Proofs and Wolfram Alpha

In my Introduction to Proofs course, I discussed the proof the following: The cube of an integer is of the form 9k, 9k+1, or 9k+8, for some integer k. The problem is from the text I use, How to Think Like a Mathematician: A Companion to Undergraduate Mathematics

The big idea here is to to note that any integer can be written as 9q+r, q some integer, r=0,1,2,..,8. Then simply find (9q+r)^3 and examine the form of the expression of the nine different cases for r. The algebra is a bit tedious - and so Wolfram Alpha comes to the rescue in the form of the command

expand((9q+r)^3) for r from 0 to 8

The output is here:

Note that the proof itself consists of the big idea of writing an integer as a multiple of 9 with a remainder. W|A simply did the grunge work for us of expanding the polynomials and substituting the values of r. In a course such as this, W|A can be a great timesaver in doing these types of calculations and students see the value of such software as efficient helpers in solving larger problems.

Teaching teachers to teach

An article in last Sunday's New York Times discusses the complex issue of teacher preparation. The writer of the article includes a lengthy discussion of math teaching in particular. I found it to be quite interesting - the work of math educators doesn't usually make it to mainstream media. Hopefully the Times will continue its coverage of key education issues such as this one.

Getting students to be quantitatively literate

One of the main aims in teaching a Math for Liberal Arts course is to get students to have a better appreciation of general mathematics. And what better way to do that than to have them read math articles or blogs aimed at a general audience? Sounds like a good plan, but I needed to make it a graded assignment so it would get done.

In my online course I am now making up questions related to articles in Steven Strogatz's NY Times blog. Steven Strogatz is a mathematics professor at Cornell University. He makes basic math so clear and brings in so many connections that anyone who reads it will get something out of it, no matter what their education level is. Using his post on algebra, I asked my students the following as part of a longer homework assignment:

Suppose the length of a hallway is y when measured in yards, and f when measured in feet.  Write an equation that relates y to f.

I gave them the wrong answer of y=3f and asked them to figure out why it's incorrect. Not exactly rocket science, but having them read something about math other than what's in a textbook is a big thing for many of my students. When we get to the probability and stats part later in the semester, I plan to do something similar with the abcnews.com articles by John Allen Paulos, math professor at Temple University. These articles are not as direct as Strogatz's, but do aim at probabilistic understanding for a general audience.