Too Much Content?

A friend of mine recently posted a link on Facebook to the following blog post:

http://garyrubinstein.teachforus.org/2013/10/19/death-of-math/

This has the provocative title "The Death of Math."  Side note:  The use of the word "math" tends to bug me.  In formal writing and public writing, I always try to use the term "mathematics."

This is a fairly long post and what I want to focus on now is one of the two recommendations Mr. Rubenstein makes to "fix mathematics."  This one is:  Greatly reduce the number of required topics and to expand the topics that remained so they can covered more deeply with thought provoking lessons and activities.  (The second recommendation is to make mathematics beyond the 8th grade into electives.)


While the second recommendation could make for some interesting discussions, it was the first one that I started thinking about in terms of the introduction to proofs course.  My first thought was that I agree with this recommendation but it is quite difficult to implement.  This seems to be a fairly constant struggle that I have to balance the work students do to start "thinking like a mathematician" to the content they may need in future courses.

At Grand Valley State University, our introduction to proofs course is MTH 210 - Communicating in Mathematics, and I believe our first priority in the course is to develop students' abilities to construct mathematical proofs and then to write them in a coherent fashion according to guidelines established by the mathematical community.  Since we need content to "learn to think like a mathematician," what content do we use?  Using content from previous course can be difficult because not every student has the same mathematical background, and it is quite likely that the "previous" content that we would select will have been forgotten by many of the students.  So here is what we do at Grand Valley.  I welcome comments, suggestions, and criticisms.

Elementary number theory dealing with properties of even/odd integers, divisors and multiples, and congruence.  This is the material that is used to introduce the various proof techniques studied in the course.

After dealing with the proof techniques, we continue to use the methods of proof while introducing the following content:

• Elementary set theory.
• Functions.
• Relations and equivalence relations.

This may not seem like enough content for a 4-credit course, but content is not the main goal of the course.  In fact, I often skip some sections of the text that I have written.  For example, the chapter on set theory includes a section on the Cartesian product of two sets and a section on indexed family of sets.  I skip both of these except for introducing the definition of the Cartesian product because it is used in the chapter on functions.

The last thing studied in the course is equivalence relations and when we had a 3-credit course, we barely had time to introduce the important idea of equivalence classes.  I am hoping that we will now have some more time to spend with this now that we have a 4-credit course.

The textbook, Mathematical Reasoning: Writing and Proof, also includes a chapter on more number theory (basically greatest common divisors and prime factorization) and a chapter on finite and infinite sets.  We do not cover these chapters in our course at Grand Valley but I know others like to include these topics in their course.  So they remain in the textbook.

One good thing about writing this is that it reinforced my opinion that our approach for the introduction to proofs course is reasonable and that we really do not have too much content in the course.