Friday, May 2, 2014

"The Heart of Algebra"

I'm working on a couple of bigger pieces on the SAT and one of the things that I've been looking at as part of the background work is this statement from the College Board discussing the changes in the math section of the test. Board president David Coleman quotes extensively from this and I'd be very much surprised if he hadn't been extensively involved in its writing. (the press releases very much have Coleman's voice.)

Reading these official statements after closely reviewing the old SAT test produces a couple of strange reactions. The first is a disconnect that comes from a list of changes that, with one or two exceptions, seem to describe the test we already have (work with systems of equations, analyze data, use percentages and ratios) and/or contradict other proposed changes (reduce the scope and add "trigonometric concepts").

The second  is a strange lost-in-translation feeling, as if the passages were almost saying something meaningful, but some key words had been omitted or put out of order. Perhaps the best example is this discussion of  linear equations and functions as "the heart of algebra." Coleman seems particularly enamored with this phrase -- he uses it frequently in interviews about the SAT -- but when I read through the press statement, I didn't see anything that made linear functions more important or fundamental than other polynomial functions (or rational functions or logarithmic or exponential functions for that matter).

Here's a little experiment. Read the passage below extolling the importance of equations and functions based on linear expressions. Then read it again but mentally strike out every occurrence of 'linear' except for the parenthetical phrase. I think you'll find it actually makes as much sense.
Heart of Algebra: A strong emphasis on linear equations and functions
Algebra is the language of much of high school mathematics, and it is also an important prerequisite for advanced mathematics and postsecondary education in many subjects. Mastering linear equations and functions has clear benefits to students. The ability to use linear equations to model scenarios and to represent unknown quantities is powerful across the curriculum in the postsecondary classroom as well as in the workplace. Further, linear equations and functions remain the bedrock upon which much of advanced mathematics is built. (Consider, for example, the way differentiation in calculus is used to determine the best linear approximation of nonlinear functions at a certain input value.) Without a strong foundation in the core of algebra, much of this advanced work remains inaccessible.
You might make a pretty good case for the central importance of polynomials (particularly if you want to get nerdy and bring in Taylor). You can make a great case for the central importance of functions. You can even make a crawl-before-you-walk case for focusing on linear expressions. But you have to make some sort of coherent argument.

Even the part about finding the slope of the tangent at a given point (that is what they're talking about, right? or am I missing something?) has an odd quality. It's difficult to see how using a derivative to help find the equation of a line makes linear equations the 'bedrock' of more advanced math. There are certainly examples where linear equations are used to find formulas and prove theorems in calculus and other more advanced fields, but the example in the parenthesis actually goes the other way. To me, the passage as a whole and the parenthesis in particular read as if the author had asked someone knowledgeable "where do we use linear equations and functions?" and had paraphrased the answer with only minimal comprehension.

What's so strange and somewhat sad about that possibility is the extraordinary pool of mathematical talent that was hanging around the halls when this was written. If you take a tests and measurements class, you soon realize that most of the good examples come from the SAT. The people who put the exam together are exceptionally good in a highly demanding field of statistics.

Not listening to people with experience and expertise is a noted characteristic of and perhaps even a point of pride with Coleman, who came into the field as a McKinsey & Company consultant and had no relevant experience in education or statistics.
When Coleman attended Stuyvesant High in Manhattan, he was a member of the championship debate team, and the urge to overpower with evidence — and his unwillingness to suffer fools — is right there on the surface when you talk with him. (Debate, he said, is one of the few activities in which you can be “needlessly argumentative and it advances you.”) He offended an audience of teachers and administrators while promoting the Common Core at a conference organized by the New York State Education Department in April 2011: Bemoaning the emphasis on personal-narrative writing in high school, he said about the reality of adulthood, “People really don’t give a [expletive] about what you feel or what you think.” After the video of that moment went viral, he apologized and explained that he was trying to advocate on behalf of analytical, evidence-based writing, an indisputably useful skill in college and career. His words, though, cemented his reputation among some as both insensitive and radical, the sort of self-righteous know-it-all who claimed to see something no one else did.
Coleman obliquely referenced the episode — and his habit for candor and colorful language — at the annual meeting of the College Board in October 2012 in Miami, joking that there were people in the crowd from the board who “are terrified.”
Given some of the changes we've seen in the test the College Board worked so hard to get right (the loss of orthogonality, the shoehorning in of "real-world" data), we may have some idea what they were scared of.

1. I don't know if this helps, but . . . as a practicing statistician, I think that polynomials are way way overrated, and I think a lot of damage has arisen from the old-time approach of introducing polynomial functions as a canonical example of linear regressions. There are very few settings I can think of where it makes sense to fit a general polynomial of degree higher than 2. I'm not sure how I'd change the high school math curriculum to deal with this, but I do think it's an issue.

1. Andrew,

Pedagogically there are a number of excellent reasons for covering polynomials in depth -- they can provide insight into number systems, exponents, non-linear relationships; they give students valuable practice manipulating multi-part expressions and working with functions; they can form the basis for problems that are challenging but still simple enough to check answers by hand -- but in practical terms, most of the things students learn to do with polynomials will never actually come up, even in the world of STEM.

One of my ongoing concerns is that the people in charge of the reforms often don't get the essential distinction between the practical and the pedagogical. You can use high-order polynomials as an example of certain modeling concepts but when I see them used in an actual model I assume that the modeler is either overfitting the data or doesn't know enough math to use the appropriate transformation.

From the curriculum side, I'd probably spend less time on polynomials and more time both on other non-polynomial functions and on the bigger question of deciding which family of curves is appropriate for describing which phenomena.

2. P.S. Do you really need that captcha thing for your comments. I find it really annoying to have to do this, it deters me from commenting sometimes.

1. Andrew,

To be perfectly honest, I don't remember ever looking at the settings for comments (I believe we just used the defaults). I'll talk to Joseph about making the process easier.

Thanks for letting us know about the problem.

3. I was not aware there were captcha's. I am a bit tech limited as I am still recovering from surgery but will try and puzzle this out once I am using a better tool than an iphone

4. Polynomials are awfully useful for programmers. They give one a way to tell how well one's algorithms will scale.

1. Kaleberg:

I think you're talking about power laws: y = A*x^b. That's fine. My problem is with polynomials with multiple terms: y = a*x^3 + b*x^2 + c*x + d, that sort of thing. I think that millions of students have been brainwashed into thinking of these as the canonical functions and that this has caused endless trouble later on.