This post was inspired by a conversation with Rick, at Exploring Binary.
Rick: I have to learn to go to Wolfram Alpha for mathematical queries such as the one I mention in the article. Typing 4, 24, 124, 624, 3124 into Wolfram Alpha gives the answer I sought directly: an = 5n – 1.
Sue: But then you wouldn’t have had as much fun! And I’d say that’s the problem with Wolphram Alpha. Having fun with math is often hard work. It’s so much easier to click your way to an answer. But just not the same.
Rick: For me, it goes beyond Google and Wolfram Alpha — it’s the availability of computers in general. I often find myself slapping together a small program or script or spreadsheet before I sit down and think about a calculation first. Sometimes this leads to serendipitous discoveries; other times it leads to more “Well, Duh!” moments .
Maybe it’s good I started my serious study of math before the computer era really got underway. When I started high school ('70), calculators weren’t yet in common use. When I started college ('74), we didn’t have graphing calculators. I remember drawing hundreds of graphs while I was in calculus. For a long time, I felt like I wasn't a real mathematician because I never learned how to use a slide rule. All the math people I knew could use one, because they'd been doing difficult calculations for a few years more than I had, and hadn't had calculators. There was no internet (in my personal life, anyway) until well after I was done with my formal education (’89).
I saw a game for learning factoring, called Divisor Miser, at the Colorado NCTM conference in ‘82, I think. Someone had programmed it on a Vic-20 or TRS-80, or something like that. But I don’t remember computers being used to solve math problems in the computer course I taught in the mid-8o’s.
I taught junior high for a bit, and one of the math teachers wanted me to help him write a program to solve a probability problem. But he had the calculations all wrong, and the right calculations were simple enough that a computer program would have been silly. I was shocked at his bad understanding of math. He was our department chair. I knew almost no probability at the time, but I learned enough from the student materials (extra credit stuff) to figure out that problem. Yikes!
And yet, I love how technology can help us see. Here are some examples that come to mind:
- Dan wanted to simulate an amusement park ride he calls a scrambler, and pulled up Geometer's Sketchpad. Nice results.
- I used the matrix solution capabilities of my TI-83 when I was solving the regions in a circle problem. (Put n points on a circle, connect each to each with a straight line, how many regions in the circle?) If you already understand the solution to that, you may think this was a totally unnecessary use of technology. But it helped me to see. I'd like to post about that problem soon.
- At last year's Math Circle Teacher Training Institute, this problem was posed: "Can you tile a rectangle completely, if you require that only squares are used and each square must be a different size from all the others? How?" We used Geometer's Sketchpad to play with our sketches of how it might work. I dropped out of the group thinking about this. When they presented their solution the next day, they mentioned having used Mathematica to solve some equations.
- Back to my conversations with Rick. The one above is on his blog. We've been having another conversation at the same time in the comments to my post about the Carnival of mathematics. I wondered about the pattern in decimal fractions, .99 in particular, when they're converted to binary. He gave me .99 in binary, and pointed out that it repeats. I started playing with the conversion by hand, got frustrated, and turned to Wolfram Alpha for assistance. (My first time using it for a serious purpose.) Without any more tedious calculations, I could think about patterns. Turns out .9 is a repeating decimal in base 2 with a four-digit repetition, but .99's representation in binary takes 20 digits. Weird. I haven't seen the light yet on that one.