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.
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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 🙂!