“What is mathematics?” Students usually answer, “Numbers.” No! Instead, mathematics is logic. Let us examine geometry for example.
Geometry is based upon postulates. A postulate is often incorrectly defined as “Something taken to be true without proof.” A postulate is also defined as “A proposition regarded as self-evidently true without proof. The correct definition is this. A postulate is an arbitrary statement, consistent with the other postulates of the system. Arbitrary means you can say whatever you please.
A mathematical system is a collection of arbitrary statements that do not contradict each other. Of course, the statements in different mathematical systems do not have to agree.
Euclidean geometry is a consistent system. A modification of Euclid’s fifth postulate will also generate a consistent system. Given the postulates, we apply logic and prove theorems. If we can prove two theorems that contradict, then we have shown that the postulates are not consistent, and so that mathematical system is not valid.
Truth in mathematics means logical consistency. Contrast this with truth in science, which means experimental verification or observation. Reminding students of this will help them stay on their toes, as they have to be self-critical, and cannot rely upon authority.
The teacher can encourage this behavior by asking, “Do you think this is correct?” If they reply that they are not sure, then tell them that they have to be sure one way or the other, and be able to defend it.
This is very different from politics. Each side gives arguments, responds to the other side’s comments, but does not have to be completely consistent.
How are we supposed to study mathematics? Students must understand what is going on in class. They need to be encouraged to ask questions. The teacher must ask students to be sure they understand the principles and the logic.
Of course, students need to do homework. First, they must review the material for understanding. Secondly, they must review just the highlights. Finally do the problems, paying attention to the principles. Problems are usually based upon the few principles of the problems.
We must all check our work.
Checking does not mean doing it again. If you did it again, you did it again, but you did not check. Check that you answered the question in the problem. Check that you used the tools correctly. Ask yourself how you feel. If you feel good, you are probably right. If you feel bad, you are probably wrong, and look more carefully. Check the numbers and signs separately.
For word problems, one has to translate from words to numbers and expressions. It is like translating from French to English. This is a hard step. Part of the checking should be to check the translation.
We have to make use of concrete models, preferably relating to their own selves, as much as we can. Here are two examples:
2 divided by is 4. To demonstrate, take two sheets of paper, tear in half, to get 4.
Another example: When discussing equations with two variables, take colored objects, say, red and green, to help understanding.
It is critical to do problems one step at a time. One is for comprehension, the other for ease in checking.
Math work must be neat. If not, redo the work.
For more discussions like this, see “Teaching and Helping Students Think and Do Better” on amazon.