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.

Final exams (and learning how to swim)

It’s that time of year… I just took my final exams in Mathematical Economics. One more exam to go.

The experience of taking that finals reminds me of the swimming lessons I took before my teens. (This was in the sixties, if you must know.) Every summer, the Bernardo Park in Quezon City offered formal swimming lessons for the youth. The park had a clean and well-maintained public swimming pool (25 x 50 meters, I think). Since the pool was just a walking distance from our house in Kamuning, it was very accessible and convenient. I went to swimming class with some friends in the neighborhood.

The program was run by “Sir Luna”, a swimming trainer and a real professional, who conducted it at the pool’s shallow end (4 feet deep).

We started with “bubbling”, learning to breathe in out of the water, and breathe out in the water. Exhale through the nose, inhale through the mouth. The key was in the rhythm. Exhale, inhale. You had to pace yourself. Exhale, inhale. We must have done the bubbling routine tens of thousands of times by the time we finished the course. Exhale, inhale. Note very well: exhale first, then inhale.

After bubbling, it was paddling. Standing chest-deep at the shallow end of the pool, arms stretched out in front and on the gutter, we paddled with the left arm, and then the right. Left, then right.

Then, we were introduced to the “flutter kick”. While we held on to the gutter, we started “fluttering” our feet a few inches up and down, knees kept straight. We also did the flutter kick across the width of the pool, at the shallow end, but no breathing. Just face down, arms stretched in front, and looking at the pool’s bottom as it moved slowly past us. When we ran out of breath, we stopped.

After several training days, we also did the routines two at a time: bubbling and paddling, bubbling and kicking, paddling and kicking. Then we tried all three. We held on to the gutter, facing the swimming pool edge, most of the time.

After two weeks or so of these, we got to try all three routines away from the gutter. The hardest part was the breathing. If you lost your rhythm, you breathed in at the wrong time and took water. You had to stick with the “bubbling” exercise, day after day, until the rhythm became part of you, and your muscles knew by themselves, without any conscious thought from you, when to inhale and when to exhale, just like you do out of the water.

This was, I think, the key – for one’s lungs to learn the exhale-inhale routine and to do it without conscious thought. One by one, my classmates got it, and they started to actually swim by themselves! Then, they were allowed to play at the deep end. But I still didn’t get it. My legs, arms, neck, nose, mouth and especially my lungs had not learned enough. I’d lose my rhythm and then inhale at the wrong time and take in water. I must have drank gallons from that pool.

To graduate, one was expected to dive from a diving board at the deep (9 feet) end of the pool, and swim the 50 meters to the shallow end. In full view of relatives, friends and guests. On graduation day, my friends all made that graduation dive. I wasn’t ready, so I didn’t. After graduation, graduates were granted free access to the pool for the rest of summer. Sir Luna was kind enough to give me access too.

For the rest of summer, my friends and I enjoyed that pool. They played at the deep end, showing off their new swimming skills. They loved diving from the diving board. Although I stayed at the shallow end, I enjoyed it as much as they did. Then, one day, it just came. I finally got my rhythm and I started swimming too! So, before the summer ended, I also got to play and to dive at the deep end.

One of our professors in graduate school had told us, “sometimes, you learn more from the tests you fail.” How true. “But your grades won’t show it,” he added. Indeed, failures can often teach us lessons better than success. We call it learning the hard way.

By the way, I made that graduation dive the next summer.