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## Sense-able Combinatorics: Students' Use of Personal Representations

As we move forward in the twenty-first century, information and its communication have become at least as important as the production of material goods, and the nonmaterial world of information processing requires the use of discrete mathematics (NCTM 1989). Combinatorics, the mathematics of counting, plays a significant role in discrete mathematics. It is usually described as having three parts: counting (how many things meet our description), optimization (which is the best), and existence (are there any at all). The NCTM is explicit about the importance of students learning discrete mathematics: “As an active branch of contemporary mathematics that is widely used in business and industry, discrete mathematics should be an integral part of the school mathematics curriculum” (NCTM 2000, p. 31).

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## Counting Pizzas: A Discovery Lesson Using Combinatorics

How many distinct pizzas can be made in a restaurant that has n bins of toppings if the pizza maker reaches into the bins at random and puts k toppings on the pizza?

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## Combinatorics Connections: Playoff Series and Pascal's Triangle

One major theme of the National Council of Teachers of Mathematic's Curriculum and Evaluation Standards far School Mathematics (1989) is the connection between mathematical ideas and their applications to real-world situations. We shall use concepts from discrete mathematics in describing the relationship between sports series and Pascal's triangle.

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## The Place and Purpose of Combinatorics

The union of curriculum goals intersects with math education standards.

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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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## Delving Deeper: Boundaries in Higher Dimensions and Their Kinship with Combinatorics

This is a description of a collaborative investigation by mathematics teachers into the numbers of dimensional boundaries for n > 2. Functions are fit to the patterns observed, and a relationship to Pascal's triangle is noted.

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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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## 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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## Connecting Research to Teaching: Combinatorial Mathematics: Research into Practice

For many students, combinatorics is associated with negative experiences calculating permutations and combinations, often confusing one with the other. What exactly is combinatorics? Combinatorics can be defined as the art of counting, or more specifically, as “an area of mathematics in which we study families of sets (usually finite) with certain characteristic arrangements of their elements or subsets, and ask what combinations are possible and how many there are” (Rusin 2002).

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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.