If we want our students to think then we need to give them something to think about—something that will not only require thinking but will also encourage thinking. In a thinking mathematics classroom, this comes in the form of a task, and having the right task is important. When first starting to build a thinking classroom it is important that these tasks are highly engaging non-curricular tasks. However, it is also important that the task is a good fit for the collaborative environment our students are working in. For example, if our students are collaborating through ZOOM using only voice and video without a shared virtual workspace then they are not well positioned to be working on a task that requires a lot of drawing and interpretations of drawings. On the other hand, if they are collaborating f2f at a whiteboard then tasks that require drawing are completely appropriate. In short, the thinking task needs to match the medium of collaboration in which the thinking will take place. Consider, for example the following task:
You are standing in the living room of your ground floor apartment when a giant wheel rolls slowly past your window. The wheel is huge with a radius of 100 miles and when it passes the window it completely blocks out the light from outside. The question is, as the wheel begins to pass the window how will it appear to block out the light: (a) from the top of the window moving straight down, (b) from the side of the window moving straight across, or (c) from the top corner of the window moving diagonally down and across.
This is a thinking tasks that can be used in almost any medium of collaboration. Students can work on it on a whiteboard making scale drawings or they can discuss it through a video connection using hand gestures. Even students working in a text only asynchronous collaborative environment can discuss it. The following problem is likewise compatible across many different mediums.
I buy a video game for $10. I then sell it for $20. I buy it back for $30. Finally, I sell it again for $40. How much money did I make or lose?
On the other hand, a task such as the following really needs a collaborative medium that allows for a shared workspace where students can work together on a representation and notation to solve the task.
I have a 4-minute egg timer and a 7-minute egg timer—the kind that you turn over and let the sand run through. Can I use these to cook a 9-minute egg? If so, how long will someone have to wait for their egg?
But the goal of thinking classrooms is not to get students to think about engaging with non-curricular tasks day in and day out—that turns out to be rather easy. Rather, the goal is to get more of our students thinking, and thinking for longer periods of time, within the context of curriculum. When transitioning to the use of curricular tasks we need to be likewise aware of the affordances and limitations of our collaborative medium when giving tasks for students to solve. For example, if a shared workspace is available then we ask students to draw graphs of functions. If the collaborative medium does not have a shared workspace, we would need to pivot these questions from drawing graphs to asking questions about graphs:
What do you think happens to this graph as x gets very large? What do you think would happen to this graph if we made the leading coefficient negative?
For younger students you can have them discussing what happens to arithmetic tasks:
So, what do you think the answer would be if I doubled the first number? If we could change the first number to be anything we wanted, what is the smallest answer we could get?