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Browsing by Author "Tillema, Erik S."
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ItemA Case for Combinatorics: A Research Commentary(Elsevier, 2020-09) Lockwood, Elise; Wasserman, Nicholas H.; Tillema, Erik S.; School of EducationIn this commentary, we make a case for the explicit inclusion of combinatorial topics in mathematics curricula, where it is currently essentially absent. We suggest ways in which researchers might inform the field’s understanding of combinatorics and its potential role in curricula. We reflect on five decades of research that has been conducted since a call by Kapur (1970) for a greater focus on combinatorics in mathematics education. Specifically, we discuss the following five assertions: 1) Combinatorics is accessible, 2) Combinatorics problems provide opportunities for rich mathematical thinking, 3) Combinatorics fosters desirable mathematical practices, 4) Combinatorics can contribute positively to issues of equity in mathematics education, and 5) Combinatorics is a natural domain in which to examine and develop computational thinking and activity. Ultimately, we make a case for the valuable and unique ways in which combinatorics might effectively be leveraged within K-16 curricula. ItemAn investigation of 6th graders’ solutions of Cartesian product problems and representation of these problems using arrays(The Journal of Mathematical Behavior, 2018-04-01) Tillema, Erik S.Two hour-long interviews were conducted with each of 14 sixth-grade students. The purpose of the interviews was to investigate how students solved combinatorics problems, and represented their solutions as arrays. This paper reports on 11 of these students who represented a balanced mix of students operating with two of three multiplicative concepts that have been identified in prior research (Hackenberg, 2007, 2010; Hackenberg & Tillema, 2009). One finding of the study was that students operating with different multiplicative concepts established and structured pairs differently. A second finding is that these different ways of operating had implications for how students produced and used arrays. Overall, the findings contribute to models of students’ reasoning that outline the psychological operations that students use to constitute product of measures problems (Vergnaud, 1983). Product of measures problems are a kind of multiplicative problem that has unique mathematical properties, but researchers have not yet identified specific psychological operations that students use when solving these problems that differ from their solution of other kinds of multiplicative problems (cf. Battista, 2007). ItemStudents’ Solution of Arrangement Problems and their Connection to Cartesian Product Problems(Taylor & Francis, 2019) Tillema, Erik S.; School of EducationTwo-hour long developmental teaching interviews were conducted with each of 14 sixth grade students, ages 11–12. The purposes of the interviews were to investigate how students solved arrangement problems (APs), and how their solutions of these problems differed from their solution of Cartesian product problems (CPPs). The 14 students represented a balanced mix of students operating with each of three different multiplicative concepts that have been identified in prior research. This paper reports on the 11 students who were using the first or second multiplicative concept. Students operating with different multiplicative concepts all experienced similar perturbing elements in their solution of APs relative to their solution of CPPs, but they operated differently to resolve these perturbing elements. These differences are identified and their significance discussed in relation to other research findings on students’ combinatorial and multiplicative reasoning.