I have been around computers and calculators for much of my life. I remember mechanical calculators that looked like big industrial strength typewriters, and I remember when the first electronic calculator came out that offered a – would you believe this – square root key, a model that cost hundreds of dollars more than models without square root keys. I remember when scientific calculators became common, and I remember when four-function calculators became cheap enough and small enough to become ubiquitous. The basic calculator I have carried for several years is now available for $2.99! It even has a square root key.
You and I, in our daily lives, may not be computing lots of square roots. Yet the fact that the early square root calculator was a commercial success at its high incremental price indicates that there is a real-world demand for computing square roots. And the people that needed square roots were willing to pay the extra hundreds of dollars rather than engage in the alternatives then available to them (mostly through a method known as Newton-Raphson, described nicely here, where it is claimed to be something already known to the Babylonians.)
Looking at what sells in real markets, where people hand over real money, is an interesting way of gauging demand, and a way to gauge what it is considered useful and by whom.
In the summer of 2002 I was observing a remedial 7th grade classroom (called ‘summer school’) when I noticed a student acting suspiciously, with furtive movements and darting eyes. I was curious, it wasn’t even during a test, and I walked closer. Under his desk, he was hiding a calculator, and the calculator had a “fraction key” on it, labeled . I’d never seen one of those, and I did some investigation. The key would help reduce fractions, so that if you entered 6 / 10, it would show in the display 3 / 5. What became clear in the investigation is that calculators with fraction keys were marketed to – guess who – secondary school students. It was secondary school students, and presumably their parents, who found those keys useful. Years after, I would ask engineers if they use fraction buttons, and many would not know what I meant, and others would say they didn’t really use them.
The mathematics of fractions and rational numbers is a beautiful piece of mathematics, and I don’t wish to argue that this mathematics has no place in the K-12 curriculum. I do want to suggest that there is a large amount of myth and unchecked assumptions as to what exactly it is good for. Fractions are good for something – but what it is good for may have nothing whatever to do with what we traditionally claim it is good for. What do you think fractions and rational numbers are good for?