In this paper, we present data from two iterative teaching experiments involving students' constructions of four basic counting problems. The teaching experiments were designed to leverage the generalizing activities of relating and extending to provide students with opportunities to reflect on initial combinatorial activity when constructing…

Computational thinking and activity are vital aspects of what it means to conduct scientific and mathematical work. In light of this, some propose that students' mathematical education should include an integration of computing into their mathematical experiences, giving students opportunities to engage with computational tools as they reason…

The multiplication principle (MP) is a fundamental aspect of combinatorial enumeration. In an effort to better understand students' reasoning about the MP, we had two undergraduate students reinvent a statement of the MP in a teaching experiment. In this paper, we adopt an actor-oriented perspective (Lobato, "Educational Researcher,"…

Computational activity is increasingly relevant in education and society, and researchers have investigated its role in students' mathematical thinking and activity. More work is needed within mathematics education to explore ways in which computational activity might afford development of mathematical practices. In this paper, we specifically…

The multiplication principle (MP) is a fundamental aspect of combinatorial enumeration, serving as an effective tool for solving counting problems and underlying many key combinatorial formulas. In this study, we used guided reinvention to investigate 2 undergraduate students' reasoning about the MP, and we sought to answer the following research…

When solving counting problems, students often struggle with determining what they are trying to count (and thus what problem type they are trying to solve and, ultimately, what formula appropriately applies). There is a need to explore potential interventions to deepen students' understanding of key distinctions between problem types and to…

The multiplication principle serves as a cornerstone in enumerative combinatorics. The principle underpins many basic counting formulas and provides students with a critical element of combinatorial justification. Given its importance, the way in which it is presented in textbooks is surprisingly varied. In this paper, we analyze a number of…

In this paper we report on a survey designed to test whether or not students differentiated between two different types of problems involving combinations--problems in which combinations are used to count unordered sets of distinct objects (a natural, common way to use combinations), and problems in which combinations are used to count ordered…

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…

Counting problems offer rich opportunities for students to engage in mathematical thinking, but they can be difficult for students to solve. In this paper, we present a study that examines student thinking about one concept within counting, factorials, which are a key aspect of many combinatorial ideas. In an effort to better understand students'…

Computer scientists have reported on "computational thinking," which Aho (2012) defines as "the thought process involved in formulating problems so their solutions can be represented as computational steps and algorithms" (p. 832). We wanted to investigate such thinking in mathematics. To study this, we interviewed five…

Combinatorial enumeration has a variety of important applications, but there is much evidence indicating that students struggle with solving counting problems. The roots of such difficulty, as well as ways to mitigate such difficulty, have not yet been thoroughly studied. In this paper, one particular aspect of students' counting activity is…

To further understand student thinking in the context of combinatorial enumeration, we examine student work on a problem involving set partitions. In this context, we note some key features of the multiplication principle that were often not attended to by students. We also share a productive way of thinking that emerged for several students who…

The multiplication principle is a fundamental principle in enumerative combinatorics. It underpins many of the counting formulas students learn, and it provides much-needed justification for why counting works as it does. However, given its importance, the way in which it is presented in textbooks is surprisingly varied. In this paper, we document…

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…