For those of you that might not immediately recognize the origin of this week’s title, the Marvel Comics character J. Jonah Jameson was fond of the headline Spider Man: Threat or Menace? My real point is this.
If you cannot do simple arithmetic without a calculator, you cannot do any arithmetic with a calculator.
Occupy Math is grouchy this week, partly because its the end of the semester and things are in their usual state of chaos and disarray. The relevant contributing factor, however, is that four, times two, is not twelve. This piece of incorrect arithmetic showed up six times on the most recent test in my first-year university calculus class. I took off more points for bad arithmetic than for bad calculus. The reason my students have such an awful time with arithmetic is that the educational system they came up through carefully avoided teaching them how to do arithmetic. In particular, many of them were given calculators instead of learning how to add.
If you get access to calculators too early, it cripples you.
Back when I worked at Iowa State University, we considered making the introductory calculus course one with no calculators. We got calls from high-school guidance councilors telling us if we did this, they would advise students to go elsewhere. This points to something toxic: we want achievement without effort, and success without risk of failure. These guidance councilors knew they were turning out high-school seniors who lacked the ability to do simple arithmetic without a crutch. The problem is that, with this education, there is an excellent chance the student will use their calculator to get an incorrect answer.
My father was a member of the International Wizard of Oz Club. I enjoyed Oz stories at bedtime for years. One of the characters in the Oz stories is H.M. Wogglebug T.E.. He founded the Royal College of Art and Athletic Perfection and, living in a fairy land, he also invented knowledge pills that gave a student knowledge without having to attend lessons, so that the student’s time could be applied to athletic pursuits.
This would all be fine if calculators were magical artifacts from a fairy land, but they’re not. If you don’t understand the underlying rules and patterns, a calculator is simply a device for generating nutty answers that, for some reason, you think are right. This too-early introduction of calculators is part of a larger rush to unearned success. In an article entitled The rush to calculus is bad for students and their futures in STEM, Kevin Knudson argues persuasively that we are trying to get students into and through calculus in greater numbers without giving them a proper foundation. I’ve linked the article here.
The students I get in my course – a self selected elite that end up with pretty good grades on average – are deficient in arithmetic, algebra, and trigonometry. They are almost entirely ignorant of geometry. They have been taught a series of rituals for getting grades on specific types of problems – without learning the overarching patterns or unifying themes of mathematics. If someone asks me to multiply 123 by 37 my brain goes:
This looks crazy – but at each step I’m using mathematical rules to solve a very easy problem – like multiplying by 2 or 10 or subtracting an even multiple of 100 from a larger number. Even 3×123=369 is easy – because the multiplication happens digit-by-digit without any carrying. My brain does this sort of thing because I got impatient doing the problem “the usual way” with pencil and paper – and I swear I typed that whole sequence without help from a calculator. Sometimes when a student asks me how I did some arithmetic I will unpack what my mind does, like the example above.
A common response to my unpacking is the question: “is that the right way to do the problem?”
I think of it as part of my growth as a person that I can now chuckle at that question (instead of bellow “Arggghhh!”). I picked a whole bunch of easy steps that solved the problem. No step is intrinsically easy – those steps were “me easy”. If a person does arithmetic without a calculator, they discover the steps that are easy for them – they develop mental constructs that make them more powerful – probably in ways not connected to mathematics as well as the direct mathematical power. Finally, and this is a key point, there are many paths consisting of simple steps that solve any given problem. There is no right way here – simply things that do and do not work for you personally. If you don’t practice, then nothing works for you personally.
Achievement always has a price.
There is an entire discipline of doing math without calculators, its got a number of names, but the article I’m linking calls it mental calculation. There are contests (there used to be more) and there are wonderful stories about mental math in Richard Feynman’s biography. It is an area well worth pursuing. Do you have a favorite mental math trick? Comment or tweet about it!
I hope to see you here again
University of Guelph
Department of Mathematics and Statistics