I’ve been hard at work here at the Math Forum @ Drexel! I’ve been working with Suzanne, Max and Val to produce “interesting” solutions for their POWs. I do my best not to look at the teacher packets or Math Forum solutions while I’m figuring out my initial solution. That way, it’s more interesting when I compare my solution to the collected solutions for each POW. This is largely what has made work here fun for me.
For one of the Trig & Calc POWs I found that my solution differed greatly from the examples in the teacher packet. Though, that’s the entire goal of my work so far. The problem was to find the area of a pizza cut in an unusual way. There were only a few solutions in the Teacher’s packet, a lot of them used Heron’s Formula where I developed a system of integrals. I don’t think I ever learned Heron’s formula in any high school classes, but I know how to analyze a geometric situation using trigonometric and calculus concepts. The goal, on my end at least, was to see how my solution differed from other students’ solutions.
I feel as if, by working this way, I’ve stumbled upon the guiding principle of the Math Forum POWs. Beyond deepening my understanding of spatial and geometric analysis, comparing my solution with that of many others from various educational backgrounds helps me to better understand how to help someone else solve the same problem. And, I mean more than just being able to suggest a repertoire of various laws or theorems. In understanding how a student solves a problem, I can figure out how they were thinking, and what ideas came to them as they were reading the problem.
From an engineer’s perspective, this problem screams “integrals!”. I know in the previous post I talked about “no one size fits all solutions” but I like to have one tool does all. To me, integrals and derivatives are more valuable than a toolbox of formulas and theorems. I know that I can find acceleration through derivatives or areas and displacement through integrals, so a huge amount of formulas and equations are basically obsolete. Though, to a student who hasn’t had calculus (or won’t ever take it) the formulas and theorems are the only tools they have. By way of comparison, my toolbox hasn’t necessarily been upgraded. Metaphorically speaking, the comparison is unto a machine shop full of lathes, drills and lots of other precision tools versus a similar shop with a C&C machine. In principle one isn’t necessarily better than the other. The one is a massive collection of tools that may not see use in every problem, but is equally effective as a machine designed to “do it all.”
I also noticed, in general, that other students’ solutions were less savvy about precision. Not in that they were imprecise in their answers, but that they didn’t consider where over-precision would over-complicate a solution or was simply unnecessary given the conditions of the problem. In most sciences, it’s satisfactory to be within 10% of an accepted value while in pure mathematics the answers are more black & white. In my brief time solving POWs I’ve found myself at odds with the answers after carefully considering how precise my answer can be assuming that any measurements given to me are fallible measurements. It’s a habit I’ll have to let go of for my time here at the Forum.
Another problem I worked on was about finding a point on a line such that two other points had a difference in distance of one. I knew I could have solved it via algebra and an established equality, but I knew a more “advanced” way to solve it. (Yes, I wanted to show off.) I solved it via linear algebra and matrices. The problem suggests that there may be more than one solution, so that tipped me off to use my knowledge of linear systems. As it turns out, there were only two solutions which were resultant from the algebraic manipulation of a square root. Even still, I solved the problem correctly, even though I had to submit some clutter with extraneous solutions.
So In conclusion, Math is fun and I’m having a great time at the Math Forum!