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     The Teachers’ Multiplicative Reasoning Research Group studies the ecology of teachers’ cognition around interrelated multiplicative topics––including multiplication, division, fractions, proportional relationships, and linear functions. We are particularly interested in ways that teachers’ reasoning about multiplication, division, and fractions support their reasoning about ratios and proportional relationships and in how number lines and strip (tape) diagrams can support this reasoning. Our theoretical framework is based on two complementary perspectives on proportional relationships, one of which we call multiple batches and one of which we call variable parts. The multiple batches perspective has been investigated in past research primarily with students (see Figure 1). The variable parts perspective has been largely overlooked by past research but holds promise for making applications of proportional relationships to linear equations and other topics more accessible (see Figure 2). 

     Our methods coordinate detailed case studies with recent advances measuring teachers’ multiplicative reasoning. The psychometric work is done in collaboration with Dr. Laine Bradshaw and is based on determining profiles of multiplicative reasoning. The profiles are determined using Diagnostic Classification Models and summarize strengths and weaknesses with four core components of reasoning about fraction arithmetic in terms of quantities. The components are referent unit, partitioning and iterating units, recognizing multiplication and division situations, and forming multiplicative comparisons. Figure 3 shows the distribution of teacher profiles in a national sample of 990 in-service U.S. middle grades teachers.  



     Our project seeks to shed light on how future teachers develop understanding of central middle and secondary grades mathematics. We expect our project to inform improved teacher education content courses and to lay a foundation for studies of middle grades students’ reasoning about ratio and proportional relationships. 

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