Author: davidkeithbutler

Money and me

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.

These comments were left on the original blog post:

Paula Krieg 19 April 2017

One thing that’s really powerful about this story,David, is how a person can be struggling so intensely with a problem has little to do with the problem that is presented. About the canteen lady, it’s easy to think that she was being a jerk to little David, but the fact is, she really may not have known how to make change. Hard to know if in that case she was the one that was struggling with problems that were hard to pinpoint. A good reminder to ask questions of the person who is befuddled. but what a great string of stories. thanks.

David Butler 19 April 2017

Thanks Paula. It was a tough thing to write.
That may possibly have been true about the canteen lady. Still, 6- or 7-year-old David could hardly have helped her understand at the time!
I guess a lesson for 37-year-old David is indeed to ask and listen to people.

Nicola Petty 2 May 2017

Kia ora David
An interesting story, and almost the opposite of many people, like my father, who felt he could only do maths if it involve money.

I have an aversion to stars and anything much to do with space. The size of it all makes me feel uncomfortable. I know for many people, space is really interesting, but I would rather not think about it. I don’t know why I feel like this, and I have a hard time seeing why other people are so interested in space. We just don’t know what will make people uncomfortable!

David Butler 4 May 2017

That’s really interesting Nic. Another fine example of how we can never really predict what might be going on inside our students’ hearts.

14 March, 2017
An opening gambit for the Numbers game

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.

The small numbers, big numbers and the target are chosen randomly. We do it by choosing pop sticks from some 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.

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.

6 March, 2017
Finding an inverse function
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.

\[\begin{aligned}
f(x)&=3x+2\\
y&=3x+2\\
x&=3y+2\\
x-2&=3y\\
\frac13 (x-3)&=y
f^{-1}(x)&=\frac13 (x-2)\end{aligned}\]

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:

\[\begin{aligned}
f(x)&=3x+2\\
\text{Let } y &= f^{-1}(x)\\
\text{Then } f(y) &= x\\
3y+2 &= x\\
3y &= x-2\\
y &= \frac13 (x-2)\\
\text{But } y&=f^{-1}(x)\\
\text{So } f^{-1}(x)&=\frac13 (x-2)\end{aligned}\]

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.
These comments were left on the original blog post:

Art Duval 19 January 2017

See this blog post, which matches your ideas very closely: http://blogs.ams.org/matheducation/2016/11/28/inverse-functions-were-teaching-it-all-wrong

Stuart Reilly 28 June 2017

Why not avoid using y altogether:
By definition
f(f^-1(x))=x
so
3f^-1(x)+2=x
Solve for f^-1(x)
3f^-1(x)= x-2
therefore
f^-1(x)=1/3(x-2)
as required

David Butler 29 June 2019

That’s very slick! Of course if it was a more complicated function, then I’d probably still use y to help me maintain my focus.
19 January, 2017
Book Reading: Which One Doesn’t Belong – Teacher Guide

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 four pictures, and asks the readers to say which one doesn’t belong, and why. The fabulous thing about the book is that there is at least one reason why each of the four pictures doesn’t belong, and talking through these with children (or indeed anyone) is a rich conversation about the properties that shapes have and don’t have.

The Teacher Guide is all about these rich conversations: why it’s important to have them, what you and your students/children can learn through them, and how to facilitate them. Chris has a friendly and welcoming style which draws you easily into a new appreciation of the sophisticated thoughts of children as they make sense of geometry and the world.

There are a few key things Chris talks about that really impacted my thoughts about teaching and learning maths. I’ve organised them by quotes from the book:

Commonly in maths class, student responses are compared to a standard answer key – the measure of what’s right is what’s in the back of the book, or what the teacher has in mind. In a conversation about a well-designed Which one doesn’t belong? task, the measure of what’s right is what’s true. – page 3

I read this quote first when someone else tweeted it out of the book and it struck me as awesome then. In my job at the Maths Learning Centre, students are always asking me if things are right, as if the measure of rightness is if I say it is. But in most places in maths, correctness is measured by truth. Your vectors will either be an orthonormal basis for the subspace or not. A number is either prime or it’s not. You can tell if you’re right by thinking about whether it’s true. I very much want to see opportunities to talk about the truth of things with students, to put the measure of rightness outside an authority figure.

The van Heiles haven’t argued that it is difficult to go from level 1 thinking directly to secondary school geometry; they have argued that it is impossible. If students don’t have experience and instruction building informal geometry arguments, they will not learn to write proofs. – page 8.

Chris is referring to the van Hiele model of “how childrens’ geometric thinking develops over time”. In this model, there is a build-up from noticing that shapes look like things they’ve seen (level 0), to noticing properties that shapes have and don’t have (level 1), to relating properties between properties of shapes (level 2), to logically supporting claims about these relationships (level 3).

The thing in the above quote that really struck me is the idea that it’s impossible to learn to write proofs without experiencing informal arguments first. I see so many students at university every day who struggle with proofs, and it makes me wonder that they maybe need more experience with informal arguments. Indeed, it makes me wonder if they need more experience simply noticing properties, since that’s an even earlier level. This is essentially applying the van Hiele models to other types of maths, but certain aspects of the progression still feel right to me, especially for things vaguely geometrical like vectors or matrices or graphs of functions.

I wonder if a student struggling with proofs might benefit from talking through a progression like this, and then helping them have experiences at the earlier levels before helping them with proofs.

Of course being able to state new facts is an aspect of learning, but much more important to me is being able to ask new questions. – page 21

I had never thought of this idea explicitly before, but immediately I saw that new questions were important to me as well. I was reminded of the time someone asked me if my students were understanding my statstics lectures. I said that I wasn’t completely sure, but certainly the students were asking very deep and complex questions. Instinctively I knew that a new type of question indicated learning.

Also, in the Drop-In Centre, there’s a certain joy when a student asks new questions you’ve never thought of before. They are wondering about the connections between things, which means they are learning, because learning is all about connections.

I am excited to listen out for new questions as a sign of learning, and to tell the students that it’s a sign of learning to have new questons!

… I hope you will begin to see geometry through children’s eyes as well as through the eyes of a mathematician. Mostly, I hope you will come to understand that these two views of geometry are not nearly so distant as the school curriculum might lead us to believe. – page 37

Now, I already believe that children’s investigations and ideas are actually very close to the way mathematicians work. You can’t be married to a very excellent early childhood educator without coming to some appreciation of this! It’s so nice to have someone publish a book telling teachers and parents the same.

Even more, this whole section is all about noticing and naming things and their properties. It’s about whether properties need names at all, or whether the objects that share those properties need names. It’s about what properties are important to make a thing a special thing and what aren’t, and in what context. It’s about the relationships between things. All of these are the work of professional mathematicians both pure and applied. And they are the work of children sorting out how the world works.

The geometry of children and the geometry of mathematicians are definitely not so far removed.

I have come to understand that talking about this difference is more important than defining it away. – page 54

Along with the rest of this chapter, this quote got me thinking about a whole new way to approach definitions in mathematics. As a pure mathematician, definitions are very important to me, and I always used to start with the definition. But I know those very definitions took years and even centuries to come to their current forms, and I also know that humans don’t learn through definition but through comparison of things that do and do not fit an idea. I think this is precisely what Chris is getting at here.

By skipping straight to the definition, we’re robbing people of a key part of mathematical thought, and we’re skipping them through the van Hiele levels before they’re ready. You don’t need a definition until you have a need to distinguish a thing from the other things around it. You don’t need a definition until you’ve noticed the properties you can use to define something.

The classic example in my own teaching is subspaces in linear algebra. The properties used to define a subspace aren’t even discussed until the definition is given. Little wonder, then, that the definition is meaningless to students!

It’s not just definitions either. I help a lot of students learn statistics, and one of the things that is never explicitly taught in your traditional statistics course is how to choose what is the most appropriate statistical procedure for the situation. I have been teaching this by focussing on some specific aspects of these procedures that statisticians use to distinguish things. Reading this chapter and this quote in particular helped me realise what I was doing was exactly “talking about this difference”. To distinguish between things you need to notice the properties that make them different, and to notice them, you need to compare things. I now have a much clearer idea of what I’m doing when teaching in the way I do.

I want to spend more time putting students in situations where they notice the differences between things and have to talk about them, so that they can distinguish between things they need to, and so that the properties I use to define things make more sense.

Thanks Chris for a most thought-provoking book.

19 January, 2017
Book Reading: 5 Practices for Orchestrating Productive Mathematical Discussions
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 National Council of Teachers of Mathematics (NCTM) in the USA.

I’ll get straight to the point: everyone in any sort of classroom where maths happens should read this book. It gives a simple and practical framework for using student work and class discussion to promote maths learning. The authors have a direct, clear style that make the nuances of the practices seem almost obvious, using careful studies of classroom scenarios to illustrate. Let me say again: read this book!

In a nutshell, the idea of the book is that you can help students learn mathematical content by giving them tasks rich enough to be worth talking about and connected to the mathematical goals you have in mind, and then orchestrating class discussion of the methods students use and their connections. They give five practices, and a smattering of other strategies and ideas to guide this.

I think this book should be required reading and/or the basis of training for staff who are teaching tutorials at university. University tutors are often given no training in teaching, and even then don’t get tools to help them choose what to do in their classrooms. In some schools here at the Uni of Adelaide, they are instructed to get students working in groups. This is great, but the part where the mathematical ideas of the week are brought out is not strong. I am hoping to take these practices to these schools, and to the ones where it’s more just another lecture, in the hope I can help to improve the learning happening in the tutes. I’ll be mentioning how I think it applies to tutorials as I go.

Here’s a summary in my own words:

“Practice 0”: Worthwhile tasks and mathematical goals

You’re not going to be able to have a class discussion about a task which is routine procedure-following, because everyone will do it the same way. You need something that has some level of challenge and has decisions to make about how you do could do it – something actually worth discussing! Also, you need to have a goal in mind for what you want to achieve so that you have a chance of achieving something. This goal needs to be about the mathematical ideas involved. For example, about the connection between the different types of equations for lines, or about the distributive law, or about the relationship between squares and rectangles.

This isn’t technically one of the five practices, since it happens “outside” the context of the discussion. Plus, you may not always have total control over the tasks that students have to do or the mathematical goals. (More likely a school teacher is in control of this, but a classroom tutor at university this will be less often true.) Even so, if you do have control, it’s very important, which is why the authors call this “Practice 0” a couple of times, because it’s needed before you even start.

As I already said, in classroom tutorials, someone else often chooses the tasks. But you can add your own question to the end to make it more open to discussion. Maybe something like “What would happen if…” are good to extend learning. Someone else may set the goal, but it’s more likely the people coordinating your course won’t tell you what the learning goal is. So you’ll have to choose for yourself. It’s so important to choose the goal so that the tutorial doesn’t end up feeling like a whole lot of activity and discussion, but with nothing of substance to take away.

Practice 1: Anticipate

When you have a goal and a task, the first thing to do is anticipate how the students will respond to the task. At the very least, you need to do the task yourself, but even better, imagine as many correct and incorrect, helpful and unhelpful approaches as you can.

One reason for this is so that you don’t have to make so many decisions on the fly during the class. You can figure out in advance some of the ways you will respond to these before you get there.

I see another advantage and it is about putting yourself in the mindset of your students. We university teachers are often so blind to how our students think, and tutors are often very focused on their own way of doing things. By explicitly trying to think of multiple approaches, it can help to break down this egocentric focus we fall into.

Practice 2: Monitor

Once you’re in class and the students are working on their task, the role of the teacher is to monitor the students’ work and thinking. The anticipating you did earlier helps you to respond appropriately to them, and sets you up into a mindset where you’re focused on their thoughts, so even unexpected methods are easier to process. It’s while monitoring their work that you will make the final decision of how you want to run the discussion, and who will be involved. It’s also while monitoring their work that you’ll ask the students questions to help them learn in-the-moment.

One thing I particularly like about this practice is how it gives us a focus while the students are working. Just the other day when talking to tutoring staff, they expressed a distaste for groupwork because it meant they, the teacher, weren’t “doing anything”. This practice says you’re not doing nothing – you’re monitoring.

The authors recommend asking students two types of questions during the working (and hence monitoring) phase:
- Ask questions about student thinking
  Help students while they are working to express their thinking about the problem and the maths. Actually ask them to tell you how they are thinking. This gets them ready for the discussion to follow, and also helps them with the problem-solving too.
- Ask questions about maths meaning and relationships
  Help students to express what the maths ideas mean and what they mean to them. In particular draw out relationships between concepts. This is what your goal is ultimately, and it front-loads this discussion so students are ready for it.
I see these two types of questions as really important for classroom tutors at university. Too often the questions we ask are about yes/no correct/incorrect answers, rather than about thinking and ideas. Encouraging tutors to focus on these types of questions makes thinking and meaning the focus of the learning activity.

Practice 3: Select

The last three practices are about making the discussion part of the class happen productively. They work together to help make sure that the discussion both uses student work, but also proceeds towards the mathematical goal. Also they prevent the random show-and-tell which often just ends up with students confused or with no particular idea of what they learned.

First, you want to select what student work you want to discuss as a whole class, and whose work it will be. The authors list a few considerations here, not least of which is choosing students who up to now haven’t participated much in class. It’s worth noting that in their examples, even though students worked in groups, specific single students are asked to talk about their work, which means people can’t hide from participating! It also means that people can’t monopolise the participation either! We all know that one person who seems to think the tutorial is just there for them to show how clever they are. By preselecting students to show their work, you’re making it less likely for this person to take over.

The thing I like most about the concept of selecting student work is that it has the potential to help students feel like their work is a valid and important contribution (which of course it is). By using student work and student generated ideas to forward the maths discussion, we can help them be more engaged in the learning and feel like we care about them. This is not a small thing to consider!

I am particularly interested in applying this idea to classes where students are expected to do preparation for the tutorial in advance and hand it in (like they do in several courses here at Uni of Adelaide). At the moment, what usually happens in these classes is that students do the homework, hand it in, and then the tutor presents their own preprepared solutions. But think what might happen if the students handed in the homework, and the tutor used the homework itself as a tool for class discussion. I think it might help the students feel like their homework was actually worth all the effort!

Practice 4: Sequence

After choosing which student work to present, you need to choose what order it will be presented in so that you can progress towards the mathematical goal. The authors give a number of things you might consider with your sequencing. For example, you might want to choose to start with a solution method that a lot of people have so that everyone can get buy-in to the discussion (I did this when I did Quarter the Cross in my daughter’s classroom). You might want to start with a solution containing a misconception to get it out of the way. You might want to avoid a specific solution because it will just send everyone off on a tangent (though you might also want to talk to that student one-on-one separately). You might want to have two particular solutions in quick succession in order to be able to compare them.

The important bit is to think about what order would be most helpful to get to where you hope to go. Importantly, the way you hope to make connections between ideas will dictate how you might sequence the students’ work.

Practice 5: Connecting

Now that you’ve chosen what student work to focus on in the discussion and in what order, it’s now time to actually have the discussion. It’s important here to remember there is a mathematical goal we’re working towards, which will often be about understanding a concept, and understanding is a sensation that happens when ideas are connected to other ideas. It’s our job to help students make these connections.

The authors suggest five “moves” you can make during your discussion to make sure it stays focused on the connections you want to draw out.
- Revoicing
  This is when you repeat what a student says to make sure you and everyone else heard it and understands it. Importantly it’s not about you making what they said more correct, simply making it heard. A good phrase to end with when you revoice a students’ words is “Is that right?” This lets them know that the point is to make their thought heard (not yours) and they get to decide if it’s been voiced right.
- Asking students to restate someone else’s reasoning
  Instead of you revoicing a students’ words, you can ask another student to explain the reasoning. This includes even more students into the discussion in a more active way.
- Asking students to apply their own reasoning to someone else’s reasoning
  This time you’re not just asking the other student to explain the first student’s reasoning, you’re asking them to explicitly explain the connection between two different types of reasoning (one of which is their own reasoning). For example, suppose you’re doing Quarter the Cross and John proved the house-shape was a quarter by cutting and overlaying, whereas Jane proved the L-shape was a quarter by folding, you could ask Jane to prove the house-shape works by folding.
- Prompting students for further participation
  There are times where a student will close off with a quick answer, and it might be more productive if they stayed in the discussion a little longer. The questions listed above of asking them to explain their thinking or focus on the meaning and relationships are useful now as well. In the Maths Learning Centre, I find “Tell me more about that” to be a good all-purpose request to participate further.
- Waiting
  This may seem paradoxical, but leaving some silence can help to promote discussion. The authors say that whenever anyone asks someone else to say something, it’s appropriate to give them plenty of time to respond. Giving them this time helps to actually make the point that their answer is important to you. You giving yourself time to form your response to their question helps to make the point that their question is important to you. Waiting a bit after an explanation to let it sink in before asking people for any questions helps to make the point that it does in fact take time to process information. These last couple were new thoughts for me (though obvious in hindsight).
It’s this last practice that we often don’t do in tutorial discussions. I was talking to some tutors from the Faculty of Arts recently, whose tutorials are traditionally only discussion. They talked about how often the discussion just goes for a while and then stops at the end of the class, without coming to any conclusion the students can take away about the concepts or the process of learning them. They recognised a need to explicitly make connections during the discussion. Over in maths tutorials, I think we assume the connections are obvious, but I can attest that they are not, if all the students complaining that the tute doesn’t teach them anything are anything to go by.

Conclusion

It may seem that I’ve given you the content of the whole book, and indeed my aim was to present the ideas clearly, mostly for my own future reference! But I would still encourage teachers and tutors to actually read the book. The vignettes of actual classroom use are vitally important to come to an understanding of what the practices look like and where they are useful, plus there’s whole chapters about how to seek support for teaching and how to include it in formal lesson planning that I haven’t even mentioned (until just now).

I am excited to take the ideas here and use them to help support classroom tutors here at University. I think this book could really be a tool that people might actually get behind. Here’s hoping.

To wrap up, here’s the headings in dot point form for future reference:
- Practice 0: Worthwhile tasks and mathematical goals
- Practice 1: Anticipate
- Practice 2: Monitor
  - Ask questions about student thinking
  - Ask questions about meaning and relationship
- Practice 3: Select
- Practice 4: Sequence
- Practice 5: Connect
  - Revoicing
  - Asking students to restate someone else’s reasoning
  - Asking students to apply their own reasoning to someone else’s reasoning
  - Prompting students for further participation
  - Waiting
13 December, 2016
Book Reading: Math on the Move

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 I’ve finished, it’s time to write about my thoughts.

The book is all about whole-body learning as it relates to maths and dance, mostly focussing on pre-school to Year 6. Some of you may be wondering why I, a university lecturer with a doctorate in pure maths, would be so very interested in something to do with dance-and-maths at the primary level. My first response to that is that you clearly need to get to know me a bit better! Perhaps start by checking out the following past blog posts: Kindy is awesome and The Pied Mathematician of Hamelin.

My second response is that seeing things from a new perspective is one of the best ways to understand them better and to understand how you understand them in the first place. I was fascinated by this new medium of a moving body for thinking about maths and I wanted to get the benefit of reading the thoughts of someone who has already considered it deeply. And Malke Rosenfeld is just that person, because reading the book you can tell immediately that she has thought very deeply about it.

The book has two main parts. The first part is about the concept of movement-scale activities and the body as a thinking tool in mathematics. The second part is about the Math in Your Feet program, which is also about the body and its movement as a thinking tool, but even more than that, that dance itself is a mathematical thing worth thinking about.

The first part had me thinking from the moment I started reading. Malke argues that scaling up a mathematical idea to the scale where your whole body can interact with it or be it can give insights and understandings not available in any other way. Malke gives examples of number lines and hundreds-charts of a scale you can walk on, and building polygons out of knotted rope that has to be held by multiple team-members. My head was whirring with the possibilities. Immediately I imagined what it would be like to stand on a surface defined by a two-variable function and questions about directional derivatives occurred to me that never had before. Imagine what would have happened if I could actually stand on the surface itself!

Malke makes the very important point that meaningful moving-scale mathematics learning is not about using your body to memorise things, or to copy what is on the page. It is about using your body to make movements that are intrinsically related to the thing you are trying to understand. Stretching your arms to copy the drawn shape of a linear graph while saying its formula is not really meaningful. Perhaps more meaningful would be walking on a graph drawn on a basketball-court-sized coordinate grid and explicitly discussing how you move relative to the x and y axes. (And just now writing this, I suddenly have this cool idea to really understand discontinuities as places in the graph where the mover has to literally jump to get to the next point.) The discussing I mentioned is important too – meaning happens when the ideas are discussed and compared.

The second part of the book, as I said earlier, described the Math in Your Feet program. Children are given a two-foot by two-foot square to dance in and a number of possible ways to move. They create steps within this framework and work with partners to make dance steps the same and different, to combine patterns of steps into longer patterns, and to transform dance movements through rotation and reflection. There’s detailed information about how the program moves forward, and the ways to facilitate work and play and thinking and discussion, as well as lots of linked videos to really see the action. You could be forgiven as a high school or university maths teacher for thinking this part of the book doesn’t really apply to you as much as the movement-scale exploration of existing maths ideas. I say you could be forgiven, but you’d still be wrong.

Firstly, there is a whole heap of very deep discussion on what it means to give the students the power over their own learning. Malke discusses the importance of clear simple boundaries, of precise language, of encouraging language, of reflection, of getting students to share, and of ways to help children to focus. All of this is vividly displayed throughout the Math in Your Feet chapters of the book, and what you can learn here would translate to all sorts of other teaching situations. It is worth watching all the videos jut to revel in Malke’s skill of never praising product but always excitedly praising participation and practice.

Secondly, it is this part of the book that is the most mathematical, from my perspective as a pure mathematician. The dance moves within the tiny square space are an abstract mathematical idea that is explored in a mathematical way. We ask how the steps are the same or different from each other, identifying various properties that distinguish them. We investigate how these new objects can be combined and ordered and transformed. We try out terminology and notation to make our investigations more precise and to communicate both current state and how we got there. These are all the things we pure mathematicians do with all our functions, graphs, groups, spaces, rings and categories. The similarity of this to pure mathematical investigation in striking.

I have been changed by reading this in ways that I am not capable of processing completely at the moment. Not until I have more chances to try out movement-scale investigation of maths, and mathematical investigation of movement, will I feel I have a handle on it. But it’s a pleasant sort of feeling all the same.

One final warning: If you read this book, don’t attempt to do it in an armchair, or on the train, or while walking. It won’t work. In order to read this book effectively, you need to sit with access to a computer to watch the video clips, and with access to a 2 foot by 2 foot square on the floor to try the dance steps in. Also if you’re like me, you’ll need somewhere to write down quotes which speak deeply to you. Quotes like this:

Using the moving body in math class is about more than getting kids out of their seats to get the wiggles out or to memorize math facts. Instead, we need to treat the movement as a partner in the learning process, not a break from it. pp 1

Using tangible, moveable objects (including the moving body) can be useful in math learning as long as attention is paid to the math ideas as well as what you do with the object. pp 13

Using language in context to label, describe, and analyze this work is one of the most powerful ways to help learners create meaning and understanding. pp 112

Grading or judging a child on his or her ability compared with others’ is harmful in this creative environment. This is a place where the focus should be firmly on the ideas expressed, not on the facility or ease of that expression. pp 146

We want math to make sense to our students, and the moving body is a wonderful partner toward that goal. pp xvii

Thank you Malke.

These comments were left on the original blog post:

Joy 5 December 2016

What about people with disabilities? How can teachers and students teach/learn if they are disabled?

David Butler 6 December 2016

Malke has a section of her book specifically about how to include children with special needs. She also encourages teachers to use the children to show examples of dance, so a teacher with a movement disability I think would be very successful with the Math in Your Feet program.

Jeremy 15 January 2017

Very interesting David !! I am very glad to have found this blog,

I’ll admit that the life size maths visualisation technique seems to me to be much more helpful than the maths in your feet program. I does sound like an interesting book 🙂!

2 December, 2016
Book Reading: The Classroom Chef
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 “appetisers”, being prepared is “setting the table”, creating curiosity before giving answers is “entree”, things you do to make life more fun in the classroom are “side dishes”, and assessment is “dessert”. The commitment to the metaphor is even more impressive than that, with the contents page called the “menu” and the references section called “secret ingredients”. Just looking at the menu was enough for me to want to read the book more closely.

The major messages the book seemed to be getting at were the following:
- Don’t be afraid to really love maths in front of your students.
- Give students the chance to show you how they understand in their own way. Posters and videos don’t feel fun for everyone when they are being assessed on it.
- Set things up in your class activities so students are curious about something. It doesn’t have to be “real world” and it definitely doesn’t have to be serious, it just has to have a question that needs an answer. Some silliness and shock value will make it taste better, but the setup for curiosity is the really important bit.
- It’s a risk to try something new with your teaching, but your students will appreciate it and you can’t learn without it.
It’s only a short book, so it didn’t go too deeply into any of those, though there’s probably more examples of the ideas in action on the companion website. But still I reckon classroom teachers would get something good out of it.

Unfortunately, for me, I had trouble as I read this book because early on John and Matt described their early teaching experiences and it brought back a whole lot of unpleasant memories for me. Their description of the days when they felt that perhaps teaching wasn’t for them actually made me cry with my own memory of feeling the same way. The worst part was that I knew Matt and John stuck with it and are now writing books about teaching, whereas I left. My first school I quit before the end of my first year there (there were a number of reasons for this), and the second school I stuck out the full year, but at the end of that I went back to uni to do my PhD in geometry.

Continuing to read the book, Matt and John talk a lot about being brave enough to take risks in the classroom. I am sorry to say that all this did was make me feel like my own reaction to these early stresses was chickening out. I felt like I had been a coward and let down the students I could have had by leaving teaching. Moreover, as they describe some of the fun things they did in their classrooms, I think back to some of the similar things I did and wonder if there was something wrong with me that they didn’t make a huge difference to how I felt about teaching.

Thinking about it more, I have found one possible factor that made my experience different to Matt and John’s: support. In the book, they both describe the support they received from their school leadership and from instructional coaches in their early years of teaching, sometimes without them having to ask for this support. I had neither of those things at my first school. At my first school, I was it for maths and science and my principal was a bully who repeatedly undermined me to the students when I was not in the room and attacked my relationship with my wife. At my second school, it was better since there were more other teachers to lean on, but still I was pushing against a curriculum leader who actually said aloud that maths was “a collection of problems and a procedure to solve each one”, and a school leadership who weren’t committed to helping me improve, only to telling me I needed to. Plus, this was before Twitter, so there was no #MTBoS.

Looking back, I think the critical lack of support was one of the major causes of me giving up on teaching high school. Reading the descriptions of support in this book made me weep for poor past David. Of course I know that it has all turned out for the best because it has meant that here I am at Uni doing the best job in the world, but I couldn’t say totally enjoyed the experience of reading it. Sorry Matt and John.

UPDATE: Check out John’s reply at his own blog – follow the link below.

This comment was left on the original blog post:

John Stevens 20 November 2016:

David,

Thank you for this. Rather than dropping a big reply here, I blogged about it:
http://www.fishing4tech.com/fishin-solo-blog/shoulders
14 November, 2016
How I choose which trig substitution to do

What trig substitution 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 is to use the magic of trig identities to make the roots and half-powers go away, thus making the integral easier. (I say “easier” by which I mean “not impossible” as opposed to “simple to do”. The resulting integral is usually not simple and usually involves attacking an integral that now has a whole lot of trig functions in it. Moreover, the stage where you put your original variables back in is often a lot fancier than in other substitutions and involves trig identities yet again.) The Mathematics 1A students at my university learned it in recent weeks and are struggling with it in various ways, not least of which is the sheer amount of paper it takes to write it up properly!

Remembering which trig substitution to do

One particular thing the students struggle with is choosing which trig substitution to do. Actually, they tell me they have trouble remembering which trig substitution to do, which I think says a lot about how this topic is presented in most resources.

When I google “trig substitution”, every page that comes up presents me with a table very much like these:

From Paul’s Online Calculus Notes http://tutorial.math.lamar.edu/Classes/CalcII/TrigSubstitutions.aspx

From S.O.S Math http://www.sosmath.com/calculus/integration/trigsub/trigsub.html

From Wikipedia https://en.wikipedia.org/wiki/Trigonometric_substitution

The examples that follow these tables usually give no reason at all why the particular substitution was chosen, or just say it’s “easy to see”. This implies that in order to choose which one to do, you look it up in the table and then do the one you find.

From S.O.S Math (emphasis mine) http://www.sosmath.com/calculus/integration/trigsub/Example1/Example1.html

And if there is no table to look up (such as during the maths exam) then of course you would just have to remember.

Choosing which trig substitution to do

When I first learned trig substitution, I also struggled to remember which one to do in a given situation. Even now I can’t remember — I simply don’t do them often enough to be fluent in just knowing the right one off the top of my head, and the stimulus-response nature of looking it up in the table just seems hollow to me. As a student I never really knew why each format implied that a particular trig substitution had to be done. I could understand each step in the working for any example, but I still couldn’t see how the decision had been made. A student in the MLC last week expressed this exact frustration last week — he could see that the forms matched the forms in the table, but he wanted a reason why.

Much much later I realised that the fact the working worked at all was the actual reason why the particular substitution had been chosen! This was what eventually freed me from having to remember. Since the working working was what made the decision, and since I had to do that working in order to finish the problem anyway, I could just get going and make the decision of which one to do along the way. Note that one reason I have trouble remembering them is that I also don’t remember all the versions of the squared trig identities off by heart, I just create them starting from (cos x)² + (sin x)² = 1. This new method allowed me to just continue doing that and it actually supported my success.

Here are a couple of examples of me deciding which trig identity to do along the way that I did in revision seminars this semester.

So that’s how I choose which trig substitution to do. I try to construct a trig identity that matches what’s in my square root as closely as possible, and then by the time I’ve done this, the substitution is chosen for me and part of the work is already done.

It works for me, anyway.

8 November, 2016
Panda Squares
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 asked my daughter what it should be called and we have come up with the much better name of PANDA SQUARES.

The puzzle starts out like this.

You have 16 square tiles, each made up of four small squares. These smaller squares are coloured either black or white. In fact, the 16 tiles cover all possible colourings considering different orientations as different.

(Figuring out all the possibilities and how many there are is a puzzle in itself, really, and I imagine would make a lovely starter for this puzzle if you were to do it in a classroom.)

You can download a Word document or PDF document with each square on a different page. Printing it 16-to-a-page works well to have individual sets, or you can do what I did and print them big then laminate them!

Anyway, the puzzle is this:

PANDA SQUARES:

Arrange the 16 tiles into a 4 by 4 square so that the colours match along the edges of tiles next to each other.

There are lots of possible solutions, so I don’t think there’s any danger in showing you one here:

And that’s it. One thing that’s nice about it is that after getting a solution you can scramble the pieces and very easily get another one that isn’t the same, or someone else can have a go after you. At the Celebration of Mind event yesterday we had the attendees at the barbecue try it out, and today at our regular One Hundred Factorial session we all had several turns making our own. (The one above is my favourite from today because I think it looks vaguely rabitty.)

Only that’s not it! If you look at several different solutions, you may find that you notice a few things, and that several questions occur to you, just like happened to us! In order to help us in our investigations, we had photos of all our solutions on one of our phones so we could flick through them. We also had 8 by 8 grids on paper so we could copy down our solutions and draw all over them. Plus, as always, we had a big whiteboard to draw our thoughts all over too.

Here are some of our noticings and wonderings and where we’re up to with them. If you don’t want spoilers I suggest you don’t look too closely!
- We wondered: How many possible solutions are there?
  A student, James, asked this question yesterday and came back today having written an algorithm to find them all. He’d found 10904, not counting rotations as different. He said he hadn’t proved his algorithm actually did cover all the solutions exactly once, but I wasn’t going to let him discount the awesome work he had done!
- We noticed: The four little squares surrounding the intersection of four tiles are the same colour.
  And we did manage to prove that yes this must always be the case.
- We wondered: Is it possible to have a solution where all the identical tiles are in a different orientation?
  It is, because a student, Lewis, found a solution today with this property!
- We noticed: There are two black and two white squares in the four corners of the big square.
  Then we found some other solutions where this wasn’t the case. But Finn did manage to prove that the only other possibilities are all black or all white big-square corners.
- We wondered: Is it possible to have only one connected black region? (Connecting only by a corner doesn’t count)
  We’re pretty sure it’s not possible, but we haven’t got a proof we all find convincing yet.
- We noticed: The number of black regions and the number of white regions is either the same or one different.
  Again, we’re pretty sure this is actually the case, but again we didn’t get any argument that was convincing beyond a doubt. Lewis and Daniel had this idea about the edges of these regions having to cross at the diagonal-two-black tiles, but it wasn’t getting us anywhere yet.
- We wondered: Is it possible to draw a cross on a solution so that a quarter of the cross is black (or white)?
  Well, I wondered that anyway, but you know my obsession with Quarter the Cross, don’t you? It turns out the answer is yes you can, but not all the time – I was quite proud of my inspiration to trace the cross on a plastic sleeve and move it around on a drawn version of the solution to see if I could make a quarter. Everyone worked together to come up with the surprisingly severe restrictions on how you can do it if the squares in the cross are whole tiles.
- We noticed: At least one of the all-black and the all-white tile were always on the edge.
  And yes this does in fact have to be the case. I think it was Michelle who had the inspiration for the proof of this one. It was quite clever and involved the fact that each of these tiles forces a surrounding border of the same colour squares.
- We wondered: Is it possible for the all-black and the all-white tiles to be in opposite corners?
  We’re still not sure on this one. We did try a few times but couldn’t make it work, but then again we don’t have a proof that it’s not possible.
- We noticed: None of the solutions are symmetrical.
  And there were several different proofs of why none of the solutions could possibly be symmetrical with different types of symmetry.
I’m sure there were several other things we were working on that I wasn’t there for since there were about ten of us going simultaneously! (Also one of the reasons I may have gotten the attributions for ideas a little mixed up — sorry!) And we haven’t even started on other arrangements of the tiles or other sized tiles!

I wasn’t quite prepared for just how fabulously rich this puzzle was, with all sorts of great noticings and wonderings. On top of that, it’s just so aesthetically pleasing and it’s still a sheer joy to see all the beautiful solutions that come up.

These comments were left on the original blog post:

Matt Skoss 5 July 2021

I wonder if using line or rotational symmetry can be a useful strategy to find a solution easily.

David Butler 9 July 2021

Possibly Matt. Later, it was discovered that there is a solution that is symmetrical after all, and I proved there are exactly four symmetrical solutions. https://twitter.com/DavidKButlerUoA/status/1224087567038959619?s=20

These other blogs and webpages have referenced the original Panda Squares blog post…
- MATH = love
  https://mathequalslove.net/panda-squares-puzzle/
- Matematika Eskuetara
  http://matematiketan.eus/2020/02/17/panda-square-puzzle-espaziala/
- Thoughts in the After-Math
  https://heidilneufeld.wordpress.com/2017/11/08/panda-squares/
- Mr Kaercher’s Instructibles
  https://www.instructables.com/Panda-Squares-Truchet-Tiles/
- Hawks Makerspace
  https://www.youtube.com/watch?v=v8auoJ5rApk
19 October, 2016
Holding the other parts constant: It’s everywhere!

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 \(f(x)^{g(x)}\) s the sum of the derivative when you pretend \(f(x)\) is constant and the derivative when you pretend \(g(x)\) is constant.

Since then I have realised that this idea actually dictates ALL of the differentiation rules where two functions are combined through an arithmetic operation! It’s everywhere!

Multiplication…

Let \(h(x)=f(x)\cdot g(x)\).

Suppose \(f(x)\) is constant, say \(f(x)=a\). Then

\[\begin{aligned}
h(x)&=a g(x)\\
h'(x)&= ag'(x)\\
&= f(x) g'(x)\end{aligned}\]

Suppose \(g(x)\) is constant, say \(g(x)=b\). Then

\[\begin{aligned}
h(x)&=f(x)\cdot b\\
&=bf(x)\\
h'(x)&= bf'(x)\\
&= g(x) f'(x)\\
&= f'(x) g(x)\end{aligned}\]

Adding these:
\[ f(x)g'(x) + f'(x)g(x) = h'(x) !!!\]

Oh my goodness — it’s the product rule!

Division…

Let \(h(x)=\frac{f(x)}{g(x)}\).

Suppose \(f(x)\) is constant, say \(f(x)=a\). Then

\[\begin{aligned}
h(x)&=\frac{a}{g(x)}\\
&= a[g(x)]^{-1}\\
h'(x)&= a(-1)[g(x)]^{-2}\cdot g'(x)\\
&=-\frac{ag'(x)}{[g(x)]^2}\\
&= -\frac{f(x)g'(x)}{[g(x)]^2}\end{aligned}\]

Suppose \(g(x)\) is constant, say \(g(x)=b\). Then

\[\begin{aligned}
h(x)&= \frac{f(x)}{b}\\
&=\frac{1}{b} f(x)\\
h'(x)&= \frac{1}{b}f'(x)\\
&= \frac{f'(x)}{b}\\
&= \frac{f'(x)}{ g(x)}\end{aligned}\]

Adding these…
\[\begin{aligned}
\frac{f'(x)}{ g(x)}-\frac{f(x)g'(x)}{[g(x)]^2}&= \frac{f'(x)g(x)}{ [g(x)]^2}-\frac{f(x)g'(x)}{[g(x)]^2}\\
&=\frac{f'(x)g(x)-f(x)g'(x)}{[g(x)]^2}\\
&= h'(x) !!!\end{aligned}\]

Oh my GOODNESS — it’s the quotient rule!

Addition…

Let \(h(x)=f(x)+g(x)\).

Suppose \(f(x)\) is constant, say \(f(x)=a\). Then

\[\begin{aligned}
h(x)&=a +g(x)\\
h'(x)&= g'(x)\end{aligned}\]

Suppose \(g(x)\) is constant, say \(g(x)=b\). Then

\[\begin{aligned}
h(x)&=f(x)+ b\\
h'(x)&= f'(x)\end{aligned}\]

Adding these:
\[ f'(x)+g'(x) = h'(x) !!!\]

OH MY GOODNESS— it’s the addition rule!

So it’s everywhere! This is totally awesome!

In the comments in the last post, it was mentioned that the reason all this happens is because of the multivariable chain rule, which is all about holding all but one variable constant and adding together the different results. When I realised this I thought to myself “duh!” – I was using the same words to tell myself what to do in both situations.

The issue is that when students learn these rules, they don’t know multivariable calculus so this can’t be used as a way to explain what’s happening. But I have a good feeling that it might be able to help make sense of the multivariable chain rule when they do learn it because it will connect perfectly so something they already know. I’ll let you know if I ever try.

22 September, 2016