Describing Students' Analogical Reasoning While Creating Structures in Abstract Algebra
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Analogical reasoning has played a significant role in the development of modern mathematical concepts. Although some perspectives in mathematics education have argued against the use of analogies and analogical reasoning in instructional contexts, some attempts have been made to leverage the pedagogical power of analogies. I assert that with a greater understanding of how students develop analogies and reason by analogy, analogies can indeed be used productively for the teaching and learning of mathematics. Using abstract algebra as the primary context, I propose three papers: (1) a theoretical paper orienting analogical reasoning as a way of thinking in mathematics (and thus learnable by students), (2) an empirical paper contributing the Analogical Reasoning in Mathematics (ARM) framework for interpreting students’ activity during analogical reasoning and (3) a practitioner paper detailing a full lesson incorporating analogical reasoning as a tool for exploratory structure creation in abstract algebra.
Paper #1 identifies analogical reasoning as a way of thinking in the context of advanced mathematics. There has been critique of the use of analogies for the purpose of students learning new content because students may fail to appropriately recognize the analogical connections developed by instructors. I counter that students can productively reason by analogy to understand new mathematics when provided with settings to develop this way of thinking. In this paper, I use examples from the work of mathematicians to argue for the important role of analogy for the purpose of mathematical discovery. I then provide an illustration of an undergraduate student engaged in similar productive analogical reasoning as they develop analogs between structures in group and ring theory. Through this process, the student showed increasing awareness of how and why they were engaging with such reasoning. This observation evidences the potential for students to reason by analogy for mathematical discovery.
Paper #2 establishes the Analogical Reasoning in Mathematics (ARM) framework for describing students’ analogical activity in mathematics contexts. I first outline a definition of analogy and contrast it with the concept of metaphor. I then introduce ARM, which categorizes analogical reasoning activity that is unique to the context of doing mathematic and explicates features of analogical reasoning that are largely implicit in existing models. Constructed from an analysis of interviews with four students engaged with analogical tasks in abstract algebra using basic qualitative methods related to grounded theory, ARM includes three dimensions of analogical activity: mapping/non-mapping across domains (MAD), attending to similarity and difference (SAD), and foregrounding a domain (FAD). Built upon these dimensions, analogical activities are identified and explicated for the purpose of analyzing student analogical reasoning. I provide examples of several of these activities in the context of abstract algebra.
Finally, Paper #3 proposes a novel lesson for introducing structures in ring theory by reasoning analogically about structures already known in group theory. In this way, students come to creatively establish new structures that they may take ownership of while providing opportunities for rich discussion about the purpose of these structures. The lesson consists of four key components: (a) introducing the definition of ring, (b) introducing the idea of analogy and analogical reasoning between groups and rings, (c) developing structures (i.e., subrings, ring homomorphisms, and quotient rings) through analogical reasoning with known structures, and (d) developing theorems/proofs through analogical reasoning. Throughout this paper, I provide thoughts and insights from previous implementations and conclude by reflecting on what has (and has not) worked well in my experience with implementing these tasks.
Taken together, these papers offer insight into understanding how students reason by analogy and suggests implications for productively incorporating analogical reasoning into instruction. Directions for future research involving analogical reasoning in mathematics are outlined based on these contributions.