by Chris Robinson, The Blake School
It is no secret that every class is different. The school community can have a large influence and a single student can sway things positively or negatively. I’m most successful at establishing a cooperative classroom when (1) I am myself, (2) my students can be themselves, (3) students see themselves as a learner of mathematics, (4) the physical layout of the classroom expects conversation and (5) the mathematics is approachable and scaffolded properly.
(1) I am sarcastic. I quote movies and eagerly wait to see which students crack a smile. When I want to encourage students to change perspectives, I will stand on a chair and look at the board upside down to focus on a different base and altitude in a triangle. I chide students, and use insults that make no sense, “Mr. Robinson, isn’t that an isosceles triangle?” “Your face is an isosceles triangle.” I treat the job like it is 10% bad standup and 90% math education. I want students to laugh, and take risks comically, in hopes that they will soon take risks mathematically. This is who I am. You are not me, although I hope you have fun with your students everyday. I keep doing it because it is working for me. Students do take risks and I hear many different voices each day. My students see me as human. I make calculation errors in front of them without embarrassment, and if a student asks a question that I don’t know the answer to I tell them so, and say “let’s look it up,” or “I will get back to you about that tomorrow.“ I expect my students to have a growth mindset and I try my best to model one myself regarding my teaching and my own knowledge of mathematics. (If you are not familiar with Carol Dweck and the growth mindset, view this video immediately before continuing to read.)
(2) Despite my meaningless insults I expect students to respect both me and others in the classroom, so that students can take risks and be themselves. I make it clear in the beginning of the year, and call out students (or myself occasionally) on going too far with a joke or impatience with another student’s appropriate questions, my classroom needs to be a space where all levels of questions are accepted and we can learn and relearn together. I expect that students will listen to each other, and will remind them “don’t be the first person to speak at your table this time.” When I assign new groups I ask students to reintroduce themselves to their classmates, to make sure they know the names of the people they are working with.
(3) I make mistakes in the classroom because I expect students to make mistakes in the classroom. We learn from mistakes. If all of my students are getting an investigation question correctly than I didn’t challenge them enough, and if no one is, than I didn’t provide the appropriate level of scaffolding. All students need to feel like they can accomplish the tasks before them (again see Carol Dweck). This may involve confidence-boosting questions like, “What is an answer you know is too small, too large?”
(4) The physical layout of your classroom should encourage talking. If you want students to collaborate, they should be seated close enough to “cheat off of each other.” Partners are good, but I would suggest trying groups of 3-4.
(5) Getting conversational buy-in from all students requires comfort and interesting problems with a low start threshold. A great problem to start the year is the handshake problem. I ask students to stand up and shake hands with everyone in the room and introduce themselves. Afterwards, I ask “How many handshakes occurred?” I force them to think-pair-share. Often some student admits or questions whether all intended handshakes occurred which gives us a great chance to talk about assumptions, if no one brings it up I ask them about assumptions they think we are making. If I had instead said, “Assuming all of you actually shook hands with each other, how many handshakes occurred?” I would have removed a layer of the problem.
This is a problem that allows for easy hypothesizing. Guesses are generally rooted in some calculation like how many handshakes a single student had. The wrong methods often need only a slight modification to achieve the correct answer, and there are multiple solutions and many, many visualizations students might create. Encourage students to use Polya’s methods of solving a smaller problem, look for patterns. If students think they have the answer, have them explain their method carefully and then raise the stakes, “what if there were 100 students in a college lecture?” (or some public schools, am I right? My largest class was 43…ugg) Later on I ask them leading questions about Gauss’s method for adding the first 100 numbers.
Start the year with many problems that have multiple known approaches, and as the year moves on students may surprise you with devising a new method you haven’t seen before.
A cooperative classroom is full of people who believe that they have something to contribute and is structured physically and mathematically to entice conversation. Find good problems, and get students talking multiple times every day.