*If I have seen further it is by standing on the shoulders of Giants* – Isaac Newton

Newton was

not first to say this (and it may or may not have been

a jab at Hooke), but the idea is sound: we can get further because we don’t have to invent everything from scratch; we can build on what others have done before. So, if we need to solve a particular problem that needs calculus, we don’t have to invent calculus from scratch to do so, we can use what Newton (and Leibniz, of course) invented. Standing on their shoulders, we can see further.

But how do we get to stand on the giant’s shoulders? (I’ll keep the metaphor to a single giant, as standing on the shoulders of multiple giants sounds too much like a circus act. And I am focussing on the

*mathematical* giant.) We aren’t born up there on the giant’s shoulders. While we don’t have to grow into the giant (

*invent* calculus), we do have to climb up the giant (

*study *calculus).

And the giant is getting ever bigger. On the one hand, this is good: being so much higher we can see so much further. On the other hand, what happens when we have to spend our entire lives climbing up the vast growing giant, and never reach the viewpoint on the ever-distant shoulders?

We need short cuts up the giant. Fortunately, other are building ropes and ladders and lifts: tools to climb the giant more easily. So, we now have computer algebra packages that can solve our differential and integral equations for us; we no long need to spend years studying and practising how to do this.

But wait! cry the purists. That is cheating.

*there is no Royal Road to geometry* – Euclid

*Understanding an idea meant entangling it so thoroughly with all
the other symbols in your mind that it changed the way you thought
about everything*. – Greg Egan

There

*are *no shortcuts, the purists insist. Mathematics is not a “spectator sport”. You have to do it, be immersed in it, internalise it, in order to

*really *understand it. The youth of today, with their fancy calculators and computers, they don’t really understand arithmetic and algebra and calculus. Get off my lawn!

There is nothing new under the sun when it comes to criticism of youth, of course.
Plato, in

*Phaedrus*, has

Socrates rail against this new-fangled literacy:

[writing] * will introduce forgetfulness into the soul of those who learn it: they will not practice using their memory because they will put their trust in writing, which is external and depends on signs that belong to others, instead of trying to remember from the inside, completely on their own. You have not discovered a potion for remembering, but for reminding; you provide your students with the appearance of wisdom, not with its reality. Your invention will enable them to hear many things without being properly taught, and they will imagine that they have come to know much while for the most part they will know nothing. And they will be difficult to get along with, since they will merely appear to be wise instead of really being so. *

This sounds suspiciously similar to those modern complaints about using calculators rather than mental arithmetic, or using computer algebra programs rather than slogging through pages of pushing symbols around. These devices give only the “appearance of wisdom”.

I do have some sympathy with this view. There does seem to be a lot of blind trust in the output of calculators and computers. However, I’m not sure it is purely the fault of the calculators. There can be uncomprehending blind trust in symbol pushing, too. I remember, many years ago, being in a computer shop, buying four items. The shop assistant wrote down the prices, and laboriously added them up, with much crossing out. When they announced the total, I said “that’s wrong”. They got a bit huffy, but then I pointed out their total was too

*small*: it was less than one of the items on the list! As well as their huffiness, I detected a faint feeling of puzzled wonder from the assistant: how had I

*known*? Despite the hand calculation, the assistant had no

*feel *for the numbers. Maybe Socrates would have said that they should have added the numbers in their head? (Notice here that I didn’t know what the right answer was, but I knew the suggested answer was wrong.)

Another example comes to mind, again from many years ago. We were buying some new pillows: four for £4.99 each. The shop assistant wrote down 4.99 four times in a list, and

*added them up*. Meanwhile I was going “£4.99 is a penny less than £5, so that’s £20 minus 4p, or £19.96.” I had the right money ready by the time the assistant came up with the answer, and was again met with puzzled wonder. (I’m sure that’s the real reason supermarkets took the prices off their goods: to stop some customers freaking out the cashiers by having the right money ready!)

I recounted this pillow story to my mother, who, faster than I did my shortcut calculation, simply multiplied 4.99 by 4

*in her head*, and got the right answer. I was almost as much in awe of this feat as the shop assistant had been in mine. But which approach shows more understanding of numbers: my short cut or my mother’s brute force calculation? Is it possible that slogging through all those exercises merely enable us to do calculations quickly,

*without thinking*? And if there is no thought, then what have we actually gained? After all, one can learn by rote and merely “parrot” remembered answers.

Back to climbing that giant. What we need is a way of taking short cuts up

*and *of having the “feel” for the numbers. An approach that could work is critical thinking about the supplied results (whether supplied by computer, or by our own unthinking calculations). We can keep the feel by using even shorter short cuts and heuristics that give an

*approximate *answer, as a sanity check. Those shorter cuts and heuristics supply the feel, and when

*they *are done automatically, they

*are *the feel.

So, education shouldn’t be focussed on getting students to wade through pages and pages of exercises, pushing symbols (be they numbers or letters) around (unless they enjoy that sort of thing, of course). It should be more focussed on training in the use of short-cut tools, education on where and how to apply the tools, and meta-training in critical thinking about the results those tools give. Then we can climb the ever-growing giant fast enough to get to the top in time to see something before we die, and confident that we’ll understand what we do see when we get there.