What do you do with a calculator?

25 November 2016

When should we start using calculators at school? A good question, which I am best to stay away from! This question is loaded with too many arguments concerning the role of rote learning in a young person's development. Yes, it hurts! But there is reason to believe it's the way to build pattern recognition and to free up thinking for harder problems. I will let that debate go on.

What I can share is that while mathematicians would never use calculators (only at the restaurant to split a bill maybe?), what they do work with a lot is computational software. Around the world mathematicians are working mostly with Matlab, Maple, Mathematica and Magma. And Magma was created here in the School of Mathematics and Statistics at the University of Sydney by Professor John Cannon in 1993. Professor Cannon still heads the team that maintains and continues to develop the program. His work is an important addition to the global mathematical community. We are very proud!

So, what do these programs do? What are they for?

Matlab is primarily for numerical computing and is used a lot by engineers and economists. Say I wanted to map pressure points on a structure for different types of loads, model detailed electricity consumptions to build power grids, or run millions of simulations to pinpoint the optimal dose for a particular drug; to Matlab I would turn! For such problems you become a numerical detective to find what combination of numbers will optimise your outcome. Matlab easily lets you solve the same equation over and over again for millions of different number possibilities.

Maple and Mathematica are more attuned for symbolic computation. So for example, you could literally just type in the following symbols xe-2xdx - a question from the 2016 Higher School Certificate Mathematics Extension 2 exam - and upon the press of the 'enter' key the result would appear. These two programs could do most questions on such mathematics exams in a flash actually!

We must remember that programs don't do any thinking though. We humans still have to do that. Which is why we still need to teach mathematics! It's the grind work these programs can help with. I always think of William Shanks who in 1873 after 15 years of calculation gave us π to 707 decimal places. We wouldn't do that anymore! could if you wanted to, but I wouldn't!

Maple and Mathematica are considered healthy competitors. Mathematica is particularly good at producing 3D graphics relevant to certain mathematics questions. Maple is particularly user-friendly and makes it easy to encapsulate your ideas in its programing language. That's just my view, anyway.

This leads us to Magma. Magma is all about attempting to approximate as closely as possible mathematical modes of thought and notation. It's therefore also based on capturing and working with the exact underlying structure of the mathematics. It's used very much at the core of mathematical discovery. Magma is particularly good at manipulating enormous numbers in exact form, which makes it instrumental for cryptography and error-correcting codes. But some other of its specialities include 'Lattices', 'Homological Algebra', 'Finite Incidence Geometry' and 'Differential Galois Theory' - highly abstract mathematics! It's the tool to have at the forefront of mathematical research where, in looking for patterns in the most abstract of mathematical structures, you might get Magma to unravel patterns in a gigantic number of structures first. Seeing what is revealed by Magma, you form a conjecture on a larger pattern, that you then set out to prove formally. Because showing that something is true for a googolplex (ten raised to the power of a googol) of examples, still does not constitute unanimous proof!

It was not uncommon for these initial pattern discoveries to take decades of hand labour before Magma. Now, armed with computing power, the sophistication has grown so that even Magma, spread over a bunch of computers working together, could still take over a year! That's right. You press 'enter' and had better find something else to do!

But we still have to return to find the greater logical steps for that proof. Can't computers do that too? There is belief that in the not too distant future, computers may be able to check all proofs produced. But the consensus about the proof itself seems to be that any possibility that computers could do that as well remains utter science fiction.

So mathematicians are as highly connected to technology today as everyone else. But whether it's Matlab, Maple, Mathematica, Magma, calculators, or anything else, the message is that we must stay aware of ourselves as the user who uses, not the user who has given up!