Introduction

Everytime again, whether during exercises or at an exam, when I’m asked to manually solve a system of equations, calculate a matrix product or inverse manually I really, really get annoyed.
Because honestly, I just don’t get it.  In this post I’ll try to share my opinion by using examples that everyone is probably familiar with.

The misconception of insight

Thomas Goossens 2 July 2012, Leuven, Belgium
Engineering student computer science at University of Leuven (2nd bach)

Everything I write is my opinion.  Because of the perspective I’ve written this text in, you might think I’m trying to act like everything I say is a fact. But I cannot emphasize enough that everything you will read is how I think about it.

Why doing manual calculations?
Yes. When you think about it, most of your time at middle school, when you are “doing math”, you are in fact just doing manual calculations. Your tests and final exams will mostly consist of “solve for x”, “calculate this integral” and that sort of stuff.
Now I want you to really think about this following question. “Why did you learn all that?”.

The first thing you might want to say is “Can’t you say that for all courses during middle class?”.
The truth is that most students probably will say that during their years at school. But afterwards, at least in my opinion, I’m quite happy that I’m able to speak English and French, that I know something about our (cultural) history and also how babies are made. (pun intended). Of course there are things I’m still not seeing the point of, but hey! Lets not get distracted!

Back to the question. “Why did you learn manual calculations?”.
To get insight? (In what?)  To enjoy yourself? (Oh please..) Or to be able to solve the greatest problems in the world? (Yeah right...)
People may argue that before using a computer you must know how to do the “real” (manual) thing... learn the basics.  What exactly is meant with the “real” thing?

As it turns out it isn’t that easy to answer that question. At least not nowadays. Because times have changed and I fear a lot of people haven’t noticed it.

Why manual calculations are obsolete
The thing people are “afraid” of when talking about doing math on a computer is. That you won’t understand what you are doing anymore.

Before I continue I’m going to tell you the shocking truth about manual methods. They were developed because people didn’t have computers those days. Most of the methods you learn at school are just clever tricks developed by some clever people so that during their era, they could do the calculations more easily. As an example I will discuss the calculation of logarithms.
They are not easy to calculate by hand. However, not such a long time ago people were still using a clever trick to easily calculate logarithms.

Imagine you are trying to solve a physical problem. And at some point you must know what log(5) equals to. So what does our “oh so intelligent man” do to get the answer? Well one way to do it is by using logarithm tables.
A logarithm table using the property that a logarithm of a product is the sum of the logarithms.  Which is nothing more than a clever trick to use it. The real intelligence here lies in the invention of the table.

Logarithm Table

How is this different from “mindlessly” asking your little calculator what the logarithm of five is? I don’t know … maybe it saves you time and allows you to more important work instead of wasting time looking at the table and “mindlessly” reading the numbers out of the table which you assume to be correct.

Wait! We just encountered a very important fact. People don’t use logarithm tables anymore but instead use a calculator. Because it is faster. So without actually knowing what happens “under the hood”, you are able to do some useful work. And yet you know nothing more than when you would have used a logarithm table.

Lets talk about another of my favourite topics: integrals. Integrals are very useful mathematical operations and as an engineer I want to be able to calculate the integral of a function as fast as possible. Of course people would think I would be insane if I did it by hand.

It is here that I often get the answer that before you can use it like that, you must gain insight into the calculation.
Insight into what exactly? What have you learned about the fact that by manually calculating that x^2 is (1/3) * x^3  ( + c for the purists out there ).
Exactly, you learned nothing more than when you would have calculated it with a computer or just read the answer on the solution page. The same accounts for solving a system of equations or a matrix product. In the case of the matrix product however, it is useful to actually do it a few times yourself (manually).

I hear you cry… you hypocrite! No! You are completely and utterly wrong. The reason why it is useful to do it a few times manually (or visually) is not because it would give you insight into the matrix product. But because it will learn you how it is defined. And that is something completely different. So by doing it a few times yourself you can test whether you understand the definition.

One last topic I just want to touch very shortly: derivatives. Now be honest. Deriving a function is just mindless work. (Proving what the derivative of a function is, is a whole different story). Everyone can learn how to derive functions. It really disgusts me if people laugh at you when you don’t know the derivative of some function by heart. Those situations only emphasize my points on the misconception of “insight in mathematics”.

Conclusion
People should really think about how pressing a few buttons on a computer is different from any other method for making calculations. It’s true, using a computer is nothing more than a new and exciting method to do calculations. Because the real intelligence lies in the creation of those clever methods to make things easier for us. And absolutely not in the student who is just executing an algorithm in just the same way a computer does. People should therefore stop trying to become a computer. We already have those.

So am I saying that a great part of mathematics you learn during middle class is just plain obsolete and stupid?  In many ways, yes! I’m certainly not done on this topic yet so I’ll leave that for a next post because I might need the room for it.
Stay tuned and feel free to post any comment below.

Update 2 July 17h51

I have encountered this very interesting video of Conrad Wolfram at a TED conference. And he shares exactly the same opinion.