Browsing by Subject "combinatorics"

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    A Case for Combinatorics: A Research Commentary
    (Elsevier, 2020-09) Lockwood, Elise; Wasserman, Nicholas H.; Tillema, Erik S.; School of Education
    In 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.
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    Units Coordination, Combinatorial Reasoning, and the Multiplication Principle: The Case of Ashley, an Advanced Stage 2 College Student
    (Taylor & Francis, 2024) Tillema, Erik; Antonides, Joseph; School of Education
    The multiplication principle (MP) is foundational for combinatorial problem-solving. From a units-coordination perspective, applying the MP with justification entails establishing unit relationships between the number of options at each independent stage of a counting process and the total number of combinatorial outcomes. Existing research literature, however, has not captured, generally, how students establish these unit relationships. We provide a second order model of an advanced stage 2 college student, Ashley, who had no prior combinatorics instruction, as she engaged in solving combinatorics problems that we considered to involve the MP. Our findings suggest that Ashley began by interpreting combinatorics problems using her whole number iterative units coordination scheme. Through engagement with the teacher-researcher, Ashley constructed combinatorial composites using a pairing operation, units coordination, and units simplification. We also found that Ashley was able to create a three-level-of-unit structure in activity, and to use notation that she produced to re-instantiate the reasoning that produced this unit structure. Doing so provides novel insights into how advanced stage 2 students, especially those at the college level, can use notation to manage complex unit relationships.