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In recent decades, there has been considerable research that explores the teaching and learning of combinatorics. Such work has highlighted the fact that understanding and justifying combinatorial formulas can be challenging for students, and there is a need to identify ways to support students' combinatorial reasoning. In this paper, we contribute to research that explores effective ways to foster students' combinatorial reasoning by highlighting the role that shifts in representational registers can play in supporting students' combinatorial reasoning and activity. In particular, we present a case in which two students, aged 12 and 14, reasoned about, and developed a formula for, binomial coefficients. This occurred in a teaching experiment designed to examine their generalizing activity. In these interviews, the students first solved problems about determining the volume of a cube and then shifted to a combinatorial interpretation of this task, leveraging binomial coefficients as a way to consider growth in higher dimensions. The students demonstrated sophisticated reasoning about combinatorial tasks and were ultimately able to generate and understand a formula for binomial coefficients. In framing this case, we focus on the students' use of different representations, highlighting the power of being able to transition between representational registers as a way of supporting students' combinatorial reasoning. In this way, our case demonstrates the value of representational registers specifically as a mechanism by which to potentially improve the teaching and learning of combinatorics. [ABSTRACT FROM AUTHOR]

In this case study, we investigate one teacher's implementation of DNR-based combinatorics curriculum in their high school discrete mathematics class. By examining the teacher's practices in whole-class discussions of two counting problems, we study how they advanced a variety of ways of thinking to support the development of a set-oriented way of thinking about counting. In particular, we find the teacher worked to build shared experience and understanding of mathematical ideas by grounding her teaching in students' ways of understanding and leveraging students' intellectual needs. In doing so, the teacher promoted a set-oriented way of thinking through attending to connections between sets of outcomes, counting processes, and formulas in student representations and justifications; elevated solutions employing process pattern generalization; and advanced the beliefs that counting problems can be solved in many ways and entail several types of mathematical activity. [ABSTRACT FROM AUTHOR]

• Understanding why counting formulae are used for different problem types requires focus on the set of outcomes • Students can leverage lists of outcomes to relate counting processes to formulae. • Tasks that target the structure of a list of outcomes can help students relate counting processes and formulae. Research has shown that solving counting problems correctly can be difficult for students at all levels, and mathematics educators have sought to identify strategies and interventions to help students reason conceptually about combinatorial tasks. A set-oriented perspective (Lockwood, 2014) is a way of thinking about counting problems that emphasizes the importance of reasoning about the set of outcomes being counted. From a set-oriented perspective, one possible type of intervention is to have students focus on the sets of outcomes rather than formulas and expressions, and specifically to reason about the structure of the set of outcomes. Yet, reasoning about sets of outcomes is not sufficient for students to make connections between outcomes and counting processes. In this paper, we investigate tasks where students wrote computer code to enumerate the set of outcomes in a specific order by implementing listing processes, and they were then asked to determine a specific numbered outcome in their list by using the structure of their enumeration scheme. We clarify particular aspects of a set-oriented perspective that were productive for students, and we demonstrate that tasks that asked students to name a specific outcome in their list elicited meaningful connections between counting processes and sets of outcomes. Further, such tasks reinforce desirable mathematical practices such as leveraging structure and connecting representations. [ABSTRACT FROM AUTHOR]