Mastery of What?

A lot of conversation on one of the lists I read about spiraling vs. mastery in mathematics curricula. Nothing new about that. Just another issue that strikes me as adding more confusion than clarity by creating false dichotomies instead of seeing that most of these things go hand in hand. Frankly, I'm hard-pressed to see how it would be possible to teach or study something as enormous as merely the tiny slice of mathematics we want all students to learn and be able to use in K-12 education (which brings us up to pretty much nothing invented in the field as recently as the 17th century and still excludes enormous amounts of what was already known when Newton and Leibniz were inventing the differential and integral calculus), without doing a reasonable amount of "spiraling" (which is to say that we must revisit already-explored ideas again when students have more sets of numbers to look at, say, or have developed sufficient mathematical maturity to delve deeper into things that many of us mistakenly think of as simple, elementary, "easy," basic, etc. At the same time, it's hard to move forward (at least in the linear way we teach the subject in the US), if you haven't attained some facility with the procedures (if not the actual ideas behind them) that are often referred to as "the basics" (that's the stuff educational conservatives keep telling us we need desperately to "get back to," as if we've somehow been skipping that and teaching partial differential equations, differential geometry, and category theory to elementary students while they weren't watching us crazy progressives carefully enough, and now we must return to sensible arithmetic, a taste of geometry, some faux algebra, etc.) However, interesting questions arise about whether it is absolutely mandatory to shove every child through the same narrow funnel, and whether much wouldn't be gained by giving kids (and teachers) a lot more options about what route(s) they want to explore on their ways up and around in the tree of mathematics. (see Dan Kennedy's provocative "Climbing Around In The Tree of Mathematics"

In any event, here's some of what I've been thinking about regarding the whole "mastery" thing that so many people worry about (or say that they do):

People talk a great deal about "mastery" in K-12 math, as if mathematics was somehow like typing. You practice and attain mastery.

But that part of mathematics, while important up to a point, isn't really what mathematics is except for kids (and then, only a small piece of it).

Consider the following quotation:

"There is something odd about the way we teach mathematics. We teach it as if assuming our students will themselves never have occasion to make new mathematics. We do not teach language that way. . . the nature of mathematics instruction is such that when a teacher assigns a theorem to prove, the student ordinarily assumes that the theorem is true and that a proof can be found. This constitutes a kind of satire on the nature of mathematical thinking and the way new mathematics is made. The central activity in the making of new mathematics lies in making and testing conjectures." (Judah I. Schwartz and Michal Yerushalmy, quoted in "Geometer's Sketchpad in the Classroom" by Tim Garry, in GEOMETRY TURNED ON, p. 55).

You could readily change a few words above and talk about problem-solving as well. We pose problems to students the solutions for which are well-known. Students cry "Foul!" when confronted by: 1) problems they haven't specifically been trained to do. That is, they think it's dirty pool to be asked to solve problems that aren't identical to others the teacher/book has explicitly worked through with them, or nearly so. These, of course, aren't problems, but rather they are exercises, just like typing drills. I show you how to use the keys with your right index finger, then drill you on that, and so forth; b) problems for which the solution pushes them beyond the immediate topic, perhaps calling on general strategies they've learned, things they've worked on, but also a bit more, stretching their minds, asking them to reach a bit, speculate, imagine; and c) any sort of problem that has no known solution (or maybe just no solution with the methods or numbers they're familiar with) just so they learn that math never ends, mathematics is always being extended and invented, and part of what drives that is unsolved problems (along with new problems, new math, and so on). It could be something as simple as asking a student who hasn't learned about negative numbers what 6 - 8 equals, or someone who hasn't learned about complex numbers to consider the equations x ^2 + 1 = 0. Or it could be having students explore accessible but unsolved problems like Fermat's Last Theorem (when it was still unsolved) or the Goldbach Conjecture. Or something like the three utilities problem or Konigsberg Bridge problem which led Euler (see photo above) to invent new mathematics (graph theory). Fun stuff, really, but many kids think all problems in math class should be trivial exercises, not real problems for them to speculate about and experiment with.

I think much of the error we make in putting so much emphasis on mastery lies in cheating students of knowing what it means to think mathematically, even though they are quite capable of doing so. There is a body of work out there that suggests kids can do much more mathematical thinking than we give them credit for. But for most, by the time they get to do some, they hate mathematics (even though they actually don't know what it is, really).

Just a little late-night food for thought.