“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.”
— Rózsa Péter
Mathematics is beautiful
essay in The Mathematical Intelligencer
Rózsa Péter and the Curious Students
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.
— Rózsa Péter
quoted at the MacTutor History of Mathematics Archive
For Further Exploration
Note: When the video narrator says “Greatest Common Denominator,” he really means “Greatest Common Divisor.”
CREDITS: “Pink toned thoughts on a hike” photo courtesy of Simon Matzinger on Unsplash.
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.
Euclid’s Geometric Algebra
Continue reading Answers: Euclid’s Geometric Algebra
Picture from MacTutor Archives.
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.
Continue reading Euclid’s Geometric Algebra
[Image from the MacTutor Archive.]
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.
Continue reading Math History on the Internet
Sitting at home with a cold, tired of watching TV and playing video games, stumbled upon…
A great theorem from math history
The Infinitude of Primes
Continue reading In Between Sneezes… Multiplication Videos
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.
Continue reading Euclid’s Game on a Hundred Chart