The importance of students’ identity, power, and experiences

In this blog post, I point out that there is growing evidence that there some so-called “active learning” teaching strategies (on average) improve learning outcomes for all students while also (on average) improving learning outcomes disproportionately more for women, students of color, and/or first-generation students.
For example, in this paper, Sandra Laursen, Marja-Liisa Hassi, Marina Kogan, and Timonthy Weston investigate what undergraduate students report about their learning in Inquiry-Based Learning (IBL) mathematics courses:
…women in non-IBL courses reported statistically much lower gains than their male classmates in several important domains: understanding concepts, thinking and problem-solving, confidence, and positive attitude toward mathematics. In fact, both men and women reported higher learning gains from IBL courses than from non-IBL courses, but traditional teaching approaches did substantial disservice to women in particular, inhibiting their learning and reducing their confidence… Overall it appeared that non-IBL courses tended to reinforce prior achievement patterns, helping the “rich” to get “richer.” In contrast, IBL courses seemed to offer an extra boost to lower-achieving students, especially among pre-service teachers. Yet there was no evidence of harm done to the strongest students.
Now contrast that paper with this recent paper: Johnson, E., Andrews-Larson, C., Keene, K., Keller, R., Fortune, N., & Melhuish, K. (2018). Inquiry and inequity in the undergraduate mathematics classroom. Proceedings of the 40th Annual Conference of the North American Chapter of the International Group for the Psychology of Mathematics Education, pages 966-969.
These researchers looked at student learning in an Inquiry-Oriented Abstract Algebra (IOAA) course taught by 12 different instructors. They compared the performance of these students with those in more traditional Algebra courses at other institutions using the same content assessment instrument. They found that men and women performed similarly on the instrument in traditional courses but that men outperformed women slightly in their sample of IOAA courses. And, this performance gap was bigger for some of the 12 instructors than others.

How do we square the research of Laursen et al with that of Johnson et al?

1. First, the student sample size in the Laursen et al paper (over 3000) was much bigger than that for the Johnson et al paper (513). And, when the latter authors analyze student performance in specific instructors’ courses, those numbers are even smaller. It’s natural to have larger sampling variation with smaller samples.

And we’re talking about something much more complicated than taking samples from the same normal distribution. Two students might sit in the same course, listen to the same instructor, and do the same assignments, but might still have very different experiences in that class based on their life experiences, identities, interests, etc. Corresponding, their learning outcomes might be different too.

I see the Johnson et al paper as a warning to us all that it’s not enough just to employ some of these active learning instructional practices. We need to also be more intentionally inclusive as we teach in more active ways by paying attention to the experiences that students have in our courses. This reminder is also echoed by Elham Kazemi and Corey Drake in their plenary paper at the same PME-NA 40 conference (where Johnson and her colleagues presented their work). 

2. We should remember that these empirical papers measure student learning and/or attitudes under different conditions for learning but they don’t explain the mechanism behind why some students might benefit or be harmed by those conditions. In other words, it is still not clear why IBL courses seems to have a disproportionately positive effect on women in the paper by Laursen et al.

3. And, keep in mind that Laursen et al measured student self-reported gains while Johnson et al directly measured student learning on a content area assessment. We would expect self-reported learning gains to be strongly correlated with performance on a content assessment, but they aren’t the same thing.

Finally, how do we think about this work by Johnson et al in relation to this CBMS statement promoting active learning? How do we adopt a critical stance when studying these instructional practices without jeopardizing the movement toward broader adoption of active learning in our field?

Rochelle Gutiérrez’s dimensions of equity comes to mind (even though she is no longer using the term “equity”).

Gutiérrez, R., 2009. Framing equity: Helping students “play the game” and “change the game.”. Teaching for excellence and equity in mathematics, 1(1), pp.4-8.
(Gutiérrez’s diagram from pg 6 of TEEMv1n1)

When we tout active learning via studies that show that it can produce disproportionately positive effects for women, students of color, and other marginalized groups of students (and I have done this!), that primarily emphasizes the dominant axis of access/achievement, since those studies generally quantify equity via summative measures of learning. What is missing is a focus on the critical axis of identity and power, which can greatly affect how students actually experience the learning in our classes.

Update: Here are some other research articles that also add some nuance to our understanding of how certain instructional strategies relate to more equitable outcomes for students.

  • Brown, S., 2018. “E-IBL, Proof Scripts, and Identities: An Exploration of Theoretical Relationships.” RUME 2018 Conference paper.

  • Eddy, S.L., Brownell, S.E., Thummaphan, P., Lan, M.C. and Wenderoth, M.P., 2015. “Caution, student experience may vary: social identities impact a student’s experience in peer discussions.” CBE—Life Sciences Education, 14(4), 1-17.

  • Langer-Osuna, J.M., 2011. “How Brianna became bossy and Kofi came out smart: Understanding the trajectories of identity and engagement for two group leaders in a project-based mathematics classroom.” Canadian Journal of Science, Mathematics and Technology Education, 11(3), pp.207-225.

Defensiveness is a barrier to equity work

Defensiveness can be a barrier to effective communication, particularly when it involves receiving feedback from others or encountering information that contradicts what we believe about ourselves. For that reason, defensiveness can be a major barrier to becoming a more equitable educator. In this blog post, I’d like to dive deeper into this subject to think more deeply about what I can do to become a more equitable educator and how we can help our field move towards greater equity and justice.

A few months ago, I was teaching multi-variable calculus and drew the following vector field on the board.


When this happened during class, a student mentioned “Oh! It looks like a swastika” and I said that it was unintentional but didn’t change the diagram and just kept going on.

After class, one of my students came to tell me that another student felt uncomfortable with the swastika and with the way that I handled the situation. In that moment, I felt a rush of defensiveness. My initial instinct was to defend myself (to point out that the swastika has actually been used for centuries before the Nazis and the version that appeared in class is actually a different orientation from the Nazi swastika and is closer to the sacred symbol used in many Asian cultures (…and even now as I write this I still feel defensive and feel like I have to explain myself)). I didn’t say any of that. Thankfully, I just paused and apologized, expressed that it wasn’t my intention to be dismissive and didn’t realize that it had that impact, and said that I would be more careful with this example in the future. Later, I removed one of the red arrows above in the online version of the lecture notes.

I share this anecdote not to make any kind of judgment on the student who felt uncomfortable. I don’t want to get into a debate about political correctness or students who are fragile “snowflakes”. All that matters here is that a student felt uncomfortable, I received feedback about it, and that feedback helped me recognize how I will do better in the future. (In the future, I won’t shy away from this example, but I will take 30 seconds to share a bit of history about the symbol.)

What was the source of the defensiveness in that situation? I realize now that the first emotions I felt in that moment had to do with surprise and anger from being misunderstood: “How could anyone think that I would be so insensitive or oblivious as to use such a hateful symbol in my class? Don’t you know all the stuff I’ve done to promote equity and justice?” I really dislike the feelings that come with someone being offended or upset by something I said or did, when that was not my intention–lots of people feel this way. And, that, dear friends, is a common source of tension for many people about talking and thinking about equity. The recognition that our past deeds might have marginalized, discouraged, or discriminated against people of color, women, people with disabilities, LGBTQ people (list goes on…) runs smack into the thought “BUT I DIDN’T MEAN IT”. And yet, while the intent was not there, the impact can still be real.

How does defensiveness lead to undesirable outcomes? When receiving feedback from someone else, defensiveness shifts the focus from the issue at hand to our own emotions. In this situation, if I would have responded defensively, it would have shifted the attention away from the student who was uncomfortable to my own hurt feelings. And the reality here is that I had nothing here to lose except my own stupid ego by listening to and receiving the feedback.

I’m not saying that we instructors should not defend ourselves against baseless accusations. If it is really the case that there is no possible iota of truth in the feedback that I am given, then I can choose to reject the feedback, or perhaps actually defend myself. However, in this situation, there really was something that I could have done differently so the feedback was helpful. And, the power difference in the student-teacher relationship is important here. Because it takes so much more courage to say something to someone who has power over you, I believe that when instructors receive feedback like this we should listen carefully and err on the side of being generous.

Here’s another example of how defensiveness can be an impediment to thinking about and working toward greater equity. An imagined conversation snippet:

Person A: “I think when our department enacts policy X, that is a form of institutional racism.”
Person B: “What are you saying? I’m not racist!”

The words “racist” and “racism” can conjure up strong feelings and sometimes bring conversations about equity to a halt. The historical significance of the word “racist” (that it came into popular use in the 1930s in response to the Nazis) helps to explain why some people react so strongly to that word.

This kind of defensive reaction often stems from the contradiction between unjust outcomes or discriminatory behavior and a belief in oneself as a good and decent person: “How could students possibly suggest that I could be racist/sexist/*-ist? Don’t they know that I have been fighting for [XYZ cause] for longer than they have been alive?” As with the previous swastika anecdote, the defensiveness here has the effect of shifting the conversation away from the topic of discussion (a policy) to a person’s feelings.

To get move past my own defensiveness, I want to work toward having a thick skin but tender heart. I want to become less sensitive to the effects of any feedback and criticism on my ego, or my conceptions of who I am or want to be, and instead be more sensitive to the experiences of others. I would rather err on the side of being misunderstood than to misunderstand others. My problem is how to disrupt the surge of defensive emotions so that I can exercise enough cognitive control to react gracefully in that situation.

And as an educator who tries to help others move toward greater equity too, I think this deeper understanding of my own defensiveness will help in situations when I encounter defensiveness with others. Defensiveness often begets more defensiveness, so having a thick skin and tender heart can help to disrupt that cycle.

(Many thanks to Dylan Kane for encouraging me to think about this topic, and to Robin Wilson and Dagan Karp for their helpful feedback.)

The Ever-Present Challenges of Group Work

I have written previously about how group work is both great for students and difficult to do well. Today I had a conversation with a student and faculty member that reinforced these ideas again.

Prof. X, a faculty member that that I know, is trying to incorporate more small group work in an upper division mathematics class. Through a set of circumstances that are too complicated to explain here, today I happened to talk to Student S who is in Prof. X’s class. Student S shared some difficult moments that she experienced in the class recently.

(1) In one situation, Student S said that she was in a group of four students. The students were asked to answer questions on the back of a worksheet, which involved parsing some definitions on the front of the worksheet. Three of the students chose to forge ahead on the questions without reading the definitions first, referring to them when they needed them. She wanted to read the definitions first and understand them before moving on. They left her behind and she basically worked on the assignment by herself.

(2) In another situation, Student S was in a group of four and was the only female student. Each person in the group was given a different task to complete and had to explain how to do that task to the rest of the group.  When it was Student S’s turn to explain her task, a student in her group was a bit confused and another student jumped in and explained Student S’s task instead of letting her explain it. And, that student didn’t explain it correctly! But, Student S diplomatically corrected the situation by giving the correct explanation to her group and also helped the student who jumped in realize that it wasn’t right that she wasn’t given the chance to talk about her assigned task. Nevertheless, Student S said that this “mansplaining” happened two more times that day.

Experiences like these caused Student S to feel frustrated in these group work experiences. Student S also shared that she has had a difficult semester full of experiences that have caused her to doubt her mathematical abilities. I don’t know all of the details, but I do know that one of the reasons why active learning (and group work in particular) is so risky is that it puts students at risk of being judged by each other and the instructor as being incapable, slow, unprepared, less than.

I also happened today to talk to Prof. X, who genuinely wants to make his class a more welcoming and inclusive space by using group work. He shared with students some research showing that that group work helps improve student learning overall but also tends to have disproportionately large benefits for women, first-generation students, and students of color. That transparency is very important and most of us could probably benefit from being more transparent about the rationale for our instructional choices. However, there is a slight danger here that I hadn’t considered before: these research studies show positive results of using group work on average, but that research does not guarantee that individual students will always benefit from the effects of group work. In fact, my hunch is that the variance of experiences in active learning settings may be higher than in a more controlled lecture classroom format. So, while transparency is important, I also wonder about whether in sharing evidence for these instructional approaches we also need to be careful to avoid making students that don’t have good experiences as a result of those approaches feel bad about that.

Let no one misinterpret what I’m saying here as a pronouncement against active learning or group work–on the contrary, we should double-down on group work and active learning, recognizing that it is simultaneously beneficial for students and difficult to implement well. We instructors need to think carefully about how we deploy group work and other forms of active learning so as to mitigate any potential negative effects, as much as is possible. Here are three questions to ask ourselves to do so:

  1. Who is likely to benefit (as a result of me using group work or some other form of active learning)? Who is not?
  2. Who is likely to feel included? Who is not?
  3. How would I know if I need to intervene in some way?

To answer questions #1 and #2, I suggest we think of groups of students such as first-generation students, introverted students, students of color, students for whom English is not their first language, students with learning disabilities, students who have had bad experiences in a mathematics classroom, etc. For example, in the case of (1) above, it might be the case that the group work was inadvertently structured in a way that students who needed or wanted more time processing definitions were at a disadvantage. It would then be a good exercise to think about ways to mitigate the difference in experiences caused by the way the activity was structured.

Note about questions #1 and #2: Almost certainly students who come to class prepared are more likely to feel included or to benefit from your class activities compared to students who don’t come to class prepared–that is a difference that is expected and that you may or may not want to mitigate.  It’s those other unintended and unwanted consequences that we need to look for.

Regarding Question #3: Student S is a remarkable person who has enough self-assurance that she was able to have a long conversation with Prof. X about these difficult topics, which led Prof. X to make some changes to the class. But, that leads me to wonder: in the absence of having such thoughtful, well-composed, mature students as Student S, how would we find out about these kinds of problems in our class? We can’t be everywhere all at once in the room when group work is happening. If we don’t have mechanisms to gather student feedback, problems can go unnoticed and result in real harm to students.

These conversations today also remind me the important of setting and maintaining norms of behavior in the classroom. It’s easy to set up good norms. It’s much more difficult to maintain them and creating a culture in your classroom where those norms are pervasive.

The reason these conversations were so poignant for me is that I know Prof. X has the best intentions and is making a strong effort to be inclusive and welcoming and yet Student S experienced these discouraging situations. It made me wonder how ignorant I am about such things happening in my class.