Over the past few years, I have of course been teaching the introduction to proofs class at Grand Valley (MTH 210 – Communicating in Mathematics) and have taught a few sections of a 100 level trigonometry course. In both of these courses, one thing that has struck me has been the impatience of students and their desire to get a quick answer or simply to get things done quickly. One of my tasks now seems to be to try to slow down the students and instill in them the idea that doing mathematics is a slow process. I am not sure why this has been on my mind lately. Perhaps it is because our life in general seems to be continually getting faster and students today have become use to being able to get information more quickly than ever before. (Another reason may be that I am trying to slow down as I am approaching retirement.)
In the proofs course, when students come to see me about a proof of a proposition, it is often the case that they say they are just stuck and do not know how to get started. So I ask them if they have written down what are the assumptions of the proposition and what is the conclusion of the proposition and quite often, they have not done this. Sometimes I feel I should the hard-nosed mathematics professor similar to Professor Kingsfield. (For those old enough to remember, he was fictional law professor in the novel and book The Paper Chase.) But I guess that is not in my nature, and so I try to work with the students and help them get started by writing these things down on a sheet of paper. This is part of the slow process of doing mathematics. As a mathematician by profession, this is part of what I do. While it is true that for many proofs in the introductory course, I may not have to write all these things on a sheet of paper, it is still the thought process I go through in trying to construct a proof. Students, on the other hand, have not had the mathematical experiences I have had and they really need to go through this process on most of the proofs they try to do in the course.
This is one reason I put the first experience with proof in my text in Chapter 1 (Section 1.2 – Constructing Direct Proofs). After a short chapter on logical reasoning, we return to various proof methods in Chapter 3. Throughout these chapters, students are encouraged to use a method to organize their thought processes with a so-called know-show table. The idea is to work backward from what you are trying to prove while at the same time working forward from the assumptions of the proposition. However, I do emphasize that these know-show tables are not an absolute necessity but they are one method of helping them organize their thoughts. These tables force the students to stop, think, and ask questions such as:
- What am I trying to prove?
- How can I prove this?
- What methods do I have that may allow me to prove this?
- What are the assumptions?
- How can I use the assumptions to prove the result?
Asking these questions and trying to answers for them is a slow process but this often provides a framework for helping to construct a proof.