On Disrupting Hierarchies

By David Bressoud @dbressoud

As of 2024, Launchings columns appear on the third Tuesday of the month.

This is an overview of the recent MAA Notes volume by Daniel Reinholz: Equitable and Engaging Mathematics Teaching: A Guide to Disrupting Hierarchies in the Classroom.

When I used Thomas Hawkins’ doctoral dissertation Lebesgue’s Theory of Integration: Its Origins and Development for my real analysis seminar (see True Grit in Real Analysis), in each class one or two students were responsible for introducing the dramatis personae and their contributions. Their presentations usually included a portrait of the mathematician in question. When one of the students couldn’t find an image of her mathematician, she just drew a middle-aged white man with a full beard. That is how they all looked. Full beards have gone out of style, but the popular impression of someone who does mathematics is still dominated by middle-aged to elderly white men. Those are the images that have dominated the hallways of math departments and that MAA and the other professional societies in the mathematical sciences have been working to replace.

Changing society’s image of who does mathematics is harder than changing hallway portraits, and society’s expectations have a corrosive effect on the teaching and learning of mathematics. One of the most dramatic insights from the MAA’s calculus study was how easily any setback, no matter how minor, can discourage a student from an underrepresented group from continuing their study of mathematics. These hierarchies of gender, race, ethnicity, sexual self-identification, and many others are the focus of Reinholz’s monograph. He identifies practices that reinforce expectations of who does and who does not belong. This is not about leveling the playing field but rather introducing strategies that encourage and empower all students to reach their full potential. It is about addressing and disrupting deeply rooted inequities.

While active learning has an important role to play, this book does not advocate banning lectures. Reinholz spends a good deal of time discussing the strengths and weaknesses of lecture and how to think about when and how to use it. He also critiques the idea that active learning is a panacea, pointing out implementations that can reinforce existing hierarchies as well as means of overcoming its weaknesses. Reinholz and Niral Shah have developed an observation tool, EQUIP, for tracking patterns in student participation. With Brooke Ernest they have documented that in whole class discussions, men will often dominate (see Figure 1) unless the instructor adopts methods for ensuring equitable whole-class participation.

Figure 1: Results of tracking participation in an active learning undergraduate geometry course. Side talk refers to participation during small group work. Public talk is from whole-class discussion. Source: Ernest, Reinholz, & Shah (2019).

The chapter on “Hierarchies in Mathematics” lays out the ways in which microaggressions, low expectations, and stereotype threats impair student achievement for minoritized students. The chapter on “Assessment” usefully distinguishes between grades and feedback, suggesting many ways in which instructors can privilege the latter. Unfortunately, it is usually impossible to eliminate the former (though several places do that for first-year students), but Reinholz suggests a number of approaches that lessen the impact of a C, B, or even a low A on students, especially those whose self-identification as someone who belongs in mathematics has been tied to high grades in previous courses. In particular, he details standards-based grading and specifications grading, approaches that I have used with a great deal of success. Too often, grades are little more than a ranking of the students in the class, reinforcing the common impression that the course is about competing rather than learning. The use of standards-based and specifications grading makes it possible for the grade to incorporate meaningful feedback. The chapters “Setting the Stage” and “Facilitating Practice” offer a wealth of practical strategies that an instructor can use in the classroom.

The final short chapter, “Catalyzing your Learning”, lays out suggestions for practicing and improving personal implementation of any of the many strategies presented in this book. This monograph is a rich source of ideas. Trying to do everything in the book would be overwhelming. However, the point is not to change everything about how one teaches but to pick up one or two ideas that feel do-able and that you believe would help your students. The book also offers concrete suggestions on how you learn together with colleagues to make the instructional change process feasible and enjoyable. I look at my own evolution as a teacher and recognize it as a process that proceeded over decades: trying new ideas, discarding some, and adapting others to my own strengths and weaknesses.  This book is both an invitation to begin and an encouragement to continue such a journey.

References

Bressoud, D.M. (2020). True Grit in Real Analysis, Mathematics Magazine, 93:4, 295-300, DOI: 10.1080/0025570X.2020.1790967

Ernest, J.B., Reinholz, D.L. & Shah, N.  (2019). Hidden competence: women’s mathematical participation in public and private classroom spaces. Educ Stud Math 102, 153–172. https://doi.org/10.1007/s10649-019-09910-w

Hawkins, T. (1975). Lebesgue's theory of integration: its origins and development. Providence, RI: AMS Chelsea Publishing. ISBN: 978-0-8218-2963-9

Reinholz, D. (2023) Equitable and Engaging Mathematics Teaching: A Guide to Disrupting Hierarchies in the Classroom. MAA Notes #97. Providence, RI: MAA Press. ISBN978-0-88385-212-5

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