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.
@ddmeyer we model, but for real, a learning stance, curious, puzzled, conjecturing about possible next steps...— Vicki Madden (@vlmadden) September 22, 2016
@ddmeyer "I have to think about this some more. I'll get back to with what I discovered you tomorrow." Urge Ss to do the same.— Elizabeth Raskin (@elizraskin) September 22, 2016
@ddmeyer Admit it and learn from it with your Ss. Teach taking risks by example!— Ann Gaffney (@annmgaffney) September 23, 2016
@ddmeyer hopefully model how to figure stuff out - rare chance for ss to see how to learn; dead ends, conjectures, testing assumptions, etc— TheChalkface (@the_chalkface) September 22, 2016
@ddmeyer Be honest in front of your students & model the power of "I don't know but I want to know". Someone in the room needs to hear that.— Sheila Hardin (@smh8129) September 22, 2016
.@ddmeyer Be human. Apply good problem solving skills. Model what to do when challenged. If stumped, research rinse repeat.— Jim Deane (@jim_deane) September 22, 2016
@ddmeyer Think. Out loud and on the spot. Sometimes Ss ask Qs that perplex me. I congratulate them and let them hear my thought process.— R Eichholtz (@YorkMathE) September 22, 2016
@ddmeyer coach in room? take a T timeout do discuss in FRONT of Ss! so powerful for Ss to see Ts don't have all the answers.— Jody (math coach) (@jodymathcoach) September 22, 2016
BTW. David Coffey has answered the same question about college mathematics, where students are sometimes very unforgiving of mathematical errors and lapses.
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.
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.
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.
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.
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!
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.
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.