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.

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.

How I would teach Math

Having become a student again, I recently got this urge to teach, and have been applying around for a teaching position. If I were to conduct a class in math (or a subject that uses math extensively), I’d keep reminding myself of the following, which all came to me sitting through lectures:

1. I’d choose a good textbook. Students need very good/excellent texts, so that they can study at their own pace. Every lecture should be covered by text that the student can go back to if they miss a point or don’t understand things at first hearing. Lectures are a bad way to introduce hard concepts. Bullet-point presentations are even worse, because the teacher can now go faster through the prepared text. Distributing bullet-point presentations as course materials is unsatisfactory. They should be treated only as course outlines (see next point). There is no substitute for a course textbook. Spend time and effort choosing it (them).

2. I’d be very specific about the course curriculum. Students who want to read ahead or even finish the course materials earlier should be able to do so. The students should always be informed in advance about the coverage of the next class session, so that those who want to read ahead, or who need to miss class, may do so without missing the lessons.

3. I’d make students spend their class and off-class hours learning course concepts by reading through the text carefully and applying course concepts by solving problems/exercises. The only way to learn is to accumulate “flying hours”, i.e. hours of practice. Here are some ways to encourage stepping through examples and problems: a) give easy, graded take home exams after each class session, which call help pull up the students’ average; b) compile a set of, say 100-200 problems that will cover the entire course curriculum, with 3 or more questions per topic, and distribute this problem set at the start of the course; c) announce to the class in advance that all exams (midterms, finals, others) will be picked at random from this problem set (i.e., if they take the time to learn to solve the entire set, they will surely get excellent marks.)

4. I’d encourage students to ask questions. Most students are shy about asking questions for fear of sounding stupid. To encourage them, make them write their questions on a piece of paper, signed with an alias (so the teacher may follow the progress of the alias.)

5. I’d help clear away obstacles to the students’ self-learning. In particular, the teacher should be immediately available to help them get out of dead-ends. This occurs when a concept is so intractable to them that they can not grasp it without help, and they are unable to make any further advance until this learning block is cleared away. Thus, in class, students should either be working on problems or asking the teacher’s help in clearing away such blocks. The teacher can use the questions and the written answers to exercises to identify which concepts are most difficult for students and to conduct special lectures about these concepts.

6. I’d conduct a math class like a swimming class. The coach stays dry, standing by the pool side. The students are in the water, learning to swim. In math, the pool is the chalk and blackboard, pencil and paper.

7. I’d give exams that are fair. Stick closely to the text as much as possible. Do not ambush students with surprise topics or trick questions. As a rule: ask many easy questions, a few hard ones. Give students a second chance; if most did not get a mid-terms problem, ask it again in the finals. Think of it as a bonus question.