Discussions in Introductory Quantitative Courses
When I teach introductory math courses, students sometimes tell me that what they like about math is that the answers are unambiguously right or wrong. That’s not the impression I want my students to have: like any other subject, the most interesting parts of math are the more ambiguous questions about why things work and how to think about them. I wanted to incorporate class discussions to bring those less clear-cut parts of the subject back to the course, but the perception that math courses always focus on objective material—and the other design choices about a course that lead students to believe it—was a challenge; no one gets much out of a discussion when everyone knows the professor will announce the one true answer at the end.
The first issue to resolve was figuring out what I wanted class discussions to be about. One approach was to turn some of the more difficult problems from the homework into topics for class discussions. That required some modifications—I quickly found that the very features that make a good homework problem often make a poor discussion topic. I’ve spent a lot of time carefully streamlining homework and exam problems, making sure they can only be interpreted one way so that students don’t waste time or miss the point over a misunderstanding. Taking those problems as written, there isn’t much to discuss.
The questions that come up outside of a classroom (or even in classes in other subjects) are never as cleanly formulated, however. There’s always some judgement and discretion in taking an actual situation and figuring out how the techniques from class apply to it. The benefit of rethinking the course format isn’t just covering the same content better; it’s an opportunity to revisit topics that have slipped out of the curriculum. Instead of always protecting students from that ambiguity, discussions gave us a chance to confront it head on.
Not every question I tried led to a good discussion. Some of our hardest homework problems require coming up with the right trick to get started. In class, that just meant that the few fastest students got a chance to shine, while the rest had no way to participate. A better choice was problems that invited lots of different attempts, but require a lot of tedious work when students try to identify the right approach at home; I could act as a calculator, letting them engage and trial and error much faster than they could on their own, but still demanding that the ideas come from the students.
The best questions I found, however, came when I looked again at the topics that don’t get enough attention in lectures. These were better questions precisely because they can’t be packaged neatly into homework problems. There are times when what I really want to do is to tell my students is “go home and spend 30 minutes meditating about this definition”—to go off on their own to ask and then answer for themselves lots of easy questions about it. Trying to distill that into homework misses the point—answering some list of questions I give them won’t teach the habit of asking questions on their own. Facilitating a class discussion was a more structured way to get students to do this: it gave them a space to ask those questions and answer them for each other. The discussion had to start with something a bit more directed than just a topic; I usually came in with a list of two or three questions I wanted to be sure were asked, and opened by asking the first one. My students took it from there, addressing that and then moving on with their own ideas and responses. More often than not, the questions on my list, and many more, got asked without further prompting.
A second issue was how to organize the discussions. In large classes, having a single, instructor-led discussion for the whole class isn’t always going to work well: 120 students can’t all have a say on a given topic, let alone have a chance to respond to each other. Instead, the natural thing to do was to have students first discuss things in small groups, and then have some of those groups summarize their conclusions for the whole class.
The variation among groups was noticeable—the students in the back of the classroom weren’t having the same quality of discussion as the students in the front, and students tended to work with friends (and get distracted) and to self-segregate by gender and race. When the discussions were a small part of the class, that was imperfect but acceptable: it wasn’t worth the disruption of trying to assign seating or move people around to make better groups.
However if I wanted to make the discussions a bigger portion of the class, I’d incorporate the lessons I learned making groups when I taught a Structured, Active, In-class Learning class (SAIL) that is, a partially flipped course. Random assignment worked fairly well, but I found I could get a big improvement with just a small amount of fiddling. Most groups I created randomly worked well on their own, and the ones that didn’t tended to stem from a small number of students who did much better when put in the right groups—shyer students who benefitted from being matched with people who wouldn’t talk over them, or less focused students who did better in a group that was determined enough to keep them on track.
One surprise was how well it worked to group students loosely by how well they were doing in the course. Over a series of group assignments over two semesters, I went from assigning groups where students had a wide range of test scores (the approach sometimes summarized as “one A student, one B student, one C student”), to having groups where the spread was much narrower. It was heartening to see several “C students” go from sitting quietly hoping no one would notice they weren’t keeping up with their group, to having plenty to say when I put them in a group together. The group of “C students” might go a bit more slowly, but it gave everyone in the group a chance to contribute, and they all did have plenty to contribute when other students weren’t getting to everything before them.
The biggest mistake I made was thinking that discussions could stand by themselves as an activity—that I could let students hold most of the discussion in their groups, make sure that the most important points got said in the whole-class part of the discussion, and move on immediately to the next topic. My students were much happier when I wrapped things up with some sort of conclusion, helping them sort out how they should think about the activity and what they should take away from it. Even when all I did was highlight a few things students had already said, getting the professor’s imprimatur made a big difference in their confidence about what the main points were.
For those of us taught and trained in lecture courses, interactive content in our courses means a lot of new issues to be considered, and it inevitably takes some trial and error to find the right approach. The reward, however, is a new tool for getting at the parts of our subject which can’t just be marked right or wrong: for putting back the more interesting, less objective, questions about “why” and “how” back even in introductory courses.
Henry Towsner is an assistant professor of mathematics in the School of Arts and Sciences.
This essay continues the series that began in the fall of 1994 as the joint creation of the College of Arts and Sciences, the Center for Teaching and Learning and the Lindback Society for Distinguished Teaching. See www.upenn.edu/almanac/teach/teachall.html for the previous essays.