Two Key — but ignored— Steps to Solving Any Math Problem
By: James Tanton
Tags: Mathematics

Here’s a strange question.

How many degrees are in a Martian circle?

And I am serious. I want you to answer it. Go!

The Nature of Mathematics

Mathematics is a human endeavour, created (or discovered?) by humans for humans, and as such is a fundamentally human experience. It is a glorious subject chock full of passion and emotion. Why else does Sir Andrew Wiles cry on camera in the 1996 BBC Horizon’s documentary when describing his journey in solving Fermat’s Last Theorem?

Every challenge or problem we encounter in mathematics (or life!) elicits a human response. The dryness of textbooks and worksheets in the school world might suggest otherwise, but connecting with one’s emotions is fundamental and vital for success — and of course, joy — in doing mathematics.

So… Experience mathematics as a human! Help your students do so too!

Answering Strange Questions

There are two key first steps to solving any given challenge. These steps are so fundamental, so important, and really will help you and your students make serious progress. I’ve never seen them explicitly stated — perhaps another consequence of the disconnect between school curriculum culture and the actual practice of mathematics — so let’s remedy that now.

STEP 1 to PROBLEM SOLVING: Have an emotional reaction.

We can’t help it. We are each human and we react emotionally to challenges. So acknowledge your human self by pausing to acknowledge your human response to a problem. That is, explicitly take note of your internal state and give it voice. Doing so gives one’s emotions a place to sit and simply be, and not overwhelm.

If a challenge looks scary, say to yourself, “This looks scary.” If it looks fun or intriguing, say “Wow! Cool!” or “Wow. Weird. Could this be true?” If you are suspicious, say “Ooh. Could matters be this straightforward?” If you are flummoxed and don’t have a clue what to do, acknowledge that and say “I don’t know what to do!”

Next step is to take a deep breath and …

STEP 2 to PROBLEM SOLVING: Do something! ANYTHING!

The key is to work past any emotional block that might hold you back. Turn the problem page upside down, draw a diagram, draw a tree, circle some words, answer a different question that might or might not be related. Simply force yourself to actually do something with no expectation of it leading you down the right path — or down any path for that matter. Really, just do something!

I cannot underestimate the power of taking a piece of action of any kind. (In fact this, I think, is my greatest wish for mankind, for everyone to have the confidence and sense of agency to do something — anything! — in reaction to a problem or challenge in life and not to sit completely stymied and stuck.)

An Example: Putting these Steps into Action

I chose the Martian question here as it is so “out of this world” that you really might be flummoxed by it. (Did you say, “I don’t know what to do!”?) What could the question possibly be asking?

In order to DO SOMETHING, let’s not answer that question and answer an easier question instead. Why not?

How many degrees are in an Earthling’s circle?

Well, we Earthling’s say that there are 360 degrees in a circle.

This now begs the question

Why that number? Who chose the number 360 for the count of degrees in a circle?

And when you sit with this question for a while you might realize that this number is very close to a count in a regular, cyclic phenomenon we humans experience: the count of days in a year.

Babylonian scholars of 4000 years ago were very much aware that the count of days in year is 365¼. Shouldn’t we be saying then that there are 365¼ degrees in a circle?

The answer to this question might be clear, and very human: Who wants to do mathematics with the number 365¼? It’s a very awkward number! The natural thing to do is to round it to a friendlier value.

If we round the number 365¼ to the nearest five or the nearest ten we get 365 and 370, not the number 360. Why did we humans decide to round down to 360?

Let’s continue to be very human.

Thousands of years ago there were no calculators and all arithmetic had to be conducted by hand or in one’s head. It is natural to work then with a number that is amenable to easy calculation.

Often in mathematics we want to divide numbers by two and we see already that choosing 365 as the count of degrees in a circle is unfriendly. Both 370 and 360 are even at the least.

We often want to divide things by three as well, and 360 is now looking good! In fact one realizes that 360 is a much friendlier number for arithmetic over 370: it is divisible by three, four, five, six, eight, nine, ten, twelve, fifteen, eighteen, twenty, and more! Whoa!

So for two very human reasons — what we experience on this planet and our desire to avoid awkward work — we settled on the number 360 for the count of degrees in a circle.

Can we now answer how many degrees are in a Martian circle? What do we need to know?

Martians might follow same reasoning we humans did, but in their context. So we need to know how many Martian days (we call them sols) are in a Martian year.

Each sol is 24 hours and 37 minutes long and Martians experience 667 sols in their year. So we might argue that Martians might initially say that there are 667 degrees in a Martian circle. But given that this an awkward number for basic arithmetic, they too will likely round that count to a much friendlier number.

So … What number do you think that might be?

The Moral of this Story

You might be wondering in this essay: Where’s the math? Why is there no analysis of an actual mathematics problem?

The point of this essay is to simply illustrate the power of being true to one’s human self. We have each encountered math problems in our work that have stymied us and have threatened to shut us down. That happens and that is okay.

And the way to get unstuck is to acknowledge that you are stuck — acknowledge your emotions — take a deep breath and just do something, anything! Astonishingly, that alone can often be enough to break through an impasse and get you going on some fun thinking. And thinking is always fun!

The gist of this essay is courtesy of the Global Math Project.