Why is maths different?

Humans struggle at a lot of intellectual endeavours. But for some reason, of all of them, maths seems to have an unreasonable level of arcane-ness (arcanity?). People who haven't spent much time thinking about maths are not only unable to appreciate mathematical arguments, but seem to be unwilling to. The same isn't true of other disciplines.

Why is this?

Here's a metaphor: Imagine that the only person who was able to hear a piece of music is the person playing it. So someone is playing a piece of music on the piano. You're watching, and you can only watch their hands move over the keys. If you're observant you may start to get an idea of which keys they press at the same time, and which ones tend to follow each other.

But then they give you the score, and you sit down at the piano and start to play through it yourself. Now you can hear it, and you begin to understand how it fits together and what makes it beautiful.

That's what maths seems to be like. You can't understand or appreciate the complexity or beauty of a piece of maths unless you're creating it yourself. And you can't create it yourself without having learnt a bit about it. And you probably won't want to learn anything about it until you've had a small experience of hearing it.

Algebra

Reading the comments to this article. There's more than one comment with the sentiment "I cannot recall the last time I used algebra!"

If you've ever been in a situation where you had to try to calculate something without having all the information, you've used algebra. Shopping without knowing the prices. Deciding what to spend your tax return on before you get it. Figuring out how big a home loan you can afford.

Let's say you're organising an event, and you want to get an idea of how much profit you're going to make. You know what your expenses are, and what the ticket price is, but you don't know how many people are going to turn up. What do you do? I'm a mathematician, and I'll tell you what I do: I work out a pessimistic estimate, and an optimistic estimate, and keep those two numbers around. I know that the real ticket sales will be somewhere in between, and now I have a rough idea of what's possible, and what's likely. I don't work out the precise linear relationship between ticket sales and profit.

You don't need to write any of this stuff on paper in order for you do be doing things mathematically. Of course if you do, you can get much more accurate results. But most of the time we don't need more accurate results, we just guesstimate. But it's still mathematical thinking.

Confusion

I don't think it's always necessary to avoid confusion. At least when teaching someone. If just trying to communicate then yes, confusion is bad, but teaching is different. At some point you need to learn how to deal with confusion. Not being exposed to it means you're being shielded.

Maths activities/problems

Buckets of fish! - http://jdh.hamkins.org/buckets-of-fish/

Fractivities - http://fractalfoundation.org/resources/fractivities/

I've just been reading this article by Adam Spencer, which has some interesting (one might say scathing) things to say about ATAR scores.

At the end of high school, in 2002, I went through a process that I haven't thought a lot about since then but has just now struck me as significant.

The QLD system gives eligible students an OP, which ranges from 1 to 25, with 1 being the highest score. The calculation of the OP is different to that of the ATAR, so they aren't easily convertible. But once you have a score the process for entering uni is similar.

I had an OP1. Very proud and all that, but then it came time to apply for a university course. The QTAC form allowed you to choose 6 options, and if you didn't get an offer from your first choice you would then be able to try for the second, and so on. Each course was associated with an OP, which was the lowest OP (biggest number) that they had accepted the previous year. Some courses were fairly stable but others could fluctuate 2 or more points from year to year.

I wanted to do science and maths in Brisbane. There were only two courses I found in the QTAC handbook that I liked the sound of: one was a Bachelor of Science at UQ, the other was a Bachelor of Applied Science (accelerated) at QUT. The first had an OP9, the second an OP1. The second required potential students to go for an interview before being accepted.

The first significant thing about my process was that these two courses were the only two I applied for. Common wisdom said to use up all six options, because the low OP of 9 does not guarantee that the course won't suddenly be hugely popular amongst OP1 students! But I couldn't find any other courses I actually wanted to do. The above article says that some students will apply for courses based on their ATAR: "I really want to do med. If I miss that, I'll do law...". I couldn't bring myself to apply for a course I wasn't interested in.

I put the QUT course first (because it had the higher score). After a few weeks I received a letter from them saying that I had been accepted for an interview, which was very nice.

But then came the second significant thing. I had realised by this stage that I wasn't so interested in applied science, because I wanted to do maths for maths' sake. But on the other hand I really strongly felt that I would be 'wasting' my OP1, which I had worked hard all through high school to get, if I 'settled' for a course that only needed a 9 to get into. There was prestige associated to the QUT course which wasn't attached to the UQ course.

But I reminded myself that the score for the UQ course was only low because it is so broad that they accept a lot of students, reminded myself that I wanted to do maths and not applied science, (and to be honest the idea of the interview scared me a little bit), and declined the interview. I was then offered a spot in the BSc at UQ and didn't look back.

Even at the time I was surprised by how strong the pull of prestige was. The idea of having 'wasted' my OP1 on a lower-prestige course, though it niggled at me, is baloney: If I'd only learnt enough to scrape through getting into the course then I wouldn't have had learnt as much at school. As it was I found myself struggling at uni, and being less prepared might have led to failing a subject or two. Or quitting before I did.

In retrospect I know that I made the right choice, but funnily enough for a different reason to the one I had at the time. The course was OP1 because it was accelerated, which would have challenged me. That might have been a good thing. But I don't think you learn as well if you try to learn in a hurry. The brain needs time to absorb knowledge. I did a 1-year honours course instead of a 2-year masters (because it was easier to get into the masters), and I now regret it.

Bottom line - not every opportunity will be good for you, or what you actually want to do.

Multiplication tables

Everyone remembers the multiplication table, right? It's that grid of numbers that you can find on the back of almost any school workbook, with numbers 1 to 10 (or 12) along the top and left side, and products of numbers filled in the grid.

You may remember staring at this while trying to memorise the 7 times table. Well, have no fear, there will be no number facts quiz as part of this article. But there are a few pretty facts about the times table that most school lessons miss.

Let's get the obvious out of the way first. (I'll save the best for last — maths always seems to do that.) By drawing a straight line down the main diagonal of the table, it becomes clear that there is a 'symmetry': The numbers above the diagonal line are like a mirror image of the numbers below the line. So the number in the 5th row and 7th column is the same as the number in the 7th row and 5th column (35).

So if your aim is to memorise the table, you really only need to memorise half of it. The reason for this symmetry is simple enough — 5 × 7 is equal to 7 × 5. But why is this true?


To show this in a simple way, first we need to take note of something that's easier to understand but maybe less obvious. Take the same entry we just looked at (5th row, 7th column) and colour in all the entries left of and above it:

Now we have a rectangle, of length 7 and width 5. The area of the green region is given by 5 × 7 (by the definition of multiplication). The number in the blue circle (35), the bottom right coloured square, is therefore equal to the area of the entire green region. This will always be true no matter which number in the grid you pick.

This gives us a visual understanding of what the numbers in this grid are telling us. Back to the symmetry — why is 5 × 7 equal to 7 × 5?

If we flip the above green rectangle about the main diagonal, we get the following picture:

All we did was flip the rectangle, so the number in the blue square should still be 35. But now the entry is in the 7th row and 5th column, meaning 7 × 5. So the fact that a × b = b × a ('multiplication is commutative') means that flipping a rectangle doesn't change its area.

But there's more we can do with a multiplication table. Pick a number (say 10), and then colour in all the entries in the table equal to that number.

No matter which number we pick, we will always have to colour in a square in the first row and in the first column, because every whole number is a multiple of 1. In 10's case, there are two more coloured squares — in the 2nd row and in the 5th row, since 10 is a multiple of 2 as well as 5.

The number of coloured squares tells you how many factors a number has — if it's prime only two squares are coloured. Here is an animation of the occurences of all the numbers from 1 to 40:

Now we can see a pattern emerging. The blue squares always seem to form a nice curve — and there's a reason for that! Let's say that 'a' says which column a square is in and 'b' says which row it's in. The blue squares are the ones that satisfy the equation a × b = n, where 'n' is our chosen number (n=10 in the previous example). If we were to draw a smooth curve through all these points, we would get the line corresponding to this equation, where 'a' and 'b' are not necessarily whole numbers. The blue squares are the points that the curve passes through that are whole numbers.

This curve is known as a hyperbola.

There's one more thing you can do with multiplication tables that results in pretty pictures — like I said, I've saved the best for last. Again we'll colour certain entries of the table, but this time we'll colour every entry that is divisible by a certain number. Thus, for 1 every entry is coloured, whereas for 5 the picture is:

Clearly any row or column that is a multiple of 5 will be entirely coloured. But something slightly different happens for 4:

We get the same sort of lattice pattern in columns and rows that are multiples of 4, but there are also solo entries whenever both the row and column are even. This happens because 4 is not prime, so (for example), 4 also appears as 2 × 2.

The pattern inside each lattice square gets more intricate as you choose numbers with more and more factors. So you can use the pattern as a way to get a feel for how 'composite' a number is. For example, 26 has the following pattern on a 40 × 40 grid:

Inside each lattice square is a very simple pattern, implying that 26 is not very composite. In fact, 26 = 2 × 13, both prime numbers. By contrast, here is the pattern for 36:

36 can be written in terms of prime numbers as 2 × 2 × 3 × 3, leading to this impressive pattern.

I should mention here that the outer boundary of the pattern is not a circle; in fact the upper left quarter of the boundary is just the occurrences of 36 in the multiplication table, so the boundary is made of four copies of the hyperbola.

I'll leave you now with an animation of these patterns of multiples for the numbers from 1 to 40.