A series of events the other day struck a chord with me. They all centered on a simple little phrase, offered often enough in math class to not be unusual. That phrase is, “I don’t know.”

The first “I don’t know” came during my 6^{th} grade class. A girl was stuck on a problem (building polygons with different numbers of sides out of as few triangles as you can). When I came over to see how she was doing, all she could say was, “I don’t know how to do this.” And, “I don’t know,” was quickly followed, as it often is, by, “I can’t do it.”

It’s funny how those two phrases so often go together. There’s no inherent reason why a student should think that just because he or she doesn’t know something right now, he or she will never know it. I mean, this is school. You’re supposed to be learning about and figuring out things you don’t already know. And I’ve seen this particular girl puzzle her way through tricky situations numerous times before, whether in a game (a context in which “I don’t know” is very rarely followed by “I can’t do it”) or on other math problems.

Furthermore, she could do it. “Doing it” for this assignment was pretty open to interpretation. There was no prescribed method, no real “right answer.” You made a pentagon out of three triangles? Awesome. You made a different pentagon out of two triangles? Well, that’s awesome, too. You now decide to move on to a different shape? Or to stick with two triangles and see what you can do? Or try to make the nicest looking pentagon out of two triangles that you can? There are lots of ways to play with this question and lots of places to have success.

She knew that—partly because I told the class when we started the assignment and partly because that’s how investigations like this always go in in our class. But despite the open-ended nature of the assignment and the wide-reaching opportunities for success and personalization, she was having trouble seeing the possibilities as real…

…Or maybe the open-endedness and wide-reaching possibilities were the problem. Huh. Freedom can be just what some students need to engage with math. But for other students, who maybe hesitate to act ambitiously in math for reasons deeper than feeling stifled or bored by conventional lessons, freedom isn’t enough. They don’t feel confident enough that they will come up with something in this new, open environment, and that the thing they come up with will be valued by others.

In my purely anecdotal and observational experience, the problem of feeling intimidated by freedom in math seems to impact girls more than boys. For many of my male students, being given the opportunity to pose problems, choose their methods, and shape class with their judgments is enough to spark increased engagement. But for a good number of my female students, an environment open to personalization is less welcoming than a more structured environment. And this makes complete sense. In a learning environment with narrow goals, where it’s a matter of procedure to learn how to do something right and easy to know if you have, success is obvious. For students with low confidence, this can be comforting.

And now it’s time for a confession: I never did the “fun” math puzzles when I was in grade school. I did really badly with “enrichment.” My fourth grade math teacher was really “creative,” and I hated it. Opportunities to personalize my math education were around when I was in school, and I turned them down. No thank you. I’ll stick to my procedural, memorization math, which I find exceptionally BORING and POINTLESS— but at least it isn’t scary. I know how to achieve within the structure. If I say, “I don’t know,” in a conventional math environment, my teacher will tell me how to do the problem. And I’ll devote lots of effort remembering it for next time. Success achieved! But if I say, “I don’t know,” in a more creative math environment, my teacher will tell me to… just try something? Use my intuition? I have no faith that either of these approaches will bring me success.

How do we get our less confident (and often female) students to take advantage of opportunities for creativity and personalization in math class?

I don’t have a system. Right now I’m just operating on a case-by-case basis. But here’s what I did with this girl and often do.

My student said to me, “I don’t know… I can’t do this.” I tried asking her where she was stuck and, as expected, she wouldn’t-couldn’t articulate the problem. So I:

1) Asked her to show me what she had done, and praised it by saying that I’d seen another student do something like it (so she knows that she’s on track with her peers) and that her approach was special in some particular way (so she knows that she has valuable original thoughts).

2) Using a tone that (hopefully) conveyed that I was curious and didn’t know the answer, gave her a pretty narrow suggestion that was absolutely certain to get her somewhere. Basically, I gave her a solution part of the problem.

Yes, I gave away an answer. I told her how to do something. “Gah!” you (and part of me) may say. “That’s not the point! She’s supposed to figure it out herself!” But, NO, says another part of me. A hallmark of the creative, open-ended approach to math education is that the answer is not the point. It’s the process. If giving her the answer to one part of the problem gives her the confidence to tackle the rest on her own, this is preferable to not giving her an answer and watching her flail around in the dark. And if I’ve designed the problem properly, there will be plenty of room for her to be creative even if I give away a little piece of the game.

Well, it worked. She quickly got a result—and was happy. Then when I gave her a more open-ended prompt (How could you extend this approach? Could you use it on other shapes?), she was able to proceed on her own. She was the first with her hand in the air when I asked her and her classmates to share their findings, and she was clearly proud of her work.

On her way out the door at the end of class, I pulled her aside and reminded her of how class had started for her. She smiled a little and hurried away, clearly not excited to talk about this; but the experience seems to have had an impact. She’s now more likely to ask me questions if she gets stuck, rather than moving straight to despair, and more likely to stick to it even when other people have given up.

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This is a much shorter story. The second “I don’t know” came during my Symmetry and Algebra elective for high schoolers. I think what happened stuck out more to me because of what had just happened with my 6^{th} grader in the morning—because I’m pretty sure this happens all the time and I just don’t notice. Especially in this class, which has segregated itself dramatically by gender. (Which I’ll have to write about in more detail later when it’s all over—they’ve been a big struggle.)

Here’s what happened: It was the end of the period. In parting, in the last few minutes, I asked the whole class a pretty new question. It was related to what we’d been talking about, but was new enough that there was no way anyone was going to answer it in the next three minutes. I knew that, they knew that. The idea was to set up something to investigate in the following class.

I ask the question. I pause. I say, “What do you guys think—just instincts, right off the bat?” A little silence. I make eye contact with a girl at the back of the room. She looks a little shifty and says, “I don’t know.”

I make eye contact with a boy a few seats away. He says, “Hmmm…” and launches into a purely speculative, stream-of-conscious-like series of ideas—all of which miss the mark. He knows that they do, and so do the boys sitting next to him—but that doesn’t stop them from chiming in. And why should it? They don’t know, either. But they do know plenty of other things, and they know that a good way to go from not knowing to knowing is to just start putting ideas together. Try things, fail a bit, try more things, fail a little less. They are perfectly comfortable with this.

Meanwhile, the girls in the back sit in dead silence.

As soon as all of this settled in my head, I paused the discussion. I think I said something like, “Guys, please indulge me in a little meta-discussion analysis.” I reminded them that the girl at the back had answered my really open-ended question by saying, “I don’t know.” Then I turned to the boy who had started talking first.

“Lucas,” I asked, “do you know the answer to my question?”

He made a little smirky face. “No! I have no idea.” The boys next to him laughed a little and shook their heads, agreeing.

I shared a little knowing glance with the girls and said something like, “So Lucas doesn’t know, but he’s answering anyway.” And then the bell rang. No time for further reflection. But everyone was laughing a little on the way out—hopefully at themselves as much as at their classmates.

I worked at an all-girls school for three years and these stories REALLY resonate with me. In that environment, I did see – as I hoped I might – that the girls were more willing to open up and offer conjectures. However, what I saw more of is what you describe in the first story. I gave my Geom class a problem on a test at the end of a unit on circles. I presented them with a blank cartesian grid and the standard form of a circle ((x-h)^2 + (y-k)^2 = r^2) and asked the following:

Think of a circle, pick a radius and center of your choice. Write the equation of this circle and sketch it on the grid below. Does this circle have any intercepts (x or y)? If so, identify them.

Most of the girls did a great job on the problem, but almost all of them complained about it when we reviewed the test later. I joked with them that they should have enjoyed this because I gave them freedom of choice. A chorus of voices responded ‘We don’t want freedom of choice, we want you to tell us what to do!’

Thanks for the comment! That sounds a lot like what happened in my class– and I’m sure if we both thought for a bit, we’d think of more stories like this. I like how you put it that it’s not necessarily the case that girls shy away from participating in general, but rather from participating when the “right thing to do” is less clear. This seems like a much gnarlier challenge.