Tuesday, October 6, 2009

smarter than 2

this whole conversation started when ben blum-smith read me the preface of mariana cook's new book of photographs of mathematicians:

"Mathematicians are exception. They are not like everyone else. They may look like the rest of us, but they are not the same. For starters, most of them are a great deal smarter."
i totally nodded right along with him reading! what is that? confession: i walk around thinking that my math friends ben and justin and all these people are just smarter, more capable, better than i am.
and the thing is, it doesn't matter if that's true. the thought that it is, the possibility that it is, is so debilitating that i have felt (in the last three weeks!) that i should stop teaching, despite the fact that i know i'm doing this about as well as people do it and the people that have seen me do it would hate to see me stop. or whatever, even if i was bad at it, i won't get better if i think i'm just not smart like mathematicians are smart. or whatever. the problem is the very idea that true hierarchy exists. the notion of hierarchy is limiting because then some people are just smarter, and then the thought "i am one of them" or "i am not one of them" are equally problematic.

why do we (or a big portion of the general we) see mathematicians as different? and how is it affecting my students? kids (plus me and unapologetic too, it seems) are walking around feeling defined by either having smarts or a lack of them. i question the truth of these ideas in the first place, but definitely the usefulness of these beliefs when kids get attached to these self-definitions. i wonder how to keep the identity question balls up in the air during high school, rather than supporting kids in pinning themselves down.

i'm proposing that ALL kids can feel empowered and smart, not because all kids are the same, but because kids aren't all one thing. i want to give kids opportunities to perceive themselves in different roles so that they don't ever feel limited, in anything. i want my kids to be able to bite into the experience of success (which will look different for different kids) and struggle (a very useful thing for everybody to know how to work through). i want them to see each other as resources, sometimes surprising and unexpected. i want anyone in the room to be able to be the one that has the insight, no matter what has happened in their pasts.

good luck, right?
well i'm trying.

seriously, big thanks to you big debaters out there. i love it.


  1. I'm glad I'm not the only one struggling with the misconception that math is like a magical power that you are born with. My students' parents even treat it like an inherited trait "I was never good at math, so..."They don't treat reading that way. If everyone can read, everyone can factor polynomials. It just takes some people longer.
    So, sometimes when they all do very well on a test or quiz that was very hard, I let them teach me something new, so they can watch me struggle a little. I learned to do "the jerk" this week. I was lucky not to fall over.

  2. Jesse you know I'm psyched our conversation had such resonance. Here's my 2 cents on the matter itself:

    First of all, to all you doubters out there, there's research backup for the idea that our culture's understanding of math talent as innate and unchanging is bad for math education. One source is the cognitive psychologist Carol Dweck, who's done a bunch of experiments showing that kids who are told their success comes from being "smart" are less willing to take risks or engage in struggle than kids who are told it comes from their hard work.

    Second, a personal story that resonates with this:

    By the time I was in 6th grade I was a "math whiz." It was a heady thing. I accrued a fair amount of admiration from my peers and a lot more from adults. I watched some PBS show about Fermat's Last Theorem and fantasized about solving it. I mean I really felt special.

    I didn't take too much math in college; I majored in anthro. I took a couple math classes, and they were awesome. Partly because the math was really awesome, but partly because I got to be the anthro major skating circles around the math majors. I was troubled by the occasional fear that I'd wake up one day and not "have it" anymore, but at least at the moment I felt like I was all that.

    Then I headed to grad school to become a teacher, and took an independent study in abstract algebra to fulfill the math requirement of my teaching degree. With no one to compare myself to to assure me of my power, suddenly I started noticing how difficult I was finding the material. It was a bit of a crisis for me. Since I was this legendary math whiz, wasn't I supposed to get it quickly? At the very least, it was easy to imagine other people getting it far more quickly. Had I missed my chance by not majoring in math, and squandering the critical college years? The source of my heady sense of uniqueness as a mathematical mind was in question. The experience of struggle, which is at the heart of every great mathematical breakthrough, was making me doubt my worth.

    I mean it worked out, I went ahead and became a teacher anyway, and then later I started studying math again in a serious way. My experience since then has made me look at mathematical accomplishment very differently. The first time I spent 2 months on a problem and then solved it, for example, and realized that just because you can't see something right away, a day or a week later, doesn't mean you'll never see it. Or the hundreds of conversations I've had with students in which the moment I got them to them to stop thinking about how dumb they are and start thinking about the problem, they had a brilliant insight. Those fears and doubts were hard to shake though. My point is that they were hand-in-hand with the heady thing. If you feel special because it comes easy, you feel like crap when it comes hard. But the best stuff is the hardest.

  3. Blogger seems to be eating my comments.

    Ben, I didn't say that ability was unchanging, and I certainly didn't advocate telling some students that they're smart and that it would always come easily to them. In fact, quite the opposite: I would advocate identifying strength and ramping up the difficulty precisely so that they wouldn't be able to coast.

    And it doesn't have to be in the form of harder material or tracking. Kate talks about many great lesson plans of hers which arrange for the students to try answering each others' questions, which does put more of a strain on the better students' abilities and understanding than just working problems does.

    Yes, every student will hit a wall at some point where the material becomes difficult. But we should be actively seeking that wall instead of pretending it's not going to be farther along -- sometimes much farther along -- for some students than for others. And it's not like mathematics is as linear as public school curricula might make it seem. It's easy to step off to the side and direct a student into other aspects of the current material than those they're showing proficiency in at the moment, like switching from algebraic to geometric interpretations of a concept and vice-versa, or how to communicate a concept to a student who's struggling where they didn't.

    Hand-in-hand with the nonlinear nature of real mathematics: I'll agree with you that there's probably no single "mathematical talent", because of how many different aspects there are to mathematics to begin with. How many students have struggled with manipulating algebraic expressions, only to display an intuitive grasp of planar and spatial relationships when they get to geometry? I, personally, can't stand more than just so much of those messy epsilons and deltas that used to pervade high school calculus, but I've met analysts who simply can't grasp the difference between "equal" and "isomorphic". There are many skills, and a student mowing down one concept without breaking a sweat may be just around the corner from another one he isn't as good with.

    It seems apparent that natural ability exists, but (like in any other area) it requires effort to actualize the potential. There's enough improvement available through work that can easily swamp idle talent, but that doesn't mean that talent doesn't exist. And that talent, encouraged and worked at the point where it finds difficulty and real improvement can be made, will achieve even more. I'd rather tell a student "Yeah, you're finding this easier than most (all?) of your peers, but believe me when I tell you that there will come a time when even you have to work as hard as they are now. I'd like to help you find that place where you are challenged and you can make some actual progress."

  4. Hi Unapologetic,

    Thanks for your thoughtful engagement. My "to all you doubters..." comment wasn't actually directed at you, although retrospectively that was the natural reading so it was reasonable for you to hear it that way. I'm new to the blogosphere and am just feeling out how to be understood.

    After reading your "nonlinear" comments I'm excited to be having this conversation with you. I hope you'll read this reply generously.

    I don't think anyone in this conversation is suggesting that we ignore the obvious differences in the ease with which different kids complete different tasks in math class. That would be ridiculous and false. But I believe there's a very important difference between "this is easy for you" and "you have a natural talent at this" and this difference is at the heart of the conversation.

    Nobody told me "you are smart and this will always be easy for you" but somehow I was communicated an understanding of my "talent" that left me freaked out at my first experience of intense struggle with no one around struggling more to reassure me of it. How did this get accomplised?

    All they had to tell me was that I was special because it was easy for me. Over and over through my schooling, I was treated as "talented" when I did things that were easy for me. This was all it took. I had the message (loud and clear) that my specialness came from the ease with which I was doing stuff. So difficulty (with no one around having more difficulty) was a setup for a crisis.

    My experience is borne out by Carol Dweck's research. All it took was telling 5th graders "wow! That's a really good score. You must be smart at this." versus "wow! That's a really good score. You must have worked really hard." to get them to become fearful and defensive in the face of future challenges.

    So I'm not advocating we ignore the ease with which certain students do what we ask of them in math class. I'm advocating we avoid praising them for this ease. This communicates that being smart is defined as things coming easily to you, with the impact that the students learn to avoid and hide struggle. I vote that we should reserve our praise for things like persistence, focus, resourcefulness in the face of frustration, cultivated skill, and generally anything that contributes to success that the students actually have control over.

    I want every kid ("weak", "strong", "good at math", "bad at math" or whatever other ideas about themselves they have) coming to class a) believing that they have something to contribute, and b) excited about growing. If I let kids get the idea that they're "low-ability" then they think they have nothing to offer. If I let them get the idea they're "high-ability" they they (like me back in 2001) will be afraid of a challenge unless they have some sort of reassurance that most other people would see it as _more_ of a challenge. Relish of a challenge shouldn't depend on this.

    Hope this has been at least thought-provoking. All the best.

  5. I'm advocating we avoid praising them for this ease.

    Raw praise? No. But what about having a number of side roads at hand towards which the students who are getting the main subject faster can be pointed. Instead of "you did great at that", what about "you did great at that; what about this?"

    It shouldn't be too hard to come up with good side roads. Even asking a student to try generalizing an example or a problem would serve a great educational purpose of its own. For example, if you're covering the quadratic formula you might ask them to think about how the two solutions change when each of the polynomial's coefficients is altered separately.

    The student still gets the sense that their talent is valued (the value that society reinforces for athletics, among others) but it doesn't happen in a vacuum. Facility with the mechanics has, as a reward, nudges towards deeper understanding behind the scenes.

  6. I like to tell my students that they are budding mathematicians. I mainly teach calculus, which makes that statement easier for my students to swallow. When making the switch to radians, I tell them that they are about to start using the angle measure of mathematicians. I give them projects that require them to research: I know that sequence, it's the Fibonacci sequence, but is there an explicit formula for it? When they ask a particularly insightful question or see a pattern that took others years to find, I praise them for thinking like a mathematician. By the time they leave my classroom, they believe that I am mathematician and that they can become one too.

    On the other hand, I was sitting in one of my geometry courses for my masters this summer and my teacher (a math ed PhD, no less) stated that "mathematicians, unlike you students..." She totally lost my respect at that point.

    I tell my students that a mathematician is someone who studies and uses mathematics. Isn't that what we, the teachers of mathematics, do as well? Must you have a PhD to be a mathematician? Is there a special club you have to join with an initiation problem to solve? (I am a member of the MAA- does that make me a mathematician? Give me a problem and I will solve it.)

    I love math. It is large, beautiful, diverse field. Just because we have not written a thesis does not mean that we can't or we won't. We use math and share our admiration of it with others. Let students- in fact encourage students- to believe that this path can be theirs as well. All it takes is hard work and dedication. I believe there is a field of math out there for everyone.