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Tuesday, August 7, 2007

A Blessing and a Curse

Have you ever worked with someone that had been promoted to his/her level of incompetency? This idiot that you're thinking of was once a model employee. It was proven excellence at a lower level that got this person promoted in the first place. Unfortunately, for both of you, this person just tried to meet one too many challenges. When I took Calculus III in college, that incompetent was me.

I have always liked math, and I fancied myself pretty good at it once. I sailed through Algebra, Geometry, and Pre-Calc without having to do much homework. As I moved into Calculus, things started to break down. My intuitive understanding was slowly eroding and I found myself struggling to grasp what looked like simple concepts. I kept getting A's, but I was working harder for them than ever before. I remember the first week of calculus in infinite dimensions. It was the hardest I have ever slammed into an intellectual wall.

Since I am going back to school in less than a month, I have started thinking about what may have caused this roadblock for me, and I have come to two conclusions. First, students determine their feelings toward and perceptions of a subject very early on. I felt math was easy and should require next to no effort on my part, because that's the way it had always been. Once it started to get tough, I began to have low-grade self-esteem issues that affected my work. Perhaps I should have been challenged more in earlier grades, just enough so I knew that some work was necessary.

The second revelation is more intriguing to me, and it has come through several years of reading popular math books, the ones with no formulas or equations, just a lot of metaphors and lay-person explanations. They have helped me learn some things that I didn't know in school. Now I understand the difference between applied and pure mathematics, terms I didn't hear of in school. Pure mathematics doesn't have to have any practical applications, as Hardy was fond of pointing out. It is very common for tools of pure math to sit on a shelf for decades or centuries before someone finds a good use for them, and sometimes one is never found. In many ways, the correspondence between chalkboard and reality is accidental. For example, it is a fortunate coincidence that Euclid's geometry so strongly correlates to life in flat space. At the scale the Greeks were used to working, it was flawlessly accurate. For space-traveling moderns like us, Euclid will not suffice. We live in a world where space and time curve, and we have had to use other non-Euclidean geometries, geometries fortunately constructed long before Einstein took his mind-trip on a beam of light.

If you've ever tried to explain non-Euclidean geometry to someone, you may have experienced how much a metaphor can help and hinder understanding. That's exactly what mathematical constructs are; they are metaphors or models for reality. If you are too wrapped up in the similarities between metaphor and reality, you may be blind to differences. That's what happened to me back in college. I had grown accustomed to thinking of all math spatially. Whenever I heard the word dimension, I was thinking of height, length, width, etc. Most of the examples in my textbooks applied the lesson to measurement of space, so when I got to Calc III, and the dimensions grew beyond the familiar three, I was lost.

I am not suggesting that we abandon spatial examples and metaphors. Student's inherent understanding of space is strong and math education is wise to piggy-back of of it. But maybe there ought to be more examples in the texts that have nothing to do with space. Comparisons of color to light/heat absorption, or age to bone density, or whatever. This way students will begin to understand that dimension can refer to any variable characteristic, not just space. This realization has certainly helped me, and I can't wait to get back into class and prove my competence.


Anonymous said...

The challenge lies in finding metaphors that your students can relate to immediately. High school students, while I'm sure they think they know everything, have a relatively limited world view and set of life experiences to fall back on.

I've often heard from educators that it takes a certain "maturity" level to comprehend certain types of math. Perhaps by "maturity" one means "set of life experiences to use as metaphors/abstract thinking aids?"

Jackie Ballarini said...


I'm glad to hear that you are going into your coursework with a change in attitude - I'm sure you'll do more than just prove your competence!

Interesting idea in your second revelation... I like Andy's comment that a certain maturity is needed. In my mind this means the ability to throw out preconceived ideas and grapple with the uncertainty ... until a deeper understanding can be reached.

Now, how do we convey that to students?