Before I get to worrying about algebra, Andrew Hacker’s essay in the Sunday NYT made me worry about writing and research. As in, “This is poorly written” and “I don’t think he did much research”.
If I were marking the column up like a first-year student’s paper, I’d immediately be all over the meandering, confused structure of the essay and its tone-deaf alternation of tremulousness and tendentiousness (a combination that a lot of Hacker’s writing in recent years has demonstrated). And then I’d mark it up for the weakness of the research behind it. All of the questions he’s asking have been asked before and debated at length in the history of American education in general and mathematics education in specific, but much of the affect of Hacker’s own essay is of the discovery of some long-ignored or never-asked question. Most crucially, he never really asks (or looks into) the basic question, “So why do most mathematics educators believe so strongly that algebra is an important objective in K-12 education?”
The way that Hacker frames the issue is consistent with the corrosive form of populism he’s been peddling lately, that he is uncovering a sort of “educators’ conspiracy” which has no real explanation other than the self-interest of the educators. If he were to frame it as, “This is an interesting on-going debate where the various sides have coherent or well-developed arguments that have both technical and philosophical underpinnings, and here’s the side that I’m on”, he’d be doing a public service. As it is, he’s just yanking some chains, either calculatedly or out of feeble cluelessness.
If you were going to reassemble the column so that it built up to a genuine argument, I think it might look something like this:
1. Algebra is a common part of the mathematical education of most Americans as well as in other school systems around the globe. By way of introduction, here’s what algebra is. (Baseline definition.)
2. Algebra is a common stumbling block for American students, far more than other subject they study in K-12 education. (Evidence thereof, which Hacker cites fairly well.)
3. Why do we believe algebra is an important educational objective? What do mathematicians, educators and others say about this? How did it get into the common K-12 sequence?
3a. Because algebra is believed to be an important conceptual precursor to every other form of advanced mathematical thought and inquiry.
3b. Because algebra is believed to be an important practical precursor to mathematical skills used in many professions. E.g., because “you will need it later in life”.
3c. Because algebra is believed to be “good to think”, a way to get high school students to regard mathematics as a form of critical and imaginative thought rather than an area of rote calculation.
Here’s where Hacker really falls down: these questions aren’t evaluated or explored a remotely systematic or coherent way.
4. Are these assertions true?
4a. Could you learn other fields of advanced or practical mathematics without any knowledge of algebra? Or is there a simple knowledge of algebra that is sufficient for certain kinds of progression?
4b. Do many careers really use algebra, or have knowledge of algebra assumed in their use of quantitative skills and data? Are there everyday uses of algebra that are important to an educated citizenry?
4c. Is algebra really useful for quantitative forms of critical or imaginative thought?
5a. If 4a. and 4b are in fact true, is there a different or better way to teach algebra that would allow more students to progress successfully through it, or at least to mitigate or excuse their inability to do so? If 4a and 4b are not true, why do we believe them to be true?
5b. 4c presupposes that the goal of high school is progression towards critical and imaginative thought. Are we sure that should be the case? If it’s not the case for math, shouldn’t that be true for everything? Maybe this is an argument against high school in toto, at least as it commonly exists?
5a is where Hacker’s essay seemed to me to just crash and burn. He fumbles around in the dark when he concedes that yes, it’s important for people to be quantitatively literate both as citizens and for their employment prospects but that no, you don’t need algebra for either. I’m not particularly quantitatively literate myself, but I think trying to read and work with statistical data with no knowledge whatsoever of algebra would be very difficult. I don’t know how you’d do anything with algorithms without having some conceptual grasp of algebra. (Just to mention two of the things that Hacker agrees citizens and employable people ought to be able to do.)
It might be that there is a different way to approach algebra that helps high school students glean some of its conceptual value, that there is a problem with how it is commonly taught or imagined. I’m sympathetic to that general question about most high school education. For example, while I think the study of literature or history and the craft of analytic writing should have a progression throughout high school, there’s plenty of room to question what kinds of literature students should read, or what ways they ought to study and know history. But this sort of terrain is way too sophisticated and subtle for Hacker, who is really doing a lot to degrade the brand value of expertise lately.