I love asking students questions that I don’t know the answer to.

Sometimes, they are meta-cognitive, asking students how they think about and approach math: Why did you chose this method? What have you tried so far? What do you understand about this problem? How could you change this problem to make it easier/harder?

Other times, they are extension questions for kids who are ready for a bigger challenge: how would this graph change if you used a 3D graphing calculator? What would happen if you tried this in base 12? How could you solve this without algebra? What other type of function better models this data?

In both cases, what is most important is that I don’t know the answer. Early in a school year, students learn that I probably don’t know the right answer (sidebar: I often fake not knowing to really hammer this home), and it shifts the onus onto the kids. They are responsible for both the thinking, and the testing of the answer. For students who are struggling, questions in the former category help them meta-cognate about the problem, and can open access to prior knowledge. I don’t know what the kid is thinking, and they might be totally on the wrong track. By answering these meta-cognitive questions, students help me help them in the way that they need.

For students who are advanced, having a bunch of extensions in my back pocket keeps them at their edge. I have learned that I am much better off with a lot of really juicy questions and few answers, than having one set extension for every high-flyer that has a known solution. When I’ve talked with other teachers about this they often worry about not knowing the answer. How will kids know if they got it right? How will they respect me as a Math teacher if I don’t know the answers?

To that I have a two-part argument. First of all, the goal of these questions is not for students to get them right. The goal is for them to think and work at their edge, which actually has nothing to do with the answer. Second, real math rarely has clear answers. Ask any mathematician- their work does not have solutions in the back of the book, or an authority figure that can instantly validate their work. Real mathematicians are adept at trying something, testing, failing, and iterating. We can serve our high flyers much better by modeling how we approach a hard problem and helping them puzzle through it, than by holding the right answer over them as they work to impress us.

I want to close with a few tips for asking these questions on the fly.

Meta-cognitive questions: respond to what you see in front of you. If they’ve tried something so far, ask them about it and see what you can tease out. I see you erased something here, what did you try and why did you erase it? What’s this picture you drew? If their page is blank, ask about their thinking. What did you notice about the problem? Are there any words or symbols you don’t recognize?

Extension questions: understand where the math can go. When graphing in school, we mostly focus on 2D graphs. If they are proficient in this have them ponder about and explore 3D graphing. I don’t know how to graph a parabola in 3D, but I’m sure a student could explore this and then teach me! We also mostly work in base 10, using base 10 arithmetic algorithms. Having kids convert numbers to a new base and then deriving new/modified algorithms pushes students at all levels. I don’t know how to simplify fractions in base 5, but I would love for a kid to show me what they find! Another way to push almost any problem is to make the numbers really big or really small and see how kids adapt their approach.

Don’t be shy to ask students questions you don’t know the answers to! In my experience, it levels the playing field in the classroom and gives all students more opportunities to succeed, while also taking a lot of pressure off of myself to be a perfect human calculator.