; NCTM 2000).

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Michael D. Steele and Amy F. Hillen

In the majority of secondary mathematics teacher preparation programs, the work of learning mathematics and the work of learning to teach mathematics are separated, leaving open the question of when and how teachers integrate their knowledge of content and pedagogy. We present a model for a content-focused methods course, which systematically develops a slice of mathematics content in the context of typical methods course activities. Three design principles are posited that undergird the design of such a course, addressing the nature of the mathematics content, the sequencing and design of activities, and the ways in which the course addresses the needs of diverse learners. Data from an instantiation of one such course is presented to illustrate the ways in which the course design framed teachers' opportunities to learn about both content and pedagogy.

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Margaret S. Smith, Amy F. Hillen, and Christy L. Catania

The capacity to reason algebraically is critical in shaping students' future opportunities and, as such, is a central theme of K–12 education (NCTM 2000). One component of algebraic reasoning is “the capacity to recognize patterns and organize data to represent situations in which input is related to output by well-defined functional rules” (Driscoll 1999, p. 2). Geometric pattern tasks can be a useful tool for helping students develop algebraic reasoning, because the tasks provide students with opportunities to build patterns with materials such as toothpicks or pattern blocks. These materials help students “focus on the physical changes and how the pattern is being developed” (Friel, Rachlin, and Doyle 2001, p. 10). Such work might help bridge students' earlier mathematical experiences and lay the foundation for more formal work in algebra (English and Warren 1998; Ferrini-Mundy, Lappan, and Phillips 1997; NCTM 2000). Finally, the relationships between the quantities in pattern tasks can be expressed using symbols, tables, and graphs, as well as words. Thus, pattern tasks can also give students opportunities to make connections among representations—a key component in developing an understanding of function (Knuth 2000).

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Melissa D. Boston, Margaret S. Smith, and Amy F. Hillen

Middle-grades students' understanding of proportional relationships should be fostered through problem solving and reasoning (NCTM 2000). Toward this end, instruction in proportionality should expose students to a variety of strategies and allow students to gain experience modeling proportional situations (Langrall and Swafford 2000). Students should be given ample opportunities to develop intuitive strategies based on factor- of-change (“how many times as many”) relationships (Cramer and Post 1993). Research has shown that middle-grades students are more successful at method is appropriate to use” (NCTM 2000, p. 221). We begin our discussion by focusing on the events that unfold in Marie Hanson's sixth-grade classroom during a lesson on understanding ratios and proportions (Smith et al. forthcoming), and use this lesson as a context for considering how factor-of-change relationships might be used to assist students in understanding why cross multiplication works.