To Active Learning and Beyond: Attending to Student Thinking AND Student Experience in Active-Learning Math Classes

By Jess Ellis Hagman, Contributing Editor, Colorado State University

On a recent trip to Santa Fe, New Mexico, I met a really cool woman named Anna Sale who runs a podcast called Death, Sex, and Money (check it out). In this podcast she interviews people about things she is curious about. We talked about how her work is similar to research (come up with something you want to know more about, then go learn about it), except much less rigorous and you get answers much more quickly.

One thing I am very curious about is how students from marginalized populations experience active-learning classes. I believe deeply in teaching in a more active way, and I also believe deeply in teaching so that all of my students have the best opportunity to succeed, and sometimes I wonder if all my active-learning moves are enough to support all of my students. So, taking inspiration from Anna, I decided to interview some experts. (I am also working on a grant proposal to look at this in a much more rigorous/slow way).

The interviewees: I chose four people to interview who I have either talked to about active learning and issues of equity or who were recommended as someone good to talk to about these things. This led me to: Brian Katz, Gail Tang, Darryl Yong, and Christine von Renesse. At the end of this post I introduce them in more depth, and I will admit that I didn’t realize I had asked only people from small, teaching-focused colleges (Augustana College, University of La Verne, Harvey Mudd College, and Westfield State University, respectively). If this were a research project, this may be an issue. However, for this blog-post, I take this as indicative of who is spending time thinking critically about their teaching and developing reputations as thoughtful teachers of undergraduate mathematics. Either way, they provided different and interesting perspectives, and answered everything so well I no longer need to submit my grant proposal looking into these questions a larger scale (just kidding!).

The interview: After BK, CvR, GT, and DY agreed to participate in my interview, I sent them each a link to a Google doc containing the questions and space for each to answer. We did it this way so that each person could react to the others’ responses, which led to fun and interesting exchanges. For this post, I will present a summary of some of the main themes to each answer and share some especially illuminating excerpts. Questions 1-3 provide some background on the interviewees approaches to active learning, while questions 4-6 get to the meat of my interests. Before I move on, I must say: Interviewing people who are experts in something you care about is SO FUN and I highly recommend it!

Questions 1-3 asked what classes they have used active-learning approach to teach, why they use active learning, and how they typically implement active-learning.

What: Between the four of them, my interviewees listed literally every course taught in a typical undergraduate mathematics department, from general education courses such as “Math in Society” and “Quantitative Reasoning” to upper-level mathematics courses like Numerical Analysis and Algebraic Geometry.

Why: Four main reasons for implementing active learning stood out to me in the responses.  Active learning:

  • Gives the teacher access to their students’ thinking
  • Results in both more cognitive and non-cognitive gains for students; BK went as far to say that “this kind of teaching can grow and empower the kind of adults that I hope to see in the world.”
  • Allows students to actually engage in the mathematics and with each other
  • Is more interesting (and less boring) for both the students and for the teacher

CvR referenced a comprehensive list of benefits of active learning for students on the Art of Mathematics website.

How: The interviewees were hesitant to describe a typical day because their teaching is so responsive – they say it depends on the content (“Everything about the active learning depends on the learning objectives for the course as a whole and for the specific class session.” – DY) and on the students (“I come up with a loose plan to do small-group work (let’s say). But then as they’re working in groups, I may have students present depending on what happens during that activity. Sometimes we’ll break in whole-class discussion, also depending on what happens during the activity… It all depends on what is happening in the class!” – GT). As a whole, the interviewees utilize small groups (purposefully structured, will say more later about this), whole-class discussion, student presentations, partner work, and individual work followed by sharing with others. BK and CvR also referenced the importance of the work they have students do before and after class, emphasizing that the active-learning class does not live only within the time and space of class.

Here is where things get meaty:

  1. There is some research that shows active learning can lower achievement gaps between women and men and typically low versus high achievers (Laursen et al., 2014). Has this been your experience? Please explain.

 All four interviewees said that this resonated with their experiences. GT has noticed that in her department women tend to get better grades in the active learning courses compared to traditional courses, and BK has had students from underrepresented groups (both women and students of color) tell him that they feel able to talk in his active learning classes when they haven’t felt comfortable in other classes. CvR noted that she has seen particular benefits for women in her classes, provided that she “find[s] a good group for them.”

She also noted that with active learning, she can more easily differentiate instruction to support students who are lacking prior knowledge and students who are ahead in their knowledge and so has seen benefits for “typically low achievers.” [Note: A few of the interviewees took slight issue with this phrasing in my question. I took it directly from the Laursen et al. paper, which grouped students by previous mathematics GPAs.] DY said “Active learning is awesome and I have seen it create a more equitable playing field for everyone,” but qualified this statement a bit: “I just feel like we shouldn’t assume it to be an automatic thing.” (Which perfectly set up my next question – thanks for the smooth transition DY!)

  1. Last year I taught an Abstract Algebra course using Inquiry-Oriented materials ( and often had my students working in groups, with whole class discussions, and groups sharing their ideas. Overall, I thought the class was a success. However, I noticed that the men and women participated very differently in my class, and that generally the women contributed much less to whole-class discussion, even though I saw them sharing ideas in groups. This made me wonder a lot about ways that active learning may potentially further marginalize students from marginalized populations. What have been your experiences with this?

The interviewees shared that this did resonate with their experiences, and shared specific instances where a student from a marginalized population shared with them that they were not comfortable either in their groups or contributing to their whole-class discussion.

DY summarized a few of their ideas well: “I really think it is important to give students some instructions and models for how to work well with each other. We cannot assume that they know how to work well with each other and in fact we need to expect that students will bring with them all of their prejudices and unconscious biases to class (just like we do) and that we need to set up the working environment to mitigate these things…I am also now more conscious about making sure my instructions to students are crystal clear and that I give them enough individual think time before they have to answer or work together. I do this because our introverted students and students whose first language is not English need a little more time to put together their thoughts. And, I don’t want students to have to read my mind about what exactly I’m asking them to do—that privileges students who might have had more experience working in more active classrooms and I want to create a more level playing field for my students.” 

  1. Finally, I think of teaching comfortably using active-learning approaches as step one. We know there may be things beyond basic active learning approaches that foster a class environment where all of our students can benefit equitably from active learning.  I think of this as step two, or active learning+. What are some easy-to-implement practices that you use that are active learning+?
  • Attend to how students are experiencing class. One way to do this is with anonymous “exit tickets” at the end of class to provide feedback on their class experiences, specifically related to working in groups and also more generally related to their whole class experience.
  • Assign competence. Intentionally name ideas for the students who contribute them, especially if an underrepresented student’s idea is being credited to someone else or abstractly. (Check out this video of Jo Boaler discussing this as part of Complex Instruction, starting at 4:30 until 5:20).
  • Call students by their names. Expect them to know each other’s names, and use some class time for them to learn these names.
  • Say something if something awkward happens. This can be in class by addressing it in the moment, saying that you will address it at a later time, discussing one-on-one with students, or even returning to an awkward moment after the time has passed. The point here is: if something happens in class that may make a student feel bad, say something.
  • Ask yourself if all students have easy access to the resources needed to succeed in the course.
  • Be purposeful in choosing who talks and who doesn’t in your class. Some specific moves to help this
    • Asking a student before a discussion if they would be willing to share an idea;
    • Sharing for a student if they don’t want to talk
    • Inviting other students to talk than the usual 5 that would run the show
    • Asking students to restate someone else’s explanation in their own words
  • Have students reflect on readings related to learning. The interviewees have had their students reflect on readings and/or videos about mindset (e.g. Jo Boaler and Carol Dweck), about pedagogy (e.g. Cirillo (2013) and Laursen et al. (2014)), learning, and the nature of mathematics (e.g. Lockhart’s Lament).
  • Your own positioning as the teacher is important: smile, sit at the tables with students rather than hovering above them, be upfront about why you are doing things in class, and be vulnerable (this may depend on your own identity in the department; as a young woman I may be less comfortable being extremely vulnerable with my students compared to a tenured man).
  • Purposefully structure groups. Again, this is a key component of Complex Instruction (Jo Boaler discusses this a bit in the video from around 1:00-1:25). A recent study found that if students are allowed to group themselves, they will group with students of the same race/ethnicity, gender, and ability level. The authors of this work encourage instructors to avoid letting students default to sitting in heterogeneous groups, but also caution against “structuring…groups in a naïve way that could either add to the barriers faced by underrepresented students or encourage struggling students to be passive.”

My interviewees shared specific ways that they structure groups, and emphasized how important this was to their class functioning equitably:

    • CvR: If I group 3 male students with one female student it can result in the female student not verbally participating. I therefore spend a lot of time thinking about good group and explicitly talking with students about while groups might be a good fit and why. (Research points to the benefit of “minority-majority grouping” for students – meaning try to organize groups so that students from a minority population are grouped so that they aren’t always the minority in their group. Try to have two women in a group together of four rather than one women and three men, for instance.)
    • GT: [I] create groups based on my perceived notions of their levels of empathy…. Recently [I had a class where the students didn’t know each other well and I didn’t know them.]  It was a very challenging experience. I usually let them work together and I observed their interactions and then reformed groups either right away, or the next class.
    • BK: Personally, I like groups to stabilize after a few days of class so that I can teach each group how to work together effectively, rather than rotate or randomly generate them. I have rearranged a couple of groups over the years, but students almost always find a group that works for them by the second day of class. I worry a little about the beginning of this process because I know that the first time groups form can be hard on students from underrepresented groups; if I sense that someone might struggle to join a group of the people near them on the first day, I will do a little subtle organizing to keep them from being left out.

I hope you enjoyed this interview as much as I did. I am thrilled to know equitable active learning is becoming a topic of more discussion – during this years’ IBL workshops we had some great discussions related to this post, and recently discussed related ideas at this year’s MathFest in Chicago.

However, there is still much more for us to learn. As mentioned briefly above, in the area of K-12 mathematics researchers use the term “Complex Instruction” to describe instruction that supports equitable experiences in active learning mathematics courses. As college educators, we can absolutely learn from this work, but there are also reasons why not all of this work is easily transferrable. For instance, typically K-12 teachers spend much longer with their students than we do as college teachers, providing more time to develop a classroom culture that supports the practices they encourage. So – while more discussion is happening related to equity and active learning, as a community there is still much for us to learn. While we learn more, here are some resources for some things we do know:

Charles Duhigg’s Chapter about Groupwork in Smarter, Better, Faster

Elizabeth Cohen’s Designing Groupwork: Strategies for the Heterogeneous Classroom

Laura Rendón Sentipensante (Sensing/Thinking) Pedagogy: Educating for Wholeness, Social Justice and Liberation

Discovering the Art of Mathematics website

Tim Erickson and Rose Craig’s United We Solve: 116 Math Problems for Groups, Grades 5-10, (DY: even though this is for younger kids, this is still a great book to use for undergraduates because they get your students to cooperate in productive ways)

Biographical Information:

Brian Katz BK is an associate professor of mathematics and computer science at Augustana College, with a PhD in mathematics from UT Austin. His research interests include proof, the evolution of shared meaning, epistemology, and equity/justice in the context of inquiry-based mathematics classrooms. For BK, all teaching is political and hence should be about justice. He is currently the Chair of the IBL SIGMAA  and coeditor of the AMS Blog on equity and social justice, called inclusion/exclusion.

Christine von Renesse CvR is an associate professor of mathematics at Westfield State University with a Ph.D. in Mathematics at the University of Massachusetts. CR is originally from Germany, and has a Master’s Degree in Elementary Education, a Minor in Music and a Master’s Degree in Mathematics from the Technical University Berlin, Germany. She is an author and principle investigator for the Art of Mathematics project, and a passionate teacher, who loves teaching at all levels — from elementary school through College. In her free time Christine loves to explore nature, sing in harmony and go dancing, especially with her daughter. To learn more about CvR and the Art of Mathematics, see the website.

Gail Tang Passionate about broadening the participation of women in mathematics, GT relies upon asset-based mentorship and teaching. Her research interests include equity in mathematics and fostering mathematical creativity. GT is an Assistant Professor of Mathematics at the University of La Verne. Additionally, she is the Mathematics Curriculum Lead for Guided Pathways to STEM Success, a DoE-funded grant aimed at providing inclusive opportunities for more students in STEM. In her spare time, GT loves to spend time with her partner, their two dogs, three cats, and five chickens.  Frequently she can be found in their garden among the bees, ladybugs, and butterflies, watching plants grow.

Darryl Yong DY is a Professor of Mathematics at Harvey Mudd College. This year, he is on leave from Harvey Mudd to serve as the Director of the Claremont Colleges Center for Teaching and Learning.  Previously, he served as Associate Dean for Diversity at Mudd from 2011-2016 and has served as associate chair of the mathematics department. He received his PhD in applied mathematics at the University of Washington.  His scholarly activities focus on the retention and professional development of secondary school mathematics teachers and improving undergraduate mathematics education. 

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