# But, Algebra is f(u)=n!

Putting real politics aside for a moment, I have a few comments on Andrew Hacker‘s op-ed in today’s New York Times, entitled “Is Algebra Necessary?” I will first answer his question.  Then I will discuss a few logical weaknesses of Hacker’s argument.

(In the interest of full disclosure, I am very proud to be a Unicorn, class of 1992.)

1. Wait, did you expect an answer?  Well, in a nutshell, the appropriate answer to Hacker’s tantalizingly ambiguous question is “yes and no.”  Clearly, algebra is not necessary for potty training, survival swimming, navel-gazing, or even fantasy football (though it helps). Strictly speaking, algebra is necessary for an admittedly much smaller set of life tasks.

The more important point is rejecting the false dichotomy put before us by Hacker.  Implicit in his piece is the presumption that something is either “necessary” or it is in need of serious, urgent reform.

The proper way to address whether algebra should be required is to ask what its mastery  does provide.  This is question of sufficient conditions.  In this case, one relevant conclusion is the fact that understanding algebra implies that one knows how to logically solve a problem.  Hacker might have a point (though it would require a lot more work than is evident in this piece) if he made a more measured argument that requiring algebra is too costly a means by which to ensure that high school graduates know how to logically solve a problem.  But his argument is not of that form.  Rather, he implicitly takes the position that “if something learned in a math class is not directly evident in everyday actions, it should not be required.”

Accordingly, Hacker has provided an argument against requiring that people learn about anything other than:

1. sitting,
3. blogs,
4. internet memes involving
5. Amazon Prime,
6. keyboard shortcuts, and
8. And Amazon Prime.

2. Oh, you meant “Is Unnecessary Algebra Necessary?”  Much of Hacker’s argument reminds me of this great correction. See, Hacker doesn’t want us to think that he thinks that we shouldn’t require, you know, useful math.

I’m not talking about quantitative skills, critical for informed citizenship and personal finance, but a very different ballgame…

Ummmm.  Okay, so the deal here is….what? Oh, yeah…Hacker wants to get rid only of the math that is not “critical for informed citizenship and personal finance.”  My brain hurts…perhaps because of all that math society made me take.  Exactly what are the bounds of “quantitative skills?”  This is never made precise, though apparently long division is a component.  As will become clear below, Hacker would have students learn how to understand where statistics and quantitative data come from and how they are constructed without having students learn about equations and fixed points.

For example, why is some data best described by the mean?  Why is it sometimes best described by the median?  What purpose does the mode serve?  What the hell is a variance?

Consider this interchange in the future.

Teacher: Suppose we flip a fair coin. If we let “Heads” equal 1 and “Tails” equal 0, the mean, or average, flip is equal to one-half.

Student: But, teacher, what does that mean? I’ve never seen a coin land on its edge.

Teacher: Ahh, don’t you worry.  Andrew Hacker assures us that you don’t need to understand that.  Now shut up, go balance your checkbook, and vote.

In short, it doesn’t appear to me that Hacker has thought through one of the central  persuasive distinctions in his argument.  He frames it as a practical offering, but there’s very little practical guidance on how to decide what to keep and what to chuck from the curriculum.  On that note…

3. No, seriously, keep the important math. There may be another explanation, of course, but as far as I can make out, Hacker’s argument is either unintentionally incoherent or simply disingenuous insofar as he pretends to still have the cake he just ate.  For example, consider this snapshot of his stream of consciousness:

Being able to detect and identify ideology at work behind the numbers is of obvious use. Ours is fast becoming a statistical age, which raises the bar for informed citizenship. What is needed is not textbook formulas but greater understanding of where various numbers come from, and what they actually convey.

This makes no sense.  Let me rewrite this in sailing terms:

Being able to detect and identify the direction the ship is moving is of obvious use. Ours is fast becoming a seafaring age, which raises the bar for informed seamanship. What is needed is not concise summaries of how to sail a ship prepared by experienced sailors, but greater understanding of how various parts of the boat work, and how to actually work them.

4. I went to the bathroom and all I got was this lousy NYTimes Op-Ed. My final salvo is aimed at this passage:

What of the claim that mathematics sharpens our minds and makes us more intellectually adept as individuals and a citizen body? It’s true that mathematics requires mental exertion. But there’s no evidence that being able to prove $(x^{2} + y^{2})^2 = (x^{2} - y^{2})^{2} + (2xy)^{2}$ leads to more credible political opinions or social analysis.

Notice the sleight of keyboard here: Hacker does not address the claim that mathematics sharpens our minds or makes us more intellectually adept.  Instead, Hacker asserts that there’s no evidence that knowing how to expand a quadratic equation leads to more credible political opinions or social analysis.

It is undoubtedly true that mathematics sharpens one’s mind and makes one more intellectually adept.  Indeed, it’s “so true” that one might challenge it as tautological.

In addition, and finally, Hacker’s claim that we should revisit the role of algebra and higher mathematics in the curriculum is based upon his assertion that there is no evidence that such training “leads to more credible political opinions or social analysis.”  Even if one grants Hacker’s concise summary of empirical evidence, this is still sleight of keyboard: neither of these conclusions is “necessary” for requiring algebra, at least no more so than it is for any other component of the curriculum.

As the world comes together to bash algebra and Michael Phelps, I leave you with this.

## 4 thoughts on “But, Algebra is f(u)=n!”

1. The main problem I had with his thesis was more selfish than the more logical arguments presented here. As someone who fell in love with mathematics studying Geometry and Algebra, I wonder if I would have ever been exposed to these topics if they were not required. Even if we grant his (shaky) proposition that some high school mathematics should not be required because it fails to meet some test of practicality, unless he is also positing that a surfeit of numerate citizens is a problem, he must address the reality that many who go on to study and work with these ideas might not have done so if it were not a requirement early on (especially as it seems most parents can forgive a students contemning mathematics).

I’ll end in saying that some of the comments did mention a possible common ground in that the actual teaching of some of these concepts should begin much earlier than is now common, thus providing early exposure and not forcing those without aptitude (though even they would be aware of the existence of such tools, if not made to attempt mastery).