niffyjiffy

I first learned in elementary school of a proverb: the greatest quality in a mathematician is laziness. When I introduce newcomers to pure maths, laziness is the first concept I explain. It comes as a surprise to most—many react with “if I was lazy I would simply not bother to do maths.” The fact is that until one learns to like maths, it is impossible to do it lazily.

As a mathematics tutor, I am enthusiastic but fearful to bring this proverb to the classroom. The mathematics which my students bring to class can be quite lazy in a sense, generally prepared so as to minimise setup and get straight to the calculator. The calculator is becoming a much more prominent part of the mathematics curriculum. At school, students are told which calculator to buy, and whole lectures are dedicated to which buttons to press to solve all your problems. In later years, students are introduced to terrifically powerful tools such as Desmos and WolframAlpha which trivialise the problems they've been solving for years.

When I try to deprive them of these tools, most students appeal to what their teachers permit them to use. However, students who hope to win argue that in the real world, nobody would go to the trouble of working a problem out on paper if the internet can solve the problem as fast as it can be typed. It's certainly the best counterargument, but it's the one I'm the most prepared to deal with.

In reality, most adults are far too lazy to use a calculator, and rightly so. If any power is desired beyond the four basic functions, calculators suck. They cost an unjustifiable amount of money. Expressing problems more complicated than trivial computation takes practice, practice which doesn't pay off unless you're completing math problems as often as a high-school maths student. Put it this way: if in the middle of a conversation you became intrigued by a simple derivation of a sports stat, pulling out a calculator would totally kill the conversation. Even among my mathematically inclined friends, calculators are avoided by referring to tedious-to-compute numbers as “some number”.

What place, then, does mathematics have in the real world? To illustrate the kind of problem that can, and should, be solved in daily life, I'd like to introduce one of my students, who is not called Mark. Mark is a student that I can easily bait into attempting math puzzles, mainly because he enjoys taunting his teachers with problems they can't solve. He came to me with the problem in the illustration below (thanks to the Scriptorium for inspiring me to put some damn illustrations in these things), bragging that it could only be solved using calculus. Mark is an Algebra II student, but the funny thing is, I'm quite sure a person who had taken calculus would have the same reaction: that it could be solved using calculus, but that they couldn't do it. I'm damn sure that no farmer I've met would set up linear equations to represent the path to the river and the path to the turkey, then take the derivative of the lengths of the lines in order to determine the optimal strategy.

So I showed Mark what he hadn't seen: the beautiful, lazy pig sunning herself across the river. This pig is going to save us many thousand years of mathematical rigor. She will let us be lazy with her. The reason is that she is just as easy to water as the turkey. As long as we assume the river is simple to ford, any path that reaches the turkey can be reflected to reach the pig, as seen in the example.

But the quickest path to the pig is simple to find—it is simply the straight line which passes through the river. Seeing this, we observe that the river and the path are two straight lines, and the path meets the river at two congruent angles. We reflect back the portion of the path after the crossing, and the problem from here is simple geometry.

Let's compare the two approaches. In particular, we have showcased two different kinds of lazy. The calculus approach minimises the amount of work done—the student sees that it is a calculus problem, and quickly determines that it is not worth doing. In comparison, the paragraph I've written and illustrations I've drawn took real work. The only reason I cared to do it is because I could see the pretty picture in my head. I would never have finished this problem if it looked like a miserable slurry of algebra and calculus. Unless the problem shows some promise, the hard work is not worth it. Instead I got to draw pictures of piggies, and the monkey-work[1] took only a brief moment. Rigor, worksheets, and formulas tend not to survive in the real world, but I like to think this piece of reasoning would survive a casual conversation. I ask, therefore, what this other “laziness” has to show for its hundreds of hours of work.[2]

Footnotes

[1] I would like to apologise to the monkey community, who are very capable of problem solving, and very incapable of algebra. [2] I will answer this soon.