The Mistaken Monolith of Math

By Gillian O'Reilly

“I don’t like math.”...“I’m not good at math.”...“Math is hard.”

Beautiful math:
a Moebius strip

In light of the current discussions around falling math scores in Ontario, I am reminded of how much these comments from kids - and, even more so, from adults - drive me crazy!

It’s not because I think everyone should like math or be good at it. It’s because the speakers are treating math as one huge monolithic subject instead of many fascinatingly interconnected strands.

Contrast this with our attitude to English literature. If a student loved short stories and poetry but was left unmoved by plays, would we say that she/he didn’t like English? No. If a winner of the $65,000 Griffin Poetry Prize was incapable of writing a science fiction novel, would we say they were “not good” at English? Hardly. But we allow ourselves to think we aren’t good at math or don’t like it if we aren’t accomplished at every aspect.

This gorgeous image is math - an algebraic fractal called a Mandelbrot set.
A fractal is a mathematically constructed pattern of shapes that are miniature
 versions of the whole shape and that echo themselves endlessly. Algebraic fractals,
like this Mandelbrot set, may look wild but are still symmetrical at heart.
Image credit: Wolfgang Beyer [GFDL, CC-BY-SA-3.0 or CC BY-SA 2.5-2.0-1.0],
via Wikimedia Commons

I used to be as guilty of this as anyone. Going through school, I had very different impressions of my ability in math, depending on what subjects we were learning. I eventually discovered that my lack of enthusiasm for division didn’t arise from missing 40 days of school in Grade 4. In fact, it is not uncommon to find addition and multiplication easier than subtraction and division.

I loved math the years I encountered geometry (nothing like encountering math with letters to make a bookworm like the subject) and algebra (reducing an unknown to one possible answer intrigues a mystery lover). Fortunately, while other years weren’t so positive, I managed to retain a fondness for the weird, the cool and the paradoxical in math.

When Cora Lee and I wrote The Great Number Rumble: A Story of Math in Surprising Places (Annick Press), we both wanted young readers to find that same pleasure. It didn’t matter whether they found it in fractals or Fibonacci numbers, topology or tessellations, or in the semi-prime numbers which I find inexplicably cool. All that mattered was that they found some part of mathematics to engage them.

If they can do that and can stop seeing math as one big indivisible mass, students - and maybe even adults - can start being able to say, “I like math,” “I’m pretty good at most of it,” or  “This is a bit hard, but it is really fun.”

Eureka!

By Simon Shapiro

I had a "Eureka" moment last night. It was probably triggered by my recently reading a comment by Isaac Asimov that the most exciting phrase in science is not "Eureka" but "That's funny ...".

Unable to sleep, my mind turned to the apocryphal story of Archimedes getting into a bath and noticing that the water overflowed. Supposedly he realized that this effect would allow him to determine (without destroying it) whether the king's crown was made of pure gold or whether the maker had cheated the king by replacing some gold with silver. All Archimedes had to do was immerse the crown in a full container of water and measure the water that overflowed. That would give him the volume of the crown. Then, by doing the same with a lump of gold which was the same weight as the crown, he would know whether the gold crown had been adulterated with a less dense metal.

Archimedes runs naked into the streets of Syracuse

And, supposedly Archimedes was so excited by this insight that he ran home naked, shouting "Eureka" ("I have found it"). My Eureka moment? The sudden thought that it seemed unlikely that Greek technology of the day would have made it a slam dunk to measure the differences in overflow water between pure gold and adulterated gold.

Even less able to sleep, I got out of bed (not naked) and stumbled (not ran) to my computer. I looked up the densities of gold and silver and calculated what the difference in volume would be if a two pound crown were made of pure gold or 90% gold and 10% silver. It would be 2.25cc - less than half a teaspoon! Maybe not impossible for Archimedes to measure - but definitely not easy. Seems more like a "Maybe I can do this with a lot of careful work" than a "Eureka" moment.

Something else suspicious about the story. It's not such an amazing insight that when you dunk something into water, the water level rises by the volume of the dunked object. (Assuming the object doesn't absorb water - dunking a doughnut doesn't do it). In fact crows are smart enough to use that knowledge to raise the water level in a jar. That's the basis of one of Aesop's fables, which were told about three hundred years before Archimedes was born. Archimedes would have heard those fables, which were written down in his lifetime - the third century BCE.

The Crow and the Pitcher, illustrated by Milo Winter in 1919. The crow drops pebbles into the jar
until the water surface is high enough for him to drink from the jar.

I did some more research.

Archimedes wrote a lot about his discoveries but never mentioned the crown. The "Eureka" story was first written down two hundred years after his death by a Roman writer, Vitruvius.

Others have also been skeptical about Vitruvius' story. 430 years ago, at the age of 22, Galileo writes a short paper called "The Little Balance". He not only writes about his skepticism but he describes how Archimedes most likely would have solved the problem. Galileo describes a hydrostatic scale which could have used to determine the composition of a crown. It's based on Archimedes' insight (not, as far as we know, shared by crows or even chimpanzees) that the weight of water displaced by an object exerts an upward force on the object.

Galileo floats a guess at Archimedes' hydrostatic scale.

The object being measured is on the right. It's balanced initially by the weight, d, on the left. When the object is immersed in water, it "becomes lighter" and the weight d must be moved to position g to balance it. The weight difference (which is the weight of the displaced water) can be read directly off wire coils wrapped from e to f.

And finally I could fall asleep.