What Should Math Teachers Do When They Don’t Know the Math?

In a comment on my last post, Tracy Zager wrote about a childhood math teacher who responded to one of her questions with, essentially, “Just go with it, Tracy, okay? That’s how math works.”

How do we handle the moment when it becomes clear, in front of the class, that we don’t understand some math we thought we understood? Do we handle it like Mr. Davis did, and view kids’ questions as challenges to our expertise and authority? Or do we say, “You know, your question is making me realize I don’t understand this as deeply as I thought I did. That’s awesome, because now I get to learn something. Let’s figure it out together.”

You don’t transition from a novice teacher to an expert in a day. The transition isn’t obvious and it isn’t stable. You become an expert at certain aspects of teaching before others and some days you regress. But one day you wake up and you realize that certain problems of practice just aren’t consistent problems anymore.

One strong indicator for me that I had changed as a teacher in at least one aspect was when I no longer felt threatened by students who caught me in an error at the board or who asked me a question which I couldn’t quickly answer. I knew some of my Twitter followers would feel the same way, so I asked them a version of Tracy’s question above:

What do we do when it becomes clear, in front of a class, that we don’t understand the math like we thought?

Here are my ten favorite responses. If you have a response that isn’t represented here, please add it to the comments.










BTW. David Coffey has answered the same question about college mathematics, where students are sometimes very unforgiving of mathematical errors and lapses.

Featured Comments

Raymond Johnson:

The most lasting memory from my Modern Algebra I class: It was a Monday, and the instructor was about 20 minutes into his lecture when he got stuck in the middle of a proof. He stopped and stared at the board, then down at his notes, then back at the board, then back at his notes. The class paused their notetaking as the instructor (who was well-respected and always prepared) mumbled and tried to sort things out. After an awkward few moments, he said, “I know there’s something wrong here and I can’t figure it out, and my notes aren’t helping. We really can’t go on before we’ve proven this, so you are all dismissed and we’ll start here again on Wednesday.” We left, returned two days later, and the instructor enthusiastically explained what had caused the problem, how he worked past it, and we moved on. The episode might not have represented great pedagogy, but it was a refreshing example of humility.

Ethan Weker:

I have a space on my whiteboard for questions that come up that I don’t have answers for at the moment. So far this year, I have a couple of favorites:
“What do you call quadrants in 3d?”
“Why do we use p and q in logic? Is it the same origin as ‘Mind your p’s and q’s’?”
Students can find answers or I find answers but either way, it reminds students (and me) that I don’t know it all and I don’t have to.

Elizabeth Raskin:

I hated making mistakes in front of students when i first started teaching. I became conformable with not appearing perfect when my classroom culture transitioned from being myself as the expert and students as the learners to all of us learning from each other.

Mr K:

I’m teaching a small, highly gifted class this year. One of the things we’ve started doing is solving 538’s Riddler each Friday.

For the first couple of weeks, they always looked to me for an answer. It took them a while to realize that I didn’t know it either. Today, while we were doing it, they treated me more as a colleague than as an authority. They’d propose ideas, I’d ask them to justify the ideas, we’d try them out, and decide whether it got us closer to the final answer or not.

It’s really fun modeling my thinking process, and narrating it at the same time. I’ve started identifying when I have interesting things to look at, aha experiences, and most importantly how I test out my suppositions rather than just assuming that they’re correct.

Ruth:

Students who catch my mistakes at the board receive a prize: a mechanical pencil. They become sought-after tokens by the end of the year, and keep students following my reasoning as we work through complex problems!

Diane Way:

In my middle school classroom, I also use 24, WODB, and Set as daily warm ups. Because I don’t “automatically” know the solutions, when students don’t find them, we are able to reason them out together. They observe me trying things out and persevering, and are often inspired to “beat” me which keeps the engagement level high. They all become more comfortable risk takers over time.

Maria Rose offers similar thoughts to Diane’s, right down to the activities they use.

Corey Null:

There is a certain amount of excitement in not knowing. I try to translate that to the students. We wouldn’t be in this game if we didn’t want to know an answer to a question but had no idea where to begin! That’s the beauty of both mathematics and of teaching. Share that enthusiasm for the chase with them. Some questions are unknown to the teacher but easily answered. Others are not. Try your best to answer them, but more than that, try to engage them with your excitement for discovering the unknown.

2016 Oct 18. An excellent companion post from Dan Teague: Demonstrating Competence by Making Mistakes.

About 
I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. More here.

29 Comments

  1. Perhaps it’s because I’m an English major, but I’m shocked this is even a thing. I have at least two different posts where I worked with students when I didn’t understand the underlying math: 1) https://educationrealist.wordpress.com/2015/12/07/tales-from-zombieland-calculus-edition-part-2/ 2)https://educationrealist.wordpress.com/2014/05/31/learning-from-mr-singh/

    In fact, I always tell my students I might make mistakes while working board problems because it’s hard to walk and chew gum. I never work problems ahead of time and often just make them up.

  2. I have a space on my whiteboard for questions that come up that I don’t have answers for at the moment. So far this year, I have a couple of favorites:
    “What do you call quadrants in 3d?”
    “Why do we use p and q in logic? Is it the same origin as ‘Mind your p’s and q’s’?”
    Students can find answers or I find answers but either way, it reminds students (and me) that I don’t know it all and I don’t have to.

  3. First of all, I guess I hadn’t visited in a while/love the new theme. Secondly ironically I wrote about something similar this week (when I admit I judged some fellow teachers for not thinking about the big picture). http://www.dormanmath.net/math/developing-growth-mindset-in-teachers/ Love Ethan’s questions to himself – shows both humility and to the students that we don’t know everything! Dan love your comment that being a master teacher is definitely not a linear rate of change from novice to expert!

  4. I hated making mistakes in front of students when i first started teaching. I became conformable with not appearing perfect when my classroom culture transitioned from being myself as the expert and students as the learners to all of us learning from eachother.

  5. I think there are two distinct ways in which mistakes happen. But before I get to that we need to remember the original question was about being in front of students when it happens, not ideas that might occur later.
    First type is the momentary lapse, where as a teacher I might need to collect my thoughts for a few seconds or even minutes before moving on. This really is no big deal as it just shows the students that teachers are fallible people, which might even be a good thing.
    The second way in which a teacher’s mathematical inadequacy surfaces is not in a mistake, but in a lack of explanation. It’s what occurs when the teacher demonstrates a correct procedure but feels like there is a power struggle when asked why something is done. It represents the classroom where power isn’t centered around knowledge and truth, but by control of the correct answers.
    The teacher’s edition has the power of the correct answers and I control the teacher’s edition, therefore “I have the power”. *Enter He-Man imagery, but instead of the Sword of Power, it is a teacher in khakis and a polo holding the solutions manual.*

  6. I’m teaching a small, highly gifted class this year. One of the things we’ve started doing is solving 538’s Riddler each Friday.

    For the first couple of weeks, they always looked to me for an answer. It took them a while to realize that I didn’t know it either. Today, while we were doing it, they treated me more as a colleague than as an authority. They’d propose ideas, I’d ask them to justify the ideas, we’d try them out, and decide whether it got us closer to the final answer or not.

    It’s really fun modeling my thinking process, and narrating it at the same time. I’ve started identifying when I have interesting things to look at, aha experiences, and most importantly how I test out my suppositions rather than just assuming that they’re correct.

  7. The most lasting memory from my Modern Algebra I class: It was a Monday, and the instructor was about 20 minutes into his lecture when he got stuck in the middle of a proof. He stopped and stared at the board, then down at his notes, then back at the board, then back at his notes. The class paused their notetaking as the instructor (who was well-respected and always prepared) mumbled and tried to sort things out. After an awkward few moments, he said, “I know there’s something wrong here and I can’t figure it out, and my notes aren’t helping. We really can’t go on before we’ve proven this, so you are all dismissed and we’ll start here again on Wednesday.” We left, returned two days later, and the instructor enthusiastically explained what had caused the problem, how he worked past it, and we moved on. The episode might not have represented great pedagogy, but it was a refreshing example of humility.

  8. Thanks, team. Awesome anecdotes and strategies from Ethan, Elizabeth, Mr. K, and Raymond added to the post.

  9. For me, this situation is easy because of the base understanding we’ve already established. In part:
    (1) I am not the source of truth in the class. Math is about the power of reasoning over the power of authority.
    (2) I am still a student, still learning. To make this clear, I show them things I am studying and puzzling over.
    (3) Mistakes and confusion are good (for too many reasons to list here).

    When I’m stuck or make a mistake, it is just another example that reinforces these ideas.

  10. I make plenty of mistakes unintentionally, but I often enjoying making a small, intentional mistake and then moving forward as if it never happened. Eventually, when the math doesn’t work out, I’ll act flustered and see how long it takes for a student to point out my error. When I do make these intentional mistakes, and when I make the slightly more embarrassing unintentional mistakes, I’m amazed by how few students are willing to correct me. I’ll say something wrong, and suddenly even my most confident students become unsure of their own work. It takes a while, but somewhere about the middle of the school year, I can usually get the kids to understand that I know a lot, but I don’t know everything, and I am just as prone to silly errors and mistakes as they are, if a little more practiced at catching them.

    Towards the effort of getting my students to understand that I don’t know everything, I like to try to highlight students who come up with a novel approach I had never considered, and make a big deal about the fact that I had never considered it. Just a few weeks ago I gave my students an assignment to write an equation that matched a growing quadratic pattern. I was expecting an explicit area model, but one of my students managed to color code each figure in the pattern in a way that perfectly generated a recursive equation. When I saw it, it even took me a good couple of minutes to understand what she had done. During the next class, I (perhaps over-enthusiastically) shared her approach and we preceded to revisit several of the quadratic patterns we had been working with to apply it.

  11. I am a math coach. When I first started, I was terrified of making mistakes in front of TEACHERS, especially high school teachers. Now, I savor those moments as opportunities to learn with and from my peers – isn’t that what my job is really about? At least once a month, I go to the high school common planning meeting with a math question. They are amazing. We always end up doing the math together.

  12. Teaching is a fine balance of strength and vulnerability. Most of the time, we’re like that moment in the movie U-571 where the first mate tells actor Mconaghey, “you may not have any idea of what to do, but you always tell the crew that you have a plan!”
    But then comes that moment of not knowing, in the words of G. Sheehy, we are, “… yeasty and embryonic again, capable of stretching in ways we hadn’t known before.” my first year teaching was all about U-571; now, I LOVE that I am confident enough to model “yeasty and embryonic”!

  13. As a math teacher with no formal math training, this happens to me on a regular basis. Thankfully, the practice of “faking it” dropped off quickly. One story comes to mind;
    First period was a sharp and spritely bunch of 8th graders tackling exponent rules. We went through the progression (2^3, 2^2, 2^1, 2^0) to show that anything with a zero exponent is 1.
    “Anything?” asked Leah. “Like… what about zero to the power of zero?”
    I had no idea and said so. I called the math coach, who claimed it was Undefined, which led to a good talk about what conditions generate that answer. There was an underlying skepticism; how come it’s true just because that voice on the phone said so? Can we figure it out ourselves?
    That was a great year.

  14. As a future math teacher, this post is incredibly interesting and helpful! I have never had my own classroom, and the reality of not knowing how to answer questions freaks me out a little bit. I know it is going to happen, and I know that it is a good thing, but it is the unknown in this situation that is scary. This post, along with reading the comments, has given me a lot of new things to think about in regards to this topic, and has made me more excited than anything for these questions to be asked! It is just such an amazing learning opportunity for me, as the teacher, and it is an incredible opportunity to build an honest environment in the classroom. Looking back on my own experiences in high school, I realize that I had a lot more respect for teachers who admitted not knowing everything, which makes me want to admit this as well. This post has really transformed my mindset around this subject, and I am now more than ever eager to teach

  15. I teach a math support class for all grades in my middle school. One of the biggest issues for these students, why they fall behind, why they hate math, why they don’t engage, is because of their fear of being wrong. We warm up most days with two different activities (we trade off days) that I don’t know the answers to…the game “24” (4 numbers, how do they make 24) and the Which One Doesn’t Belong resources (shapes and numbers). We practice sharing reasoning for why they believe their answer is “right” (WODB has at least 4 different correct answers, “24” has at least 1, but often has many more) OR what strategies/thinking they used before they got stuck, whether or not they came close to an answer. They get a real kick out of the fact that I have to work just as hard as they do to figure it out. Often, they find the answer before I do, which they find hilarious…and then that becomes a teaching point about why we don’t laugh at people for struggling AND that everyone has strengths and weaknesses. Sometimes NONE of us find the answer to “24,” and we all look it up on the Internet.

    I also have one group of students who have so much fear of being wrong in front of their peers (6th graders, 10 young ladies), that these two games cause frozen looks and uncomfortability. The possibilities are too open. They don’t have any parameters for what a “correct” answer looks like. I started using the pattern/relationship game called “SET” with them, and all of a sudden, they are all over explaining their thinking AND they are OK with being wrong, as long as someone backs up their evaluation with a reference to the rules.

    Try SET out as a class warm up. If you go to the SET website, they have a daily game on line for free (so you don’t have to buy the cards). They’ll tell you if your choices are correct, but not why, so the students have to explain. Could be a way for them to deduce the rules based on their guesses, but site also has a help function.

    I got “24” on Amazon, but you can find it in independent toy shops as well. There are tons of different versions, including exponents, variables, double digits… I was surprised at the variety.

  16. In my middle school classroom, I also use 24, WODB, and Set as daily warm ups. Because I don’t “automatically” know the solutions, when students don’t find them, we are able to reason them out together. They observe me trying things out and perservering, and are often inspired to “beat” me which keeps the engagement level high. They all become more comfortable risk takers over time.

    Furthermore, while it’s not really an answer to the question, I also teach kids to play backgammon, cribbage, Yahtzee, and Chinese Checkers in our daily warm ups. There is so much math and logic in these and other “old” games, but many kids haven’t learned nor have an opportunity to play games in their homes. Card and board games teach kids so much about predicting, planning, computation, and risk taking. Nothing is more gratifying than to have a parent tell me that their child’s interest in game play has prompted more family time doing that. I want my students to understand that math happens in places and times other than in my classroom for 83 minutes a day.

  17. Students who catch my mistakes at the board receive a prize: a mechanical pencil. They become sought-after tokens by the end of the year, and keep students following my reasoning as we work through complex problems!

  18. There is a certain amount of excitement in not knowing. I try to translate that to the students. We wouldn’t be in this game if we didn’t want to know an answer to a question but had no idea where to begin! That’s the beauty of both mathematics and of teaching. Share that enthusiasm for the chase with them. Some questions are unknown to the teacher but easily answered. Others are not. Try your best to answer them, but more than that, try to engage them with your excitement for discovering the unknown.

  19. I myself don’t see that there’s an issue at all. As long as you set your ego aside and be willing to admit that you have to dwell a bit to get the answer, students will understand your position pretty well. That’s said, I think that dismissing the class for being stuck on a proof is an overkill, the professor could have asked the students for help, or he could have just asked everyone to dwell on it a bit, and move on to the next topic. There’s really no big fuss here as long as a person handles the situation in a relaxed, mature manner.

  20. What’s interesting about this is that I often answer student questions with, “I’m not sure. Maybe this will help you answer your own question [gives hint or suggests similar problem]. What do you think?” Even when I do know. I’m not interested in answering questions for students who can reason through things themselves. They catch on pretty quickly that I likely do know the answer/reason. But it means if I really don’t know, students are used to my response of “I don’t know.”

    Either way, it’s not a big deal to me to admit to students I don’t know something. I can’t know everything, always, every day.

  21. Really helpful perspectives, all the way around. I’ve added responses from Diane, Ruth, Maria,, and Corey to the post.

  22. I really needed this post. As a university student aspiring to become a math teacher, I feel like being on the wrong end of a question is fear-inducing, and can be an obstacle that won’t be overtaken right away. During my time in practicum, the fear of making an error and having a student ask a question that goes against what I am saying is always in the back of my mind. I end up criticizing myself for not making a concept clear, and can be very hard on myself.

    At the same time, I want to learn to be honest in front of my students, and after reading the tweets above, I want to show my class that I am more than willing to learn alongside them. Having years of education above these students does not change the fact that I am still an active learner myself. Thanks for the encouraging post!

  23. I am hoping to have a math classroom of my own sometime in the near future (I can see from the comments I am not alone here) and I can say that this is something that I, along with many of my peers, worry about often. It is a relief to see so many responses to that Twitter question include admitting to students that the teacher does not know everything. Since I was a high school student not long ago, I can remember a few times (although not the exact situations) when a teacher was unable to provide a concrete answer to a question, and I also remember never holding it against them. I think that this is something to always remember as a teacher. Students have a lot going on, and a couple of teacher slip-ups are not going to be the primary occupiers of their minds.

  24. Joe:

    Students have a lot going on, and a couple of teacher slip-ups are not going to be the primary occupiers of their minds.

    Nice! And you can aim even higher! Rather than “I hope they forget my mess-up,” you can say to yourself, “I hope they watch their teacher struggle and not lose his cool and realize that struggling is a natural part of challenging yourself academically.”

  25. I honestly look forward to being asked questions that I don’t know the answer too. I really do because it gives me opportunities to engage with students thoughts and also show them that I am not just a teacher, but I am also a Mathematician! I am not simply a teacher teaching them math, but I am a Mathematician who wants them to learn the way mathematicians learn, by not simply knowing the answer, but by working hard to find it out.

    I think that students should feel comfortable to ask any question they desire, and I think that a natural way for this comfort to develop is to admit that you are unsure when a student asks a question we don’t know the answer to. This shows them that even we can ask questions and model for them how we can go about finding answers to them. We must model being an active and life long learner and mathematician if we want students to see what they need to do to become the same.

  26. I’m actually so glad I stumbled upon this post. Earlier in my student teaching I was trying to prove why you couldn’t divide by zero to my students, but one of the students countered my proof. It was a bit embarrassing, but I thanked her for showing me that and told the class that even this proof doesn’t suffice. Reading some of the comments, I never thought about myself as being a model for my students on how to handle a problem I didn’t know, that the struggle and unknown itself can be a great learning opportunity for both students and teacher. As I soon will be heading into the teaching world, I need to remember that I myself am still a learner as well, and that questions and “not knowing” is not something to be afraid of but instead something to celebrate about because it now puts both the teacher and the students in the same role of problem solvers. Thank you for this post, though it’s still kind of scary to think about being in that position of not knowing again in front of all my students, I can now face it as a great opportunity not just for myself, but for my students as well.

  27. Use your judgment of which of two approaches to use:

    1. Figure it out on the board.

    2. Tell the kids you will get back to them later.

    Obviously choice 1 is preferred to the extent that you think you can figure it out, question is critical, and class is on track time/content wise. To the extent the opposite is the case (don’t think you can nuke it out, less important issue, class does not have spare time), than you should pick choice 2.

    Other choices (trying to snow them, belittling the question, etc.) are not acceptable.

    If the question is truly something super-advanced (and presumably not relevant to the content) than you may be forced to get back to them the next day with an answer that you tried, but can’t answer it.