This post is about Tracy Zager’s most excellent book, Becoming the Math Teacher You Wish You’d Had. I actually finished reading it back in January, and I live-tweeted my reading as I went. The process culminated with this tweet:

That’s what I thought about it at the time, but I haven’t sat down to organise my thoughts on it. Until now.

I was first drawn to the book based entirely on its contents page.
Check this out:

Chapter 1: Breaking the Cycle
Chapter 2: What Do Mathematicians Do?
Chapter 3: Mathematicians Take Risks
Chapter 4: Mathematicians Make Mistakes
Chapter 5: Mathematicians Are Precise
Chapter 6: Mathematicians Rise to a Challenge
Chapter 7: Mathematicians Ask Questions
Chapter 8: Mathematicians Connect Ideas
Chapter 9: Mathematicias Use Intuition
Chapter 10: Mathematicians Reason
Chapter 11: Mathematicians Prove
Chapter 12: Mathematicians Work Together and Alone
Chapter 13: “Favourable Conditions” for All Maths Students

Is this not awesome? Here was a list articulating things about maths that I know are important and yet that I’ve struggled to articulate all my life as a mathematician and maths educator. Many of them cut straight to the heart of the difference between how I experience mathematics and how it usually is experienced in a classroom.

“Mathematicians use intuition” you say? Well, yes. Yes we do. But many a maths classroom is about following rules and avoiding the need for intuition.
“Mathematicians work together” you say? Well, yes. Yes we do. But so many students think maths is only a solitary activity.
“Mathematicians make mistakes” you say? Well, yes. Yes we do. But mistakes are feared and avoided in most maths classes.
“Mathemaicians connect ideas” you say? Well, yes. Yes we do. But so many maths curriculums are just so many piles of disconnected procedures, even here at my own university.

The contents page promised a book about the most important aspects of mathematical work and thinking, and a hope that it would give ways to bring these into the experiences of students in all maths classrooms.

And the hope was made real.

Each chapter starts out comparing how mathematicians talk about what they do and what students’ experience of it is. Then it moves on to detailed examples of the aspect of maths thinking in action in real classrooms, as well as strategies to encourage it both in your students and in yourself as a teacher.

I didn’t expect to see this last point about encouraging these attitudes and thinking in yourself as a teacher. Yet it is the most compelling feature of the book for me. Indeed, I don’t think the book would have had nearly the impact it had on me (or the impact I see it having on others) without this constant message that to help your students experience maths differently, then you yourself need to experience it differently too. More than this, Tracy doesn’t just make this need clear, but actively and compassionately empowers us to seek out ways to fill it.

“Somewhere inside you is a child who used to play with numbers, patterns and shapes. Reconnecting with your inner mathematician will improve your teaching and benefit your students, and it will also benefit you.” — Tracy Zager, Becoming the Math Teacher You Wish You’d Had, p39

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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 […]

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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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