contributed by: Luke Wolcott
go back to category: Dialogue

Summary


A role-playing partner activity. After working independently on the same problem, one student explains their work while the second listens; next the second re-articulates the explanation; finally they compare their work. Based on a dialogue exercise done during a weekend workshop with Dan Barbazet, Mirabai Bush, and Rhonda Magee at the Omega Institute in August 2014.

Class information


I've done this exercise in five different courses at Lawrence, a small liberal arts college: Applied Calculus I and II; Calculus II and III; and Foundations of Algebra, the intro group/ring theory course. We have 3 ten-week terms each year; courses meet MWF for 70 minutes. The Applied Calculus students are often planning to be psychology, economics, or other non-science majors; they aren't inclined to like math much, but they are very good communicators and they responded particularly well to this activity.

Logistics


In the Calculus courses, the activity was done once a week, and was part of a weekly quiz. The other courses were more ad hoc, averaging once every two weeks probably. The activity takes 10-20 minutes, longer the first few times because the instructions are somewhat complicated. Students seemed to "get it" and buy in (mostly) by the third time.

Description of activity


I explain to the students that we will do an activity about explaining and, more importantly, listening. I make it clear that this is not just a conversation, but a role-playing. While I write three or four problems on the board, they are told to find a partner to work with. Then the pair decides, together, which of the problems to work on, ideally making a list, e.g. "We'll do Z first, then X, W, and Y."

The students then work independently on their chosen problems, for 5-8 minutes. If they finish the first one on their list, they move on to the next one. While they work I write (more or less) the following steps on the board, and when they finish I explain the steps.

Steps:
  1. Choose who will explain first (Person A), and who will listen (Person B).
  2. Person A now has 1-3 minutes to explain what they did on your first problem (e.g. Z). Imagine that Person A is a tutor, or the teacher. If you got stuck, explain what you could do, and what you were trying to do. Person B listens in an engaged way, carefully and silently.
  3. When done, Person B re-articulates Person A's explanation. "So what you did was... and then you..." Do not correct mistakes or add your own comments, only work with Person A's explanation. Aim for the big picture. During this time, Person A listens in an engaged way, carefully and silently.
  4. When done, you are free to discuss the problem together, in a conversation. Person B can explain what they did, or Person A can correct mistakes they may have realized while explaining. Generally have a back-and-forth about the problem, until you feel that you both understand it and could explain it correctly.
  5. Repeat this process with the second problem on your list (e.g. X), but switch roles of explainer and listener. If you need time to work independently, do this first.

Comments


It was helpful if I modeled the role-playing steps, before asking them to do it. Step 3 is the most confusing, at first.

Letting the student pair decide which of the problems to work on gives them the chance to break the ice a little and build some camaraderie, and also have some agency in what they work on. While they're working independently, they know they have a buddy, and are already solving their problem in the wider context of knowing they will then be explaining, or hearing an explanation, about it. This encourages metacognition while they work. Having several problems to work on means that they can work at their own pace and everyone has plenty to do.

Letting them choose who will explain and who will listen, the first time, gives some flexibility in case one student is really anxious or struggling.

When the time comes to explain and listen, the students seem to go along well with the role-play. They are both familiar with the problem, and relatively comfortable with each other. The explainer takes their job seriously, and the listener is focused and attentive. The more structured the conversation -- separating who is speaking and who is listening -- the more space the students have to reflect on the pedagogy. They realize that they are teaching each other not just the mathematical content, but also what makes a good mathematical explanation.

Students were encouraged to differentiate between small mistakes (snags) and big mistakes (real confusion). By listening to an explanation without immediately stopping to correct small mistakes, students could focus on the big picture of "What I'm trying to do here is..." or "What I would do next, with this, is...".

The activity can be adjusted to be long (30 minutes) or short (10 minutes). Pairs can complete one cycle, discussing only one problem, or they can repeat again and again until all problems (W, X, Y, and Z) are done.

Usually around Step 4, the role-playing devolves into a simple conversation, albeit a lively one about the problem at hand. I see this as a welcome phase of the activity. If repeating, it may be important to reiterate the importance of returning to role-playing for Steps 2 and 3.

Background/Theory


As mentioned above, this was inspired by an activity facilitated by Dan Barbazet, Mirabai Bush, and Rhonda Magee at a weekend workshop at the Omega Institute in summer 2014. That activity was very prescriptive, with a bell to mark time for each step. My experience of that activity convinced me that being more flexible with timing would help. The steps above are designed to let students go at their own pace, and also allow time for the natural collapse into dialogue.

The Insight Dialogue method, developed by Gregory Kramer, could easily be woven into this activity. Depending on the students, that process -- Pause, Relax, Open, Trust Emergence, Listen Deeply, Speak the Truth -- could deepen the exchange.

Variations/Related Activities


In two courses this was done as part of a weekly quiz. The students were told to read a few (1-3) pages of the textbook, and focus on understanding a particular example in the text. The quiz was a single problem, almost identical to that textbook example. After working independently on the problem for 5-8 minutes, the students paired up and did this Listening and Explaining activity. They were graded out of two points, and the quizzes counted for 10% of their course grade. These basically amounted to participation points; if students were in class and engaged in the activity, and from their work seemed to have read the text pages, they received two points.

Aims


  • To practice formulating good mathematical explanations, actively
  • To practice formulating good mathematical explanations, receptively, by listening and following a classmate's explanation
  • To develop camaraderie by watching fellow classmates struggle to articulate their understandings, and by having a chance to look over each others' work
  • To differentiate between small mistakes and big mistakes, and practice seeing the big picture of the problem in spite of small snags.
  • To get more practice working on specific math content

Feedback/Assessment


Overall students seemed to enjoy this activity. It took some practice and some repetition before they bought into it, but once they did, they seemed to appreciate the educational goals. More importantly, they seemed to have fun (particularly in Step 4 when the role-playing devolved into conversation, and they often devolved further into gossiping). This was especially the case with my Applied Calculus students, who were non-math majors and very good communicators; this activity would often continue for 30 minutes in a wonderful chaotic active learning environment.

Non-contemplative variations


Some pairs of students seemed to ignore Steps 2 and 3, and simply discuss, conversationally, the problem they had just been working on independently. This is still dialogue and still useful for building understanding, but it seemed to lack the additional level of reflecting on what makes a good explanation.