“When I began my college education, I still had many doubts about whether I was good enough for mathematics. Then a colleague said the decisive words to me: it is not that I am worthy to occupy myself with mathematics, but rather that mathematics is worthy for one to occupy oneself with.”
I would like to win over those who consider mathematics useful, but colourless and dry — a necessary evil…
No other field can offer, to such an extent as mathematics, the joy of discovery, which is perhaps the greatest human joy.
The schoolchildren that I have taught in the past were always attuned to this, and so I have also learned much from them.
It never would have occurred to me, for instance, to talk about the Euclidean Algorithm in a class with twelve-year-old girls, but my students led me to do it.
I would like to recount this lesson.
What we were busy with was that I would name two numbers, and the students would figure out their greatest common divisor. For small numbers this went quickly. Gradually, I named larger and larger numbers so that the students would experience difficulty and would want to have a procedure.
I thought that the procedure would be factorization into primes.
They had still easily figured out the greatest common divisor of 60 and 48: “Twelve!”
But a girl remarked: “Well, that’s just the same as the difference of 60 and 48.”
“That’s a coincidence,” I said and wanted to go on.
But they would not let me go on: “Please name us numbers where it isn’t like that.”
“Fine. 60 and 36 also have 12 as their greatest common divisor, and their difference is 24.”
Another interruption: “Here the difference is twice as big as the greatest common divisor.”
“All right, if this will satisfy all of you, it is in fact no coincidence: the difference of two numbers is always divisible by all their common divisors. And so is their sum.”
Certainly that needed to be stated in full, but having done so, I really did want to move on.
However, I still could not do that.
A girl asked: “Couldn’t they discover a procedure to find the greatest common divisor just from that?”
They certainly could! But that is precisely the basic idea behind the Euclidean Algorithm!
So I abandoned my plan and went the way that my students led me.
Remember the Math Adventurer’s Rule: Figure it out for yourself! Whenever I give a problem in an Alexandria Jones story, I will try to post the answer soon afterward. But don’t peek! If I tell you the answer, you miss out on the fun of solving the puzzle. So if you haven’t worked these problems yet, go back to the original post. Figure them out for yourself — and then check the answers just to prove that you got them right.
After the Pythagorean crisis with the square root of two, Greek mathematicians tried to avoid working with numbers. Instead, the Greeks used geometry to demonstrate mathematical concepts. A line can be drawn any length, so straight lines became a sort of non-algebraic variable.
You can see an example of this in The Pythagorean Proof, where Alexandria Jones represented the sides of her triangle by the letters a and b. These sides may be any length. The sizes of the squares will change with the triangle sides, but the relationship is always true for every right triangle.
The story of mathematics is the story of interesting people. What a shame it is that our children see only the dry remains of these people’s passion. By learning math history, our students will see how men and women wrestled with concepts, made mistakes, argued with each other, and gradually developed the knowledge we today take for granted.
In a previous article, I recommended books that you may find at your local library or be able to order through inter-library loan. Now, let me introduce you to the wealth of math history resources on the Internet.
Math concepts: subtraction within 100, number patterns, mental math Number of players: 2 or 3 Equipment: printed hundred chart (also called a hundred board), and highlighter or translucent disks to mark numbers — or use this online hundred chart
Place the hundred chart and highlighter where all players can reach them.
How to Play
Allow the youngest player choice of moving first or second; in future games, allow the loser of the last game to choose.
The first player chooses a number from 1 to 100 and marks that square on the hundred chart.
The second player chooses and marks any other number.
On each succeeding turn, the player subtracts any two marked numbers to find and mark a difference that has not yet been taken.
Play alternates until no more numbers can be marked.
The title which I most covet is that of teacher. The writing of a research paper and the teaching of freshman calculus, and everything in between, falls under this rubric. Happy is the person who comes to understand something and then gets to explain it.
No peeking! This post is for those of you who have given the trisection proof a good workout on your own. If you have a question about the proof or a solution you would like to share, please post a comment here.
But if you haven’t yet worked at the puzzle, go back and give it a try. When someone just tells you the answer, you miss out on the fun. Figure it out for yourself — and then check the answer just to prove that you got it right.
[Feature photo above by Michael Cory via Flickr (CC BY 2.0).]
One of the great unsolved problems of antiquity was to trisect any angle using only the basic tools of Euclidean geometry: an unmarked straight-edge and a compass. Like the alchemist’s dream of turning lead into gold, this proved to be an impossible task. If you want to trisect an angle, you have to “cheat.” A straight-edge and compass can’t do it. You have to use some sort of crutch, just as an alchemist would have to use a particle accelerator or something.
One “cheat” that works is to fold your paper. I will show you how it works, and your job is to show why.