The uncrossable line of math


Darren at Right on the Left Coast has spot-on commentary on a spot-on piece in the Seattle Times about math instruction. The part I want to highlight for now is this quote from the Times article:

[B]y high school, kids have spent years marinating in a culture that disses math. Few people in this country boast about being illiterate. But it’s long been a laugh line to declare “I’m not a math person.” Not so in countries such as Japan and Singapore, where students are expected to conquer math — and keep trying until they do.

And in America, where are the math bees, the volunteer math tutorial corps, the math-is-fundamental public-service campaigns? As a society, we root for reading. But we expect success in math to just happen … or not.

Ilana Horn, associate professor of mathematics education at the University of Washington, says that makes a big difference. “We have a belief in innate ability. It perpetuates this idea that you either have it or you don’t, instead of that you aren’t trying hard enough.

“It allows teachers to not question student failure in the same way, and it allows parents to excuse the kids’ poor performance, and kids to excuse their own poor performance.” [Emphasis added]

I think that’s right, although I don’t think Prof. Horn goes far enough. “Innate ability” suggests that some people have more to work with than others — that for whatever reason, some people “get” math faster or more deeply than others. Frankly, we don’t believe in that when it comes to math. We believe in a sort of mathematical predestination — that there is a select group, a chosen few, who have been endowed by their maker with mathematical skill; and that the rest of humanity does not have that skill and cannot attain it by our own efforts, no matter how hard we may work. We believe in an un-crossable line when it comes to mastery and fluency with mathematics — the chosen ones on one side, the rest of humanity on the other, laughing and cracking jokes about how bad we are at math and how nerdy the people on the other side of the line are.

There may well be differences in innate ability with math. Regular readers will remember this comment thread that features a discussion by virusdoc about possible evolutionary reasons for differing innate math ability. Sure, OK, those may exist. They certainly appear to exist, although by the time I see students when they are 18 years old, the layers of evolutionary or biological causes on the one hand and social and cultural forces on the other are so conflated that you simply can’t tell why one student might be doing well and the next one poorly.

But assuming those differences actually exist, and whatever the causes may be, what do classroom instructors make of them? In other words, what do we do about it? I fear that the majority of instructors, high school and college and elsewhere, assume the existence of the uncrossable line and proceed in their classes to separate the sheep from the goats. We make excuses. Instead of setting clear goals for all students that aim toward mastery of the material, and creating classes that push students toward those goals, we mitigate those goals and let the lower-acheiving students settle for less. Because, after all, they just haven’t got whatever the higher-acheiving students have; they are not chosen.

No more excuses, everybody. All of us — math teachers and students, and the culture as a whole — need to come to terms with the importance and centrality of mathematics throughout all of education, and start expecting more of students and of ourselves. We need to erase that line that we have so conveniently drawn.

10 Comments

Filed under Education, High school, Higher ed, Math, Teaching

10 responses to “The uncrossable line of math

  1. Your commentary is spot-on as well!

  2. I noticed that I tended to think I was not a “math” person while in public school. Yet I can remember a brief time when I loved it. I believe, after homeschooling my children and seeing their responses to math, that it is largley due to a technique of passing and failing a student in a test. If I fail, I think I am not a “math person. At home, our children didn’t have the threat or pressure oof passing or failing. We just showed them how it worked and how they could figure it out, and worked with them til they “got” it. If they ever reached a point of frustration, we put it away and went back to something simpler. Math is something that cannot be pushed. It has a lot to do with the doors of the mind opening up, and some people develop at different levels than others. I can go back to math I couldn’t figure out when I was a teenager, and now I can do it just fine. It has somehting to do with maturity, I thnk. We didn’t have pass or fail. If they didn’t figure out a problem correctly, they were shown how to do it over and over until they did. There was no mystery and we didn’t make them memorize anything. They could look up the tables and formulas to the problems any time they wanted, because we wanted them to learn, not trick them or punish them. If this teaching method had been used on me, I would have been a “math person.”

  3. Mrs. S: There does appear to be some rather disastrous interaction of the factory-like, production-oriented setup of education in this country one the one hand and the patience-intensive nature of learning mathematics on the other. I often tell my students that a sense of humor is a prerequisite to any math class, but seriously, I think the stakes are set so high in school that it sucks the fun and wonder out of it all, and students would rather not try at all then risk failure.

  4. virusdoc

    Robert, could I get you to flesh this claim out a little:

    the importance and centrality of mathematics throughout all of education

    Because I’m just not convinced math is all that important and central to all education. Nor do I think that my own field–biology–is important and central to all individuals. I think everyone needs to understand germ theory enough to wash their hands and wear a condom, but I don’t think they all need to understand the inner workings of the cell or the genetic code.

    The same I think is true of math: we all need to be able to do simple arithmetic to survive in this culture, and probably some other skills. But rarely geometry or trig or calc. Perhaps I overestimate the extent of math content mastery you think is fundamental. How much do you think is required for everyone?

  5. Doc: To give a short version of my reasoning here, there are two sides to mathematics — the content (calculating a standard deviation, factoring polynomials, graphing lines, etc.) and the process (problem-solving through structured heuristics; the process of experimentation, conjecture, and proof; etc.). When I talk about “mathematics” I mean BOTH sides of the game, and that’s important for getting my drift here.

    As to what content ought to be required for everyone, that is a moving target. A few decades ago, nobody really needed much knowledge of statistics, whereas now I think you have to know rather a lot to participate in a democratic society. The opposite is true about things like, say, being able to find a square root by hand or even using the quadratic formula — that is, mechanical processes that are typically automated by computers now. (NOTE: I am not suggesting we stop teaching the quadratic formula!) But in general I think the overall amount of math content needed by an educated person is growing as the amount and complexity of information that our society depends upon grows.

    And here’s where the process comes in. Because mathematics basically teaches you how to extract information from data — or more bluntly, extracting order from chaos — and because we are now and forever will be awash in data and in need of information, mathematics is now central to all of education, even if it didn’t use to be. Education is successful when the person being educated can think freely for her/himself, which today means being able to process the input around themselves, turning it into information, and then using the information wisely. The humanities, I think, help us to convert information to wisdom. Mathematics helps us to turn input into information. The content of math is just a means to that end.

    You could make the same point about science. I’m sure you agree (because you’ve written about this before, I think) that not everybody needs to know all the facts about biology, but everyone ought to be fluent with the scientific method and be able to distinguish scientific knowledge from folk knowledge.

    And yes, that is the short version.

  6. A well reasoned reply. I’ll think about it some more, but here are some off-the-cuff responses:

    1) I concur on the content/process division. But I believe that the process you describe is not unique to mathematics. In fact, this type of thinking is central to my discipline and is a core element of scientific training. I don’t find critical thinking to be especially mathematic in nature, although it must be quantitative and evidence-based to be rational. So there is a certain element of math competence required to think critically in the sciences, but I think many of the same thinking skills apply in the humanities (philosophy, most especially) and take the form of logical/rhetorical analysis as opposed to quantitative analysis. And from my experience, most college mathematics courses give “process” lip service while dwelling mercilessly on content.

    2) You are definitely correct about the massive data sets problem affecting the biological sciences. However, the way most biological sciences departments have handled this issue is by hiring a small number of bioinformatics specialists who can assist the rest of in mining our data for information. Most of these people live in “core facilities” where we bring our biological samples for analysis. I clearly need to be able to think quantitatively enough to know if these people know what they’re talking about, but I don’t need to know how to crunch the numbers. (And in my experience, many of them don’t know how to do the raw statistical analyses either–they just know their way around the software!). Perhaps this specialization is merely the end result of the goats/sheep segregation at the grade school level and should be corrected. But content specialization is the norm for any discipline at our level, and I would argue that the system ain’t broke.

    So, although I concede the “process/critical thinking” aspect of your argument, I’m still left wondering whether the math department is the only or even the best place to teach this.

    Off to Vandy for a job interview, so if I don’t comment again for a few days it’s not because I’m offended or disinterested!

  7. I think there are certain aspects that are mathematical in nature and not anything else. (Whether the math *department* should teach that is another question; for instance, when I was a psychology major, I got all my stats courses from the psych department.) Taking raw data and gleaning statistical information from it, for instance, is something uniquely mathematical at its core. Yeah, I know lots of natural and social science fields to this, but I am thinking of the process itself, not which department it may be located in. And in the K-12 level, that sort of training has to come from mathematical instruction in order for the later content specialization to happen.

    Likewise, there are some elements of critical thinking that science is best equipped to handle, which is why I have a similar concern to what I mentioned in the article when it comes to science instruction in this country as well.

    Basically I am thinkiing about this from a liberal arts perspective — what are the general intellectual skills that liberally educated people (=people who can think for themselves) ought to have? And how do we get them there?

    Good luck at Vandy.

  8. JimMc

    I won’t argue the merits of nature vs. nurture but I’ll offer this theory: the more math you learn, the more math you retain. For instance, I think those who have had more math instruction over their lifetime find it easier to recall the basics. I find it much easier to help my daughter with her math homework than my wife does even though I know she’s had the same basics as me, and is every bit as smart as me. It’s just that I got an engineering degree and she got a psych degree. She had physics for poets too!

  9. The real question is this: Why is it that everybody believes that a grounding in literature and the humanities, beyond just a couple of classes, is part of a well-rounded classical education, yet when it comes to math, suddenly the question becomes “but what good is it?”

    The answer, of course, is that math — and not just what you “need” — is as important a part of a well-rounded education as is Shakespeare. After all, few ask what good Shakespeare is.

    Just something to think about.

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