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The use of computer programming is ubiquitous for research mathematicians, and two common practices are facilitation of experimentation and testing of conjectures. These practices align with empirical re-conceptualization, which is the process where empirical data is used to identify patterns, form related conjectures, and then re-interpret the conjectures from a structural perspective. However, literature on empirical re-conceptualization has only examined student work done by hand. We present case study data of one student, Allen, using the programming language Python to facilitate empirical re-conceptualization as he found the closed-form solution for binomial coefficient C(n,k). We discuss why the computational environment was productive for empirical re-conceptualization and provide avenues for future research. 

In this paper, I discuss undergraduate students’ engagement in basic Python programming while solving combinatorial problems. Students solved tasks that were designed to involve programming, and they were encouraged to engage in activities of prediction and reflection. I provide data from two paired teaching experiments, and I outline how the task design and instructional interventions particularly supported students’ combinatorial reasoning. I argue that emergent computational representations and the prompts for prediction and reflection were especially useful in supporting students’ reasoning about fundamental combinatorial ideas. I argue that this particular mathematical example informs broader notions of disciplinary reflexivity and representational heterogeneity, providing insight into computational thinking practices in the domain of mathematics. Ultimately, I aim to explore the nature of computing and enumeration, shedding light on why the two disciplines are particularly well-suited to support each other. I conclude with implications and avenues for future 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 differentiate meaningfully between such problems. In this paper, we investigate undergraduate students’ understanding of sets of outcomes in the context of elementary Python computer programming. We show that four straightforward program conditional statements seemed to reinforce important conceptual understandings of four canonical combinatorial problem types. We also suggest that the findings in this paper represent one example of a way in which a computational setting may facilitate mathematical learning. 

Students often struggle with issues of order – that is, with distinguishing between permutations and combinations – when solving counting problems. There is a need to explore potential interventions to help students conceptually understand whether “order matters” and to differentiate meaningfully between these operations. In this paper, I investigate students’ understanding of the issue of order in the context of Python computer programming. I show that some of the program commands seemed to reinforce important conceptual understandings of permutations and combinations and issues of order. I suggest that this is one example of a way in which a computational setting may facilitate mathematical learning.