Mathematics teachers and educational researchers ordered arithmetic and algebra problems according to their predicted problem-solving difficulty for students. Predictions deviated systematically from algebra students' performances but closely matched a view implicit in textbooks. Analysis of students' problem-solving strategies indicates specific ways that students' algebraic reasoning differs from that predicted by most teachers and researchers in the sample and portrayed in common textbooks. The Symbol Precedence Model of development of algebraic reasoning, in which symbolic problem solving precedes verbal problem solving and arithmetic skills strictly precede algebraic skills, was contrasted with the Verbal Precedence Model of development, which provided a better quantitative fit of students' performance data. Implications of the findings for student and teacher cognition and for algebra instruction are discussed.

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- Author or Editor: Mitchell J. Nathan x

### Mitchell J. Nathan and Kenneth R. Koedinger

### Elizabeth L. Pier and Mitchell J. Nathan

In *Mathematics and the Body: Material Entanglements in the Classroom*, Elizabeth de Freitas and Nathalie Sinclair present an approach to embodiment that they term *inclusive materialism*. Their aim is to radically disrupt notions of “the body,” primarily by decentering the body in accordance with an ontology categorizing physical matter, mathematical concepts, diagrams, sounds, gestures, and technological entities as an assemblage of “entanglements” constituting mathematical activity. Their perspective is explicitly influenced by feminist, queer, and critical race philosophies, which they channel to redefine what is considered human, to redraw the boundaries of what has historically been described as material and embodied, and to “rescue the body, so to speak, from a theory of discourse that denies its materiality in order to grant the body some measure of agency and power in the making of subjectivity” (p. 40).

### Mitchell J. Nathan and Kenneth R. Koedinger

Think about this question: Why are algebra story problems considered to be the most difficult tasks facing algebra students? Teachers generally regard them as difficult (Nathan and Koedinger forthcoming), textbooks typically place these problems at the ends of chapters (Nathan and Long 1999), students find them least favorable, and even comic-strip folklore presents story problems as the bane of formal education. (See **fig. 1**). Are these perceptions held by teachers and textbook authors justified? This article examines teachers' judgments of the difficulty of algebra problems. Our findings may surprise readers, and we hope that they will motivate readers to reexamine some long-standing assumptions about mathematics learning and instruction.

### Caroline (Caro) Williams-Pierce, Elizabeth L. Pier, Candace Walkington, Rebecca Boncoddo, Virginia Clinton, Martha W. Alibali and Mitchell J. Nathan

In this Brief Report, we share the main findings from our line of research into embodied cognition and proof activities. First, attending to students' gestures during proving activities can reveal aspects of mathematical thinking not apparent in their speech, and analyzing gestures after proof production can contribute significantly to our understanding of students' proving practices, particularly when attending to dynamic gestures depicting relationships that are difficult to communicate verbally. Second, directing students to produce physical actions before asking them to construct a mathematical proof has the potential to influence their subsequent reasoning in useful ways, as long as the directed actions have a relationship with the proof content that is clearly meaningful to the students. We discuss implications for assessment practices and teacher education, and we suggest directions for future research into embodied mathematical proof practices.