An Inside-Out Course on Number Theory (Pt 2)

Link to previous post: Pt1

Teaching a math course inside of a prison is surprisingly unremarkable. Once we get into the classroom and starting doing mathematics, it’s easy to forget that you’re in a prison. We’re just a bunch of people learning mathematics together.

There are only several real complications I’ve encountered so far: (1) logistics, (2) limits on what we can learn about each other, and (3) much greater heterogeneity in my students’ prior experiences with and attitudes toward mathematics.

There are some significant logistical issues to figure out. My “outside students” and I have to drive from our campus to the prison each week (costs covered by Harvey Mudd College), make sure we’re dressed accordingly, don’t have cell phones or any other contraband, have proper identification, and follow any rules dictated by the prison.

There are some limits on what we can learn about each other. My outside students and I never ask for sensitive information from our inside students: the length of their sentences, what they did to wind up in prison, etc. It’s wonderful if an inside student choose to share that kind of information, but we don’t inquire. Both the inside and outside students are asked not to use their last names in this class. This is so that outside students aren’t tempted to look up information about the inside students, and the inside students don’t continue having a personal relationship with the outside students after the course is over (a rule of the Inside-Out Prison Exchange Program).

This past week, I was reminded about the great heterogeneity of mathematical experience that exists in our classroom. As you can imagine in any group of adults, there were are some people with positive attitudes toward mathematics, and some with negative views. Some of these stories came out during our first class meeting when we shared our experiences with each other about mathematics.

This past Friday, I got a bit of a shock. A student asked me, “What’s that little number next to that number?” He had not seen an exponent before. I was taken aback, not by his lack of experience with exponents, but that I had been so oblivious to that up to this point. There have been some exponents that have appeared in the course materials so far, but this question didn’t come up yet. We quickly talked about exponents and the student was fine and continued working. But, I was a bit shocked and still am a bit unsettled.

In the description for this number theory course, I wrote that some fluency with high school algebra would be required. It could be that some people are taking the class anyway, even if they aren’t fluent with high school algebra. Or it could be that some students have taken high school algebra in the past but have forgotten it. Either way, the reality is that there are students in this course that have very different experiences with mathematics. Some are just (re)learning about exponents and others are making sophisticated connections about what they’re learning. It’s my job as the instructor to make sure that we are all learning mathematics together regardless of our prior experiences; that is a pretty big challenge, however.

Other complications I’ve experienced so far are not germane to teaching Inside-Out. For example, if a student misses a class, I have to find ways to help them find out what they missed so that they can participate in class. My outside students and inside students have both missed classes for various reasons. Outside students have missed class due to illness or travel. Inside students sometimes cannot come to class because their dormitories are on “lock down” during our class.

There is one more significant difference between teaching courses at the Harvey Mudd and teaching through the Inside Out program. The pedagogy that we use in this course is quite different. As I mentioned in the first part of this post, I am not lecturing in this course. We do mathematics together through carefully sequenced problem sets that take into account what we learn during each class meeting and what students are wondering about. I do enjoy teaching in this way and wish that more of my Mudd courses could be taught this way. The pressure of having to “cover” a pre-determined body of material in a prescribed amount of time prevents me from fully teaching in this problem-based approach. One day, I would like to figure out how to teach in this humane way in my Mudd courses.

An Inside-Out Course on Number Theory (Pt 1)

This semester, I’m teaching a course entitled “Introduction to Number Theory” through the Inside-Out Prison Exchange Program. In short, what that means is that every Friday, I take a group of students from the Claremont Colleges (the “outside students”) with me to the California Rehabilitation Center to join 15 incarcerated students (the “inside” students) in learning some introductory topics in number theory.

The California Rehabilitation Center (CRC) is a medium-security state prison for men, located in Norco, California. Here’s a painting of CRC by Sandow Birk (held by the Pomona College Museum of Art).

“For [this] project I visited every one of California’s 33 state prisons and painted a picture of them. The idea was about the changes that California has gone through over its 150 years–from being seen in the 1850’s as an American Eden, where you could go west and dig out gold out of the ground and eat the oranges from the trees and it was always sunny and warm and you could strike it rich, to becoming the most incarcerated population on Earth. It’s shocking. And the more you learn about prisons the more nasty it all becomes.” — Sandow Birk

For over 20 years, the Inside-Out Prison Exchange Program, based at Temple University, has brought campus-based college students with incarcerated students for semester-long courses held in a prison, jail or other correctional setting all around the world. What I appreciate most about the organization is the way it approaches education as a collaborative endeavor and not one in which higher education professors and students go to a carceral organization to “help inmates” out of a sense of volunteerism or charity. Our local Inside Out program was started by Pitzer College and is run in part by a group of incarcerated men at CRC who make up our “Think Tank”. The truth is that I and the Claremont Colleges outside students are learning just as much as inside students are, if not more.

How are students selected? All students (inside and outside) are asked to fill out a questionnaire to find out why students want to take this course and what they hope to gain from the experience. There are several Inside Out courses that the Claremont Colleges offer each semester, and all of us instructors figure out how to allow the greatest number of students as possible to take our courses.

What are the goals of this course? While I do want students in this course to learn some interesting mathematics, the underlying goal of this course is for students to learn something about themselves and others through doing mathematics with each other. In particular, I am hoping that students in this class will have a more nuanced and complete understanding of what it means to be mathematically brilliant so that they can recognize that in themselves and others. This is one of the ways that I am hoping to create a rehumanizing mathematical experience for me and my students.

What is the course like? The Inside-Out Program is very particular about the kind of pedagogy we are to use. Lecture-based courses don’t provide for the kind of mutual engagement and co-learning that the program is trying to encourage. Therefore, I’ve structured my course using materials based off of my work with Bowen Kerins, Al Cuoco, Glenn Stevens in 2009 at the IAS/Park City Mathematics Institute Teacher Leadership Program.

On the first day, I tell students that this course is likely to be very different from any other mathematics course they’ve taken. The class is designed so that students learn from and with each other, not directly from me; I spend almost no time lecturing. Instead, the students work in small groups on a set of mathematical tasks during each class period. I’ve designed the tasks to pique curiosity and encourage students to make conjectures and look for patterns—in other words, the tasks are designed to engage students in doing mathematics the way that professional mathematicians do mathematics.

We basically spend almost all two hours of our time together doing math. I interrupt the work from time to time to facilitate students sharing their observations with each other. We close out the time by having a whole-class discussion and share-out about the (1) questions that we’re still wondering about, (2) interesting mathematical observations that we made, and (3) our gratitude toward one another for the contributions that they made to our learning.

Why number theory? Number theory is a wonderful area of mathematics that has a low threshold for entry and high ceiling for exploration. I have designed the course materials so that only experience with high school Algebra is required. Also, I am not at all an expert in number theory, so that allows me to approach things with a fresh perspective and to be surprised along with my students.

What’s it been like so far? We’ve already had four class sessions. We started by looking at the divisors of numbers and we’re currently thinking about modular arithmetic. Both the inside and outside students have been fantastic. Everyone seems to be deeply engaged in the mathematics and in working with each other.

Ideally, I would have had an equal number of inside and outside students, but right now I have 4 outside students and 15 inside students. We have been arranging ourselves in four groups of 4-5 students. This has worked out really well so far.

Unpredictable things happen all the time that prevent people from attending class. For example, during the first class session, parts of the prison were on lock-down so some students were not able to get to class. I have to be flexible and find ways to fold in students when they are able to attend class.

I hope to write more about my experiences throughout this semester. These are just some preliminary thoughts that I wanted to jot down.

This teaching and learning experience would not be possible without (1) the training and support I received in May 2018 from the Inside-Out Program, (2) support from administrators at the CRC, (3) the amazing students that are currently in the course, (4) and logistical support from the Claremont Colleges, made possible in part by a grant from the Andrew Mellon Foundation.

Next… Part 2

Research on class size and participation of women in undergraduate STEM courses

This 2019 paper by Ballen et al in BioScience entitled “Smaller Classes Promote Equitable Student Participation in STEM” is worth reading if you’re interested in equity, teaching, and STEM.

This large study (with 26 co-authors!) attempts to determine which features of undergraduate STEM classrooms correlate with more equitable vocal participation of women through a careful analysis of 5300 student-instructor interactions in 44 courses (observed over a full term) at six different institutions (4 in the U.S., 1 in Egypt, 1 in Norway). Answer: (1) class size was most impactful, followed by (2) the number of different strategies that the instructor used to elicit students’ vocal participation in class.

Educational literature up to this point has been very mixed about the connection between class size and student learning (see OECD 2012 report, CampbellCollaboration 2018 report, and many others). This is the first paper in which I’ve seen a strong argument for reducing class size: lowering class size increases the likelihood that women will participate in undergraduate STEM classes. In their data, increasing class size from 50 to 150 students decreased the likelihood of a woman participating by 50%.

The researchers were also able to reject several alternative hypotheses with their data. These alternative hypotheses included connections between equitable vocal participation and (1) abundance of student-instructor interactions per class period, (2) instructor gender, (3) proportion of women in the class, and (4) whether the STEM class was lower- or upper-division. (In other words, none of these things significantly correlated with more equitable participation.)

But since class size is often out of the instructor’s control, what can one do to make participation more equitable? The researchers found that instructors using a large repertoire of methods of eliciting vocal participation from students also got more equitable participation. (This makes me wonder, however, whether the size of an instructor’s repertoire of methods for eliciting student participation might correlate with instructor’s overall skill.) The article gives at least 7 different strategies that instructors can use to elicit students’ contributions in a classroom:

1) increase wait time between posing a question and selecting someone to answer in front of the whole class
2) using think-pair-share before selecting someone to answer
3) letting students work in small groups
4) having students write first before sharing out loud
5) soliciting multiple volunteers and calling on students only after a certain number of students have raised their hands
6) assigning student groups a number and using random number to select a group to answer
7) assigning a student in a group to be the “reporter” based on some arbitrary characteristic (e.g. random number or who woke up earliest)

I have a few more:
8) in addition to having good wait time, when posing a question to the class that you want students to respond to, select concise and clear wording for your question prompt, display the question on the screen (if possible) while reading it, and avoid that awkward continuous rephrasing of the question as a way of filling the silence
9) using vertical non-permanent surfaces (like white or chalk boards) to make student work visible before having students share ideas verbally to the whole class
10) positioning incomplete, partial, incorrect answers as being valuable to classroom discourse so as to lower the social risk for students to participate in whole-class discourse
11) using Google docs or equivalent online platforms to allow students to simultaneously contribute their ideas in an online space (this idea doesn’t involve vocal participation like the others, but it still involves students articulating their ideas in front of others)

Because the researchers’ posited reasons for women participating in class less than men involve imposter syndrome and social identity threat, that means these results should also to students of color in undergraduate STEM courses as well. In addition, I can see that many of the same arguments will also apply to secondary schools.

Characteristics of Effective Teacher/Faculty Professional Developent

Note: This blog post is based on a presentation that I gave at the 2019 MathFest in Cincinnati during a contributed paper session entitled “Professional Development in Mathematics: Looking Back, Looking Forward, on the Occasion of the 25th Anniversary of MAA Project NExT” organized by Dave Kung, Julie Barnes, Alissa Crans, and Matt DeLong.

For over 15 years, I have designed and led professional development for K-12 (mainly PCMI and MfA LA) and higher-education faculty (mainly Claremont Colleges Center for Teaching and Learning), primarily to help others enhance their teaching and learning.  I never received any formal training in how to do this kind of work, but I was fortunate to have worked alongside other educators in the field (Ginger Warfield, Gail Burrill, Peg Cagle, Pam Mason, and many others) and I have tried to learn as much as I can from the education literature.

Professional development for both K-12 and higher-education faculty is crucial if we want to continually improve the quality of education for our students and to reduce the loss of talent and resources that comes from faculty turn-over. And yet, anyone who has gone through professional development trainings and workshops knows that the main problem with professional development is that not all of it is good. In fact, some professional development is just plain awful. Bad professional development not only turns people off from wanting to continue to advance their skills, but it also muddies the waters about whether money spent on professional development is worthwhile.  However, the reality is that we are continually learning more about what effective teaching looks like and that information needs to be disseminated to teachers and faculty, so we will never stop needing good professional development for teachers and faculty. Moreover, “one constant finding in the research literature is that notable improvements in education almost never take place in the absence of [teacher] professional development” (Guskey, 2000, p. 4).

There is much less published research on the professional development of higher-education faculty than there is for K-12 teachers. This makes a lot of sense because there are many more K-12 educators than there are higher-education faculty and much more is spent on K-12 teacher professional development than for higher education faculty. I believe there is a lot that folks doing professional development for higher-education faculty can learn from what has been written about in the K-12 world.

Recently, I did an extensive literature search to find research on what effective K-12 teacher professional development looks like (not limited to mathematics). I found over 30 years of research commentaries, empirical studies, and meta-analyses that try to characterize effective professional development (Banilower, Boyd, Pasley, & Weiss, 2006; Birman, Desimone, Porter, & Garet, 2000; Blank & de las Alas, 2009; Borko, Jacobs, & Koellner, 2010; Darling-Hammond, Hyler, & Gardner, 2017; Desimone, Porter, Garet, Yoon, & Birman, 2002; Garet, Birman, Porter, Desimone, & Herman, 1999; Garet, Porter, Desimone, Birman, & Yoon, 2001; Little, 1993; Loucks-Horsley et al., 1987; Loucks-Horsley, Stiles, Mundry, Love, & Hewson, 2010; Stein, Smith, & Silver, 1999; Timperley, 2008; Timperley & Alton-Lee, 2008; Wilson, 2013).

There is a remarkable amount of consistency among all of this scholarship. To demonstrate this, I’ve selected three papers from the 16 papers listed above and summarized their lists of characteristics of effective PD below.

Loucks-Horsley, S., Harding, C. K., Arbuckle, M. A., Murray, L. B., Dubea, C., & Williams, M. A. (1987). Continuing to Learn: A Guidebook for Teacher Development. The Regional Laboratory for Educational Improvement of the Northeast and Islands; National Staff Development Council.

Characteristics of effective K-12 teacher professional development:

  1. Collegiality and collaboration
  2. Experimentation and risk taking
  3. Incorporation of available knowledge bases (by this they mean that teaching practice should be informed by research and validated in model programs and practices)
  4. Appropriate participant involvement in goal setting, implementation, evaluation, and decision making
  5. Time to work on staff development and assimilate new learnings
  6. Effective leadership and sustained administrative support
  7. Appropriate incentives and rewards
  8. Designs built on principles of adult learning and the change process (andragogy—the practice of teaching adult learners; includes opportunity to try new practices, guided reflection and discussion, time for significant change, balancing support and challenge)
  9. Integration of individual goals with school and district goals
  10. Formal placement of the program within the philosophy and organizational structure of the school and district (by this, they mean that it cannot be the effort of a few energetic individuals, it must be embedded in the organizational structure and culture)

Garet, M. S., Porter, A. C., Desimone, L., Birman, B. F., & Yoon, K. S. (2001). What Makes Professional Development Effective? Results From a National Sample of Teachers. American Educational Research Journal, 38(4), 915–945.

Characteristics of effective K-12 teacher professional development:

  1. Focuses on subject-matter content and how students learn it
  2. Includes opportunities for teachers to become actively engaged in meaningful discussion, planning, practice
  3. Professional activities are coherently organized around goals that align with state and district standards and procedures
  4. More contact hours over a longer time span allows for learning to sink in
  5. Collective participation of people from the same school, department, or grade level is more helpful than participation of individuals from many different schools

Darling-Hammond, L., Hyler, M. E., & Gardner, M. (2017). Effective Teacher Professional Development (p. 76). Retrieved from

Characteristics of effective K-12 teacher professional development:

  1. Is content focused
  2. Incorporate active learning
  3. Supports collaboration
  4. Uses models of effective practice
  5. Provides coaching and expert support
  6. Offers feedback and reflection
  7. Is of sustained duration

I hope that you see many connections between the items on these three lists. And the same is true if you look across all 16 papers.

Based on this survey of the literature and my own experiences doing this work, here are a few important takeaway messages for people who lead and design professional development for both K-12 teachers and higher-education faculty. These ideas are oversimplifications, so you’ll need to think about how they might apply in your own context.

First of all, learning takes time and being able to see evidence of change takes even more time. We should not expect much to happen from a one-time 90-minute workshop. Programs that happen over longer periods of time are more likely to lead to real change in behaviors. This seems pretty obvious and yet a lot of professional development programs rely on the one-time workshop model.

Second, we need to make sure that the work that we are doing is aligned with the realities of the institutional (schools, districts, colleges, universities) and departmental contexts faced by participants in our programs.

Third, authentic community is important because it supports collaboration. Having some shared context is one way to create an authentic community.

Fourth, program evaluation is crucial. This is not an item that we see in the lists above, but it is one that I have found to be true based on the work I’ve done so far. Effective professional development efforts are ones that can document growth and success over time, both for ourselves and for our stakeholders and potential funders. That documentation requires us to be strategic about program evaluation and assessment. We have to get much better and smarter at how we evaluate our programs. Not everything can be quantified, but we can’t let the challenge of measuring progress keep us from constantly improving through program evaluation..

I don’t think these characteristics are necessary and sufficient conditions for professional development to be effective. I suspect they are only necessary at best. There are probably other conditions that are required too.

What do you think are essential characteristics for teacher/faculty professional development to be effective?


  • Banilower, E. R., Boyd, S. E., Pasley, J. D., & Weiss, I. R. (2006). Lessons from a Decade of Mathematics and Science Reform: A Capstone Report for the Local Systemic Change through Teacher Enhancement Initiative. Retrieved from Horizon Research, Inc. website:
  • Birman, B. F., Desimone, L., Porter, A. C., & Garet, M. S. (2000). Designing Professional Development That Works. Educational Leadership, 57(8), 28–33.
  • Blank, R. K., & de las Alas, N. (2009). Effects of teacher professional development on gains in student achievement: How meta-analysis provides evidence useful to education leaders. Washington, DC: Council of Chief State School Officers.
  • Borko, H., Jacobs, J., & Koellner, K. (2010). Contemporary approaches to teacher professional development. International Encyclopedia of Education, 7(2), 548–556.
  • Darling-Hammond, L., Hyler, M. E., & Gardner, M. (2017). Effective Teacher Professional Development (p. 76). Retrieved from Learning Policy Institute website:
  • Desimone, L. M., Porter, A. C., Garet, M. S., Yoon, K. S., & Birman, B. F. (2002). Effects of Professional Development on Teachers’ Instruction: Results from a Three-Year Longitudinal Study. Educational Evaluation and Policy Analysis, 24(2), 81–112.
  • Garet, M. S., Birman, B. F., Porter, A. C., Desimone, L., & Herman, R. (1999). Designing Effective Professional Development: Lessons from the Eisenhower Program [and] Technical Appendices (No. ED/OUS99-3). American Institutes for Research.
  • Garet, M. S., Porter, A. C., Desimone, L., Birman, B. F., & Yoon, K. S. (2001). What Makes Professional Development Effective? Results From a National Sample of Teachers. American Educational Research Journal, 38(4), 915–945.
  • Guskey, T. R. (2000). Evaluating Professional Development. Corwin Press.
  • Kennedy, M. M. (2016). How Does Professional Development Improve Teaching? Review of Educational Research, 86(4), 945–980.
  • Little, J. W. (1993). Teachers’ professional development in a climate of educational reform. Educational Evaluation and Policy Analysis, 15(2), 129–151.
  • Loucks-Horsley, S., Harding, C. K., Arbuckle, M. A., Murray, L. B., Dubea, C., & Williams, M. A. (1987). Continuing to Learn: A Guidebook for Teacher Development. The Regional Laboratory for Educational Improvement of the Northeast and Islands; National Staff Development Council.
  • Loucks-Horsley, S., Stiles, K. E., Mundry, S., Love, N., & Hewson, P. W. (2010). Designing Professional Development for Teachers of Science and Mathematics (Third).
  • Stein, M. K., Smith, M. S., & Silver, E. (1999). The development of professional developers: Learning to assist teachers in new settings in new ways. Harvard Educational Review, 69(3), 237–270.
  • Timperley, H. (2008). Teacher professional learning and development (No. Educational Practices-18; p. 32). International Academy of Education.
  • Timperley, H., & Alton-Lee, A. (2008). Reframing Teacher Professional Learning: An Alternative Policy Approach to Strengthening Valued Outcomes for Diverse Learners. Review of Research in Education, 32(1), 328–369.
  • Wilson, S. M. (2013). Professional Development for Science Teachers. Science, 340(6130), 310–313.

Effect of Vertical Non-Permanent Surfaces (VNPS) on Task Cognitive Demand: A Case for Pedagogical Reasoning

This blog post has to do with vertical non-permanent surfaces (VNPS), but it really is the need for more pedagogical reasoning and sense-making in our community. I don’t mean to pick on VNPS in particular–I am using it as an example only because I happen to have some data about it.

About VNPS: (I apologize for using the acronym VNPS, but “vertical non-permanent surfaces” is lot of typing.) In a nutshell, this instructional strategy involves having students working at a whiteboard/blackboard (mounted on wall or easel) or nifty erasable plastic sheets that stick to the wall instead of working at their desks. The instantiation of this instructional strategy can vary: students can be asked to work solo or in small groups; all the students might be working on the same task or different tasks; the teacher might have students stay at their stations the whole time or combine the activity with a gallery walk. There are endless variations. Search Twitter to learn more. VNPS can bring lots of wonderful benefits to a classroom, and I’m not disputing those things here. You can also find some research (here’s one example) on VNPS.

In this blog post, I would like to make the argument that VNPS makes student thinking visible in the classroom and sometimes that visibility can lower the cognitive demand of the task. This likely happened in some classrooms that I visited a few months ago.

Information about the classrooms:

  • Four different teachers at the same school were implementing the same mathematical task in an Algebra 1 class and all asked students to work at whiteboards in groups of 2 or 3. The four teachers work closely together and decided on the task and implementation of the task via VNPS together.
  • This task was given near the beginning of the school year, when students were beginning to develop their understanding of linear growth.
  • I observed four periods on a day when the periods were about 50-minutes long. The teachers had students working at the board roughly between 15 to 30 minutes.
  • I should also admit my own bias here in that I know these four teachers well and respect them highly. I look to them to gain insights about teaching mathematics.

Here is the task that students were working on.

Person A starts off with 400 widgets and gains 20 new widgets every day. Person B starts off with 50 widgets and gains 70 new widgets every day.

1) For each of the next 10 days, who will have more widgets?

2) Will there be a time when both people have the same number of widgets?

3) Is it possible for them to have the same number of widgets again past the ten day period you have explored? Explain.

Make sure that all of your answers are thoroughly explained and justified on your boards. Use any and all mathematical methods/models you can think of to explain your answers and reasoning.

Overall, the quality of the student work was very high. All four teachers were careful to avoid telling students to use specific procedures (to use a graph, or a table, etc.) or to “help” students in a way that lowered the cognitive demand of the task.

(Modified version of task to protect anonymity of the teachers and students involved. The original task had a context that was much more interesting that “widgets.”)

As you might expect for this open-middle task, students used a variety of approaches for the task including tables, graphs, and symbols. Across all four of these Algebra 1 classes, I observed 39 different student groups. Out of the 39 groups, 31 of them displayed a table of numbers similar to these:

Some tables listed number of widgets for A and B separately…
… while some students created one table for both A and B together. Notice this group above circled the day in which both A and B had the same number of widgets and used that to answer Question #1.

Out of the 39 groups, 11 of them displayed a graph similar to these:

Some students created graphs for A and B separately…
.. while two groups graphed A and B together, showing the point of intersection more clearly.

Out of the 39 groups, 16 of them showed some form of algebraic expression. Eight groups set up an equation and solved for the day when the number of followers was the same. Here’s a table summarizing the frequency of various approaches that students used.

What strategies were used in all four class periods? This table shows the number and percentage of groups in each class that attempted a particular strategy, based on observations of students’ work during class.

Tables were the most common approach taken up by students. That is not surprising given that it was the beginning of the year and not students might have been fluent with graphing quantities and using algebraic expressions yet. In addition, the wording of question 1 led to most students to start off by listing the number of widgets by days.

Algebraic approaches were the least common. However, Period 3 stands out in that 5 out of 8 groups used an algebraic approach that also involved solving an equation. Here is the student work for those 5 groups. (As with the other photos here, I’ve edited the photos slightly to protect the identities of the students and teachers.)

Algebraic approach #1
Algebraic approach #2
Algebraic approach #3
Algebraic approach #4
Algebraic approach #5

Did you notice how all five examples of student work from this class period bear a striking resemblance to each other?

  • All of the student groups used “x” as the independent variable. There were other examples from other classes where the variable used was “d” (for days). Similarly, when a dependent variable was used, it was always “y” even though the objects being counted in this problem didn’t suggest the letter “y”.
  • All of the student groups solved for the variable using the strategy of adding or subtracting quantities from both sides of the equations at first. And, they all showed their work in the same way. Four of the groups chose to subtract 20x as the first step.

Based on these observations and my own notes taken on what students were doing during this period, I am almost certain that students looked around the room and copied this algebraic approach to their board. One group first came up with the approach, then I observed other groups copy it from the first group and from each other. It was fascinating to watch how ideas spread throughout the room in this and other periods.

(By the way, the teacher of this Period 3 class also communicated that a significant proportion of students in that class all had the same very demanding 8th grade teacher the previous year. By all accounts, that teacher apparently had remarkably inflexible standards for how to “do math” and reinforced them using methods such as public humiliation. We wonder how much of the very similar boardwork from this period came from having had this particular approach to solving equations drilled into students the previous year, and how much a fear of public humiliation might have contributed to students copying each other.)

Evidence of students getting ideas from each other in other periods was not as strong, but I believe it also happened to some extent. For example, I observed students who saw that another group had a graph and so that led them to also include a graph. It is also interesting to me that all of the groups never had any disagreement on whether the number of widgets was 400 and 50 on day 0 or day 1. And, the fact that there were no arithmetic errors at all across all 39 work samples leads me to suspect that perhaps were checking their numbers with each other.

It seems like a pretty obvious point to make that if you’re going to have students working publicly on a task using VNPS, you might get students getting ideas or directly copying work from each other. But, I think this is an important point to make for several reasons.

First of all, the possibility of students’ revealing ideas to their peers during the enactment of a mathematical task, voluntarily or involuntarily, means that the use of VNPS has the potential to affect the cognitive demand of that task. The QUASAR project has helped our field understand how the cognitive demand of mathematical tasks tends to decrease as they are successively taken up by teachers then students.

A representation of how a math lesson unfolds during classroom instruction. Taken from “Implementing Standards-Based Mathematics Instruction: A Casebook for Professional Development” (1st ed, 2000) by Mary Kay Stein, Margaret Schwan Smith, Marjorie A. Henningsen, Edward A. Silver.

In this case, all four teachers were very careful not to lower the cognitive demand of the task through their interactions with students during instruction. However, they might not have anticipated that the cognitive demand might have been affected by the choice of using VNPS.

Second, the possibility of students being able to see each other’s work during a mathematical task has implications for whose work gets valorized and ignored in the classroom. Imagine a hypothetical situation in which Students A, B, C are working together in a small group. And, imagine that Student X (not in their group) is a student that most others in the class perceive (accurately or not) to be mathematically competent. Student A shares a good idea with Students B and C, but they ignore her contribution and defer to Student X’s work instead, which they spy on from across the room. This hypothetical example is one possible way in which the public sharing of student work in progress might affect whose ideas get taken up.

Borrowing the language of complex instruction: VNPS could exacerbate any status issues that you might have in your classroom because interactions with any high status students are no longer restricted to the small group that those high status students belong to.

Third, the possibility of students being able to gain ideas from each other by looking around the room limits a teacher’s ability to use tasks to assess students’ abilities and understandings. Let’s return to the implementation of the widgets task described above. If the Period 3 teacher above wanted to know which solution strategies for modeling linear growth students remembered from their 8th grade math class, the VNPS enactment of this task might have led to an erroneous conclusion that many of them could proficiently work with algebraic expressions and solve equations.

I am not criticizing these four teachers for lowering the cognitive demand of the task. And in fact, there is one aspect of their use of VNPS that I think is quite brilliant: Because they did not specify whether to use tables, or graphs, or algebraic expressions, the fact that students had different approaches and then copied them to their work areas means that they had opportunities to make connections between these different representations of the scenario. And, I observed these teachers encouraging students to make these connections when they talked to individual student groups.

The purpose of highlighting these issues with VNPS is also not to demonize this very useful instructional strategy or to criticize these four teachers. Rather, I highlight these subtleties with VNPS to illustrate a more general point: when we educators talk about our teaching with each other, we tend to focus on the “hows” and “whats” rather than the “whys” and “whens”, and that incomplete transmission of ideas limits the effectiveness of those conversations.

A lot of what gets presented at conferences, written about in the Math Twitter Blog-o-Sphere, including a lot of what I have written myself, focuses on the “hows” and “whats”: how I create performance tasks when I use standards-based grading, what formulas I use in my grading scheme, what I do to handle re-dos, etc. What gets talked about less is why I use standards-based grading in the first place, when might I use that strategy over a different strategy, etc. We tend to focus on the “hows” and “whats” because that practical information is, um.., practical, so it helps us implement ideas quickly. However, the “hows” and “whats” likely don’t help us understand why something doesn’t work the way we thought it would, or how we adapt an idea to fit within our specific teaching contexts.

To effectively spread instructional innovations across a community, members of that community need to not just explain how to do X, but why they do X, when they do X instead of Y, what aspects of their teaching contexts lead them to implement X in the specific way that they have.

For example, when we talk about VNPS, let’s also talk about why we use VNPS as opposed to other instructional strategies. We shouldn’t use VNPS just because they are trendy–there should be some specific instructional purpose behind those decisions.

VNPS can be an effective strategy for getting students out of their seats and actively engaged in mathematical tasks with each other, but you could do something similar by setting up stations around the room and having manipulatives or handouts on desks as opposed to having mathematical work shared publicly around the room. What are the affordances of each in your specific context?

VNPS can be an effective way to get students to try out incomplete ideas in draft form because you can erase your work easily. However, you can achieve something similar by having personal whiteboards or shared whiteboards placed horizontally on tables. If you want student work to be shared publicly, that can still be done after the work is completed by having the personal whiteboards lifted up or placed along the wall for a gallery walk.

The specific arrangement of your classroom also matters quite a lot. One of the four teachers I mentioned above has a rather large room and even though students were working on white boards mounted around the room, they were far apart from each other that the only transmission of ideas that I observed were by nearest neighbors. In other teachers’ rooms, white boards were packed closely together and it was much easier for students to see each others’ work.

So, instructional goals, classroom culture, students’ predilections, even furniture arrangement all matter a great deal when we talk about our teaching ideas with each other. We should aim to situate our conversations about teaching innovations more thoroughly so as to enable others to make the greatest use of those ideas. If we don’t, then we increase the risk of unexpected consequences when we try out each others’ ideas and we miss out on opportunities to strengthen our own and others’ pedagogical reasoning and sense-making skills.

Think about it this way: if a student tells you the answer is 42, you will probably also ask them to justify their answer. Why then, should we as educators, only tell each other the things we’re doing in our classrooms but not also justify our ideas with pedagogical sense-making and reasoning?

What Must Teachers Possess to Support Their Equitable and Inclusive Teaching?

What must learners of mathematics possess so that they can learn mathematics on their own? For example, once a student graduates from my college, what does she need to continue learning mathematics on her own? Besides having some body mathematical content knowledge, this student would probably need to have some mathematical habits of mind, some productive dispositions about learning mathematics, a sense of self-efficacy, a growth mindset about learning mathematics, perseverance, etc.

So, now let’s ask an analogous question: say we are trying to help teachers (of all levels, primary, secondary, tertiary) learn how to teach in more equitable and inclusive ways. What must they possess to do that? They might need to know some instructional strategies for equitable and inclusive teaching, but clearly they need more than that.

Why is it, then, that most professional development is focused on helping teachers acquire instructional strategies, but not much more than that? It would be as silly as if we only focused on helping students acquire content knowledge in our fields and none of the other things that they might need to continue their own learning.

To all of you who think deeply about equity and justice in education and are looking for ways to help teachers teach in more equitable and inclusive ways, here’s an important question: What must teachers possess to continue making their classrooms more equitable and inclusive long after they have left our workshops or training sessions?

Clearly they need some of the “whats” and “hows” of equitable and inclusive teaching–they need to learn some equitable instructional strategies. Pick your favorite ones: using “equity sticks” to call on students more equitably, finding ways to remove gendered language and examples from your course content, active learning strategies (like inquiry-based learning, POGIL, etc.), justice-oriented curricular materials, etc. Those things are important, but what else?

With input from Sumun Pendakur and Peg Cagle, I’ve compiled this list of things that teachers need to continue making their classrooms more equitable and inclusive. Will you help me edit/add to this list?

  1. A Sense of Purpose: What’s the force that compels me to do more and grow? It could come from a personal moral and ethical stance, spiritual practice, determination to make the world better for future generations, or affinity with your institution’s mission, etc.
  2. Productive Dispositions about Teaching and Equity:
    • (a) A belief that time and effort I spend on improving teaching and learning can result in fruitful changes in my classes that benefit students
    • (b) Similar belief that time and effort spent learning about equity and inclusivity can benefit my classes and students
  3. Humility: Knowing that there is something that I can learn from everyone and that I will never really “arrive” at the perfection in my teaching but that I still strive toward that ideal
  4. Growth Mindset about Equity: Believing that my capacity to be more equitable and inclusive in everything that I do can improve with effort, rather than insisting that people are just “good” or “bad” or “racist” or “woke” (I wrote about this in a previous post.)
  5. Perseverance: Having the courage to continue trying to teach in better, or more equitable and inclusive ways even if I don’t attain success the first few times
  6. Equity-Oriented Habits of Mind:
    • (a) Having an asset-oriented, as opposed to deficit-oriented, way of thinking about students. Knowing that I and my institution have the responsibility to help students succeed and that there are things within my control that can make a difference
    • (b) Asking not just about the “whats” and “hows” of teaching, but also the “whens” and the “whys”; having an understanding that classrooms and people are too complex to boil things things down to “best practices” and instead being able to think of the classroom as an ecology of people, environments, and their relationships with each other.
    • (c) Having a habit of self-reflection about one’s teaching practice.
  7. Community: Participation in a community with other professionals who are struggling through similar problems of practice helps me continue to grow. But it’s not just having folks around, but also knowing how to engage in effective professional conversations. Julianne Vega recommends the book Talk About Teaching! Leading Professional Conversations by Charlotte Danielson.
  8. Joy and Humor: This kind of work is too difficult to sustain without self-care. The ability to find joy in your work and to take work seriously, but not take yourself too seriously, helps us pick ourselves up when we make mistakes and continue learning. I also highly recommend the “Killjoy Survival Kit” in Living a Feminist Life by Sara Ahmed.

There is a great deal of overlap here between things one might need to improve as a teacher in general and things one might need to teach in more equitable and inclusive ways. Items #4 and #6a above seem to me to be specific to teaching in more equitable and inclusive ways.

What else is missing from this list? And the big question, of course, is: how do we cultivate these things in ourselves and help others cultivate them too?

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.

  • Hicks, M., Tucci, A., Koehne, C., Melhuish, K. and Bishop, J., 2021. Examining the Distribution of Authority in an Inquiry-Oriented Abstract Algebra Environment. RUME 2021.

  • Johnson, E., Andrews-Larson, C., Keene, K., Melhuish, K., Keller, R. and Fortune, N., 2020. Inquiry and gender inequity in the undergraduate mathematics classroom. Journal for Research in Mathematics Education51(4), pp.504-516.

  • Maries, A., Karim, N.I. and Singh, C., 2020. Active learning in an inequitable learning environment can increase the gender performance gap: The negative impact of stereotype threat. The Physics Teacher58(6), pp.430-433.
  • Stone-Johnstone, A., Reinholz, D.L., Mullins, B., Smith, J.L. and Andrews-Larson, C., 2019. Inquiry without equity: A case study of two undergraduate math classes. In Proceedings of the 22nd Annual Conference on Research in Undergraduate Mathematics Education.

  • Smith, J., Andrews-Larson, C., Reinholz, D.L., Stone-Johnstone, A. and Mullins, B., 2019, March. Examined inquiry-oriented instructional moves with an eye toward gender equity. In Proceedings of the 22nd annual conference on Research in Undergraduate Mathematics Education (pp. 1035-1041).

  • Andrews-Larson, C., Can, C. and Angstadt, A., 2018. Guiding whose reinventions? A gendered analysis of discussions in inquiry-oriented mathematics. In 21st Annual Conference on Research in Undergraduate Mathematics Education (pp. 1548-1554). San Diego, CA: Mathematical Association of America, SIGMAA on RUME.

  • Brown, S., 2018. E-IBL: An Exploration of Theoretical Relationships Between Equity-Oriented Instruction and Inquiry-based Learning. In Proceedings of the 21st Annual Conference on Research in Undergraduate Mathematics Education (pp. 1-15).

  • 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.


Why should faculty care about diversity, equity and inclusion?

Sumi Pendakur and I recently published an article in the Notices of the American Mathematical Society entitled, “Advocating for Diversity and Inclusion in Faculty Hiring,” in which we offer some best practices for faculty searches. Though it was written for people in the mathematical sciences, the strategies and tactics we offer can be used by any faculty search committee.

We wanted to share with you a section of the paper that didn’t make it into the final printed version, due to space constraints. We both consider this section to be an important piece of the puzzle.

I’m going to label this missing text as “Section 0” because it comes before Section 1. (This text will probably make more sense if you read it in the context of the rest of the article.)

Section 0. Why should faculty care about diversity, equity and inclusion?

Ultimately, the degree to which our institutions are successful at increasing diversity, equity and inclusion depends on how our institutions and those connected to it see these things as vital to their mission (Smith, 2009; Taylor, Milem, Coleman, 2016). The more that we can articulate our own reasons for promoting diversity, the better we can share those with others and see how they are connected with the missions of our institutions.

One of the most important steps that departments can take to support diversity in faculty hiring is actually something that takes place long before the position is approved. That step is to come to a shared understanding as a department about what is meant by words like “diversity” and “inclusion” and how those things relate to the mission of the department. If your institution has a strong vision for diversity, that is always a good place to start. Coming up with a department diversity statement can be helpful, but written documents like these are helpful only as much as they are supported and enacted by individuals in the department.

There are many specific reasons why your department or institution might value diversity in the mathematical sciences. The following economic argument is cited by publications such as “Rising Above the Gathering Storm: Energizing and Employing America for a Brighter Economic Future”: minority groups are increasing as a percentage of the U.S. population and if these groups are not well-represented in the STEM workforce, we will suffer from a shortage of innovators who will spur economic growth.

Another argument has to do with the quality of work that we produce as a discipline. Without a broad representation of people, we cannot be sure that our work in the mathematical sciences addresses all of the important issues that need to be addressed, or that we are addressing these issues having heard all of the voices that need to be heard. More equitable participation will increase the robustness of our discipline.
Leadership is a common educational goal at many institutions, and that can often be a way to connect the mission of the institution with diversity. In today’s global society, multicultural competence and awareness of the societal impacts of one’s work is an increasingly important component of being a leader.

The student success imperative is also tied to the vision of diversity held by the institution and the search committee. Recruitment, retention, and successful graduation of an increasingly diverse (racially, sexually, and socioeconomically) student body is directly impacted by multiple factors, including quality of faculty interaction and whether students see themselves reflected in the curriculum and in the faculty body. The powerful mentoring relationships of faculty of color and students of color are well-documented in the literature and contribute to the feeling of a welcoming and inclusive learning environment. In addition, critical mass of minoritized communities in both the student and faculty bodies reduces stereotype threat, thereby enhancing academic success.

Then, there are a whole range of arguments for diversity having to do with social justice. If your institution or department has a goal of graduating ethical people who contribute to society, one can make the argument that it is important for our students to understand their mathematical work in a historical context of exclusion and underrepresentation in the mathematical sciences in the U.S. Hiring faculty who have an understanding of the historical and sociological context of patterns of exclusion in our discipline and are able to integrate that knowledge into their teaching benefits ALL students.

If your department can find consensus around a set of arguments for diversity, equity and inclusion, it will be much more likely that its actions, including teaching practices, curricula, and faculty hiring practices, will support these outcomes. It will also make it less likely that a search committee will fall into the trap of making a false dichotomy between excellence and diversity; instead search committees will see that diversity is part of what it means to reach excellence by fulfilling its goals.

References mentioned in this excerpt:

Smith, Daryl. Diversity’s Promise for Higher Education: Making it Work. John Hopkins University Press, 2009.

Taylor, Teresa E., Jeffrey F. Milem, and Arthur F. Coleman (2016). Bridging the research to practice gap: Achieving mission-driven diversity and inclusion goals. New York,
NY: College Board. Available at