Grouped, Practical Assessments

I’ve been working through some ideas about assessment in high school physics. The goal is to assess students in a way that is more meaningful, more engaging, more effective at analyzing a student’s ability to do real physics tasks, and more likely to result in useful learning experiences. At the same time, with my IB classes, I cannot take my eye off the inevitability of high-stakes standardized exams and the concomitant need to prepare students for these.

I’ve been curious about authentic assessment for a while, but this specific work is inspired by Joss Ives‘s work on two-stage collaborative exams and by conversations and collaboration with Kelly O’Shea, with whom I will be presenting a workshop on the topic this summer.

In this post, I will outline and analyze one assessment I have attempted.

First, the students worked in groups to build background by creating some review notes about double-slit interference, which was the topic of this assessment. I encouraged them to “use their resources”, which in most cases meant their notes and the textbook, although a couple also used the internet for translations and definitions. Below is the prompt and a typical response.

Next, I regrouped the students (always groups of 3) and gave them their tools: a laser, a double-slit slide, a ruler, and a tape measure. I told them that their task was to determine the wavelength of the laser. Here, there was a couple minutes of uncertainty: one group launched into a debate about whether the laser was emitting light, another forgot about their notes from a minute before and tried thinking of ways to measure the laser’s frequency (planning to use the wave equation). Without nudges or hints, however, all the groups converged on the same idea: shine the laser through the double slits onto a distant wall, and measure the various distances. Below is a typical example of their work.

 

Finally, I asked the students to reflect on their experience. The first question aims to get them to think about their role in the group. Most of the answers here were descriptive (“I held the laser”), and few tackled the second part meaningfully. The second question aims to get them to reflect on their experimental design. The majority discussed something related to random error and the need for repeated trials.

The third question is inspired by something inspired by Ilana Horn and aped from Kelly: different ways of “being smart” or “doing science” in our class. Here are the results:

I like the diversity of approaches that were used, and that are sought. Working systematically seems to have been viewed by most of the students as key in this assessment, which is something I agree with.

Finally, is the question of whether the students preferred this type of assessment to a traditional test. Overwhelmingly, they preferred this grouped, practical approach. Even the disadvantages they suggested were quite positive. Here are some of the responses:

The educational idea of authentic assessments dates back at least to the 1990s, and of course the theory-vs-practical debate in science education predates the work of the Committee of Ten who laid the foundation for public education in the USA back in the 1890s. For me, the challenge is finding a way to do meaningful, practical assessment in a way that upholds the rigor of our contemporary courses while also being more engaging and meaningful for students.

If this is of interest to you, then please check out Kelly’s blog post and stay tuned for our workshop this summer. I’d also appreciate hearing about any ideas, feedback, or experience if you have a story to tell.

Studying StudyIB

The wonderful Chris Hamper has been working on a new educational idea over the last year. Housed on StudyIB, the Virtual Tutor is an attempt to recreate the experience of working one-on-one with a tutor as you go through a multi-step physics problem. There is a network that draws in resources and reminders for students, depending on their progress. It’s a good idea and, with the current web technologies available, just about due. Here is a video where Chris explains how the Virtual Tutor works.

I introduced the Virtual Tutor to my students as a way to study for their IBDP Physics exams. The response was generally along the lines of “this is interesting”. However, it wasn’t clear whether or not this approach was effective, so my students and I devised a small study to try to answer that question.

First, the students wrote a pre-test on a particular topic. Second, they worked through one of the learning networks on the Virtual Tutor (we did Forces 3). Third, they wrote a post-test on the same topic (but with slightly different questions). The pre- and post-tests have three questions.

The first question is about something we have practiced extensively, and that they should know how to do: drawing a free-body diagram. The average pre-test score was 2.45 (out of 3), and the post-test score was an increase by 0.27 points. This corresponds to a small number of students forgetting or misdrawing one of their force vectors. It seems that the Virtual Tutor was an adequate reminder. Below is a sample or pre- and post-test work that shows this.

 

The second question is about something we have not practiced very much: drawing force diagrams, where the forces are drawn at the place where they originate, rather than applied to a hypothetical center of mass. Here, the Virtual Tutor helped some students (as shown in the pre/post examples below), while two students had a lower score on the post-test for this question. The overall effect was an increase of 0.46 points to 1.37 (out of 3).

 

The third pre/post question is something more akin to what the students saw on the Virtual Tutor: a standard physics problem where students need to move through several steps, doing mathematics, in order to find a numerical answer. This is the type of question the Virtual Tutor was designed for, and here it was most effective: the average student score increased from 1.00 (out of 4) to 2.82. The below work is typical: a student was able to start the problem, but got “stuck”: the Virtual Tutor reminded or taught him the necessary steps for this type of problem, and he was able to transfer that knowledge and finish the problem.

 

I will follow-up with my students after their mock exams next week, to see if and to what extent they found the Virtual Tutor useful. From this small study, however, a few conclusions emerge:

1. The Virtual Tutor probably does about as well as any sort of studying for reminding students about fundamentals that they already know.
2. The Virtual Tutor isn’t particularly effective as an expository tool. If students need to learn some new ideas or facts, their textbooks, videos, or classroom learning experiences are better (I should add that the rest of the StudyIB site is quite good for this).
3. The Virtual Tutor is effective at reminding students of the difficult, complicated processes involved in solving multi-step problems. As seen on the third question of this study, one session with the virtual tutor was sufficient to get about half the students in this study from a low score to a high score on the problem.

I’m pretty impressed with the Virtual Tutors. If you’re a physics student reviewing for exams, consider giving it a try.

Here’s the (averaged) data:

Enculturation and Acculturation

I’ve been writing my M.A. thesis over the past couple months, and have been thinking a lot about the role of culture in how students learn, how teachers teach, and how we prepare students for the world.

Aikenhead distinguishes between Assimilation, Enculturation, and Autonomous Acculturation. These three approaches toward education, with a focus on cultures, need to be distinguished and understood.

Assimilation is forcing a new a culture onto a student whose worldview diverges from that of the culture. This is what was being done at Canada’s residential schools.

Enculturation is an attempt to bring students into a new divergent culture. This is what universities do for science students.

Autonomous Acculturation is finding ways for students to adopt a new culture under their own power. This would be like lending a student a popular science book.

Softer approaches are labeled “anthropological”, and are akin to taking a trip to the zoo. The teacher might say, “this is how scientists work”, and the students learn about scientific culture, rather than being turned into scientists.

The chess videos of Ben Finegold are a great example of enculturation. GM Finegold identifies heroes (Morphy, Carlsen), trades in quips (“never play G3”, “put it in H”), establishes values for the community (high ELO ratings, clever play), and relates the mythology of the field. The people who attend the lectures, or watch online, are submitting themselves to the enculturation provided by GM Finegold, and thus adopting the culture of chess as their own. There is little doubt that the children who attend his class (like the ubiquitous Arjen) see themselves as chess players, idolize chess grandmasters, and trade in the culture of chess.

What I am seeking to understand is how society should bring children into the culture of science. In the former Soviet Union, children went to schools that placed great emphasis on maths and the sciences: students who did well won prizes, and could be assured of successful careers within the Soviet technocratic apparatus. This is enculturation, as part of mainstream education, starting from young ages.

In the West, students who develop a love of science generally do it outside of their classes — through extracurricular activities, through popular science books and websites, or because of teachers who inspired them to continue thinking about science outside of school. This view helps to explain some of the continued disproportional representation of students from poor school districts, in spite of efforts to ensure high-quality classes for these students. These students, because of their socio-economic situations, and because of a lack of extracurricular programmes through which they can autonomously acculturate themselves to science, are less likely to adopt the culture of science as their own before the critical point of applying to university.

My question, then, is what role schools should play in connecting students with the culture(s) of science (and other cultures, like the humanities, arts, and trades). I think schools should teach using an anthropological approach, and provide plentiful opportunities for students to autonomously acculturate themselves during the course of their education. It is too much to ask that students jump wholeheartedly into a new culture every 45 minutes, but visiting new exhibits in a cultural zoo, followed by some time for students to deeply acculturate themselves via projects, and under the supervision of a cultural transmitter like Ben Finegold — now, that would be great.

AAPTsm16 Reflections

After tweeting incessantly for the past six days, it’s time to reflect on the 2016 summer meeting of the AAPT. I saw a few themes emerge from the research and work that was being presented, but some important ideas also presented themselves through the structure of the meeting itself. What follows are just the opinions of a myopic, unread teacher.

Social Justice. On the final day of the PERC, symposiarches Moses Rifkin and Amy Robertson kicked off a discussion about social justice by asking the audience to define what the term meant in physics education. This was surprisingly difficult: is it levelling the playing field? it is educating students about inequality? are we talking about social reconstruction? In any case, we used a lot of different words through the conference(s) to talk about how we physics educators can address injustice. Geraldine Cochran called for us to think of education debt, rather than achievement gaps: I hope that is the language and mindset we adopt going forward.

There were multiple sessions and workshops that aimed to address issues of equity, including two that were unfortunately scheduled at the same time. The highlight, I think, was a session organized by Mel Sabella that included complementary and interwoven talks by Konstantinos Alexakos about safe spaces, by Ximena Cid about poor targeting of PER studies, and by Geraldine Cochran challenging the deficit model and proposing the debt concept mentioned above. Hearteningly, the scheduled 30-minute discussion session turned into a 90-minute huddle. There was a spirit of passion and community present that I have never seen before at AAPT.

Session DD morphed into this discussion that carried on for an hour past the scheduled end

That final session at PERC was a good conclusion for those who were present, I think, because it resulted in the production of some documents, and a call to action. I am looking forward to a round of “this is what I tried” posters at the meeting next year.

Gender. In the past, the discussion about gender has centred on the gender gap, with plenty of work showing that it exists, and some insightful research trying to understand the origins. To an extent, that continued this year, such as Melissa Dancy’s excellent work, below.

Melissa Dancy shows that female students who leave physics (yellow) are more likely to prefer interactive classes, but were also more likely to have experienced only lectures

However, given the relative power of white female physicists, it seems as if gender issues might be subsumed into the broader struggle for social justice. To be clear, I am not saying that the gender issues have been resolved, even theoretically; however, I think many people have started to view gender as a small piece of a bigger issue. Identity is a part of that big issue, and Katherine Rainey’s work — which is probably on the leading edge at the moment — shows the importance of developing a sense of belonging.

Katherine Rainey's work shows clearly that white/men feel more belonging; less likely to drop b/c identity #aaptsm16 pic.twitter.com/Il7dgqcvCP

— Danny Doucette (@danny_doucette) July 20, 2016

Progressive Policies. We saw a number of advances this year by the AAPT, including the adoption of a code of conduct, the establishment of gender-neutral bathrooms, and the creation of child care grants. These should be understood as commendable first steps, and I understand there is a proposal to extend the child care grants to include caregivers more generally, and several other positive steps.

I think the real benefit of the AAPT making these important and symbolic gestures is that it tells the members that this meeting was a safe space, where everyone is valued. Especially for those of us who work at markedly less-progressive institutions, that support is valuable and heartening.

@AAPTHQ knows what's up. 👏🏻 #hb2 #AAPTSM16 pic.twitter.com/l9xFK7IGT3

— Colleen Countryman (@FrizzlePhysics) July 18, 2016

I saw two key areas in which I would like to improve the accessibility of the conference. First, I saw that many people stayed in the youth hostel, about 5 blocks from the convention centre. The price of USD 37 per night for a dorm bed is markedly better than the rates for the suggested hotels (USD 99 and up). The popularity of this option, especially for visitors from abroad (like me), for high school teachers, and for those attending the conference out-of-pocket, suggests that the AAPT should think carefully about providing affordable options in the conference planning. Cincinnati doesn’t have a youth hostel, and I haven’t been able to find cheap accommodation in the downtown.

Second, the child care grants are nice, but not a complete solution given the difficulty of finding care locally, and the cost of bringing along a caregiver on a cross-country flight. Instead, I think we could come up with a child-minding option that (a) make the conference more friendly for families, and (b) bring some more joy and life to the conference for the rest of us. I understand there might be insurance issues in such a case.

Spaces. Interpreted both physically (ie: classrooms) and more broadly (ie: the learning context), the concept of students’ learning spaces arose several times. For one, we need to provide spaces free of gendered or racial cuing. The spaces also need to be designed to be comfortable for students. There is the question of how to establish safe spaces. And there is also the matter of the analysis of spaces, as a way to examine non-content learning.

Standard Problems Suck. Steve Kanim spoke at the Physics Teacher Camp about the role of math and problem-solving in physics. The slide speaks volumes. Kanim’s solution is the TIPERs, and he generously arranged for a book of these for all the attendees.

On typical questions: "There's no physics going on. There's no learning going on" -Steve Kanim #aaptsm16 #aaptcamp pic.twitter.com/lhj6EGoIA5

— Danny Doucette (@danny_doucette) July 17, 2016

On the other side of the conference, Beth Thacker reported on traditional learning and problems in unequivocal language.

High School. The AAPT didn’t promote the excellent physics teachers camp as widely this year, unfortunately, and I met several high school teachers at the AAPT that didn’t know about it (and likely would have attended if they had). It is a wonderful initiative, and certainly the best PD available for teachers like me.

Sadly, high school teachers seem to be fading from the scene. I saw fewer neat-teaching-idea talks than ever before, as the PER tradition continues to dominate the sessions. Steve Nixon’s talk about strategies in high-needs classrooms was a particularly effective exception.

Steve Nixon with solid advice on teaching high-needs students at #aaptsm16 pic.twitter.com/iXNvbEU2FZ

— Danny Doucette (@danny_doucette) July 20, 2016

Competition. A few speakers extolled the virtues of competition in the classroom, but I didn’t hear the counter-arguments anywhere. I’m not sure what to make of that.

Accelerometers Everywhere. Rebecca Vieyra was on hand to show us more of the Physics Toolbox, and Colleen Countryman had a poster about MyTECH. Lab4u had a booth in the exhibition hall demonstrating something similar. UIUC debuted a device called the IOLab that holds sensors and communicates wirelessly with a USB stick. PASCO’s new Smart Cart was a hit, too. I think accelerometers and other smartphone-based sensors have hit the mainstream now. Surprisingly, I didn’t hear anything about Google’s new offering, Science Journal.

Other educational technology seems not to have caught on quite as well: many people professed ignorance of Desmos and I didn’t see even a single presentation or poster about PhET.

Labwork. At the end of the PERC, the audience was asked to suggest growth areas for the community. Saalih Allie (actually on the panel) suggested that lab work, more complex than theoretical physics, deserves attention in the future. Perhaps, after last year’s focus during PERC, the AAPT community decided to give the study of practical skills a rest.

 

Two Submission Types. Perhaps I am becoming cynical, but I found that most of the posters I saw fit into one of two categories: (a) those that attempted to document some sort of project that was undertaken, but don’t generate many take-aways, and (b) those that address questions that could be answered in a single sentence, but don’t actually say that conclusion clearly. I think the history of PER comes from the desire to improve the physics 101 course, but just as we’ve moved on from that target, it’s also time to move on from that methodology.

Demos. Physicists still like demos. This lecture hall had a rotating stage, to allow for twice as many to be set up!

In my recent TEDxRiga talk, I called demos “snake oil”, unless they are being used carefully to provoke curiosity and engagement. I think we risk being too enamoured with the big booms (apparently there were some of these at the PASCO picnic!) and lose sight of the fact that students need active engagement in order to develop understanding.

Dianna Cowern (Physics Girl) gave a talk about demos. Her view is that they stimulate curiosity, which is what she aims to do with her channel.

IB and AP and A-levels. I attended a session on three college-at-high-school curricula. It seems that all three have gravitated toward the same curriculum and assessment model, with subtle distinctions (although it is unclear the IB and A-levels were ever very different). Of the three, the AP is the more progressive, with the recent revision sacrificing breadth for depth, and resulting in an unpopular drop in grades at many schools as the exams stiffened up.

I think that all three programmes suffer from a lack of transparency in how they design their curricula, and in how they design and implement their assessments. The assessments all attempt to go slightly beyond standard problems, but are nonetheless comprised primarily of the same. The AP does better than the others here, too, having documented much of their revision, and releasing the free-response portions of the exams. I saw that Boston University is doing a program that brings AP physics to students in schools that don’t offer it, by blending online learning with a weekly on-campus session.

Physics or Physics Education? I missed much of this discussion, but gather that there have been some existential questions raised about PER. Essentially, I think, is the question of whether the techniques of physics research are going to provide much more insight into physics education, or whether we need to turn fully to the learning sciences in order to make progress.

Saalih Allie advocated for “humble theories”, which need not attempt to describe phenomena fully, but nonetheless provide contextual insight and can be pulled together to make a patchwork understanding at some point in the future, if needed. This is probably a reasonable stance to promote, given (a) the physicist’s obsession with finding fundamental truths, and (b) the complete lack of anything like absolute truth in the social sciences.

Culture. I talked briefly about border-crossing, but I don’t think the cultural dimension is on many people’s minds right now.

Community. In a broad sense, the community of the AAPT gives me faith in the power of human institutions to come together and make progress in our field. At the same time, I felt like the PER community was missing some faces this year, Joe Redish among them. Perhaps we are witnessing the changing of the guard, and will see the emergence of leaders for the second generation of PER over the coming years.

I think the most valuable thing the AAPT does is provide a community for its members. I am over the moon to have had the opportunity to spend time with people I deeply respect and admire, and I cannot wait to continue that conversation online over the coming months.

Latvian Physics Exam

Today was the pilot administration of Latvia’s new 12th-grade physics exam. Next year, students will be required to write either the physics or the chemistry exam as part of their secondary school leaving requirements (for those students who pursue the academic track of studies). In this post, I will take a close look at the exemplar, which can be found (in Latvian) here.

Part 1.1 (multiple choice): Assumptions and Memory

The exam opens with a question about the applicability of the accelerated motion model. It’s a promising sign, but all of the answers — a launched javelin, a parachutist in the first minute of fall, a person in an elevator, and a hammer dropped on the moon — are possible, and the question is really about the degree to which the model can be applied to these situations. For example, a javelin has a very narrow cross section, so it would be reasonable to assume it experiences very little drag force. In fact, the aerodynamics of javelins are interesting, and although their motion is non-parabolic, the reasons are not obvious (an example).

The fifth question is identical to one from an IB physics exam from a couple years ago, where the motion of a piston is presumed to be uniquely responsible for the temperature increase of a compressed ideal gas. The twelfth lists four core equations about electricity and asks which explains the need for high voltage in long-distance power lines. These types of questions are frustrating because multiple answers are reasonable, and the students are forced to try to guess what the exam-writer is thinking.

Another type of question that appears is simple recall. Question 6 requires that students remember that isochoric processes have constant volume. The 13th question seems to ask that students remember how the technology of electromagnetic inductive charging works. Question 20 is about the origin of the atoms that make up our bodies.

The third type of common question is about relationships between variables. Students are expected to recall how electric fields depend on the distances from point charges, how photon energy is related to wavelength, and how the energy of photoelectrons is related to the frequency of incident light.

These multiple choice questions are pretty standard, with questions similar to IB Physics and the GRE. The difficulty level is probably appropriate, in that students will show a nice Gaussian distribution of responses. But as a tool for educational assessment, I think these multiple-choice questions have only marginal utility.

Part 1.2 (short answer): Not For The Meticulous

The second part of the exam consists of 10 short-answer questions. In reality, it is a mix of question types that is sure to frustrate students. There are 6 multiple-choice style questions where students must choose all the correct answers, 2 ranking tasks, and 2 calculations. Problematically, some of the distractor options on the multiple-choice style problems could conceivably be relevant. For example, problem 27 shows possible light paths refracting and reflecting around a glass block. A line path that is not visibly refracted by the block is possible, depending on the index of refraction (and the refraction is exaggerated compared with what would be observed in the lab).

The biggest problem with questions like this is that, while they are accessible to very weak students and straightforward to students who have a teacher’s level of understanding of physics problems, they will not be very successful at differentiating between a student who is at 90% of a robust understanding, and a student who can intelligently guess 10%.

I’m happy that the test-writers are attempting to expand beyond multiple-choice questions, but this section will be hard for the students, and will provide little feedback about students’ ability to use their physics knowledge and skills to meaningfully create knowledge.

Part 2 (long answer): Physics Problems, With a Dash of Discussion

The long-answer section begins with the ominous note that this section will involve judging pace, but with 135 minutes to solve 5-7 problems, the question likely will not be pace so much as sustained concentration.

I want to like these problems: they combine calculations (kinetic and gravitational energies for the first, circular motion for the second) with diagrams and comparison of quantities. This fixes the problem with the first half of the exam, and allows students the opportunity to explain their answers.

However, the structures for the problems seem set up to greatly restrict the students from any sort of individual actions, knowledge construction, or creativity. The first problem requires that students draw the velocity and acceleration vectors for a pendulum — something that is usually recalled, rather than figured out.

The rest of the problems mix standard physics things-that-must-be-remembered with little calculations. There is a clear effort to make the problems meaningful and/or relevant: one concludes by asking students for a conclusion about heat loss in a house, and another is about the laser on a Mars rover. However, the overwhelming sense with these questions is that of standard physics regurgitation: draw a voltage divider, calculate the diffraction angle, identify this range of the EM spectrum, etc.

The final question (at last!) asks that students work with some data, but this is an illusion: after finding the slope from a graph, it’s the same deal as before: find the efficiency of the electric kettle.

By the time you get to the end of these problems, it is clear that they are merely standard physics problems, with just a few “explain why” questions. This is better than the mechanistic rigamarole of the IB physics exams, but not by much. Even here, there is no chance for students to construct their own meaning or use genuine critical thinking, and I suspect that most students will get most of their points by reiterating standard physics solutions to these standard physics problems.

Implications

There is a suspicion that the new physics and chemistry exams are an attempt by the Ministry of Education to add more rigour to a national curriculum that has recently come under fire after Oxford University decided not to recognize the Latvian grade 12 diploma as adequate preparation for undergraduate studies. Thus, this exam will probably give universities a better way to sort students. I’ve heard that, after promising they wouldn’t be required, many universities are asking to see students’ results for this pilot year for the exam.

However, like all standardized testing of this sort, what we will primarily find is that students attending better-funded schools, in Rīga, and with higher socioeconomic class will do better on these exams. Additionally, these exams will be felt backward into the 11th grade and earlier, as physics and chemistry teachers are increasingly under pressure to prepare students for the exam, and thus are forced to sacrifice good science teaching in favour of test preparation. The quality of meaningful science education will fall, resulting in weaker critical thinking and scientific reasoning skills across society, while universities will increasingly rely on these test scores to make admissions decisions, and disadvantaged students will be left behind.

If I’m wrong about some or all of this, or if you have perspective or insight to share, please reply or email me (danny.doucette at gmail).

IB Physics vs GRE Physics

Can you tell which problem is from a test designed to assess the skills of high school students, and which problem is from a notiously difficult test used to rank applicants for graduate school?

A student needs to get 70 to 80% of these questions correct to be eligible for the top graduate schools in the USA (MIT, CalTech, Harvard, Stanford, etc). The same percentage is required to achieve a “7” (the top grade) in IB Physics.

The GRE consists of 100 of these multiple-choice problems. For IB physics, it is 30 or 40 (depending on the level) and then two more papers with longer-form questions. Students generally find the multiple choice questions easiest.

So which are IB and which are GRE questions? (hint: the GRE questions have five possible answers)

Gender Balance

This is the story about how I learned the real meaning of gender balance.

A couple months ago, I heard of this great way to integrate the Theory of Knowledge (the philosophy hub of the IB diploma) into my math classes. I took it a bit further, and ended up creating posters for eight different ways of knowing. These include reason, emotion, sense perception, language, faith, etc. My posters featured a picture of a famous mathematician whose work epitomized that way of knowing, along with a (carefully sourced!) quotation where they describe their process.

Making these posters was challenging but fun. Some were easy: Bertrand Russell’s work was pivotal to the language of math, Cantor’s work with infinities was deeply religious, and Edward Frankel’s great emotional love for the field is a great modern example. I sought mathematicians the students might encounter (Venn, famous for his diagrams), attempted to include those whose work is accessible to students, and tried including various cultural backgrounds where possible (Ramanujan for imagination).

Throughout the process I knew that I was failing to include women. Noether almost made the cut, but I couldn’t find the right quotation. Partly, I felt that no female mathematician deserved to be on the wall beside Hilbert, Poincaré, and von Neumann (except for, perhaps, Noether).

I don’t think this was a sexist viewpoint, but rather a reflection of the reality of a past in which women were barred or strongly disincentivized from participating. I provide plenty of female role models for my students, do interventions when necessary, and build discussions about the culture of sexism that is currently poisoning much of STEM into my classes. But these posters were historical, and so I figured it would be okay.

When I first put them up, I asked my classes why they thought there were no women, and whether they thought it was fair. They understood: we can celebrate the history of mathematics while also recognizing its faults.

After a couple months, a colleague challenged me to create posters with women. We played a game on the bus ride home of coming up with role models for the eight ways of knowing. I tried to provide only female physicists (physics is my other subject) but found it challenging.

So I took a couple weeks to work through it, reading biographies and wikipedia articles when I could find time, searching for the origins of pertinent quotes. Finally, I had a set. But these women were not like the men.

Three of the women were still living (compared with only one for the men). Several had contributed in ways that would not arouse the suspicion of the Fields Institute. One, a lecturer on hyperbolic surfaces, was better known for her crocheting than her equations. Few had names that appear in math textbooks. Several were unmarried throughout their lives. All had experienced sexism that slowed or derailed their careers.

So, what did I learn from the inclusion of these women? That math is about people, not theorems. That contributions to the subject come in many shapes. That, contrary to my expectations, the history of mathematics is rich with female role models. And that gender balance is a necessary condition for cultural balance.

You can download PDFs of the 16 posters here.