Domino Principle

In the mathematics instruction colloquium, the presenter Phil raised the question of how to teach mathematics induction to students. He gave examples of how students answered the question “What is mathematics induction” after taking a course on that. The discussion went on and one of main issues became why math induction was so hard for students to learn and, as JP put it, how we could make this subject easier for students.

In the mean while, JP mentioned a Minimal Counter Example method of reasoning the mathematics induction for students. It then became interesting to ask, which of the following was our primary goal to teach mathematics induction, and in which order students usually process them:

1. accept mathematics induction (being convinced by an argument such as using the Minimal Counter Example)
2. understand mathematics induction (after being convinced, or through other means)
3. be able to use mathematics induction (usually the main goal of teaching, but has the defect of having students mimic the procedures without understanding what they did this or that, and hence causing a problem such as not being able to answer the question “What is mathematics induction” even though the question is vague.)

Following an example given by Nate, who mentioned math induction as a sequence of dominos, once the first one is pushed down, and we know that the next one goes down after the previous one, we have all the dominos down. Ted clarified the statement into something like

1. a domino means a statement
2. a domino is down means a statement is true for a certain number
3. in order to prove that the statement is true for all natural numbers, we prove all dominos are down

I recalled how easy Pigeon Hole Principle sounds, and it was verified by others that students usually don’t have as much trouble with Pigeon Hole Principle. So, as the first step of making the mathematics induction sound easier, I suggested we call it Dominal Principle instead. The hope is to bring a pictorial memory to the students as imagining dominos falling down one by another after the first one is put down.

However, this effort does not solve one of the main difficulties for the students, as well as for other un-trained people who tried to understand mathematics and mathematics reasoning, which is to state what is in their minds correctly in mathematical terms. For mathematicians like us, it is a habit of doing so, the sixth sense. But for un-trained people, it is mission impossible. They could sometimes even state mathematics totally correctly in words but when trying to put it down on a piece of paper they are in trouble. It is like a mathematics illiteracy, similar to the case when people without good education could speak without being able to write.