Saturday, December 6, 2014

Puzzle Math, II: abstract pattern recognition

I’ve already blogged about the abstract patterns in math that are vanishing from today’s algebra classes. But my daughter just completed a particularly elaborate set of problems in her 1920s algebra book (Wentworth's "New School Algebra") for which abstract pattern recognition is absolutely essential:

To solve these equations, it’s massively helpful to recognize that each has the form of a quadratic—i.e., of ay2 + by +c.

For example, the first problem is of the form a(x2)2 + b(x2) + c, so if you let y = (x2), then you have y2 - 5y + 4 = 0. And if you’ve had enough practice with factoring, you immediately recognize the resulting pattern of coefficients and see what the roots of y are.

Similarly, in 8, the underlying quadratic structure is evident if (after adding 20 to both sides and putting the terms in descending order of their exponents) you take y = x1/4. The result: 2y2 - 3y + 20 = 0.

For 9, let y = xn, yielding 5y2 + 3y – 6 ¾ = 0.

For 11, let y = sqrt(x2 - x + 1), yielding 2y2 - y - 1 = 0, which, in turn, can be factored in a snap.

For 17, you see that you almost have a quadratic pattern. So, after adding 3 to both sides, express the 3 on the left hand side as 9 + -6. Then you have, with y = sqrt(2x2 + 3x + 9), y2 - 5y - 6 = 0—which (assuming you recognize the pattern of coefficients) is blissfully easy to factor.

Finally, for 20, which might at first seem despairingly removed from ay2 + by +c form, you notice that the coefficients of the x and x2 terms outside the radical are twice those of the corresponding terms under the radical. So you don’t despair, but subtract (4x + 9) from both sides of the equation. Then, as in 17, you split the constant term, expressing -9 and -6 + -3. Then you have 2x2 - 4x - 6 + sqrt(x2 -2x - 3) – 3 = 0. Let y = sqrt(x2 -2x - 3), and you get 2y2 + y - 3 = 0—which solves itself in a snap by factoring--assuming, once again, that you recognize the friendly pattern of coefficients.

Naturally, many American “math education experts” would look at this and, especially if they don’t actually try to do the problems (wouldn’t that be fun to watch!), write it all off as mindless manipulations of meaningless symbols. But what we have here, among other things, are wonderully abstract, multi-layered patterns—of the sort that are completely absent from Reform Math and Common Core-inspired math problems.

So many people wax poetic about how mathematics is all about patterns, and how doing math is all about pattern recognition. But how many of them realize what that really means? How many self-proclaimed math enthusiasts realize that the patterns of concrete quantities and 2-3 dimensional shapes so often used to illustrate mathematical beauty and mathematical depth are only the beginning, and that the most powerful patterns in math extend far below these shiny surfaces?

And how many self-proclaimed math experts realize, in the spirit of Puzzle Math, just how much fun it can be to unearth or sculpt out the underlying patterns and use them to disentangle what at first glance seems hopelessly disordered, unaesthetic, inelegant, and, in short, unpatterned?


Barry Garelick said...

Excellent. The "math is about patterns" meme is usually limited to IQ test type problems that depend on inductive (rather than deductive) reasoning. Supposedly this is to develop the "habits of mind" of algebraic thinking outside of the algebra class (like in 6th grade). In real math, the appropriate habits of mind are developed simply by doing the work. The fluency and the skill in recognizing key "patterns" comes from continual exposure to the material and a grounding in appropriate tools of the trade, like subject content, experience, knowledge and skill.So it's one thing to say "math is about patterns" but you still need the subject content and skill to know where to look and what to do with what you find.

By working from routine problems, students can then progress to non-routine problems and beyond. See also this article.

Niels Henrik Abel said...

Unfortunately, a lot of students have trouble with these kinds of exercises. The ones I run across tutoring get stumped with problems that look like #1, which is really rather easy if you know your stuff. The latter ones (e.g., with the radicals) would not even be attempted by such students, I'm sure ~

Anonymous said...

What book is this from?

Katharine Beals said...

It's Wentworth's New School Algebra--just updated the post with that info.