What does it mean to “Prove” something?

Some time around grade 11 or 12, you start to see questions with start with the word “prove”. (There are closely related questions that start “Show that….” and “Verify that…” These questions are harder because rather than trying to find the correct answer, you are given the correct answer and asked to explain why it’s true. And explaining always takes more work than doing.

This video by Derek Muller at Veritasium is a great introduction to the concept of “proof” and why it’s so hard. There is nothing hard to understand about the math involved; just making lists of numbers following rules. But it shows the problem of “proof”. You can’t “check” the answer for all possible numbers because there are an infinite number of possible numbers and that would take literally longer than forever.

At the end they also talk about something important which is the idea of a “counterexample”. That means one exception that breaks the rule you suspect of being true. When you think about it logically, any statement is true for all numbers, if you can fine just one number where it’s not true, all minus one is no longer all. So you if it turns out the thing you suspect is true really isn’t, you can show it with a single example, which is of course much faster. (Science works like that, sort of. You have things you believe are true and you send people out — that is do experiments — looking for a counteerexample, and if loads of smart people fail to find one, then you can continue to feel confident that the thing you think is true really is true.)

Knotty Mathematics

It’s actually very hard to define exactly what mathematics is. On top of all of the number-heavy things like arithmetic and algebra, mathematicians tend to indulge their curiosity and apply their problem solving strategies to all kinds of things. Including the properties of knots and tangles.

It’s quite different from the kinds of things you might see in math class, but still interesting.

The Mathematics of Counting Votes

Do you pay attention to the news? Or does someone in your household pay attention to it and you just happen to hear about stuff as you go through your day?

If you’ve been listening to the news over the past year, most of it has probably been about one of two topics: the pandemic, and elections! As you go through high school and get older you start to think about your soon-to-be right to vote.

Thinking about numbers during voting can be confusing. On the one hand, it’s hard to really feel like just one person’s vote counts for very much at all. On the other hand the elections you vote in seem to matter an awful lot. And that can be hard to reconcile.

It turns out that from a math perspective, elections can get even more confusing? Did you know that it is possible for someone to get fewer votes than their opponent and still win? HOW?!? It turns out that our elections have complicated middle steps. You don’t just ask everyone who they want to be prime minister (or president) and count up all the votes and whoever gets the most wins. The country is broken up into many small districts, and each district counts its votes and then reports who got the most in their district. That means there’s a flattening effect. If you won with 90% of the vote, or you won with 53% of the vote, it counts the same. And that means that there are clever ways to organize the boundaries of the districts themselves to possibly still win, even if fewer people are voting for you.

That’s one of the many topics in this long math-heavy podcast with Jordan Ellenberg and Sean Carroll:

Nothing to do with this episode, but Jordan is a great math communicator and advocate. He wrote the math book with the greatest title ever: How Not To Be Wrong – The Power of Mathematical Thinking.

Exponential Growth in Bacteria

(Definitely check this out if you are taking biology and math at the same time)

At some point, typically in Grade 11 in Ontario, you learn about exponential growth. This shows up twice. Once when exploring exponential functions, and also as part of geometric sequences and series. (The math behind the two is similar, the difference comes typically in whether or not we’re interested in graphing the results.)

But one of the most important real world examples is the growth of bacteria. Here you can see some of the numbers involved. In particular, look at the length of time that it takes a size of container of water undergoing exponential growth to get as large as the earth, or as large as a galaxy. Most people find exponential growth extremely confusing, or at least surprising because the answers you get tend to be very far away from any answer you would estimate. (That’s why, in this difficult last year, you notice many people can’t seem to imagine how a small chance of spreading COVID can get out of hand so quickly.)

The second piece of interesting math comes when looking at unlikely events. In this case, that’s the rare event that a single bacterium (singular for bacteria… although you won’t have just one for very long) picks up a mutation in its genetic code. (Single-celled organisms have DNA like we do.) The odds of that happening are very low. But you might have billions of bacteria in a single flask, and so when you look at the flask as a whole that number is quite big.

Where do the Symbols Come From?

Most of us realize as we go through school that math is its own language with its own symbols and grammar. But where did that come from? It didn’t appear overnight. It developed gradually over many centuries.

This talk by Sarah Hart from Gresham College looks at where it all comes from. Including numbers the symbols we use for plus and minus, exponents and square roots and more.

The history of mathematics is not just interesting. I think it’s important for students to understand that the material that they are being asked to take on as teenagers is stuff that took some of the smartest humans alive hundreds and thousands of years to figure out. And part of what makes that material tractable is that we evolved clear and useful symbols to express our ideas.

Can you imagine trying to do (what we would now call) algebra for two thousand years without an equals sign?

Fermi Estimates

Take a look at this video from Numberphile applying Fermi Estimates to a curious question: How much Coronavirus is there in the world?

According to Kit Yates, it would all fit inside of a Coke can. On the one hand, that’s a surprising bit of number trivia: the idea that something so small could cause so much harm.

On the other hand, it’s interesting to see the power of math to extract a lot of information even in the presence of a lot of uncertainty. Often you don’t need to know an exact answer, but you just need a general sense of how big a number is, and that can be worked out with very rough estimates as you’ll see above.

The technique is named after the American Italian physicist Enrico Fermi.

A Logic-Themed Rave at “Learning Man”

Here is a logic puzzle from TED-Ed about some logicians (people who obsess over logic… think of a cross between a philosopher and a mathematician) who attend a rave out in the desert at “Learning Man”

It’s a fun short thinking game that you might like to play… or not… and that’s what today’s post is about. One thing that struck me about this video was even though it was enjoyable, it didn’t resemble anything that I saw in school.

If you watched the video, you realize there are two ways to do an activity like this. One is to watch the video straight through, which means the question is immediately followed by the answer.

The other way would be to pause the video and try to answer it yourself and see if you can do it. Therein lies the problem. How long do you pause for? A minute? Ten minutes? An hour? A week? Secretly built into the way the question is phrased is that it’s possible to answer the question, and not just answer it by making a lucky guess, but by actually justifying your answer so you can know you’re right with or without a teacher with a red pen to tell you.

But what if you’re a teacher with that red pen. How long do you let them sit and ponder? And what does that time look like? Do you make them sit quietly at their desks and yell at anyone who speaks? Do you let them ask for hints? Work in groups? Or do you send it home as a kind of free-time assignment and go on to teach other material? After all, time in class with a teacher is a scarce resource. Sitting still staring at a blackboard doesn’t strike anyone as a productive use of that time.

And then after they’ve had their best shot, when do you tell them the answer? After one student gets it? After half of the students get it? After 90% of the students get it? If no one gets it, do you just leave it forever as an unanswered question that they never know the answer to?

And that points to another problem. Once you know the answer, what does that get you? This problem with demons and masks and colours and standing in a circle with the random rule that you’re not allowed to hold up your phone and look at the reflection in the glass to know what colour your own mask is… rather singular. If I explain how this problem gets answered, does that empower you to answer other logic problems? Or only was this just a one off puzzle, that you either get or don’t and the only advantage to getting it is the smug look you get to give to the kids who don’t?

Does this just lead people to learn that some people can do math and some people can’t and shouldn’t waste their time trying?

Now you can start to see why your classroom isn’t like that.

For one thing, there’s the time constraint. If you want to be able to use math in your adult life, there’s a certain amount of it you need to learn. And if you’ve noticed the speed at which we move from lesson to lesson, especially in high school where we are basically trying to teach you a new thing every class from now until the end of your school life. We’re in a real hurry.

And we also tend to work on very general problems. When we teach you how to solve an equation for x, the technique we try and teach you is supposed to work for all similar equations. If we change a four to a seven, or make a five into two-thirds, the technique keeps working.

Part of that mask problem is if I change the number of people in the circle up or down one, that problem becomes very different, and probably unsolvable. The puzzle solving aspect of taking in each new problem and staring at it for an hour or two hoping for a flash of inspiration is the sort of activity you’re likely to hate (except for a small fraction of people who like that sort of thing.) And as a teacher, if a student wants “extra practice” I don’t have a bottomless reservoir of those questions that I can keep giving them so that they can get the hang of it.

On the one hand, I think these activities are fun, provided you have an escape hatch. The answer is there when you decide you’ve had enough. And if you do enough of them, you start to notice a set of “useful hints” that might be transferrable. But they aren’t good “classroom activities” and I think by now you can see why.

Distributive Surprise

Early on in high school, we all learn the distributive property. Most of the time it’s just a mechanical tool designed to get rid of brackets in expressions. But we were blown away by this video from the Mathologer which gives one of the most startling applications of the distributive property we’ve ever come across.

The problem being solved is a strange one. For a given amount of money, how many different possible ways are there to make change.

The video starts out simple, so if you have been working with the distributive property for a little bit, try to follow along. Like most of his videos, the material gets more complicated the more you watch, and usually by the end you are into some serious university level stuff. But hang in for as long as you like and see how far you get.

The Confusing Origins of Calculus

Are you getting into calculus this quadmester? It’s a subject that many people find confusing at first. Here’s a short video that might help.

Calculus was created because there were a number of logical problems that, at the time, seemed impossible to solve. Most of what gets taught in math class originates because at some point, someone had a problem they wanted to solve, and we’ve been collecting the more interesting and/or useful solutions for hundreds and thousands of years.

Some students think that math is just “made up” because it’s nice to have some random rules to make students memorize. But we are genuinely interested in solving problems. (I’ll admit, not all of the problems are practical. Sometimes we just play with ideas because we find them interesting, elegant, or like many people we enjoy a challenge.)