From the current issue of *The New York Times Magazine*:

One of the most vivid arithmetic failings displayed by Americans occurred in the early 1980s, when the A&W restaurant chain released a new hamburger to rival the McDonald’s Quarter Pounder. With a third-pound of beef, the A&W burger had more meat than the Quarter Pounder; in taste tests, customers preferred A&W’s burger. And it was less expensive. A lavish A&W television and radio marketing campaign cited these benefits. Yet instead of leaping at the great value, customers snubbed it.

Only when the company held customer focus groups did it become clear why. The Third Pounder presented the American public with a test in fractions. And we failed. Misunderstanding the value of one-third, customers believed they were being overcharged. Why, they asked the researchers, should they pay the same amount for a third of a pound of meat as they did for a quarter-pound of meat at McDonald’s. The “4” in “1/4,” larger than the “3” in “1/3,” led them astray.

Heh. It reminds of something I read in John Allen Paulos’s book *Innumeracy*. He described watching television with a group of people when a commercial came on. The ad claimed that its prices were a fraction of the other guy’s prices. Paulos quipped that the fraction was probably 4/3, and was mostly met with blank stares from the other folks in the room.

That’s the high point of the article, though. The rest is just the standard clichés of education reporting. American math education is terrible because it focuses too much on rote learning. We should be emphasizing reasoning and problem solving and blah blah blah. These two excerpts will set the mood:

Instead of having students memorize and then practice endless lists of equations — which Takahashi remembered from his own days in school — Matsuyama taught his college students to encourage passionate discussions among children so they would come to uncover math’s procedures, properties and proofs for themselves. One day, for example, the young students would derive the formula for finding the area of a rectangle; the next, they would use what they learned to do the same for parallelograms. Taught this new way, math itself seemed transformed. It was not dull misery but challenging, stimulating and even fun.

And later:

As soon as he arrived, he started spending his days off visiting American schools. One of the first math classes he observed gave him such a jolt that he assumed there must have been some kind of mistake. The class looked exactly like his own memories of school. “I thought, Well, that’s only this class,” Takahashi said. But the next class looked like the first, and so did the next and the one after that. The Americans might have invented the world’s best methods for teaching math to children, but it was difficult to find anyone actually using them.

It’s not so much that the main claims in the article are wrong. It’s just that this article gets written over and over again, each time presented as though the reporter has made some great discovery. Yes, if you present math to students as strictly a matter of rote memorization then, in addition to driving them mad with boredom, you are going to leave them with little ability to apply what they have learned to practical situations. After all, life does not present you with clear-cut problems in the manner of textbook exercises. This is news?

There are all sorts of reasons for why it is so difficult to change the habits of math teachers. Here is one:

Consequently, the most powerful influence on teachers is the one most beyond our control. The sociologist Dan Lortie calls the phenomenon the apprenticeship of observation. Teachers learn to teach primarily by recalling their memories of having been taught, an average of 13,000 hours of instruction over a typical childhood. The apprenticeship of observation exacerbates what the education scholar Suzanne Wilson calls education reform’s double bind. The very people who embody the problem — teachers — are also the ones charged with solving it.

I see this every semester in my own classes, especially the ones that are directed toward future teachers. Students are happy to memorize formulas, and they will apply those formulas with machine-like precision at test-time. But if you ask them to prove anything, or to explain why something is true, they feel betrayed. They will regard you as unfair for asking such a thing.

Shortly after you present a new concept to students, you can be certain they will ask, “How will you ask us about this on a test?” They are not amused by the correct answer, which is that they should focus on understanding the ideas, because if they do that they will be able to answer any question I ask. When it becomes clear that you are serious, they often become visibly nervous, even scared.

It’s very hard to break that mindset in a semester, or even several semesters. Inevitably, even at the college level, they can get pretty far just by memorizing formulas. A typical test will have a mix of computational questions and proof-oriented questions. The better students will clean up on the computational questions and, through the miracle of partial credit, generous grading, and steep curves, will pick up enough points on the proofs to get a decent grade.

For all my snark, the article is worth a look. It’s very supportive of the Common Core math standards, which I do not really know much about. I do think, though, that there are two big elephants the article ignores. One is class size, and of school resources more generally. When you have, say, fewer than twenty students in a classroom, you have the freedom to be very interactive and to get everyone involved in the discussion. Once class sizes get above thirty, and when the classroom is physically very crowded and unpleasant, your pedagogical options are far more limited.

The other problem is that, in this country, teaching is not a respected profession. Frankly, teachers are usually treated with contempt. People who wouldn’t last five minutes in a classroom protest that teachers have it easy, since they only work ten months out of the year. In most districts they are poorly paid and not given adequate supplies for their classroom. The only reason they get the crumbs that they do is that the unions fight tooth and nail to get them, and that is why teachers unions are uniformly reviled by the media and politicians. Meanwhile, there has been a wholesale abandonment of the public schools in this country, and for once it’s not just the Republicans who are responsible for it. In that environment, should we be surprised that talented people with options rarely decide to go into teaching?

In that regard, permit me a personal anecdote. My mother was a public school teacher, spending much of her career in high school special ed. Of course, when she was in college, teacher, nurse or secretary were pretty much the only options for women. Early in my junior year of college, when I was starting to think seriously about what to do after graduation, I suggested to her that I might be interested in teaching high school math. She pitched a fit. Off the top of her head she tossed off about a dozen reasons why that was a bad idea. She was so convincing that I decided that graduate school was worth another look.

I’ll close with my own favorite example of innumeracy. In 2000, in my first semester at Kansas State University, I taught a course for future elementary school teachers. Manipulating percentages was something we spent a lot of time on, and since 2000 was a Presidential election year I decided to put an election-themed question on the test. I pointed out that, at that time, Kansas accounted for six electoral votes out of 535. I then went to the census bureau and got the population figures for both Kansas and the United States generally. The problem was then to work out the percentage of the electoral college represented by Kansas as compared to the percentage of the population that lived in Kansas.

Well, it turned out that .97 percent of the population lived in Kansas. Several people, alas, made a decimal point error and arrived at an answer of 9.7 percent. Now, anyone can make a decimal point error. No shame in that. I pointed out, though, that they could have caught the error had they realized the implausibility of suggesting that one in ten Americans lived in Kansas. I noted that around three million people lived in Kansas, as compared to eight million who lived in New York City. They just stared at me blankly. It had never occurred to them that the answer should actually have some connection to reality.