Tag Archives: learning

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.

Learning arithmetic

My attitude towards learning and education has been shaped by my own experience as well as by authors like A.S. Neill and John Holt, who represent what might be called the “free school” (“free” as in freedom) approach. Taking a cue from software, this might also be called the “free/open school” approach, an approach that basically gives students much more freedom to choose by themselves what to learn, how to learn, when to learn, and even where to learn. Teachers play a more passive role in the background as advisers. Traditional schools, in contrast, are so regimented that John Holt has called this traditional system a virtual prison for children and youth.

As an example of the free school approach, I’d like to quote the first few paragraphs of Chapter 1 (And ‘Rithmetic) from the book Free At Last (The Sudbury Valley School) by Daniel Greenberg:

Sitting before me were a dozen boys and girls, aged nine to twelve. A week earlier, they had asked me to teach them arithmetic. They wanted to learn to add, subtract, multiply, divide, and all the rest.

“You don’t really want to do this,” I said, when they first approached me.

“We do, we are sure we do,” was their answer.

“You don’t really,” I persisted. “Your neighborhood friends, your parents, your relatives probably want you to, but you yourselves would much rather be playing or doing something else.”

“We know what we want, and we want to learn arithmetic. Teach us, and we’lll prove it. We’ll do all the homework, and work as hard as we can.”

I had to yield to them, skeptically. I knew that arithmetic took six years to teach in regular schools, and I was sure their interest would flag after a few months. But I had no choice. They had pressed hard, and I was cornered.

I was in for a surprise.

My biggest problem was a textbook to use as a guide. I had been involved in developing the “new math,” and I had come to hate it. Back then when we were working on it — young academicians of the Kennedy post-sputnik era — we had few doubts. We were filled with the beauty of abstract logic, set theory, number theory, and all the other exotic games mathematicians had played for millennia. I think that if we had set out to design an agricultural course for working farmers, we would have begun with organic chemistry, genetics, and microbiology. Lucky for the world’s hungry people that we weren’t asked.

I had come to hate the pretensions and abstruseness of the “new math.” Not one in a hundred math teachers knew what it was about, not one in a thousand pupils. People need arithmetic for reckoning, they want to know how to use the tools. That’s what my students wanted now.

I found a book in our library, perfectly suited to the job at hand. It was a math primer written in 1898. small and thick, it was brimming with thousands of exercises, meant to train young minds to perform the basic tasks accurately and switfly.

Class began — on time. That was part of the deal. “You say you are serious?” I had asked, challenging them; “then I expect to see you in the room on time — 11:00AM sharp, every Tuesday and Thursday. If you are five minutes late, no class. If you blow two classes — no more teaching.” “It’s a deal,” they had said, with a glint of pleasure in their eyes.

Basic addition took two classes. They learned to add everything — long thin columns, short fat columns, long fat columns. They did dozens of exercises. Subtraction took another two classes. It might have taken one, but “borrowing” needed some extra explanation.

On to multiplication, and the tables. Everyone had to memorize the tables. Each person was quizzed again and again in class. Then the rules. Then the practice.

They were high, all of them. Sailing along, mastering all the techniques and algorithms, they could feel the material entering their bones. Hundreds and hundreds of exercises, class quizzes, oral tests, pounded the material into their heads.

Still they continued to come, all of them. They helped each other when they had to, to keep the class moving. The twelve year olds and the nine year olds, the lions and the lambs, sat peacefully together in harmonious cooperation — no teasing, no shame.

Division — long division. Fractions. Decimals. Percentages. Square roots.

They came at 11:00 sharp, stayed half an hour, and left with homework. They came back next time with all the homework done. All of them.

In twenty weeks, after twenty contact hours, they had covered it all. Six years’ worth. Every one of them knew the material cold.

That’s the free school approach.

A well-known school that uses this system is Summerhill School in the U.K. The school was set up by A.S. Neill, who wrote about his experiences in his book Summerhill: A Radical Approach to Child Rearing. The books How Children Fail and How Children Learn by John Holt are also illuminating.