Making Connections from the Secondary Classroom to the Abstract Algebra Course: A Mathematical Activity Approach

Abstract

Connections between Abstract Algebra courses and high school mathematics teaching commonly start with identification of abstract algebra topics in secondary school mathematics. We take an alternative path. We start in the daily practice of teaching school mathematics to determine how Abstract Algebra course experiences might be useful to teachers. Our approach benefits from our experience with the Situations Project (Heid et al., Mathematical understanding for secondary teaching: A framework and classroom-based situations. Charlotte, NC: IAP, 2015). The project team worked from specific incidents that they witnessed in the practice of teaching secondary mathematics to inform a framework for Mathematical Understanding for Secondary Teaching (MUST). One of its three perspectives is Mathematical Activity, which includes Mathematical Noticing (noticing structure of mathematical systems, noticing symbolic form, noticing argument form, and connecting within and outside mathematics), Mathematical Reasoning (justifying/proving, reasoning when conjecturing and generalizing, and constraining and extending), and Mathematical Creating (representing, defining, and modifying/transforming/manipulating). We revisit some of the specific instances from the Situations Project and analyze them anew. We foreground mathematical activities as we draw parallels between mathematical activity in School Algebra classrooms and mathematical activity in Abstract Algebra. We argue that the mathematical activities in which Abstract Algebra students engage transcend the two course settings; we suggest strategies for incorporating Mathematical Activity into the teaching of Abstract Algebra; and we explain new ways in which Abstract Algebra students benefit in ways that serve well those students who become School Algebra teachers.