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## Common Errors in Counting Problems

Analyzing combinatorics problems involving flags and playing cards presents common pitfalls for students—pitfalls that can be avoided.

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## Revisiting Some Counting Problems in Discrete Mathematics

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## Solutions to the Counting Counts! Problem

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## A New Look at an Old Triangle Counting Problem

The Matchstick problem of counting equilateral triangles is modified to allow for additional side lengths and summation formulas, which lead to a result represented with combinatorics.

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## Problem Solvers: Solutions to the Cheerio Count Problem

The problem appearing in the December 2005/ January 2006 “Problem Solvers” section was stated as follows:

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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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## Informing Practice: Counting Using Sets of Outcomes

### research matters for teachers

A branch of mathematics—combinatorics—is explored through counting problems.

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## The Back Page: My Favorite Lesson: Why Do We Need Proof?

A well-known counting problem convinces students that proof is necessary.

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