Math is Hard, Let's Just Quit

Mon 30 July 2012 by psu

Andrew Hacker has pissed me off before. A few years back he wrote a book where he claimed that the standard humanities-oriented "Liberal Arts" curriculum was the only one worthy of actually being taught at the university level. Now, he might have just been making an oblique argument that our current college programs tend to be too vocational, and I'd agree with that. But he wrapped his point in a lot of inflammatory bullshit that to me meant that he really didn't think that mathematics and its related fields were really worth anything.

Today he has proven that my instinct was right. Mr. Hacker has written another piece of inflammatory bullshit with the exciting title Is Algebra Necessary?". Reading this piece it's clear that Mr. Hacker didn't like math when he was in school and I think he still doesn't really think that it's worth all that much at the college level. He seems to think that math is all about numbers, computation and a lot of tedious busywork.

But this is not what math is about at all. Math is the systematic study of the patterns and rules that govern various aspects of our lives including, but not limited to, counting things (arithmetic), arranging things in space (geometry), using fixed sets of rules to compute interesting answers (computation) and of course, the subject of the article at hand: algebra.

Algebra is a bit harder to describe in a short sentence. Algebra is about understanding the basic rules that govern things like arithmetic in the same way that Euclidean geometry is about understanding the rules that govern objects in space. Algebra is a game of puzzles. You take a situation where all the answers are given to you and you see how much you can take away while still being able to get to the right answer. This is why you get those awful word problems in algebra class. The problem is a little story with one piece missing, and your job is not only to compute that answer from what you know, but also to understand the principles involved in knowing beyond any doubt that your answer is right. The point of the exercise is to begin to learn about abstraction.

In mathematics, abstraction is the process by which you go from single examples to universal truth. So, back in the day Pythagoras looked at a bunch of right triangles and thought "maybe there is a relationship between the lengths of the sides." Then he proved his famous theorem, which is famous because it is true for all right triangles and because it's simple enough that you can use it to teach ninth graders what it means to prove something to be true. Ultimately this is what mathematics is about. Not numbers, or simple formulas. It's about making up rules, observing patterns, and then proving that the patterns are true.

This is a powerful tool and ultimately it is this kind of puzzle solving and hole filling that leads you to not only all the other branches of higher abstract mathematics (calculus, abstract algebra, number theory, geometry, etc) but also to any sort of field that has its basis in rational thinking. This includes the obvious ones like physics, engineering, chemistry, and also less obvious ones like law, philosophy, or any important area of human thought in the last few thousand years.

Now at this point you, or Hacker, might say something like "This is all well and good, but why do we have to learn from the bottom, why not just teach the good stuff?". The answer to that is also more complicated. In my experience, the only way to learn these things well is with a three stage feedback loop:

  1. You gain experience with the concrete operations. You roll balls down planes. You learn how to do basic arithmetic, probably by rote. You write simple programs in BASIC.

  2. You use your experience to hypothesize that other more general rules might apply, and you use the mechanisms of logic and proof to figure out if you are right or not.

  3. You take the true ones and apply them to more interesting concrete problems.

I think most people who have spent any time in this world at all will agree that you need all three of these stages to really learn something well. Skipping step (1) makes it hard for you to understand the motivations and rationale behind any higher level rules you might learn in step (2). Skipping step (2) to work directly in "applications" is equally hopeless. If you provide people tools without any real understanding of where they come from, you just create an army of drones who can solve a fixed set of problems ... and we can make software to replace those people.

Which brings me back to the article at hand. The first half of the piece basically argues that "Algebra is Hard." Various statistics and anecdotes are used to show that a terribly large number of people seem to get hung up on math. For example:

Algebra is an onerous stumbling block for all kinds of students: disadvantaged and affluent, black and white. In New Mexico, 43 percent of white students fell below “proficient,” along with 39 percent in Tennessee. Even well-endowed schools have otherwise talented students who are impeded by algebra, to say nothing of calculus and trigonometry.

On the face of it this is not too disagreeable. It seems to be backed up by the facts. But it also misses the point. Algebra is not hard in any fundamental way. You can teach some of the basic ideas to a bright 8 year old. Learning algebra is no harder than learning to write complete sentences, or learning why Henry V is a pretty good play. Algebra is hard for many people because math is taught badly. In fact, I'll claim that for the most part math is taught in a way that is singularly designed to suck any possible joy and interest out of the subject. Maybe English and Shakespeare are also taught this way. I'm not sure. But I'm pretty sure it's true for math.

But this is not the point that Hacker wants to make. His main claim is:

  1. Math is hard.

  2. No one needs to learn the core principles, instead we should just teach "math-like" skills without making them learn the awful boring machinery underneath.

So for example, he writes:

Quantitative literacy clearly is useful in weighing all manner of public policies, from the Affordable Care Act, to the costs and benefits of environmental regulation, to the impact of climate change. 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.

Hacker tries to make this claim that we should replace actual mathematics with something that teaches math-like skills over and over again. Here is one more example:

Thus mathematics teachers at every level could create exciting courses in what I call “citizen statistics.” This would not be a backdoor version of algebra, as in the Advanced Placement syllabus. Nor would it focus on equations used by scholars when they write for one another. Instead, it would familiarize students with the kinds of numbers that describe and delineate our personal and public lives.

Finally, he pulls out the old "but you won't use it in your job anyway", trope:

Of course, people should learn basic numerical skills: decimals, ratios and estimating, sharpened by a good grounding in arithmetic. But a definitive analysis by the Georgetown Center on Education and the Workforce forecasts that in the decade ahead a mere 5 percent of entry-level workers will need to be proficient in algebra or above.

Here is my problem with all of this. Remember my little feedback loop up above? Each one of these proposals is nothing more than a lazy attempt to take a shortcut around one or more of those steps. You just can't do this. What good is it to teach people how the consumer price index is computed if they don't have the algebraic and statistical basis to understand what weighted averages are? For that matter, how do you even teach statistics without the machinery necessary to explain its basic concepts (i.e. algebra, and some basic calculus). Finally, who cares if you use the material itself every day for the rest of your life? If this were the only basis for determining the curriculum at a college or university then we'd just shut all the schools down.

The important thing is teaching you the feedback loop, not making sure that in your job it's very important that you remember what the quadratic formula means. If you want to teach people math-like skills then the way to do that is to figure out how to teach them the math in the first place.

It's important to remember that these skills form the basis of the rational systems of thought that are at the core of so much of our modern society. It's simply not possible to claim that these skills should not be taught in the standard liberal arts education. Such an education is supposed to represent the most important ideas that we have yet developed, and certainly basic mathematics qualifies.

But maybe Mr.Hacker is not arguing this. Maybe he is arguing that there should be different tracks available for people who, for whatever reason, are not prepared to deal with the vagaries of learning high school algebra. I can get behind this idea, as it points out one of the great weaknesses in our system of higher education. That weakness is that "college" has turned into a catch-all for every kind of learning that you might do past high school rather than having its older, more rigorous meaning. But if Hacker thinks this then either his thinking or his writing is not very clear. He never comes out and says that this is what he would propose. What he says is that we should stop requiring algebra at the university level because it's too hard and besides, it's also useless to most people in their later lives. This is an idea that I obviously cannot abide.

My little feedback loop is important to me. Perhaps I am irrationally attached to it, but I think it's clear that people do not see it enough while growing up and going through school. In my work with computers and the people that they confuse I find it abundantly clear that the reason people have a hard time with the machines is that they have a hard time building a model in their mind of what the machine is going to do to them next. Therefore, every day with the computer is full of new and exciting surprises. Surprises that tend to do things like destroy all your work. But it does not have to be this way. We can teach people the tools they need to not be surprised by the machines. Or to not make ludicrous assumptions about unpredictable risk factors like mortgage failure rates. Or to understand random sampling and why election polls are mostly stupid. But there are no short cuts. We have to teach them the math first. To do otherwise is to continue to fail.

Category: Things