http://mathedseminar.pbworks.com/w/file/fetch/70289809/A%20model%20of%20students%E2%80%99%20combinatorial%20thinking.pdf

A model of students' combinatorial thinking

We focus on a major difficulty in solving combinatorial problems, namely, on the verification of a solution. Our study aimed at identifying undergraduate students' tendencies to verify their solutions, and the verification strategies that they employ when solving these problems. In addition, an attempt was made to evaluate the level of efficiency of the students' various verification strategies in terms of their contribution to reaching a correct solution. 14 undergraduate students, who had taken at least 1 course in combinatorics, participated in the study. None of the students had prior direct learning experience with combinatorial verification strategies. Data were collected through interviews with individual or pairs of participants as they solved, 1 by 1, 10 combinatorial problems. 5 types of verification strategies were identified, 2 of which were more frequent and more helpful than others. Students' verifications proved most efficient in terms of reaching a correct solution when they were informed ...

Students' Verification Strategies for Combinatorial Problems.

For the Learning of Mathematics 34, 2 (July, 2014) FLM Publishing Association, Fredericton, New Brunswick, Canada The purpose of this article is to present and describe one particular perspective on solving counting problems (a setoriented perspective) that emerged during a study involving post-secondary students. The notion of a set-oriented perspective is presented both theoretically and empirically and is integrated with Harel’s (2008a, 2008b) ways of thinking. I make the case that such a perspective offers a valuable lens through which to study the learning and teaching of counting and may be an effective factor in helping students correctly solve counting problems. There is clear evidence in the literature that students at all levels struggle with solving counting problems correctly. Many researchers have identified student difficulties with combinatorial topics and reported low success rates (e.g., Eizenberg & Zaslavsky, 2004), and others speak to the overall difficulty of teaching and learning counting (Annin & Lai, 2010; Martin, 2001; Tucker, 2002). Some researchers have taken strides toward helping students become more successful at solving counting problems (e.g., Batanero, Navarro-Pelayo & Godino, 1997; Eizenberg & Zaslavsky, 2004; English, 1991; Lockwood, 2011; Maher, Powell & Uptegrove, 2011), but there remains a need for further examination of the ways of thinking that students bring to counting problems. In particular, the current combinatorics education literature has not explicitly highlighted the importance of focusing on sets of outcomes in students’ solving of counting problems. A set of outcomes is the set of things (objects, arrangements, partitions, etc.) a student is trying to count, the cardinality of which is the answer to the given counting problem. I call this focus on outcomes a set-oriented perspective on counting, which is a way of thinking about counting that involves attending to sets of outcomes as an intrinsic component of solving counting problems. The point is that by adopting such a perspective, in which facility with outcomes is a natural part of how they approach and solve counting problems, students may avoid common pitfalls such as incorrectly applying a formula, inadvertently overcounting, or confusing issues of order. I elaborate this perspective throughout the remainder of the article, illustrating the ideas with examples from my research.

A Set-Oriented Perspective on Solving Counting Problems.

Counting problems provide an accessible context for rich mathematical thinking, yet they can be surprisingly difficult for students. To foster conceptual understanding that is grounded in student thinking, we engaged a pair of undergraduate students in a ten-session teaching experiment. The students successfully reinvented four basic counting formulas, but their work revealed a number of unexpected issues concerning justification in counting. In this paper, we describe the students’ successful reinvention of the four counting formulas, we critically examine their combinatorial reasoning in terms of Lockwood’s (2013) model of students’ combinatorial thinking, and we offer several directions for further research.

https://link.springer.com/content/pdf/10.1007%2Fs40753-015-0001-2.pdf

Patterns, Sets of Outcomes, and Combinatorial Justification: Two Students’ Reinvention of Counting Formulas

Although counting problems are easy to state and provide rich, accessible problem-solving situations, there is much evidence that students struggle with solving counting problems correctly. With combinatorics (and the study of counting problems) becoming increasingly prevalent in K–12 and undergraduate curricula, there is a need for researchers to identify potentially significant factors that might have an effect on student success as they solve counting problems. We tested one such factor among undergraduate students—their systematic listing of what they were trying to count. We argue that even creating partial lists of the set of outcomes led to significant improvements in performance in students’ success on problems, implying that systematic listing may be worthwhile for students to engage in as they learn to count. Our findings suggest that more needs to be done to refine instructional interventions that facilitate listing. We discuss these findings and suggest avenues for further research.

Combinatorial tasks and outcome listing: Examining productive listing among undergraduate students

In this paper we examine typical difficulties involved in solving a combinatorial problem, and demonstrate them throughout the solution of the “Problem of the misaddressed letters”. We focus on difficulties of two sorts: Those which stem from the fact that a combinatorial problem is usually an infinite set of problems, and others which are involved in finding a systematic approach to their solution.

The Road to Solving a Combinatorial Problem is Strewn with Pitfalls.

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The Road to Solving a Combinatorial Problem is Strewn with Pitfalls.