One conversation with a colleague this year has stuck with me. We were chatting after the first class of the year, and he made an offhand comment about how math was unambiguous and logical, with only one right answer.
I’m fascinated by this perception, and I need to remember that while I think of mathematics as a discipline grounded in struggling with ambiguity, resolving complexity, and working through confusion and uncertainty, most humans do not. I want to create structures in my class to communicate to students that math can be ambiguous, and that seeking out ambiguity can be an important way to learn about mathematics. I realize that many students say that what they like about math class is being able to find the one right answer and I want to value that aspect of mathematics — I definitely feel satisfied after finding and confirming the answer to a hard problem. At the same time, I want to expand student conceptions of what mathematics can be. They’ll have lots of experiences valuing right answers in my class and beyond; I want to make sure I also value ambiguity.
One way I try to value ambiguity and use ambiguity as a teaching tool is in my warmups. Which one doesn’t belong, visual patterns, number talks, and between two numbers are four great low-prep resources for tasks that students can look at in lots of different ways. Other folks have written about the finer points of each of these tools. I want to think for a minute about three teaching moves common to all of them that I try to use to communicate my values to students.
Who Thought About It Differently?
This is my favorite question, and I get to ask it every time I do one of these warmups. I phrase it purposefully to assume that students approached the task in different ways, rather than saying “did anyone think about it differently?” I want students to understand that looking at a problem from a unique perspective is valuable for everyone’s learning, and to highlight those perspectives each day.
Rough Draft Thinking
I think the most valuable part of these warmups is rough draft thinking — hearing students reason through a problem out loud and share ideas that might be wrong in front of the class. In all four structures I’m likely to start with individual think time and a partner share. Students are unlikely to offer rough draft thinking on their own, but I can listen in and ask students who have valuable but unfinished ideas to share. I’m not trying to find students making mistakes for the purpose of mistakes, instead seeking out partially-formed strategies that offer a new avenue of approaching the problem and creating an opportunity for the class to help. Creating a space where students feel comfortable sharing this type of thinking is hard, and involves celebrating mistakes every time as useful ways for the class to learn. But it’s worth all the effort to allow students to share ideas more freely and feel more comfortable taking risks.
Value Divergent Ideas
Students often say things that, strictly speaking, are wrong. Highlighting rough draft thinking is one example of this. But ideas that students share often have important grains of truth. They might think about the step number of a visual pattern differently than the rest of the class, make a computational error despite sharing a unique strategy for a number talk, or misuse vocabulary while describing a new idea for why a certain graph doesn’t belong. I have tried to actively cultivate the habit of looking for the valuable ideas in everything students share. This has been hard. For the first few years of my teaching, I spent a lot of time asking the class questions, calling on a student, and then telling them, implicitly or explicitly, whether they were right or wrong. I have had to practice slowing down, unpacking what a student has to share, valuing their contribution, and building off of it to create an opportunity for the class to learn.
These three practices build off of each other. First I need to create a space where students see a task as an opportunity to compare and contrast approaches rather than to guess the right answer hiding in the teacher’s head. Then I need to help students see rough draft thinking as worth sharing and valuable for learning. Finally, I need to approach student ideas with a disposition to build off of strengths rather than point out mistakes. It’s an iterative cycle that, hopefully, over time, creates an environment where students see math as a discipline grounded in communicating, working with ambiguity, and connecting ideas. And all for five or ten minutes for each warmup.