## Monday, May 30, 2011

### Mental Math: My Calculus Story

Here's an interesting quote from an even more interesting article about mental math (HT: Mark Thoma)
But the mental arithmetic gap has more subtle implications. Mental calculations often require intuition about, and comfort with, the use of fractions. Pre-calculator: 1/3+1/3=2/3. Calculator era: 0.3333....+0.3333....=0.6666.... Pre-calculator: "To multiply by twenty-five, divide by four and add two zeros (25*Y=1/4*100*Y)" Calculator: Multiply by twenty-five. Back in the day, fractions were easier than - or at least not much more difficult than - decimals. Calculators make fractions obsolete.
This is an interesting point that reminds me of my high school calculus class. When I took calculus, it was the first year that the class was taught using fancy TI-92 calculators. Because of some grant or benefactor, the teacher had enough fancy calculators for everyone to use and we were required to "work" some problems using the calculators.

I didn't like this policy one bit because I had my own less-fancy calculator (TI-85), which was a mere three years old at the time. Even though my calculator wasn't the most state-of-the-art piece of equipment, it seemed like such a waste to use a school-provided calculator when I had my own. I took it as a challenge to myself to figure out how to produce all of the plots, limits and calculations we had to perform using the calculator.

Because my calculator was less than ideal for this purpose, I had to learn calculus better than my classmates to figure out how to get the right answer. This is because we were given step-by-step instructions to use the TI-92 calculators. To use my inferior calculator, I had to convert the TI-92 instructions into math concepts and then figure out how to implement those concepts on a foreign piece of technology. I learned a lot about calculus in the process.

From this story, I think there are two observations to be made. First, calculators can spell doom for learning if they are misused in the classroom. Had I followed the leader to the TI-92, I could have avoided learning calculus whenever the problem involved (or allowed) the fancy calculators. Second, calculators can be extremely beneficial to learning if you use them the right way. I learned quite a bit about calculus when I had to translate the instructions out of TI-92ese into mathematics. I also learned quite a bit when I translated back from mathematics into TI-85ese. If this was part of the curriculum, everyone in my class could have benefited from having to be a translator of calculus.

Based on my calculus story, I take the linked article to be more of an indictment of mathematics teaching methods than an indictment of the calculator itself. It's true that many people have come to use calculators as a crutch and using calculators as a crutch crowds out alternative, beneficial ways of thinking. That said, new pieces of technology open up new possibilities for learning (i.e., being a calculus translator) that we couldn't imagine before the technology existed.