Tuesday, October 28, 2008

Imaginary Education

A main part of my job as a TA involves working directly with the
results of our country's education system. This has been extremely

One guy who came in for tutoring was incapable of dividing 800 by 400
in his head. He had come in to my office for help with a problem, and
our problems are usually designed so that you can do it without the
aid of a calculator. That's the theory, anyway. The problem required
finding the percentage change of something, which meant doing the
aforementioned calculation. I was guiding him through the problem,
and when we got to that part, he said, "I can't do it; I don't have my
calculator with me." I said, "You've got scrap paper; do it by hand."
The words did not seem to register. He looked at me like I had asked
him to calculate a cube root by hand.

Now, this guy admits that he is not good at math. But he got acecpted
to a big-name university, one of the top 30 in the nation. This year,
we accepted about 3,000 incoming freshmen out of over 15,000 freshman
applications. Half of the incoming freshmen were in the top 10
percent of their high school graduating class, and they have an
average SAT of 1226.

My aunt is an elementary school teacher with a couple of master's
degrees in education. I asked her at what grade students would be
expected to divide 800 by 400. Her answer: second grade.

So we have a guy who got into a high-quality, selective university,
and he can't do second-grade math reliably. I'll give him the benefit
of the doubt; maybe he could have done it if he really had to. But
this learned helplessness, this dependence on machines to do math, is
endemic among the freshmen I am teaching.

I toyed with the idea of giving a five-minute math quiz in class one
day. I'd hand out a paper with 100 simple arithmetic problems, give
them five minutes, prohibit calculators, and see what the results are.
When I mentioned this to my aunt, she pointed me to some websites
that have math testing materials for elementary school tachers. None
of the tests fit what I wanted to do, but I say something that
literally stunned me: a test on imaginary numbers meant to be given to
fifth graders. My aunt confirmed that the standardized tests do
indeed include imaginary numbers.

This is ridiculous. Nobody except mathematicians and electrical
engineers, and maybe computer programmers, needs to know anything
about imaginary numbers. I was first introduced to imaginary numbers
in an elective class in my senior year of high school, at a
math-oriented magnet school. I am studying a math-intensive research
field, and I will never, in my professional career, ever use imaginary
numbers. There is no possible justification for trying to test
elementary school students on this stuff.

Clearly this is a symptom that the people who make these tests are
losing contact with reality. What were they thinking? Maybe it was
along the lines of "Advanced mathematicians use this stuff, and it is
technically possible to make the kids memorize it, so if we teach it
we are teaching high-quality math." This is of course rubbish.
Making kids memorize the rules for manipulating complex numbers does
nothing to teach the fundamental concepts. The brains of little kids
are simply not equipped to grasp the concept of 'The square root of
negative 1'. I can barely understand the meaning of the concept. I
would be pleasantly surprised if a fifth grader even understood the
concept of the square root, and knew how to use it.

Is it any wonder that kids ignore or forget basic math lessons? If
the imaginary numbers are any indication, they have been subjected to
cargo-cult math lessons their entire life. They may be able to
memorize a routine and pass a test, but the system has clearly put
very little effort into teaching them how to actually use math. Math
is a set of tools that you use to understand reality. If you don't
understand that, then you will always be hopeless at it. If you don't
know how to connect math to reality, than the math is useless.

I have opten complained about how algebra is a requirement for getting
a high school diploma, even as there is almost no effort to teach
basic financial literacy. I would guess that only about 10% of the
people on the planet actually need to use algebra in their jobs. But
everyone needs to know about interest rates, compounding, and the time
value of money. And if you don't take business courses in college,
you will never learn about these kinds of things. One of the main
reasons for the economic mess we are in now is that people simply do
not understand what it means to borrow a lot of money.

Now, algebra does help you work with the interest rate equations. If
you don't have algebra skills, you have to learn half a dozen formulas
instead of just one or two. But people are much better at
memorization than symbolic manipulation. Most people simply do not
possss the kind of mind that is required to be comfortable with
algebra, and it is foolish to expect them to learn this esoteric skill
at the expense of things that you need to know to live a good life.

Teaching imaginary numbers in elementary school is far, far worse than
teaching algebra in high school. Instead of something that 10% of the
population needs, it is something that 0.01% of the population needs.
It is also worse than useless. The kids, if they have any sense, will
forget everything they ever learned about imaginary numbers right
after they take the test. This creates horrible habits. Not only do
they learn to forget the useless routine of the week, they also learn
to forget the basic life skills that they will really need.

I have seen more and more evidence that our education system is
completely failing to provide students with a useful, science-based
mapping of reality. Instead, it all seems to be devoted to putting
random facts and arbitrary routines into short-term memory. Now,
memorizing facts is important. Facts are the bricks that you use to
build the structures in your mind. But you need something to connect
that knowledge to, so it becomes long-term knowledge. Facts without
context are soon forgotten.

Education should be about teaching people how to understand and work
with reality. I have seen little evidence that our system
accomplishes this.

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