In the online resources for Becoming the Math Teacher You Wish You’d Had, Tracy Zager provides information about the benefits of writing a “math autobiography”. I really have tried to do this, but I am having a lot of trouble organising my thoughs and memories. However, I reckon I can track some of my memories about one particular application of maths: money.

As a child, I hated money. My mother says as a young child I would actually cry if someone gave me money as a gift. She thinks it might have been related to having to make a decision about what to spend it on, which was way too big a responsibility for little me to handle.

This sounds right, based on a very specific memory to do with making decisions. I don’t know how old I was, but I wanted to buy some lollies at the local deli. I talked to my older brother and sister about the process and had a well-formed plan. I could go up to the counter and ask the man for 5c worth of lollies, he would give me lollies and I could go home with them. I walked to the deli with my 5c and went in the door. I reached up to the counter and placed my coin on top and asked “Can I have 5c of lollies?” The man asked me “What would you like…” and proceeded to describe about five options. I couldn’t cope. He was just supposed to give me a selection of lollies and I could have them and go! I wasn’t supposed to have to make all these decisions! I left my money on the counter and ran all the way home. It turns out that using money meant making a whole lot of decisions on the fly and I did not like having to have that sort of pressure.

At some later time, I remember plucking up the courage to try and buy lollies again. I went to the school canteen and put my 20c coin on the counter and asked for 10c of lollies. The canteen lady looked at me and said, “You’ve given me too much.” I said, “But that’s all I have.” After a pause, she reiterated, “You asked for 10c and you’ve given me 20c.” I looked at her, not knowing what to say. I was certain that if you gave the lady more than you needed, they could just give you whatever was leftover. I don’t think I knew the word “change” to describe this, but I certainly knew how the process worked. Still, I wasn’t sure how to explain the concept to the canteen lady on the fly. She stared at me. Acutely aware of the line of people behind me, I took my coin and left. Looking back on this as an adult, I wonder if she simply assumed I didn’t know what the value of the coin I gave her was. Clearly it never occurred to her that I didn’t actually want to spend my whole 20c. I learned that sometimes people treat you like an idiot when you use money.

Later again — I think it was during Year 4 — the teacher had set us an assignment asking us what we would do to spend one million dollars. I couldn’t do it. I had done absolutely nothing on the assignment right up until the very last minute, because I simply couldn’t face it. My mother sat me down to talk to me about it and tears streamed down my face as we tried to figure out what the problem was. I described all the examples the teacher had given, which were about cars or houses or things that people might want for themselves, but I didn’t want any of those things. Cars didn’t interest me, I liked the house I lived in, and I simply couldn’t imagine wanting to spend that huge amount of money entirely on myself. When my mother realised what the root of the problem was, she encouraged me to think about how I might spend the money on someone other than me. How could you use the money to make life better for someone else? In the end, I wrote my assignment about using a million dollars to start an animal shelter in order to look after lost animals. Not until now do I realise this assignment was actually about the size of the number one million. For me at the time, it was about money and spending, which were emotional topics and not maths at all.

This aversion to money-related maths never really went away. In High School, I found topics on compound interest intensely uninteresting. I could “do” them, and I understood their application to my future life, but I never cared about them. When teaching financial maths to high school students, the explanations of which numbers in the graphics calculator had to be positive or negative were never natural to me. Even now I can’t process explanations of probability which define probability as “how much you’d be willing to pay” for something. How much I’m willing to pay for something is such a complex issue, which requires me deciding if I actually want the thing and is dependent on how much money I actually have to spend and my emotional state, not to mention the horror of having to manage the interpersonal minefield of actually negotiating a price.

I don’t have any particular goal in relating all of this and I wonder if it will mean anything to anyone else. For me, it has helped me realise just how much of my like or dislike of particular applications of maths has to do with emotional and interpersonal things. It makes me aware that there will be internal battles inside my students that affect how they respond to maths and its applications that I can’t see or even they can’t see.

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It was O’Week a couple of weeks ago, when new students arrive on campus to find out how uni works and the services they have access to. Our tradition for the last several years is to play Numbers and Letters on a big whiteboard out in public as a way to engage with students.

In case you don’t know how the Numbers game works, here’s an example.
Use 25, 60, 4, 4, 2, 6 and +,-,*,/,() to make 810
The small numbers, big numbers and the target are chosen randomly. We do it by choosing pop sticks from some buckets.
Buckets

Whenever we do this I am fascinated by how people approach the games. Many of them stand there staring at it apparently trying to come up with the solution by sheer mental effort. These especially seem to be people who have just revealed they are studying a degree with a lot of maths in it. Others refuse to participate at all, claiming that they are not a “maths person”.

This year I discovered a way to help both types of people engage: write something on the board that is not a solution. After a minute or so, I’d take some of the numbers and do an operation on them so that the answer was within a couple of hundred of the target (but not equal to the target) and write this calculation on the board.
2*6*60=720

Somehow, this seemed to help people get involved more. My theory is that people were thinking something like this: “The man in the maths t-shirt is doing partial answers, so it must be ok for me to do that too.” Or this: “The man in the maths t-shirt is writing things that are wrong, so it must be ok for me to be wrong too.” Or at the very least, this: “Oh! I reckon I can fix that…”

I had been announcing this move by saying, “Here’s my opening gambit”. Upon hearing this, one of my staff said it wasn’t a gambit because that’s a chess term for when you make a sacrifice to hopefully be made up later. But it occurred to me that I DID make a sacrifice: I sacrificed looking clever in order to help everyone participate. And it was a very worthwhile sacrifice, because this year’s Numbers game was a much more positive experience for our new students, and for me.

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There is a procedure that people use and teach students to use for finding the inverse of a function. It goes like this (this image comes from page 10 of this document from Edexcel, but this pic is from Jo Morgan’s blog where I first saw it):

My problem with this is that it doesn’t make any sense, in two ways.

Firstly, why should this produce the inverse function at all? Why do you switch x and y? Why do you make y the subject? Why can you just declare “f-1=” at the end? I have seen any number of students in my drop-in centre who can do this, but have no idea what is going on or indeed what an inverse even is. Others know there is a procedure but struggle to remember what step comes next.

Secondly, the written working that goes with this procedure contains things that make no sense mathematically.

inverse-func-method1-fixed

In this example, you write down y=3x+2 and then x=3y+2. I’d be wondering how it follows that if y is 3x+2 then you know also that x=3y+2. Indeed if both of these are true, then wouldn’t that mean you could sub one into the other to get that y = 3(3y+2) + 2? Which would mean that y = 9y + 8, and so y=-1. But wait, where did the function go? At the end, the “f-1(x)=” appears by magic. It seems to be replacing the y, but wasn’t y = 3x+2 at the start? This isn’t good maths writing!

I use a different procedure. It has one or two extra lines, but it makes more sense mathematically (to me anyway). Moreover, it just uses one fundamental maths problem-solving strategy, rather than being a procedure per se.

Here’s an example to show you how it goes:

inverse-func-method2

There are two important things I want to say about this:

Firstly, my working includes the mathematical reasoning: the phrases with “let”, “then”, “but” and “so”. To me, this helps to make it more like proper maths writing. Also, I think it has a better chance of interpreted as reasoning rather than symbol-shuffling.

Secondly, the key strategy here is to give the inverse function a name “y” so that you can more easily reason about it in the hope to find out more explicitly what it is. This is a common maths problem-solving strategy and I think it is worth reinforcing much more than the inverse function procedure because it can be used elsewhere too! In order to do the reasoning, I have used the fundamental property of what the inverse function actually does, rather than just “switch x and y”.

If I had to write it as a procedure, this is what I would do:

  1. Give the function f-1(x) a name, like “y”.
  2. Do reasoning to figure out y in terms of x.
  3. You already know y=f-1(x), so you have the expression you need.
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This is another post about a teaching book I’ve read recently. This one is about the Which One Doesn’t Belong Teacher Guide by Christopher Danielson.
It goes with a beautiful little picture book called “Which One Doesn’t Belong?”, which is a shapes book different from any you’ve ever seen before. In this book, each page has […]

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Writing about the teaching books I’ve read is fast becoming a series, because this is the third post in a row about a teaching book I’ve read.¬† The book I finished earlier this week is “5 Practices for Orchestrating Productive Mathematical Discussions” by Margaret S. Smith and Mary Kay Stein and coming out of the […]

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Over the last week or so, I have been reading the book “Math on the Move” by Malke Rosenfeld (subtitled¬† “Engaging Students in Whole Body Learning”). Ever since connecting with Malke on Twitter back in June or July, I’ve wanted to read her book, and I finally just bought it and read it. Now that […]

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Over the weekend, I read “The Classroom Chef” by John Stevens and Matt Vaudrey. This is a post about my reaction to the book.
The premise of the book is to use cooking in a restaurant as a metaphor for constructing teaching in a classroom. It’s a good metaphor, and executed well. Warm up routines are […]

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What trig subsitution is
Trig substitution is a fancy kind of substitution used to help find the integral of a particular family of fancy functions. These fancy functions involve things like a2 + x2 or a2 – x2 or x2 – a2 , usually under root signs or inside half-powers, and the purpose of trig substitution […]

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This post is about a puzzle I’ve been tweeting about for the last couple of days. I got it originally from a book I was given back in the 1980’s called “Ivan Moscovich’s Super Games”. In the book, Ivan calls this puzzle “Bits”, but I don’t think that’s nearly descriptive or cute enough, so I […]

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It seems like ages ago — but it was only yesterday — that I wrote about differentiating functions with the variable in both the base and the power. Back there, I had learned that the derivative of a function like f(x)g(x) is the sum of the derivative when you pretend f(x) is constant and the […]

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