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.


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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 “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 in the comments.

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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 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:

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.

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.

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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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Here in Australia, we are at the tail end of a reality cooking competition called “Zumbo’s Just Desserts“. In the show, a group of hopefuls compete in challenges where they produce desserts, hosted by patissier Adriano Zumbo. There are two types of challenges. In the “Sweet Sensations” challenge, they have to create a dessert from […]

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In Maths 1A here at the University of Adelaide, they learn the following theorem (this is taken from the lecture notes written by the School of Maths here):

It says that, given a function of x defined as the integral of an original function from a constant to x, when you differentiate it you get the […]

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I was talking to a student about his calculus last week. He was trying to differentiate xx. (Actually he was trying to differentiate x ln(x) and had decided the best place to start was to raise e to the power of it, thus producing xx.) At first he tried this:

I asked him what he thought […]

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It’s university holidays again (aka “non-lecture time”), which means I’m back on the blog trying to process everything that’s happened this term. Mostly this has been me spending time with students in the Drop-In Centre, since I made a commitment to do more of what I love, which is spending time with students in the […]

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Last week, I helped quite a few students from International Financial Institutions and Markets with their annuity calculations, which involve quite detailed stuff like this:

There were several small issues a lot of them had, which combined to stall their calculations. One of the more important problems was about how the calculator interprets what they type […]

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