On an accelerated intro to proofs course: the good, the bad, the ugly, and the good again

O, the bittersweet feeling of the end of the school year. So much happiness to be done with the exhausting cycle of teaching and grading, but so sad to see many students graduating and leaving your classes. These last five weeks were particularly brutal and particularly rewarding in different ways, since I was teaching a course that we affectionately refer to as “Math Camp”.  Today is the last day of this intense experience, and I wanted to share some thoughts on the pros and cons of teaching such an accelerated and focussed class.

Bates has an atypical schedule: two 12-week semesters in the Fall and Winter, and a 5-week Short Term from mid-April to the end of May. Other small liberal-arts schools do a short term in January, but they usually don’t teach “serious” classes. And even though not all of Bates’ Short Term classes are created equal, many are plenty serious, including Math Camp, aka Introduction to Abstraction. This is our introduction to proofs course, and a requirement for the math major and other upper level courses. We meet Monday through Thursday for five hours a day (9am to noon and 1 to 3pm), and on Fridays we have an exam. We have changed content and texts a few times, but for the last few years we have been using Carol Schumacher’s “Chapter Zero“, which I personally really like. This is the only class we team teach (three instructors plus two student assistants) which helps with the workload, but we also allow anyone who wants to enroll to take the class (we had a class of 44). Sound intense? It is! Both instructors and students are a little burned out by the end. So here is my breakdown of the good, bad, and ugly parts of the experience.

  1. The good: I am a firm believer that the best way to learn mathematics (especially the difficult kind) is to do mathematics. In that sense, Math Camp is great. This is the only class students are taking, so they can devote their full attention to learning to write proofs. More contact hours also means that we can help them through problems, and we actually assign very short homework assignments (the idea is that they’re working enough during the day already). They work in groups which we mix up daily (and sometime even from morning to afternoon), and there is little to no lecturing. All of these contact hours also bond the class as a whole (some of whom will be taking many more math classes together in the future) and  also brings the students closer to the instructors teaching it, and is a great way for us to get to know the students. It is a little bit like a language immersion class, but where the language is replaced by math. We also teach them LaTeX and a little bit of programming using Maple. They love that part of the class, and I think it’s incredibly useful and the perfect time for them to learn these skills. I also think that overall this class works the way I like my own semester-long classes to work. It is inquiry-based and collaborative, and in a way we are doing exactly what we want, which is that we are teaching our students how to do mathematics the way mathematicians do mathematics.
  2. The bad: The mood can turn very quickly. Especially last week (the end of week four), there was a lot of grumpiness and frustration among the students. I get it! They are spending the first few weeks of nice weather in Maine cooped up in a classroom writing difficult proofs. They are learning tons of new material in a short amount of time, and they may not have enough time to retain things. Because they are spending a lot of time doing difficult things, they constantly feel inadequate and don’t even realize all of the progress they are making. Even though we rig the class so that it is not the difficult to pass if you are working hard, coming to class, and participating, the scores on the exams can be a little low (by their standards — I don’t think a median that hovers around 80 is bad). I was thinking of writing this post last week, but my mood had soured too and so I decided to wait. I am glad I did because things went much better this last week. I think they all started to appreciate how much they had done and learned, and even though doing difficult things can be frustrating, it can also be very rewarding.
  3. The ugly: I worry that this style of class can be great for some students, but maybe not so great for others. In particular, the intensity and competitiveness that arises from an intense environment may not be the best place for someone who already feels inadequate or like they “don’t belong”. Had I been a student in this class, for example, I might have felt intimidated by the more vocal people (most commonly the white males, but not exclusively). If that was combined with my inability to take time to digest things on my own and low exam scores, I might have decided the math major was not for me. I had conversations with students near the end, and most of the students who mentioned that they might not want to major in math anymore were women or students of color (but not the International students, they were at home in this type of class for some reason). This made me wonder: is this a product of the type of class we’re teaching? By the way, I think I should mention AGAIN that student-centered classes are definitely my thing. I believe that these types of classes are more inclusive, more conducive to learning, and more rewarding of hard work and perseverance. But somehow the condensed nature of this particular class makes me think that we might not be serving all of our students as well as we could.

But going back to item 1 above, when this class works, it works extremely well. And I believe that for the most part it worked. We did a couple of things to adjust the course that I think worked very well. For example, we made it so that the average participation score was 100% (reducing the incentive for the people who talk too much, and increasing the incentive for the quiet ones). We also had them write an extra credit assignment were they reflected about their accomplishments and learning gains. Not everyone did this assignment, but from what I read I saw that people really were thinking carefully about their learning and that the mood was not as negative as I thought. Finally, for the most part, everyone did really well in the class. I honestly believe, if they decide to go forward, that they are now prepared to take more advanced proof-based courses. I won’t know who declares a math major until the end of next year, but I hope that the doubts they had will be dispelled once they get some well-deserved R&R this Summer.

Item 3 I think is way more complicated than just this class. I think a lot about the issues Math and other STEM fields have in attracting and retaining diverse students. I don’t have many answers, and I think we all need to keep thinking about how best to serve and teach everyone, not just the students who have been traditionally successful in mathematics. Anyway, that is probably enough for this post.

We did have fun, though, and they were a great and committed group of people. I hope I see them in my classes in the future.

Math camp students at work! The are happy because it's the last day of class and we brought donuts.

Math camp students at work! The are happy because it’s the last day of class and we brought donuts. (Photo courtesy of my fellow instructor Katharine Ott.)

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3 Responses to On an accelerated intro to proofs course: the good, the bad, the ugly, and the good again

  1. Avatar Mitch Keller says:

    Having just survived a similar experience at Washington and Lee, I found this perspective interesting and mostly in line with my experience. I’m writing up some blog posts of my own detailing my experience in greater depth and will come back and share a link when I’ve got more written. However, I’ll contribute a bit to the discussion now.

    One thing I’m very conscious of when putting students into groups is gender. (Our campus is very white, so racial/ethnic diversity is rarely something I can do much about.) I work hard to form groups of three that have two women and one man (or groups of four that are half and half). Obviously I sometimes have parity issues, but I got lucky with my intro to proofs course. (Two hours per day, five days per week, for four weeks this time around.) I found that having the women grouped in pairs with one man added to their group allowed them to feel comfortable with what they were doing. Even when grouped with a man with a strong personality, they would stand up to him. However, experience in the past has taught me that even a very strong personality woman student put with two men will tend to shut down and defer to them. I also found it helpful to group the students so they were about at the same ability level. (I changed groups weekly, but I also only had 14 students. There were two other sections with 15 students each.) By grouping by ability, I was able to give the strongest students the hardest proofs (or parts of proofs) and the weaker students could have something where they would still accomplish things but not feel overwhelmed.

    I saw a remarkable amount of persistence from the women in the class. I allowed lots of revision to homework (and tests), and there were lots of positive comments about that. I think with the break-neck pace of an accelerated course like this, students in general but women in particular benefit both in attitude and intellectual development by knowing that they can work to get something right. I think I probably looked at 10 drafts of one woman’s proof that the union is the least upper bound of a subset of the power set (and the intersection is the greatest lower bound). In the end, she had a very nice proof.

    As I said, more details to follow with links here in coming days. Perhaps we need a support group for those of us who teach a class like this!

  2. Avatar evelynjlamb says:

    Wow, that is intense! 44 students 5 hours a day! I think you’re right to be concerned that a pressure cooker environment like that might be harmful for some students. I wonder how much resistance there would be to developing a semester-long version of the class that allowed students more time to process and digest information.

  3. Avatar Kevin O'Bryany says:

    We do it because we’re human, but we shouldn’t draw conclusions from 44 students, especially about performance of subgroups. The sample is not large, the underlying distribution is not normal, the 44 are not a simple random sample, and they weren’t assigned to groups randomly, the protocols of the experiment changed during the experiment. Basically, from a science/statistics point of view, everything that could go wrong has.

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