Tag Archives: algebra

How to teach algebra in 15 minutes

Yes, I did try to teach algebra in 15 minutes, before a panel of about a dozen math professors at that of the University of the Philippines Institute of Mathematics.

In previous pieces, I had expressed frustration at the lecture-based method of teaching math and math-related subjects. Having gone back to school for an MA in economics, I strongly believed there was a huge room for improvement in teaching methods. (You can read those pieces here: “How I would teach Math“, “The only way to learn is to accumulate flying hours“, “The problem with lectures“).

I felt strongly enough about this matter that I did what seemed to some a very presumptuous thing: I applied for a part-time teaching position so I can implement the methods I had in mind. Prof. Joey Balmaceda of the Math Institute very kindly considered my application instead of rejecting it outright, and I was scheduled for interview and a demo lecture last Monday, April 27. About ten other applicants were also interviewed that day.

This is my “algebra made easy” 15-minute lecture:

Imagine a balance people use to compare weights. Yes, the same balance typically held by a blind-folded lady to symbolize that everyone is equal before the law. If you can understand how a balance works, you can understand algebra in 15 minutes.

Suppose I put an unknown weight on the left side of the balance and, by trial and error, find that a 5-gram weight (in the demo, I used kilos) balanced the unknown weight. Then, we can conclude that the unknown weight is 5 grams, right? Suppose I add a 2-gram weight on the left side, how do I keep the two sides in balance? I should also add a 2-gram weight on the right side. If I add a 1-gram weight on the right side, then I need to add a 1-gram weight on the left side two, to keep the two sides in balance. If I take 3 grams away from one side, then I must also take away 3 grams from tne other side to keep the whole thing in balance. Lady Justice

Like Lady Justice is supposed to do (this sentence occurred to me only now, I didn’t say this in my demo), both sides must be treated absolutely equally. If I triple the weight on one side, then I must also triple the weight on the other side. If I halve one side, then I must also halve the other side.

Do unto one side what you would have done unto the other side.

That, in essence, is algebra. Everything else is just details and tricks.

Solving algebraic equations is simply playing this game of balance. And that’s what we will learn the next time. (If I had another 15 minutes, that would have been enough to explain the process of solving for X.)

How did the panel of interviewers respond? They were very polite, but I also saw a few nodding heads.

I am keeping my fingers crossed.

Actually, I found it ironic trying to prove myself with a demo lecture (as required from all applicants), when in fact I spent the previous 30 minutes trying to convince the panel that my approach would be to minimize lectures and to maximize individual reading from the textbook and actual problem-solving with pencil and paper. My own demo lecture highlighted, for me, my point that lectures are a very poor way to implement the learning process. If it were a real class, I would have simply started by giving everyone a set of algebra problems, from the extremely simple to the moderately simple, and in addition assigned as homework a range of pages to read as well as another set of problems to work on, for submission in the next session.

In a lecture, I had told the panel, the lecturer learns more than the students. A math class should be run like a swimming class, where the instructor stands by the pool, out of the water, while the students are in the pool, learning how to swim.

In math and other math-related subjects, paper and pencil are the students’ swimming pool.

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An old comment haunts me

It turns out that I wrote two versions of the piece on the financial crisis I posted earlier. The other version, which you’ll find posted here, says basically the same thing but contains the following comment which now haunts me. I had written:

“Unfortunately, most economists appear to have little understanding of system design. (When I was in college, many of those who failed our engineering subjects shifted to economics.) Instead of following good principles of design, our economists repeat the most common mistake of amateur programmers: they rely on global variables.”

What a twist of irony! Although I did pass my electrical engineering course, I’m now an MA economics student in the same university where I learned engineering more than 25 years ago.

It is interesting, though, that many of the early founders of neoclassical economics, like Vilfredo Pareto and Leon Walras had engineering training.

An insightful historical analysis by Philip Mirowski (Against Mechanism) traces the development of neoclassical economics from 19th century physics, whose mathematical methods the early neoclassicals imported en toto and applied to the analysis of consumer utility, producer profit and market equilibrium. The methods of physics, Mirowski says, have changed radically since then, but neoclassical economics remains mired in 19th century physics methods of analysis. I am still trying to grasp the full meaning of Mirowski’s analysis, however.

In another piece (I don’t recall now if in the same book or another), Mirowski also commented that economics needs an algebra of its own, which also haunts me in a different way. It has challenged me to learn more about different algebras. (Aside from the more commonly-known high school/college algebra, there’s boolean algebra, matrix algebra, vector algebra, set algebra, etc.) When I mentioned this to an economist who was currently taking a PhD in Math, he thought about it for a while and then said, “actually, that’s true.”

Another area to explore.