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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.

Teaching for Tricks or Sensemaking

Here are two approaches to teaching zero exponents that are worth comparing and contrasting:

First, a Virtual Nerd video:

Once you have your non-zero number or variable picked out, put it to the zero power. Now no matter what number or variable you picked, once you put it to the zero power, I know what the answer is. One. How do I know this? Well in math, if we put any non-zero number or variable to the zero power it always equals one. No matter what.

Second, Cathy Yenca’s activity, which has students completing the following table:

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One approach will lead students to understand that math is a fragile set of rules that have to be transmitted and validated by adults. The other will help students realize that these rules are strong and flexible, and exist to make math internally coherent, with or without any adults around.

BTW. This is as good a time as any to re-mention Nix the Tricks, the MTBOS’s collection of meaningless math tricks and great strategies for teaching those concepts with meaning instead.

BTW. Check out Cathy Yenca’s own post on the comparison.

BTW. Check out the comments on that YouTube video. Interesting, right? What do we do with that?

Featured Comment

Tracy Zager:

I wrote a story about this exact moment in my book. Cliffs notes version: 8th grade. Mr. Davis told us the rule a^0=1. I questioned why. “Because that’s the rule.” I said, “But why?” Stern voice now. “Because that’s what’s I just told you.” The first boy I’d ever kissed said, “Give it up T! It’s in the book, that’s why!” Everybody laughed. Me too, on the outside. Not on the inside. On the inside I was angry and frustrated and humiliated. If I write my math autobiography, this moment goes in it. It’s one of the moments math lost me.

By the way, my favorite way to see the pattern Cathy shows here is with Cuisinaire rods. You literally go from cubes to squares to rods to single unit squares. It’s this amazing moment to see one as the fundamental unit.

But I didn’t get to see that in 8th grade. I didn’t see that until my late 30s.

The saddest part? Most kids don’t try again after they’re burned. They never come back. The not-quite-saddest-part-but-still-sad-part? I doubt Mr. Davis ever learned why either.

That’s the biggest legacy of that story to me, as a teacher. 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.”

Kent Haines:

Something that has been effective for me (that I mention in the video above) is to really emphasize that 1 is the origin and invisible starting point of all multiplication problems.

Scott Farrand offers another helpful way for students to make sense of zero exponents:

There’s a truly great old lesson on exponentiation that I believe comes from Project SEED, that has been used with amazing success in hundreds of elementary classrooms.

education realist brings up a Ben Orlin post that includes a) a beautiful technique for teaching lots of rules of exponentiation and b) this beautiful paragraph:

Math’s saving grace, though, is that it can make us feel smart for another reason: because we’ve mastered an ancient, powerful craft. Because we’ve laid down rails of logic, and guided a train of thought smoothly to its destination. Because we’re masters—not over our peers, but over the deep patterns of the universe itself.

The Explanation Difference

Brett Gilland coined the term “mathematical zombies” in a comment on this blog:

Students who can reproduce all the steps of a problem while failing to evidence any understanding of why or how their procedures work.

When I think about mathematical zombies, I think about z-scores – how easy it is to calculate them relative to how difficult it is to explain those calculations.

Check it out. Here is the formula for a z-score:

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In words:

1. You subtract the mean from your sample.
2. You divide that by the standard deviation.

Subtraction and division. Operations simple enough for a elementary schooler. But the explanation of those operations – why they result in a z-score, what a z-score is, and when you should use a z-score – is so challenging it eludes many graduates of high school statistics. Think about how easily you could solve these exercises without knowing what you’re doing.

That difference brings this chart to mind and helps me understand all of the times I’m tempted to just tell students, here’s how you do it already so now just do it. That’s where the operational shortcuts are most tempting.

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All of this is preface to a lesson plan on hypothesis testing by Jeremy Strayer and Amber Matuszewski, which is one of the best I’ve read all year.

Hypothesis testing is, again, one of those skills that’s far easier to do than to understand. As you read the lesson plan, please keep in mind that difference. Also notice how capably the teachers develop the question, disclosing the mathematics progressively, and resisting the temptation to shortcut their way to operational fluency.

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It’s spectacular. I’m struck every time by a moment where Strayer and Matuszewski ask students to model an experiment with playing cards, only to model the exact same experiment with a computer later. They didn’t just jump straight to the computer simulation!

Here is a video of an airline pilot landing an Airbus A380 in a crosswind. This is that for teachers.

Featured Comment

Amy:

I always think of z-scores as a set of transformations from one plain-vanilla normal curve to the hot-fudge-sundae Standard Normal curve. Maybe once you see it this way, you can’t unsee it. To me, that helps make sense of the “why” you would bother standardizing and the “how” it’s done.

David Griswold:

I’m not sure I agree that z-score is so conceptually difficult as to be worth the shortcut. Though I suppose it requires understanding of standard deviation, which is kind of hard. But if you think of standard deviation as “typical weirdness distance” then z-score as the idea of “how many times the typical weirdness is this point” becomes pretty straightforward. A z-score magnitude of 1 becomes average weirdness, less than 1 becomes less weird than average, etc. The bigger the magnitude of the z-score, the weirder the point.

Bob Lochel:

In introductory stats courses, much of what we do simply comes down to separating “Is it possible?” from “Is it plausible?”. We have seen a wonderful growth in the number of free, online applets which allow teachers and students to perform simulations designed to assess these subtly different questions.

NCTM Puts up a Sign

NCTM President Matt Larson wrote an essay last week titled “Curricular Coherence in the Age of Open Educational Resources.” As I read them, the big takeaways are:

  1. Coherence in curriculum is important.
  2. A curriculum is more than just a sequence of activities.
  3. Activities that are downloaded from the Internet vary in quality and often undermine curriculum coherence.
  4. If you’re going to download activities anyway, then download them from NCTM’s lesson site or download them in learning communities where there might be more accountability for coherence.

I’ll co-sign all of the above. I hadn’t thought about collaborative lesson planning as insurance against incoherence. That’s clever. All of that said, I’m disappointed at how much of the essay obliges teachers and how little of it obliges publishers.

It reminds me of desire paths.

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Desire paths occurs when people walk off the path you pre-determined and create their own. They create another route because it satisfies their needs better than yours did. In that situation, you have options. You can post signs directing people to stay on the path. You can hire security to make sure people stay on the path. Or you can admit that you messed it up on your first try and pave the desire path.

NCTM is putting up a sign.

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Aside from a brief mention of sympathy for teachers who “lack highly engaging, high-cognitive demand tasks or lessons,” the essay doesn’t acknowledge the desire that leads to online activities.

It isn’t as if many teachers are eager to spend nights and weekends cobbling together a curriculum from scratch. What gives? I asked teachers about their desire. Most prominently they want materials a) that are engaging, b) that are scaffolded appropriately, and c) that create high cognitive demand. A large number of them don’t think their core curriculum is particularly coherent.

I appreciate Larson’s leadership and support NCTM’s interest in coherent mathematical experiences for kids. But if teachers – especially at secondary levels – had access to resources that offered those features above, I suspect the desire lines, and consequently NCTM’s sign, would be unnecessary.

So an open question: What would it mean to pave the desire path here? Now that we’re at the point where people are tromping across the lawn, marching towards online activities, what would it look like to say, “Okay,” and then pave that path for teachers.

Recommended. Tyler Auer’s analysis.

Also recommended. The comments of the essay, where Larson is taking questions and adding commentary, including what appears to be an answer to my open question above.

Featured Comments

Please read Matt Larson’s comment.

Michael Pershan:

While Matt’s piece didn’t rub me the wrong way, it does seem to me that treating this as a challenge for people who want to influence what teachers do is going to be a better framing than trying to convince teachers to change their online patterns.

Steve Weimar:

One of the projects that the Math Forum at NCTM is beginning to plan within NCTM concerns the idea of an online collaboration space and supporting repository through which the community can move along the continuum from sharing good tasks to identifying and playing with sequences and instructional practices that lead to the engagement, depth of understanding, rigor, and competence we all seek for our students.

Patty Miloro:

Curricula coherence is my biggest struggle. I work in a small charter high school and we have not purchased a commercial math curriculum. Instead of only 20% of my time spent planning, I am lucky to have the freedom to spend lots of time attempting to develop a cohesive unit by vetting online materials. This autonomy is both wonderful and terrifying.

Henri Picciotto:

One way NCTM might help with both coherence and quality is to offer a really-core curricular framework (as opposed to the much-too-massive CCSSM), and within that offer curated links to high-quality online materials. A positive contribution along those lines would be a lot more useful than anxiety-provoking warnings about coherence.

Elizabeth Statmore:

One last deep thought — I sure am getting tired of being blamed for the incoherence of standards and curricula that are way above my pay grade. Unfortunately, the way all of this has been set up (or not set up), everything rolls downhill into my yard.

Jason Slowbe:

Perhaps the “next big opportunity” for NCTM is in connecting members online around its quality content. NCTM’s average member age is 55 years and fewer teachers are buying memberships, opting instead for free online connectivity with other teachers that is still quite good overall.

Christopher Danielson’s “Which One Doesn’t Belong?” Now in Print

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Stenhouse just released Christopher Danielson’s book, Which One Doesn’t Belong?

It’s a must-have if you’re a parent or a teacher with any interest in helping your children or students learn to speak mathematically.

There are few tasks that offer so much mathematical value yet require so few instructions as Which One Doesn’t Belong?

You see four mathematical objects. You ask kids, “Which one doesn’t belong?” You help them negotiate their overlapping and conflicting answers, developing vocabulary and the capacity for argument and abstraction along the way. That’s it.

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You can find loads of great WODB prompts online but you can’t find Christopher’s unique presentation, narrative, and teacher’s guide, which is its own kind of graduate-level course in pedagogy.

Highly recommended.