Flipping the Classroom

This year I’m trying out a flipped classroom while teaching Linear Algebra. I’m using the ABC Model developed by David Pengelley. His webpage seems to be down at the moment, but he wrote an excellent article for the AMS Notices describing his method. Basically, the idea is to get the students to read the textbook as their first contact with the material, rather than lecture to them. Homework consists of three parts:

• • Part A: Students do a reading, and respond to reading questions I have written for that section. The reading questions are supposed to ensure that students think about key concepts and pay attention to subtleties. Students also respond to reflection questions. These are turned in online via a Google form the day before class.
  • Part B: Students do about 10 warm-up problems for each class period. I grade these during class based entirely on completion; the students either have given them a try or not. Most of the class period is spent with students discussing these problems in groups.
  • Part C: Students do 2 more challenging problems and turn them in the next class period. This are usually not computation-based, and receive thorough feedback from me.

Class time at the moment tends to follow this schedule: We begin with a discussion of the reading questions, where I select students who had especially good answers to present to the class. I ran into a problem where students didn’t realize I was curating these answers, and had to let them know that I was only calling on people who had correct solutions! After this, I may spend some time clearing up common misconceptions and/or responding to questions from the reflections. This usually takes a total of 15-20 minutes. Then the students work in groups, and I circulate. As I go through the room, I check in everyone’s Part B work, turn back the Part C homework from the previous class, and ask if they have questions for me. This is my chance to interact with each student individually, and I’m not sure that I use it to my best advantage. During this process, I may put some problems on the board for students to write their solutions. These students then explain their solutions to the class. This continues to the end of the period.

 

What’s Going Well

What students seem to like most about this style of course is that it gives me more time to interact directly with them and answer their questions. When asked “What are three things the instructor does well to aid in your learning,” the most common themes were:

• Answers questions in class and responds to our answers in the reading questions
• Great enthusiasm
• Good explanations of harder concepts
• Focusing on conceptual issues in class

Students are working really well in groups. They have organically formed groups of about 3 students each, and they do a good job spending their time checking their work, helping each other, and collaborating on harder problems. As I walk around the room, I see them discussing possible solutions and explaining errors. I hear a lot of, “I see, I made an error in this step and that’s how I got the wrong answer.” I love that they are identifying their own mistakes.

So far, getting everyone involved has not been much of a challenge. I’m keeping track of who writes solutions on the board and ensuring that everyone does so pretty evenly, by recruiting presenters when needed. I use a color-coded spreadsheet to mark attendance, who answers reading questions, and who presents at the board.

Students mostly felt that they could participate easily in class, though 2/3 of them have expressed concerns about appearing stupid in front of their classmates. I staged in intervention with my students, where I told them that fact, and also shared that about 90% of math grad students at UChicago feel “inadequate in relation to their research.” I also told them about times that I have failed and felt stupid, and expressed the hope that this class can become a space where they learn to feel comfortable saying that they don’t know the answer. We’ll have to see if that makes a difference in their willingness to be show their confusion, or risk making a mistake.

Additionally, I think the students are learning the material pretty well. They do a universally good job of answering the reading questions, and they all seem to be conversant in the relevant vocabulary. One benefit of this system is that we are on a schedule, and at no risk of falling behind. My friend who is teaching the same class as me is about a week behind in terms of content. So far, everyone is doing well on their homework and I haven’t noticed anybody who is just totally lost. This makes me feel pretty good!

 

What Could be Improved, and How I’ll Do It

My biggest issue with this structure is that about half the students don’t really like it. Part of that does seem to be that it is harder to “coast” to a good grade. Students complain that we’re covering a lot of content, but I’m not covering more than what we are told to do. I think that other teachers end up cutting material that they don’t have time to cover, so perhaps my students will actually learn more content than their peers. But mostly, I think it just feels like more content because they aren’t zoning out during lectures. The amount of time the students report doing the reading is (on average) less than a class period. Students have said things like, “I think lectures make slightly more sense with regards to math because it is a more difficult subject.” I tend to think that even if math is a more difficult subject, that would make it a good place to apply a seminar/flipped learning style.

I’ve also had students tell me that they hoped I would lecture in class, because I’m very good at it! That people who are bad at lecturing should teach like this. I don’t really know what to think about that. On one hand, I think people should play to their strengths in teaching. On the other hand, I suspect that most bad lecturers would also be bad in a flipped classroom.

My planned solution is to be a bit more systematic in my talks at the beginning of class. I still won’t give definitions or theorems, but I’ll try to help them organize the material and relate the new material to older content. I want students to get first contact with the material through reading the textbook, not through lecture. So i don’t want my in-class talks to allow them to skimp on the reading. Luckily I’ll still have reading questions to keep them honest, and I think that if my “lecture” serves more to organize material than it does to explain or introduce it, then I’ll be helping with something the textbook isn’t as good at. Hopefully that will be enough to scratch that lecture itch for most of my students.

Another fear that I have is that, even though I am interacting more with students in this style of classroom than I would in a lecture, I am less aware of how well my students will do on the midterm now than I was in any of my previous classes. This is because in previous classes, I have used oodles and oodles of Classroom Assessment Techniques (CATs), to keep track of how the students are doing, and to give the students a heads up on how they are doing. In a single lecture, I might have three mini-quizzes for students to practice their problem solving, and to give me information on what the class does and does not understand. I don’t have that now; since each problem is only presented by a single student, all I know is that at least one of my students knows how to solve it. And while students are doing well on the homework so far, that only tells me that their group work is really paying off. I have no idea what will happen when they have to work alone on an exam.

While I love reading the responses to the reflection questions, and sometimes get surprisingly self-aware replies (“I am learning more in this class that in a more tradition lecture, but maybe that’s just because I’m working harder.”), I don’t know how much I can trust their self-assessment of their own knowledge. The Dunning-Kroger Effect famously shows that there is an overlap in knowledge required to do a task and to assess how well you did that task, so people who are struggling are less likely to be aware that they are struggling.

My solution is that I will simply ask them to spend time in class doing little CAT quizzes. I’ll decrease the number of warm up problems, and use some of the problems I would have assigned in the quiz instead. There’s no reason not to do this, except that I need to work out how best to fit this into the flow of the class. I think the best bet is to do it in the last five minutes of class, after they’ve had a chance to work on some problems with their peers and ask me questions.

Finally, the students are 50/50 on whether they like the textbook or think it is awful. My own feelings about the textbook are equally mixed. While I have already listed some resources, I know that students are not using them regularly.

My solution for now will be to make alternative resources for certain topics more obvious, perhaps by linking directly to the relevant Khan Academy video (or similar). I think the best solution in the future would be to carefully curate each section’s resource, and provide one or two quality alternatives. The downside to this is that there are a lot of resources out there, and finding the “best” one is going to be a lot of work. But then again, if I’m expecting a lot of work from my students it’s only fair that I do the same.

Book Review: Inferior, by Angela Saini

For Christmas I was given Inferior, a book by Angela Saini which investigates the current scientific consensus on sex differences in humans. Overall I found it absolutely fascinating – full of intriguing examples from foreign cultures and difference species. It makes explicit some of the fallacies built up over centuries around female inferiority, and how those cultural ideas influenced ‘unbiased’ science. The prose itself was compelling – I finished the book in less than two days, primarily because I couldn’t put it down. Saini has a skill for finding the most illustrative and compelling examples, and then gives explanations in the words of the scientists who discovered them.

My only complaint is that there is an undeniable bias in the rhetoric. Studies which support the old patriarchal view are presented skeptically, with their potential failings explicit. Work for the other side, on the other hand, is generally presented as undeniable fact. Male scientists who step outside their own realm of expertise are castigated, while female scientists who do so are lauded – as long as their conclusions are pro-feminist. On the other hand, in some situations the science is murky and Saini acknowledges this. She always presents conflicting views, both from the researchers and their peers.

Incidentally, despite this book being non-fiction, there was a villain. Dr Trivers, whose research suggests that females are naturally chaste, or at least less interested in sex than males, appears to be a complete tool. In one story about photocopying an article, he mentions the feeling of his testicles pressed against the Xerox machine – why has he included this? Would he have said it to a male interviewer? (Would a male interviewer have included it in the book?) In addition to the sexual harassment, Dr Trivers has also failed to read a well-received research paper, written by a scientist who he admits is good and reliable, which contradicts his own research. That is odd enough – that a researcher would fail to read a response to his own work – but additionally, Saini explicitly asked him to read it. He admitted that he knew about it – even denigrated it – and still has not read it. Is this because this paper is written by a woman? Saini bravely does not rise to the bait here and accuse him of sexual bias, as I do.

Below are a collection of the highlighted tidbits I learned while reading this book:

• Evidence of Sex Bias
• • • In a study published in 2012, Corinne Moss-Racusin and a team of researchers at Yale University sent out applications for a job as a laboratory manager. Every resume was identical except for the name – one versions had a typically feminine name, and the other a typically male name. Scientists – regardless of their own gender – rated the ‘women’ as less competent and less hireable. They were less willing to mentor the women, and offered lower starting salaries. (Pg 5)
    • Married women with children in the US are a third less likely to get tenure-track jobs than married men with children. (Pg 6)
    • A 2016 paper looked at how men and women perceived competency in their peers. Students in a biology class were asked to guess the grades of their classmates. Male grades were overestimated – by men – by .57 points in the GPA scale. Women showed no such bias. (Pg 7)
    • This is not genetically baked in, because other cultures have different results. In Bolivia, 63% of scientific researchers are women. In Central Asia, over half the researchers are women. (Pg 8)
• Sex hormones aren’t as binary as we might think: horse testes are the best known source of estrogen (Pg 34), and women with slightly higher than usual levels of testosterone neither look nor feel less feminine than those with normal levels (Pg 36). On the other hand, research from 1980 onward suggests that sex hormones do alter the brain as we develop, though the extent to which that effects us is unclear. (Pg 76)
• In 2011 census in India showed that there were 7 million fewer girls than boys under the age of six This is thought to be due to voluntary abortions after learning that a fetus is female. That was such a problem that it was actually made illegal in 1994. (Pg 40)
• Some hospitals in South Asia have 80% of child patients that are male, because the girls are simply not brought to the hospital when they are sick. (Pg 41)
• Across every society, women live about 5 to 6 years longer than man – and this has been true for centuries. We don’t really know why, but it has been hypothesized that female hormones play a role in strengthening the immune system. Supporting this is the fact that women are more likely than men to have autoimmune diseases – in the US, over 75% of people with autoimmune diseases are women. (Pg 46-49)
• Despite extensive research into the physiological differences between the sexes, until 1990 it was common for medical trials to be carried out exclusively on men. (Pg 58)
• Subtle and prolonged differences in the ways we are treated can affect the brain. In the 1970s and 1980s, research suggested that the ration of boys with mathematical genius to girls with mathematical genius was 13 to 1. Since then, the ratio has plummeted to 4:1, or even 2:1. This suggests that the difference is not inherent, but learned. Furthermore, in some communities women out-perform men on this scale – Latina kindergartners, for example, outperform their Latino classmates. (Pg 117-118)
• Many birds thought to be monogamous have recently been shown to have a penchant for cheating. Female bluebirds will leave their nests and fly considerable distances at night to mate with other males. Some scientists have rejected this finding, insisting that the females are being raped (though personally I can’t see how the male rapes the female into flying a “considerable distance”). (Pg 116)
• At Florida State, researchers had participants walk up to people (strangers) on the quad and propose a date or casual sex. The women were significantly less likely to agree than the men, and usually had a strong negative response (“Who do you think you are?”). The men were more likely to say yes to casual sex than to a date, and would often apologize if they refused sex. This was taken to show that men are naturally more promiscuous than women. But in 2015, two researchers questioned those results. They concocted an elaborate ruse where testees come in to a lab to participate in a dating study. They were shown ten photographs of people who “were interested in going on a date or having sex with them in particular.” If they agreed, they were told, they would go out in a safe location and the date would be filmed. In this situation – where the moral and physical risks were greatly decreased – 97% of women agreed to a date with at least one person, and almost every woman agreed to have sex. (The men all agreed to a date and sex with at least one woman). (Pg 174)
• Researcher Alan Bateman wrote an influential paper about fruit fly mating which indicated that females were passive, choosy receivers of attention, while males were promiscuous. Several years later, Patricia Gowaty began to suspect some of these findings. She looked more deeply into Bateman’s paper, and found multiple potential spots of error. She also tried to replicate his result, and found instead that females were just as likely to approach males as vice versa. She wrote a paper about this, which has yet to be read by some of the most influential researchers in the field (including Dr Trivers), even though they are aware of the paper and otherwise consider her a solid researcher. (Pg 174-178)
• Pigeons clump together for warmth in the winter. They are in mated pairs, and when they huddle together the male pigeons always sit in the middle of the set, so that they are between their mate and any other male. If there are three pairs, this becomes impossible – the middle male can’t keep his mate from both the other males at once. So he engages in ‘mate-guarding’ behavior, and pecks her until she backs up a few steps out of the huddle. The male pigeon would rather have his mate cold, tired, more likely to get ill than next to a male. If, in the night, she sneaks back into the huddle, he’ll peck her again until she is out. Question: Why go to such extreme lengths (we see such behaviors in humans as well) if the females do indeed have minimal sexual appetite? (Pg 180)
• Our two closest relatives, chimps and bonobos, exhibit markedly different behaviors. Male chimps are violent towards their partners, and will even kill a potential partner’s child to make her more likely to have sex with them. Bonobos, on the other hand, are run by the females. Though they are smaller than the males, they work together as a team to control the males. (Incidentally, the idea that female bonobos were in charge was rejected for many years by male researchers who couldn’t believe it). Females spend 2/3 of their time building relationships with other female bonobos. (Pg 200-205) What this means for humans is that it was not written in stone that males should be the dominant sex – not even with them being larger!

Edinburgh Half-Marathon

One of the best things about my dearly beloved is that he pushes me out of my comfort zone.  Well, he’s done it again: I have officially signed up for the Edinburgh Half Marathon on May 27, 2017.

This is officially way outside my comfort zone.  Even when I was at my peak fitness point, I considered myself “the anti-runner.” I even justified it with physiology – small feet + short shins + long femurs = minimal torque = bad running.  I have essentially no justification for this, but I’ve used it to rationalize my distaste for running for years.  For whatever reason, I never experienced that “runner’s high” that everyone talks about.  I knew the feeling from biking, but I never found it with running.

But now I’m going to try to find it.  I don’t think I will ever be totally comfortable running with Jeremy.  He’s fast, and my competitive drive would get in the way.  BUT I think we could both be runners, and I’d like to be more able to keep up with him.

My goal for the next 10 weeks: no complaining about running.  Maybe in my head I’ll think, “Oh, I really don’t want to do this.” But I won’t give in to moaning, and I won’t say it out loud. No sighs, no facial expressions…I’ll act like I enjoy running, and hopefully I’ll trick myself into believing it!

Women in Math: The Symposium

On Saturday, Tori and I hosted a symposium to celebrate Women in Math – sponsored by the UChicago AWM.  We had undergrads, grad students, a post-doc, and tenured faculty talking about their own research (in the case of women), and the work of female mathematicians (in the case of the men). Unfortunately, we started a little too early in the morning – attendance was very low for these talks, which were universally excellent.

In the afternoon we had a larger attendance.  Most of the talks were more about women in math than about math itself.  Kiki Zissimopoulus ran an excellent session on how to teach to an inclusive classroom. We explored what an inclusive classroom is, what constitutes ‘mathematical thinking,’ and ways that we can foster an inclusive environment in our own classes.  I was excited to apply some of these ideas to my own teaching – especially the idea of strategic learners and deep learners, and trying to encourage deep learning techniques (returning to old homework problems, repairing errors, etc).

We also had a fascinating session by Stephen Meyer, of UChicago Astrophysics, about Implicit Bias.  He gave an interesting presentation, which centered on a study of letters of recommendation for male and female medical candidates.  The statistical differences are striking, and hard to ignore: twice as many letters for women as for men included ‘doubt raisers’ – comments like, “She worked hard on projects she accepted,” which are interpreted negatively, while offering surface support. Only 3% of the letters for women referred to them by their titles, while 12% of letters for men did so – and there was no difference in the number of women who had such titles compared to the number of men. Letters for women are much more likely to address training, personal life, and teaching than letters for men.  On the other hand, male letters tend to address research much more than female letters. Perhaps the most interesting method of analysis was looking at the most common words following possessives: “her research,” “his colleagues,” etc.  Overwhelmingly these words followed social stereotypes.  All of this is true regardless of the gender of the recommender – one of the most insidious aspects of implicit bias.  And, as Professor Meyer pointed out, this effect is multiplied by the biases of the readers. On a related note, this is a cool resource (shown to me by Kiki Zissimopoulus) about gender biases revealed by word usage in teacher reviews.

We also had a great session on Impostor Syndrome, run by Claudio.  He ran it as a workshop, and at one point asked us to turn to our neighbors, look them in the eye, and tell them something that we are proud of.  I found this incredibly difficult – even though he had told us to do it, even though I knew they would not think of it as bragging, even though I knew they wouldn’t laugh or make a face. Even so, I was near tears by the time I said anything. What was interesting (and surprising) to me was that everyone in our little group had a quote-unquote “rational” for thinking they did not really belong in the UChicago PhD program. I was able to look at my fellow grad students and see that they were being absurd – they are excellent mathematicians and well suited to UChicago – but I can’t do the same for myself. Very frustrating, to know that one is being irrational and yet be unable to escape from those thoughts.

We closed the night with mathematical trivia, put together by Mathilde. The questions were perfectly chosen (though I may be a little biased since my team won).  Questions ranged from mathematical trivia (How many 4-dimensional platonic solids are there?), to historical personages (I am a French mathematician whose hand was buried apart from my body.  Who am I?), to estimation questions (How many kangaroos are there in Australia?). We had a great time, and it was fun to get competitive over something small.  My team won a bag of tropical fruit as a prize – a UChicago math department tradition started by accident in my first year.

All in all, I think we did a good job of the symposium, given that it was planned in less than two months.  We even (after adjusting slightly) had the right amount of food! (Plus some non-perishables that we’ll use in our next event.) Goodness – I was exhausted afterwards, primarily over the stress of making sure everything ran smoothly.  We luckily had a lot of volunteers to help run each classroom, deal with setting up projectors, and even clean up the food after meals. I have no idea how anyone can run a several-day conference! But it was a great experience, and next year we’ll do more early planning, get more speakers, and all in all have a great time.

Intelligence Squared: Has the political establishment failed America?

I went to a debate on February 23, hosted by the UChicago Institute for Politics and Inteligence Squared.  IQ2 hosts “Oxford-style debates”; there are two teams of two people.  Each speaker makes a 15 minute opening statement, followed by a round of questions from the audience, and then 5 minute closing statements from each speaker.

The topic on hand was “Has the political establishment failed America?” It’s a wishy-washy topic, and hard to get a handle on – who counts as “political establishment,” what does “failed” mean, and even what does “America” mean (citizens, or the idea of America). I find it hard to grapple with questions like this. On one hand, there is a sense in which any question precisely worded would have a straight-forward answer, or at least it would have a series of studies that argue for opposing views.  In this case, a debate would be either pointless or dull.  But if you go too far in blurriness of a question, then each side can choose an interpretation that best suits their case, and thus talk past each other.  There was a lot of “Just because the political establishment has not always failed America, or just because the political establishment doesn’t fail America in every way, does not mean that the political establishment has not failed America” on one side, and on the other, claims that “‘Political establishment’ is such a meaningless phrase, and is up for so much interpretation, that the only response to this is to vote no.” And so the debate, which should be a coming together of opinions and ideas to create a consensus, devolves into a battle of who can persuade the most people that their definition is the best. Both sides talk past each other.

The Pro team: Michael Eric Dyson (Georgetown University) and William Howell (UChicago).

Michael Eric Dyson, of the Georgetown Sociology department, was a fiery speaker and incredible orator.  His words were rhythmic and alliterative, emotionally powerful and compelling.  When I took a moment to step back and think about what he was saying, I realized that he wasn’t really addressing the question at hand – he was talking about Trump and the Republican party in general.  Though they are the current political establishment, you can’t address their failings and not address the system that brought them to power. Just because he was such a good speaker, here is a video of him talking about racism and Trump.

The other member of the Pro team, William Howell, is a member of the UChicago Harris School in Public Policy. He was my favorite of the debaters, because he stayed on the question the entire time.  He also didn’t attempt to evade the question by arguing that we shouldn’t be answering it all – he accepted the premise of the debate and tried to live within those confines.  Unfortunately, he did seem a bit arrogant.  He was also pushing for broad institutional reform – he even implied changing our entire form of government – which was a bit more extreme than I am ready to support.

The Con team: Jennifer Rubin (The Washington Post) and Eric Oliver (UChicago).

Eric Oliver gave the opening statement for the Con team.  Admittedly, following Mr Dyson is a big task.  However, his attempt to equate anyone who supports the motion with Donald Trump and his supporters was logically flawed in the extreme.  Indeed, while the Pro teams most pervasive problem was trying to turn the debate into a referendum on Trump and his policies, the Con team’s was over-inflating the facts and an inability to wrestle with inconsistencies.  Yes, Donald Trump ran on subverting the establishment.  That does not mean that President Donald Trump is not now a part of that establishment.

The other half of the team, Jennifer Rubin, is a conservative columnist for the Washington Post. I found her to be a poor speaker and prone to logical fallacies, but I worry that I only think that because she is (1) female, and (2) conservative.  I so fundamentally disagreed with much of what she said that I’m not sure I can be an objective judge of her abilities.

Intelligence Squared debates are won by convincing more people to join your side.  They poll the audience at the beginning and end of the debate, and compare the scores. I began as an “Undecided” voter, and ended supported the Pro team.  But much of that support came not so much from agreeing with their arguments, but disagreeing with the Con side’s approach.  Both teams made valid points throughout; the political system is failing America in terms of discrimination of all sorts, environmental planning, and protections from big companies.  But on the other hand, American quality of life goes up and down periodically, and it would be wrong to place that at the foot of the political establishment.  Especially since it is the establishment that has stepped in to stop so many of the most egregious outrages – the courts stopping Trump’s travel ban, the federal government (evnetually) stepping in for Flint, MI.

So, who won?

The final result was that 4% of respondents switched from Pro to Undecided, and the rest stayed the same.  So Con technically won, I suppose?

I got to ask a question at the end of the debate. I asked to what extent each team’s argument was historical – did the Pro team think there was ever a time when the political establishment was not failing America? Did the Con side think the political establishment had ever failed America? And if yes, when did it change?  Unfortunately, neither side really answered my question; in fact, both sides seemed to argue the opposition’s point! Afterwards, at a reception hosted by the IoP, one of the speakers noticed me and told me that I had asked an excellent question.  Very flattering – but next time, please answer it!

Women in Math

On Thursday I helped run a Wine and Cheese event in the math department, hosted by the UChicago chapter of the AWM. The goal of the event was to bring the community together, and try to encourage conversation about women in math.  We offered snacks and drinks, a venue for conversation, and filled the chalk boards with statistics about women in math, and STEM fields more generally.  Perhaps the most interesting aspect of these statistics is that men and women are approximately equally represented in math in high school and earlier, but start to disappear from the math community as time goes on.  Perhaps this indicates a problem with the attrition rate (prompting questions like, “Why are so many women leaving math?” and “How can we adapt the norms of the math community to make it easier for women to stay?”).  But on the other hand, it may indicate that we are finally turning a corner in getting women interested in math, and all that is needed now is time for the current class of high-schoolers to start to permeate the upper levels of academia. The way to tell would be to look at similar statistics from several decades ago, but I don’t know where to find such information.

Some women in STEM statistics.

We were moderately successful in prompting conversations on women in math.  I overhead several groups discussing whether representation of women in math was something that needed to be “fought” for. This reminded me of an excellent post by Vi Hart and Nicky Case. They created applets the show how even a very slight preference for living with “people like you” leads to segregated communities. It also shows that it takes an active desire for integration to overcome existing segregation. While their original intent was to use this as a parable about racial segregation in housing, I think it also applies to issues of under-representation in math (and other academic fields, more generally).  Given that math is a primarily male field, it will take an active desire to include women – women will need to embrace working in a (currently) male-dominated field, and men will need to embrace the diversity that women offer.

More statistics on women in STEM!

It’s good to get so many people together and talking about something that is usually avoided. It’s hard to talk about representation of women, or to talk about historical female figures in mathematics.  I have heard many comments along the lines of, “It’s not fair how easy it is for women to get into the top graduate programs,” , or “That historical female mathematician gets more credit than she deserves.” In addition to being incredibly discouraging personally (did I only get into graduate school because I’m a woman?), the former (1) is simply wrong if you consider the percentage of graduate students who are male versus the percentage who are female, and (2) ignores the difficulty of remaining in math long enough to even apply for graduate school. While the former is more difficult to hear as a woman in math, I think the latter is more insidious.  It forgets that much of the work women did was published (with or without permission) by men, and credit was given to the male authors. Perhaps the most famous example of this in a STEM field is the work by Rosalind Franklin that was used without citation (and possibly without permission, though this is disputed) by Watson and Crick. Part of the reason for this was at least partially due to the misogynistic views of Watson toward Franklin. It also forgets that many women were explicitly banned from membership in mathematical societies, denied university diplomas (if they were even allowed to attend university), and therefore unable to support themselves as mathematicians unless they were independently wealthy.  And this of course ignores the usual lack of education of women and societal expectations that women would be uneducated, quiet, and family-focused.  In this context, a woman producing any math is impressive, even if it is not the most incredibly unique work of the time.

Even more!

All of this is to say, I’m excited for the Women in Math symposium the UChicago AWM is hosting next week. Most of the speakers will actually be talking about historical female mathematicians or their work, and I’m looking forward to hearing how this is presented.

After the event, we found a photo of a former graduate class and their pet goldfish. He appears to have been named C. Abel.

Two Performances

On Thursday, I finally got a chance to see Hamilton, the hit musical by Lin-Manuel Miranda.  Overall, it lived up to the hype – which is a significant accomplishment!  I enjoyed the music more than I expected, and loved how intricate the text could be because of the freedom of rap. Wayne Brady performed as Aaron Burr – a surprising choice that ended up working incredibly well. The best thing about the libretto is the depth of  Alexander Hamilton and Aaron Burr. While Hamilton is of course the hero of the story, we are shown some of his flaws: hot-headedness, arrogance, limitless appetite. Similarly, the supposed villain of the show, Aaron Burr, is presented as a Salieri-esque character, a la Amadeus. He was meant to be a success, but was overshadowed by a younger, more brilliant replacement before he ever got to make his mark. It’s an interesting contrast in today’s political climate as well – the man with strong principles who is unwilling to compromise for political advantage, versus the man who compromises so much that his beliefs are unknown.

The Schuyler sisters, performed by Karen Olivo, Ari Afsar, and Samantha Marie Ware. Photo credit to TimeOut Chicago and Joan Marcus.

The performers were of course incredibly.  Karen Olivo stood out at Angelica Schuyler, especially in her performance of “Satisfied.” The role of Angelica is overshadowed by that of Eliza, but it she is a much meatier character; a woman who gives up one type of love for another, and who can engage with Hamilton on his politics. Eliza, performed by Ari Afsar, never really grows beyond her first meeting with Hamilton when she confesses herself “Helpless” over her love for him. Both women had strong voices, but I preferred Ms Olivo’s less strident tone. Miguel Cervantes was excellent as Hamilton, giving a convincing performance throughout.  Jonathon Kirkland stepped out during intermission, leaving Carl Clemons-Hopkins to replace him as George Washington.  While he got off to a rocky start, he showed off his gorgeous voice in the closing of “One Last Time” and finished the show on key. Thomas Jefferson, played by Chris De’Sean Lee (who also played Lafayette), was a hit in his elaborate purple costume. But King George, brilliantly acted by Alexander Gemignani, stole the show with each of his entrances. Attired in a red suit with ermine-trimmed robe and a crown, and brandishing a scepter, he carried himself like a ballet-dancer, tip-toeing from one part of the stage to another.  His first number, “You’ll be Back,” is full of inherently funny lines: “I’ll send a fully armed battalion, to remind you of my love.” Delivered in manner that was simultaneously serious and campy, King George was portrayed as a bumbling and dim leader who could not see history staring him in the face.

Miguel Cervantes as Hamilton, and the Chicago cast. Photo credit to TimeOut Chicago and Joan Marcus.

Then on Friday, I attended an altogether different type of concert: an evening of art songs, performed by Sandrine Piau (soprano) and Susan Manoff (piano).  The program, Apres un reve, focuses on dreams and love. It is inspired by her recent CD of the same title. Ms Piau’s soaring voice was simultaneously intimate and powerful, stylized and natural.  Her perfect technique, round sound, and controlled vibrato was all the more stunning after seeing Hamilton (which is not to say that the performers in Hamilton were unskilled – they are completely different art forms and, in many ways, incomparable). What I loved most about the concert was the apparent friendship and obvious communication between Ms Piau and Susan Manoff, the collaborative pianist. When a cell phone rang at the end of Romance (Debussy), Ms Piau and Ms Manoff exchanged grins and laughs.

Sandrine Piau

Throughout, each song was delivered directly and conversationally. The audience recognized the humorous moments by Ms Piau’s delivery, without needing to read the texts. Ms Manoff’s piano playing was virtuosic. She perfectly balanced supporting a primary vocal line with artistry and life. This was especially evident in the show-stopping Hexenlied by Felix Mendelssohn.

Susan Manoff

As a soprano, I had major repertoire envy, especially for La courte paille (Poulenc), a fun collection of child-themed songs. I was also jealous of Ms Piau’s comfort and impeccable technique.  Throughout her range, her tone remained so even and round. High notes floated and rang in the large hall. Vowels were so solid that I felt I could reach out an touch them.

They changed the order of the final set of Poulenc songs, choosing the start with the cabaret-esque (and oh-so-very-French) Les chemins d’amour, and closing with Sanglots, whose texts speaks of the unity of humanity, and the despair that comes from forgetting this. It made less sense musically, but politically was an interesting choice.  The thunderous applause caused them to return for two encores, however: Fantoches from Les fetes galantes (Debussy), and Sleep (Ivor Gurney). Sleep was especially moving; at the conclusion, there was a gap of silence as the entire audience breathed before breaking into applause.

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Most of the program can be heard via Spotify on the album Apres un reve by Sandrine Piau and Susan Manoff. I especially recommend Hexenlied and Nachtlied (Mendelssohn), and Les chemins d’amour (Poulenc). I also particularly enjoyed La courte paille (Poulenc), and Wasserrose (Strauss), though I can’t find recordings of these by Ms Piau.

Skiing in Door County

I went home for two days to get some cross-country skiing in.  I love skiing – it’s peaceful, steady, exhausting, fast…everything good. The steady action is almost meditative.  I’ve been skiing since I was tiny – my parents have photos of me in a little pink and purple snowsuit, toddling along on my miniature skis.  But before I could even walk, my parents were pulling me in a pulk – sled you attach to your waist and pull behind you.  They were traditionally used to transport supplies and people during ski journeys in the Nordic countries.  I have vague memories of getting strapped into the pulk, feeling the plastic of the sled flex under my back as we moved, hearing the roar of the sliding snow, and watching snowflakes fall from the sky and onto my face.

I never pulled a pulk until this winter, when my friend Liz purchased one for her baby Grant.  Liz is learning how to skate ski, so while she gets her feet under her I get to pull the baby.  It’s a different motion than skiing without the pulk – you have to be more smooth, move your hips differently so that you aren’t constantly jerking the sled. Grant is funny – he likes to sing along with the sliding sound for a bit, and then he falls asleep. The pulk doesn’t cause much of a problem on the flats or downhills, but going up a steep hill can be a battle! Sometimes I have to let Dad pull for me, when I am too tired.

Dad, Liz, and the Baby G at Snowkraft.

I love to ski at Snowkraft Nordic in Sturgeon Bay. A local man runs it.  He spends hours and hours making sure that the trail surface is perfectly smooth, the grass just the right softness, the trees trimmed to perfection so that you can ski there when nowhere else has snow.  He goes and grooms at 2am, when the snow is coldest, to keep the trails as perfect as possible. The result is some of the best ski trails in the state.  And because of the careful maintenance, they can be skiied on when nowhere else has snow.  He also uses up every inch of his land to make fun, twisty trails that make good use of the natural terrain. The Orchard trails are built along a ridge, so as you travel east to west you climb a surprisingly steep grade! Then you turn around and whoosh back down the hill, speeding around S-bends and banked turns. The trails in the pines are flatter, but also beautiful, fast, and impeccably groomed.  I can’t wait to get back!

Snowkraft ski trails.  The Orchard is hilly!

Solving Problems

It’s so hard to explain what math research is like.  For some reason, people seem to assume it involves doing long, complicated calculations; perhaps because that was what they found difficult about math in school.  In reality, long and complicated calculations are performed by computers.  We’ve gotten much better at automating those processes – it’s really stunning what computers can do these days.  Every once in a while I stop taking it for granted, just for a moment, and remember how powerful these ubiquitous tools really are. Anyway, this post is going to be a bit of a philosophical ramble about the nature of math reasoning followed by a (charming) anecdote about how I proved my first little theorem of my graduate career.

In fact, the difficulty of math is much more about figuring out how to prove something, and less about the actual argument.  Consider any of the theorems you learned in school – the Pythagorean Theorem for triangles, for example.  This proof is taught to,  and understood by, children under the age of eighteen! The argument itself is not so difficult – you follow some logical steps and end up at the result. There are literally hundreds of proofs.  But now imagine that you are discovering this theorem for the first time.  You have noticed a pattern in the relationships between the lengths of sides of a triangle.  You can’t find any counterexamples, so you think that it might be universally true.  But you don’t know, so you set out to prove the general concept.  How do you do it? If someone from the future were to show up and explain why it’s true, you would understand, just like the kids in school do.  But there isn’t  a time-travelling teacher – and you have this niggling worry in the back of your mind that it might not be true at all. The difficult part of the proving something is not following the logic; it’s figuring out which logic to follow.

I spent the past few days working feverishly to try to complete a proof. As usual in math research, this was an emotional rollercoaster.  Every supposed breakthrough is followed by the discovery of an error. But one nice thing about struggling with how to prove something is that as you play with a mathematical object, you start to understand more and more about it.  Eventually you can usually convince yourself that something is true, even if you have no formal proof for it. You get excited, and then discouraged, but even when you realize your argument is flawed, you can see that a similar argument ought to work.

This kept happening to me while I was solving this problem.  I had one insight – that graphs can be either fat and round, or long and skinny.  As obvious as that sounds, after that realization I knew that I would be able to solve my problem.  I’m trying to generalize an argument made by two other mathematicians, and I knew that this was the way to do it – it was too similar, too elegant not to be.  I did pages and pages of calculations and estimates, drew diagram after diagram.  But I couldn’t quite get what I was looking for.

Two pages of the notebook I almost filled over the past three days. I especially like the comments to myself, to help me navigate these notes in the future.

On Tuesday, I thought I had it.  I tested all the cases, did all the calculations, checked for obvious errors – it looked good, so I emailed my advisor that I had something to show him.  On Tuesday night, I found the first error.  Talk about emotional turmoil!  So I spent another few hours brainstorming ways to fix it. None of them seemed to work.  They were all reasonable, but just didn’t work out in the end, or relied on a claim that seemed like it should be true but that I couldn’t prove.  I went to bed disappointed (and couldn’t sleep, so I got up and did another hour of work before going back to bed).  The next morning, I went back to some of those ideas.  “Maybe one of these can be massaged until it works,” I thought. They’re all reasonable; surely something can be fixed.  But to my chagrin, none of them worked. When I went to bed on Wednesday, I thought about my problem and discovered another error.  (Luckily, when I got up to write it down, I realized that it didn’t matter, and actually made my life easier.) On Thursday, I decided to brute-force one method; to apply as much heavy machinery to it as it took to make the argument work.  Up to a single claim, I had the argument.  But I couldn’t prove this claim.  It seemed reasonable, but that wasn’t enough.  I took it to a professor, who agreed that it seemed reasonable but didn’t have an argument for it.  I took it to a fellow graduate student, who said that he might have a sketch of an argument, but wasn’t sure.  I started to feel more confident.  I took it to two post-docs, who agreed that it seemed reasonable, and then stood at a chalkboard with me for 40 minutes trying to prove it.  We couldn’t do it.  I asked my advisor, who told me that it was indeed an interesting question (which was a relief, to know that it wasn’t completely trivial!), but that he didn’t know the answer.  I started to get hopeful.  But then one of the post-docs came and showed me a counter-example.  My claim was not true, my argument failed. Ten minutes later, I had the actual solution. It was easy, straightforward, elegant. I wrote it up, checked it, and sent it to my advisor.

More of the work, demonstrating the messy parts of the process (at left) and the attempt to write this up in an organized manner (at right).

 

My understanding is that this is normal – that math research is full of ups and downs, false starts and blind alleys.  I think it also highlights two things in particular: the first is the importance of collaboration.  Throughout the process, I talked about my problem with friends.  Explaining my ideas to someone else helped crystallize them, and they could help point out potential flaws or confusions.  And for this final claim that my friend disproved, I wouldn’t have found that counterexample without her. The second thing is the importance of perseverance.  There were a lot of setbacks in this process – and I’m not done with my proof yet.  I’ve done half of the first step of (at least) three.  It’s great, and I’m incredibly pleased, but it’s just a battle, not the whole war. Luckily, I enjoy this process and I think my problem is interesting, so perseverance isn’t much of a challenge.  It’s easy to get discouraged of course, but as soon as one method fails another idea pops into my head, and I have to start exploring it right away. And it helps that  I can feel it as I get closer to a solution, so that even if I hit a setback I know that the right answer is just around the corner.

Winter Vegetable Soup

I managed today to make an attractive soup, after several months of producing only “punishment soup,” as my beloved SO refers to them.  I will not include any photos of punishment soup – suffice it to say that putting red cabbage and potatoes together in a soup does not lead to an appealing meal.

My recipe today came from this website.  The most interesting ingredient is the cinnamon. I have no idea what cinnamon is doing in a tomato based soup.  I can’t tell if I can taste it or not, but it smelled nice when I put it in.

Tasty!

Tchai was pretty sure that the pancetta was meant to be for him, but when I gave him a piece he ended up abandoning it on the carpet.  I should have predicted this, but his little cat face was so sincere.

Ingredients:

olive oil
3 oz pancetta, chopped
1 Spanish onion, chopped
4 garlic cloves
1 peeled, cubed acorn squash
6 small peeled, cubed red potatos
1/2 c diced carrots
1 tsp dried basil
1/4 tsp cinnamon
1/4 tsp thyme
1 (14 oz) can diced tomatoes
4 diced Campari tomatoes
1 qt chicken broth
4 c chopped kale
1 (15 oz) can butter beans

Preparation:

Heat olive oil in a deep pot over medium-high heat.  Add pancetta and stir until it releases its fats and starts to smell nice.  Add the onion and garlic, and saute until the onions go translucent.  Add squash, potatoes, carrots, basil, cinnamon, and thyme.  Cook for 4 minutes, stirring occasionally.

Add tomatoes, cook 2 minutes.  Stir in the broth.  Bring to a boil, then simmer 10 minutes.  Add kale and simmer until it relaxex (about 5 minutes).  Add the beans, and simmer until potatoes and kale are tender (5 minutes).