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## Multiplication Principle Problems and Permutation Problems

Distinguishing multiplication principle problems and permutation problems in the classroom helps us examine common student errors.

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## Two Undergraduate Students' Reinvention of the Multiplication Principle

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, the authors used guided reinvention to investigate 2 undergraduate students' reasoning about the MP, and they sought to answer the following research questions: How do students come to understand and make sense of the MP? Specifically, while a pair of students reinvented a statement of the MP, how did they attend to and reason about key mathematical features of the MP? The students participated in a paired 8-session teaching experiment during which they progressed from a nascent to a sophisticated statement of the MP. Two key mathematical features emerged for the students through this process, including independence and distinct composite outcomes, and we discuss ways in which these ideas informed the students' reinvention of the statement. In addition, we present potential implications and directions for future research.

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## Counting Attribute Blocks: Constructing Meaning for the Multiplication Principle

The fundamental counting principle (fcp) is a “big idea” in mathematics. It says that if an event E can occur in a ways and event F can occur in b ways, then event E followed by event F can occur in a × b ways. It is easy to understand and easy to apply. For instance, if Ng has 3 sweatshirts and 2 pairs of jeans, he has the possibility of 3 × 2, or 6, different outfits. Yet many learners whose view of mathematics is strictly procedural have little conceptual understanding of the fcp. This deficiency becomes evident when they try to remember formulas or procedures, such as those for counting various types of arrangements or subcollections. The activity described in this article offers a meaningful introduction to the principle.

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## Theoretical Framing as Justifying

In our March editorial (Cai et al., 2019), we discussed the nature of significant research questions in mathematics education. We asserted that the choice of a suitable theoretical framework is critical to establishing the significance of a research question. In this editorial, we continue our series on high-quality research in mathematics education by elaborating on how a well-constructed theoretical framework strengthens a research study and the reporting of research for publication. In particular, we describe how the theoretical framework provides a connecting thread that ties together all of the parts of a research report into a coherent whole. Specifically, the theoretical framework should help (a) make the case for the purpose of a study and shape the literature review; (b) justify the study design and methods; and (c) focus and guide the reporting, interpretation, and discussion of results and their implications.

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## Developing Mathematical Reasoning Using Attribute Games

The game of Set has proved to be a very popular game at our college mathematics club meetings. Since we started playing, the membership has grown every month. In fact, one of our members brought her sixyear- old son to a meeting, and he now looks forward to playing Set with us. As a result of playing the game in our club and thinking about the results, we created and solved a variety of mathematical questions. For example, we wondered about possible strategies for winning and conjectured about phenomena that happened when playing. These questions involve a wide variety of traditional mathematical topics, including the multiplication principle, combinations and permutations, divisibility, modular arithmetic, and mathematical proof.