Whole Class Discussions
Facilitating instead of Evaluating:
I believe that students should not only create their own mathematical ideas but also gain the confidence to evaluate for themselves whether their reasoning is correct or not. It is my job to support them in gaining that confidence instead of evaluating their thinking myself.
This semester we are video taping our IBL classes and as I am watching the videos I am reflecting (again) on all the pieces necessary for a productive whole class discussion. My goal for a discussion is to make the “Big Mathematical Ideas” visible by having students construct connections between different solution strategies or attempts. (For more goals of whole class discussions, download Goals for Whole Class Discussions)
I will try to organize the context and some of the key pedagogical ideas in a list so that it is easier to see the structure:
- Group Work: Groups of about 4 students work on the investigation(s) and find examples, conjectures and proof ideas (see the blog Our Inquiry-Based Classroom).
- During the group work I walk around, observe and support the groups using careful questioning (see the blog The Art of Asking Good Questions). My hope is to have groups present different solutions and so I will use my questions to support the groups in pursuing and completing their different strategies. It is amazing to see that for every problem there are at least two very different approaches in the class!
- While I am observing the groups I am already thinking about which groups I want to ask to share out and which order would work best. Sometimes it helps to start with a strategy that did not succeed and the other groups can add on. Sometimes it is helpful to compare and contrast two very different approaches.
Sometimes I let only one group share out and ask the other students to decide if they agree or disagree. Sometimes I ask two groups to share out that have contradicting results.
- The word share is a bit misleading since there is way more happening than just one group sharing their thoughts while the other students are listening (or not?). The goal is to get students to discuss the strategy that was shared, and compare it to their own thinking. During the discussion I am often using talk moves that I learned from Suzanne Chapin’s book “Math Talk” (Five Talk Moves).
- Wait time. Doing nothing is often the most difficult for me. It also has the most amazing results.
- Revoicing. I repeat what a student said in my own words or I have another student repeat what a student said. I do this all time; it is fascinating to see how much better students listen to each other if I make clear that listening is important to us.
- Agree or Disagree. After a student or group make a statement I can ask the other students if they agree or disagree and why. Rather than making someone “right” or “wrong” this provides more neutral language for discussing a mathematical idea. I want the whole class to decide if something is correct or not (instead of me). I only jump in (with a question) if all the students agree with an incorrect statement.
- Adding on. When a discussion stops or seems finished I like to ask if anyone wants to add on. This opens the conversation up to either more mathematical ideas or a reflection on what just happened. It certainly makes the discussion richer for me.
- After a bigger investigation (for instance a proof) is completed, all students write up their thinking as graded homework. These are real essays, not just lists of computations. See Proof as Sense-Making for some examples of student work.
Videos: Talk Moves in the Classroom
In practice, several talk moves and other pedagogical moves can be used together. The following videos from our classrooms illustrate some of our moves:
Video 1: "Revoicing & Recording"
The students are trying to prove why the finger trick of determining the multiples of 9 works.
We see one student explain her thinking. I ask if someone can repeat her reasoning. As the second student revoices, I silently represent her thinking on the board. Notice that I do not evaluate in either case whether the thinking is correct or complete. Instead I continue with a new question that connects this idea back to the original problem.
Video 2: "Repeat or Ask Questions"
Students are investigating Salsa Rueda dancing. The question is how to predict if you will dance with every leader/follower given $n$ pairs and only dancing dame $k$ (Salsa Rueda chapter in the Dance Book).
Prior to the scene in the video, I asked a student to share her conjecture with the class because she had done the greatest number of examples and I knew that her work would be a great resource for the class. We see the student explain her conjecture. I then ask whether somebody in the class can repeat her conjecture or ask her a question, which leads to an interaction between the students.
Notice that I am not involved in this interaction other than gently suggesting a particular example to look at.
Video 3: "Wait Time & Agree/Disagree"
For homework, students were trying to find an equation that connects two whole numbers, their least common multiple and their greatest common factor.
The video begins with one student sharing her findings. I revoice and represent her thinking. In her explanation, the student has made a distinction between the larger and the smaller number. Even though I know that that is unnecessary, I go along with her thinking. Another student asks "Does it matter?" and again I give that question back to the class without evaluating.
Then I'm asking other groups to agree or disagree with the first conjecture. I gave some wait time, then ask for questions and examples. The next student shares his equation, which is a minor (from my point of view) algebraic variation of the first equation. Again I go along with the student's suggestion. Later on, comparing the two versions leads to the students understanding (independently of me) that the distinction between larger and smaller number does not matter.
Video 4: "Show of Hands"
Students have been exploring the $3a + 5b$ problem: Determine all possible values generated by the Diophantine equation $3a+5b$ when $a,b ≥ 0$
(see the blog $3a+5b$ Proofs which contains a complete video of her explanation).
Even though the proof is still incomplete, I comment that the proof idea is beautiful. Next, I ask: Does it make sense to you? Are there questions? Normally I would give wait time here but since class is almost done, I use a show of hands to monitor understanding.
References and Resources
Chapin, S. H., O'Connor, M. C., & Anderson, N. C. (2009). Classroom discussions: Using math talk to help students learn, grades K-6. Sausalito, Calif: Math Solutions.
A summary of Chapin's "Talk Moves" is included here for reference.