Should we let computers do all the maths for us?
There is a post over at Mapleprimes called ‘The feeling of Power‘ which contains an extract from a short story by Isaac Asimov written in 1958. The story is also called ‘The feeling of power’ and is set in the far future at a time when humans completely rely on computers to do all of their calculations for them. In this vision of the future even the keenest human minds are incapable of performing simple calculations such as ‘nine multiplied by seven’ without the aid of a machine.
Until reading the Mapleprimes post I had never heard of this story and I wanted to read more. A quick google search gave me the full text (it is very short and well worth a read). I am not sure if putting the full text online in this manner is completely legal but I doubt that Asimov would have minded – especially since it seems to be currently out of print.
The message of the piece struck a chord with me since I spend a lot of my working and personal life working with various mathematical packages such as Mathematica, MathCad, Matlab, Octave, the NAG libraries and so on (as you may have guessed from the sort of things I write about in this blog) – all of which give humans various ways of doing Mathematics on a computers.
You need a PhD in order to do percentages in your head you know?
Thanks to all of these technological aids, it is certainly not difficult to imagine a time in the very near future when those of us who can compute 9*7 in our heads are in the minority. In fact, it is quite possible that this is already the case. Some time last year I bought a new pair of spectacles that had a marked price of exactly 300 pounds. When it was time to pay for them, the nice lady at the till told me that it was my lucky day because they had just started a 15% off promotion. She then started hunting for her calculator so she could let me know how much I needed to pay as she didn’t know how to get her fancy till to do it for her.
I patiently waited…and waited…and waited some more – her desk was rather untidy you see. Eventually I said – “The discount is 45 pounds so I need to pay 255.” I didn’t want to appear arrogant so I added “I think” to the end of the sentence. She smiled but continued hunting for her calculator. Eventually she found it – punched in the numbers and was utterly amazed that I had worked this out in my head.
This bothered me – why should such a simple calculation be considered amazing by someone who was well educated? When I recounted this story to my wife she rolled her eyes and said (and I quote) “Do get off your high-horse Mike – not everyone has a PhD like you do. You think you are so clever just because you can do a bit of maths.” This upset me even more and off I went to sulk for a bit. If I tell the same story to a mathematician or physicist then they usually respond with a knowing nod and then something along the lines of “That’s nothing – the other day I saw a student get out his calculator to multiply 32 by 10”.
So is this a bad thing? Should we be worried that many people today would struggle to do these basic calculations in their head? Alternatively, to put it in a more domestic setting, who is right – me or my wife?
Strong opinions
A lot of people have some very strong opinions on this – check out “Will it rot my students’ brains if they use Mathematica?” by Jerry Glynn and Theodore Gray for example – where they argue that skills such as mental arithmetic or being able to evaluate integrals by hand are no longer useful in today’s society. Actually they put their point a little more forcefully than this when they say
“If you are worried that your child will suffer by not learning to solve a polynomial by hand, I would suggest worrying more about not learning how to skin a rabbit, or how to start a stalled car. Of all the failures of education likely to get your child into trouble, manual polynomial solving is not high on the list.”
My own opinion is a little less polarised than that – I personally feel that maths education in the future should contain both hand calculations and the use of computers. Basic mental arithmetic, for example, gives one a certain level of intuition about numbers that would be lost otherwise (I am reminded of the story about Richard Feynman and the abacus here) but if I had a large list of numbers to add up then I would use a computer to do it.
Similarly I think that is important that calculus students learn manual techniques such as integration by parts or partial fractions and be able to apply them to reasonably complicated functions and not rely on packages such as Mathematica all the time. On the other hand, they should also know how to evaluate integrals using a computer (both numerically and symbolically) and check that the result is plausible (Even Mathematica gets it wrong sometimes you know). Ideally they would also be able to knock up a simple trapezium-integration routine in a language such as python, estimate the errors, and talk about how you might improve it.
Enough of what I think – feel free to read the Asimov story and the other articles linked to in this post and post a comment letting me know what you think.
Your wife is correct..
Mike,
I’ve been thinking about this for a while since you posted it, basically because I think it’s not an easy question, an there isn’t a “one size fits all” answer. Clearly, if anyone is going to have the benefit of programmes that will do the maths for them, whether in Python or anything else, someone else has to develop those applications – so at least someone needs to know the mathematical background in order to write the routines.
For people who are engaged in research fields that involve mathematics, it seems that an understanding of the principles behind the software operation is also desirable, in order to check the results or to be able to tell when one algorithm will produce a better result than another, and why.
For the general population, however, whether this kind of understanding is necessary is an entirely different question. Part of the problem in answering it is that those of us who enjoy mathematical problems have a subjective relationship with maths – I would say that it is almost an emotional attachment. Partly because we find it wonderful ourselves, I think we expect – and want – other people to feel the same way.
I argued over at Pissed Off that people are less numerate these days because the use of plastic cards in stead of money, for instance, has meant that we can largely get by in society without any understanding of number. This was in response to a post about calculators, a related argument that crops up regularly there as elsewhere.
Whether the decline of numeracy is so bad I’m not sure. There’s certainly less functional need for it nowadays, but my feeling of loss is more emotional than pragmatic. The one thing I can think of, however, comes from the new fashion for “Brain Trainer” type devices, and Sudoku and the like. There’s certainly some evidence to suggest that our brain function can be preserved farther into old age if it is exercised, and mathematical problems and arithmetic must surely exercise different brain areas than word based games and pursuits.