{"id":188,"date":"2023-10-30T16:33:56","date_gmt":"2023-10-30T20:33:56","guid":{"rendered":"https:\/\/mathvoices.ams.org\/teachingandlearning\/?p=188"},"modified":"2024-01-29T09:43:38","modified_gmt":"2024-01-29T14:43:38","slug":"active-learning-in-graduate-mathematics-courses","status":"publish","type":"post","link":"https:\/\/mathvoices.ams.org\/teachingandlearning\/active-learning-in-graduate-mathematics-courses\/","title":{"rendered":"Active Learning in Graduate Mathematics Courses"},"content":{"rendered":"

November 2023<\/p>\n

Carolyn Abbott<\/a>\u00a0(Brandeis University)<\/p>\n

I\u2019d like to start a conversation about how we teach graduate classes.\u00a0 In the last several decades, there has been a lot of discussion about active learning, in all its many forms, in mathematics classrooms from early elementary through college.\u00a0 There is generally a consensus among those who care about math education that engaging students actively in their own learning leads to better outcomes.\u00a0 Students take ownership over the material, think more deeply about the subject, and make more complex and nuanced connections between topics.\u00a0 Of course, not all educators have embraced active learning at those levels, and I don\u2019t mean to imply that it\u2019s always implemented perfectly.\u00a0 The conversation is ongoing and should continue to be so.<\/p>\n

Graduate education, however, is largely left out of the conversation.\u00a0 A standard graduate class is still a lecture.\u00a0 It\u2019s as if we, the larger mathematical community, think those that are strong enough to get into a PhD program should not \u201cneed\u201d active learning in their classes.\u00a0 The research shows that an actively engaged classroom encourages everyone<\/em> to engage more deeply with the material, strong and weak alike.\u00a0Certainly, graduate students can<\/em> learn the material in courses taught in the traditional lecture format; they\u2019ve been doing it for generations.\u00a0 But wouldn\u2019t they learn the material better<\/em> if they engaged more deeply with it, in the classroom as well as when working on problem sets?<\/p>\n

As a research mathematician, I think about how I engage with material I am learning, say a paper I\u2019m reading, one whose techniques or results I\u2019m trying to generalize.\u00a0 I read it, not necessarily linearly, but again and again, until I understand what the authors are doing.\u00a0 But I don\u2019t stop there.\u00a0 I engage more deeply with the text, questioning each hypothesis and each step: is the hypothesis truly necessary, or just necessary for this particular method of proof?\u00a0 Is there an alternative way to approach the problem or step?\u00a0 What\u2019s the intuitive, or \u201cmoral,\u201d idea behind the proof?\u00a0 Are there connections to other, seemingly unrelated, results I\u2019ve read?<\/p>\n

Increasing one\u2019s level of engagement with the material is one step towards becoming a research mathematician.\u00a0 Anyone who has advised PhD students knows that, for many, perhaps most, students, this change does not happen overnight.\u00a0 It is a process, often a long and difficult one.\u00a0 If this is where we want our graduate students to end up, why aren\u2019t we teaching them these skills from the very beginning, in the classroom?<\/p>\n

I have some ideas about why this is, which I\u2019ll get into later, but first let me tell you what led me to start thinking about active learning in graduate classes.\u00a0 I taught middle school and high school mathematics for five years in New York City, and during that time I embraced active learning in my classrooms.\u00a0 When I started graduate school and was a TA, I continued using it in the recitations I ran.\u00a0 I even trained entering graduate student TAs at University of Wisconsin in active learning methods.\u00a0 I don\u2019t mean to imply that I\u2019m an expert in implementing active learning in my classrooms \u2013 I still have a lot to learn.\u00a0 But it is something that I believe in, and something that I am used to incorporating into my classes.<\/p>\n

So I was surprised when I looked back on my first year of teaching graduate courses after becoming an assistant professor and realized that I basically lectured for the whole year. \u00a0I lectured; the students listened and took notes.\u00a0 There were questions from time to time, but the room was often quiet.\u00a0 Too quiet.\u00a0 I missed the activity, the discussions, the liveliness of a classroom where students were actively engaged with the material.\u00a0 What happened?\u00a0 Why did my whole teaching philosophy seem to go out the window when I started teaching graduate classes?<\/p>\n

Last summer, I spent time thinking about how to revise Algebraic Topology I, which is a required course for first-year graduate students.\u00a0 Incorporating active learning techniques presented real challenges, many of which felt more complex than when teaching an undergraduate class.\u00a0 Some of these are real, some (I believe) simply require a change in mindset, and none are insurmountable.<\/p>\n

    \n
  1. Student buy-in. Learning via lecture is comfortable for students, perhaps because it requires less engagement.\u00a0 Other first year graduate courses, as far as I know, are being taught in a standard lecture format.\u00a0 I worried that my class would be the odd one out, and not in a good way.<\/li>\n
  2. Complex material. Problems at this level are usually long and difficult!\u00a0 Worksheets, an easy and effective fallback for undergrad classes, are not a good option for a grad class.\u00a0 Even most examples (with some notable exceptions, like first applications of van Kampen\u2019s Theorem) are too long and complex to expect students to work through them in 10 minutes.<\/li>\n
  3. Lots of material. This is a challenge at every level, but the higher the level of the course, the harder the challenge seems. \u00a0There is just so much to get through every semester.\u00a0 \u00a0I did not see how to find time for interactive learning.<\/li>\n
  4. My role as a teacher. While students can read the book, the concepts and proofs are difficult and harder to follow than in undergraduate texts.\u00a0 This is somewhat counterbalanced by students being more sophisticated readers of mathematics, but reading the book is still quite challenging. \u00a0Even more than when teaching an undergrad class, I see my role as the go-between between the text and the students.\u00a0 I\u2019m there to explain, to draw the key pictures, to give the intuition behind the steps, to give the big picture, to explain the history.\u00a0 I worry that spending time on active learning will take away from this role.<\/li>\n<\/ol>\n

    In the face of these challenges, I decided to start small.\u00a0 While I always approach summer thinking it contains endless free time, the reality is quite different. There are grant proposals to write, conferences to attend, research to catch up on.\u00a0 Finding time to completely rework the course was unrealistic.\u00a0 My strategy, then, is to take small steps: each time I teach this course, I will \u00a0make a few small changes to encourage active learning.\u00a0 Maybe after I\u2019ve been doing it for 20 years, I\u2019ll have it all figured out.<\/p>\n

    The two small steps I took this year are both techniques I use when teaching undergraduate classes: \u00a0encourage discussion and questions from day one, and start each class with a problem.\u00a0 I know it doesn\u2019t sound like much, but it\u2019s what I\u2019m able to do right now.<\/p>\n

    Before starting to teach high school, I was told to start the school year the way I wanted it to continue.\u00a0 The advice was given in the context of discipline: while one can easily become more lenient as the year goes on, it is almost impossible to become stricter.\u00a0 But the advice applies in almost every aspect of teaching, as well as life, and I think of it often.\u00a0 If I want to have a lively classroom filled with discussion and debate, I need to model and encourage that from day one.\u00a0 During the first week of class, I went out of my way to say that I am happy to be interrupted with questions, and I paused more often than I usually would to ask if there are questions.\u00a0 I reguarly asked if the class understood.\u00a0 This typically resulted in stares and silence, and so I would follow up by asking if someone could explain it in their own words, and then waiting.\u00a0 Waiting until someone spoke up with an explanation or question. I wanted my students to know that when I asked a question, it was genuinely a question; I was not going to answer it myself.<\/p>\n

    Starting each class with a question or problem has made a bigger difference than I expected.\u00a0 It has not always been easy to come up with a question, and, to be honest, some days I don\u2019t manage it.\u00a0 When I do, though, the question(s) I choose either foreshadow what we will do that day or reminds students what we did the previous class.\u00a0 For example, one class we defined what it meant for a loop to be nullhomotopic, and at the beginning of the next class I asked the students for a straightforward example of a non-contractible space in which every loop is nullhomotopic. \u00a0This resulted in a lively debate, including about what counted as a \u2018straightforward\u2019 example.\u00a0 A different class ended in the middle of a proof, so I started the next class by sketching what we had done in the proof so far, but with some steps missing.\u00a0 The question was to fill in the missing steps. Once I asked for the definition of a homotopy, which we covered in the previous class.\u00a0\u00a0 In today\u2019s class, we used van Kampen\u2019s Theorem to reprove that the fundamental group of a torus is Z^2.\u00a0 To start the next class, I\u2019ll ask them to generalize this to a closed surface of genus g, and then to come up with a different way to apply van Kampen\u2019s Theorem to those surfaces.<\/p>\n

    As can be seen from the examples, the level of difficulty of these questions varies widely.\u00a0 Some simply ask the students to look back in their notes.\u00a0 These serve the purpose of reminding students what we were doing in the last class.\u00a0 I like to use this kind of question on Mondays, since it\u2019s been several days since we last met.\u00a0 Some are much more complicated, like the application of van Kampen\u2019s Theorem.\u00a0 I\u2019ll probably give them about 10 minutes to work it out, and I hope they will be eager, or at least willing, to discuss it with each other during that time.<\/p>\n

    So how active is my class this semester?\u00a0 Well, not as active as I\u2019d like, but so far, it\u2019s more active than last semester.\u00a0 Incremental changes lead to incremental results.\u00a0 Students seem to be more willing to ask and answer questions, particularly ask questions. And I enjoy starting class by hearing a couple of student voices.\u00a0 At the end of the semester, I\u2019ll look back on the changes I made and think about how I can engage the class even more.<\/p>\n

    There is nothing groundbreaking about the techniques I chose to implement.\u00a0 They are standard, even expected.\u00a0 They\u2019re certainly not the only way to make a class more active, and I don\u2019t know that they\u2019re the best way.\u00a0 They\u2019re simply what I was able to do this semester.\u00a0 I\u2019d like to hear what other professors are doing and how graduate students are reacting.\u00a0 To get new ideas and learn what has worked and what hasn\u2019t.\u00a0 To push my boundaries as a teacher of graduate students. \u00a0I want to start a conversation.<\/p>\n

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    November 2023 Carolyn Abbott\u00a0(Brandeis University) I\u2019d like to start a conversation about how we teach graduate classes.\u00a0 In the last several decades, there has been a lot of discussion about active learning, in all its many forms, in mathematics classrooms from early elementary through college.\u00a0 There is generally a consensusRead More →<\/a><\/span><\/p>\n","protected":false},"author":31,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"advanced_seo_description":"","jetpack_seo_html_title":"","jetpack_seo_noindex":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_post_was_ever_published":false},"categories":[1],"tags":[],"class_list":["entry","author-cabbott","post-188","post","type-post","status-publish","format-standard","category-uncategorized"],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/mathvoices.ams.org\/teachingandlearning\/wp-json\/wp\/v2\/posts\/188","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathvoices.ams.org\/teachingandlearning\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathvoices.ams.org\/teachingandlearning\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathvoices.ams.org\/teachingandlearning\/wp-json\/wp\/v2\/users\/31"}],"replies":[{"embeddable":true,"href":"https:\/\/mathvoices.ams.org\/teachingandlearning\/wp-json\/wp\/v2\/comments?post=188"}],"version-history":[{"count":9,"href":"https:\/\/mathvoices.ams.org\/teachingandlearning\/wp-json\/wp\/v2\/posts\/188\/revisions"}],"predecessor-version":[{"id":227,"href":"https:\/\/mathvoices.ams.org\/teachingandlearning\/wp-json\/wp\/v2\/posts\/188\/revisions\/227"}],"wp:attachment":[{"href":"https:\/\/mathvoices.ams.org\/teachingandlearning\/wp-json\/wp\/v2\/media?parent=188"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathvoices.ams.org\/teachingandlearning\/wp-json\/wp\/v2\/categories?post=188"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathvoices.ams.org\/teachingandlearning\/wp-json\/wp\/v2\/tags?post=188"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}