I came across this Seymour Papert quote over the weekend, the best part of which is below. In context, Papert is speaking about effecting real change in the content of school mathematics, and he focuses particularly on the teaching of fractions:
One theory [among educators about why we should teach fractions in school] was that manipulating fractions was actually closer to what people needed back before there were calculators. So a lot of school math was useful once upon a time, but we now have calculators and so we don’t need it. But people say that surely we don’t want to be dependent on the calculator. To which I say, Look at this thing, these eyeglasses, that make a dramatic difference to my life and the life of everybody who reads or looks at any tiny detail. Once upon a time we would have been crippled, severely handicapped. Now we’ve got these and we don’t need to go through all that suffering. So we are dependent on this little thing.
Well, so what? There is nothing wrong with being dependent on a little thing that everybody can have lots of. It doesn’t even cost much. So, that is no argument.
People float the “dependence on technology” counter-argument against the use of technology in the mathematics classroom pretty frequently. But as Papert notes, is it really all that bad if students became dependent on a technology that’s cheap and easy to come by? In fact, here in the US at least, aren’t most of us dependent on cheap and ubiquitous technologies — eyeglasses, running water, cars, kitchen appliances? (And some of those aren’t cheap!) We don’t make students in culinary school learn how to cook over a campfire out of fear they’d become dependent on ovens. Why should we shy away from calculators?
That threat of becoming dependent upon technology to do mathematics is only a real concern, for me at least, under one of two conditions.
One is if the technology we use is expensive or otherwise hard to access for some learners. This can be a real problem. But math teachers can combat it by seeking technologies that are cheap or free and easy to access — think cheap, functional, sturdy devices like the TI-30X instead of monstrosities like the TI NSpire. (That’s an order of magnitude difference in the price there, in case you missed it.) Or, as much as I love MATLAB, it’s pricey — and if accessing it is an issue for students, think instead about open-source alternatives like Octave.
The other condition is when our definition of “mathematics” becomes so restricted that it includes only those tasks that can be easily farmed out to technology. When you remove all the human elements from mathematics — modeling, problem solving, pattern-finding, written expression, and so on — and reduce the subject to nothing more than rote mechanics, of course technology poses an existential threat to the discipline. And deservedly so! Any discipline that can be replaced by software probably ought to be.
Far more of an threat to students’ long-term success is the dependency they can develop upon people, especially teachers. If a student has trouble manipulating fractions without a calculator but can read a problem thoughtfully, model a quantitative situation intelligently, and complete and validate her work independently, I feel pretty good about that student’s chances in the future. But if a student can ace all the test questions about fractions but can’t do anything with a real-world problem without external prodding and validation from a teacher or other authority (“Is this right?“, “Am I on the right track?“, and so on), that’s when there’s real trouble, and it’s got nothing to do with technology. Who’s talking about that kind of dependency in school mathematics these days?
Photo from http://www.flickr.com/photos/draggin/