In this monthly column, we speak with a notable member of the mathematics education community about their work and their perspectives on the teaching and learning of mathematics. This month, we had the pleasure of speaking with Malke Rosenfeld.
Malke Rosenfeld is a percussive dance teaching artist, math explorer, math artist, TEDx presenter, author, and editor. Her interdisciplinary inquiry focuses on the intersection between percussive dance and mathematics and how to build meaningful learning experiences at this crossroads. Malke’s interests also include embodied cognition in mathematics learning, task and activity design in a moving math classroom, elementary math education, and writing as a professional development tool. Her teaching and artistic endeavors focus on explorations of the relationship between number, rhythm, constraint, and shape in a variety of modalities. Malke delights in creating rich environments in which children and adults can explore, make, play, and talk math based on their own questions and inclinations.
First things first, thank you for taking the time for this conversation!
For the past decade or so, you have been exploring the relationship between mathematics learning and dance, which has included developing and facilitating a program called Math in Your Feet and, most recently, publishing a book entitled Math on the Move (2016, Heinemann). In most people’s eyes, mathematics and dance are an unlikely pair. So I would like to start by asking: How can the two disciplines inform and complement each other?
Both math and dance are highly creative and expressive human endeavors. Not all of math is danceable and not all dance is mathematical, but there are some really nice overlaps between the two. In particular, the moving body is best positioned to explore and express the verbs of math, literally embodying the mathematical practices. There are also certain math ideas that can literally be put into action. In the early stages of writing my book, I did an experiment with dancers in three dance forms different from mine (hip hop, modern, and belly dance!) to see how/if certain action-oriented math ideas might resonate with the dancers. I also wondered if these ideas could function as choreographic prompts for what is called “movement invention,” and it turned out I was onto something!
I created a list organized around three big-picture mathematical ideas: relationships, transformations, and rules. More specific concepts within the three categories included ideas such as dilate, algorithm, compose/decompose, scale, iteration, function, and many others; I also provided short definitions emphasizing the conceptual meaning of each term. In one exploration, a college dance professor had her students choose a mathematical concept from the list and explore it further as a way to create variations in a movement sequence. One of her students, a double major in dance and math, examined the definition of ‘function’ and felt that it could be used as a way to encourage changing the movement quality; specifically, she thought that converted ‘values’ could be manifested in movement as converted ‘qualities’. (The full story can be found here. I’m also happy to share the original list.)
One tool that you have created to bridge the disciplines of art and mathematics is Jump Patterns, which you describe as “an integration of traditional percussive dance and elementary-level mathematics” (Rosenfeld, 2011, p. 78). What kinds of concepts and mathematical processes can students engage with through Jump Patterns and other “moving-scale” activities?
[perfectpullquote align=”right” bordertop=”false” cite=”” link=”” color=”” class=”” size=””]Exploring ideas at moving- or body-scale allows novices and experts alike to get a feel for what it means to “do math,” think mathematically, and deepen one’s intuition around a variety of concepts.[/perfectpullquote]Wow! It’s hard to believe that it’s been six years since that article was published. Since then, many more math teachers and mathematicians have interacted with Jump Patterns and what we do with them in Math in Your Feet. Based on their experiences and perspectives, I have come to understand that the math in my math-and-dance classroom includes transformational geometry, working within mathematical constraints, group theory, pure mathematics, dynamic investigations of pattern units, patterning (creation of complex moving patterns), equivalence relations, and spatial reasoning. Exploring these ideas at moving- or body-scale allows anyone, novice and expert math learners alike, to get a feel for what it means to “do math,” think mathematically, and deepen one’s intuition around a variety of concepts.
Why explore these concepts through dance? What does this context afford that, for instance, paper does not?
This is a great question and I think the answer is simple. No math concept can be understood completely in just one representation or mode, including doing math at “body scale.” The late Zoltan Dienes (creator of the Dienes blocks, which are also called base 10 blocks and inhabit many elementary math classrooms) proposed a theory of Multiple Embodiments, encouraging math educators to provide their learners with opportunities to explore a math idea in multiple, varied contexts and representations and to create generalizations based on these experiences. When I first looked into his work, I was tickled to find that he had written a book titled Mathematics Through the Senses, Games, Dance, and Art (1973, National Foundation for Educational Research), which is about giving children experiences with mathematical ideas through physical games and group dances.
In addition to the math-and-dance work I do, I have begun to investigate ideas for whole-body based math lessons that are outside a dance system. These are activities where the familiar mathematical objects we generally see on the page (such as polygons and polyhedra, hundred charts, or open number lines) are “scaled up” so that learners can interact with the idea or tool with their whole body in a meaningful way at a larger scale. I am enjoying the challenge of converting math that typically lives on the page into this new body-based modality. In collaboration with some amazing math educators, in addition to my book I’ve created four scaled-up, body-based math lesson plans for people to try out.
[perfectpullquote align=”right” bordertop=”false” cite=”” link=”” color=”” class=”” size=””]Learners at all levels need a chance to (quite literally) move away from the page now and again in an effort to explore more deeply “what math is.”[/perfectpullquote]Overall, the whole-body modality for math learning is not a panacea for math education necessarily, but I’m hoping it is a reminder that learners at all levels need a chance to (quite literally) move away from the page now and again in an effort to explore more deeply “what math is.” From what I’ve seen, mathematics is as much about the process of doing as it is in the final result. In his essay A Mathematician’s Lament, Paul Lockhart speaks clearly about the need for learners to do the actual work within a creative discipline such as music, visual art, or mathematics. His example of learning to read music before playing an instrument illustrates the absurdity of thinking math is solely about notation or memorization. The whole, moving body, harnessed in pursuit of honest exploration of mathematical ideas (whether dance or non-dance) can provide learners with a sense of what it means to think and do in both disciplines.
In your book Math on the Move, you acknowledge that the Math in Your Feet program is
the perfect blend of the two most anxiety-inducing disciplines for most of us raised in American society […]. For one thing, both math and dance have a lot in common in their apparent ability to invoke fright and a flight response. They also share the deep-seated myth that we as a culture hold about learning and knowledge: that you are either good at math or dance or you’re not. (Rosenfeld, 2016, p. xiii)
How do you alleviate the trepidation that students—and, for that matter, teachers—may have when trying whole body learning activities in the math classroom for the first time?
[perfectpullquote align=”right” bordertop=”false” cite=”” link=”” color=”” class=”” size=””]Children’s bodies and psyches are primed for movement; from birth, our physical exploration of the world around us is said to be the foundation of human intelligence.[/perfectpullquote]Honestly, my experience over many years has shown that children, even through the middle grades, are mostly excited and curious about what we’re doing and what comes next, and are pretty much all-in from the get-go. This makes sense, because children’s bodies and psyches are primed for movement; from birth, our physical exploration of the world around us is said to be the foundation of human intelligence, which is what is meant by the term “embodied cognition.” Another benefit to using the moving body in math class is that it supports further developments in spatial reasoning, which is literally built through the body and is the foundation of mathematical thought. Moving is a natural activity for young people. Teachers who have begun to do this work with their own students have reported to me that the movement aspect focuses learners on the mathematics at hand and, even better, that the movement itself becomes part of the reasoning.
In terms of adult anxiety, I imagine that a teacher’s first thought about whole-body learning is that they have to be expert dancers or movers for this kind of approach to work in their classrooms, which is absolutely not true! I also imagine teachers are also wondering about what might happen if students get out of control while they’re out of their seats. In reality, the expectations for a moving-math activity follow on from others you already have in place in your classroom.
To quell the qualms, it might be helpful for educators to think of the body as a “thinking tool” with an instructional focus similar to the work and discussion that goes on during an activity where hand-scale manipulatives are put to use. The added bonus of a whole-body approach (which I don’t think is on most people’s radar until it happens in front of them) is that when you get learners out of their seats for math, you will likely see surprising new strengths emerge in your students. There are always learners who struggle with math on the page, but in this body-based modality, some learners are more able to express their mathematical thinking and/or take the lead in problem solving in a way they haven’t exhibited at their desks. Everyone benefits from changing the mode and scale of a math investigation, but for some learners, having the opportunity to realize, even for a moment, that they understand math can help keep that door open.
Also important to know is that moving math is a context for non-permanent problem solving, similar to what Peter Liljedahl has created with vertical non-permanent whiteboard surfaces (e.g., Liljedahl, 2016). This means that learners can easily change and adapt their their work and thinking and are not hindered by a final answer until they’ve decided they’ve got it the way they want it. In the movement context at least, this leads to a higher level of learner motivation and perseverance through the inevitable tangles.
[perfectpullquote align=”right” bordertop=”false” cite=”” link=”” color=”” class=”” size=””]Much of the adult anxiety around a whole-body approach to math learning can be overcome by knowing that the teacher’s role is one of facilitator, not expert mover.[/perfectpullquote]Overall, I think much of the adult anxiety around a whole-body approach to math learning can be overcome by knowing that the teacher’s role is one of facilitator, not expert mover. I’ve thought deeply and long about the elements of a moving math classroom (whether in a dance system or at body-scale) so that math can be explored in a meaningful way. To create and facilitate a meaningful context in which children can think deeply and engage in mathematical sense making with their bodies, the following elements should be present:
- The activity explores one or more math concepts at a new scale.
- The math-and-movement lesson provides a structure in which students make choices, converse, collaborate, and reflect verbally on what they did, how they did it, and what they noticed while they were engaged in whole-body activity.
- The body activity is focused on mathematical sense making, not mnemonics (memorization), often through efforts to solve a physical or moving-scale challenge of some kind, and not on illustrating a math idea as it is typically represented on the page.
- The teacher is the facilitator of the activity, pacing, and discussion, supporting learners and their collaborative relationships as they work and, later, connecting the mathematics as experienced in the moving challenge to a new mode or new contexts.
- Students experience the activity at the center of the action and as observers of others’ work, providing needed perspective to reflect on the mathematics in question.
- The math-and-movement should be explicitly connected with the same math idea as it is experienced in other scales, contexts and/or modes, often through some kind of written or schematic documentation (like a map), or a table to organize the results of the moving math challenge or investigation.
Many mathematics teachers will have encountered—often, by means of a well-meaning friend—the image of a stick figure posing its arms in the shape of various graphs, with the function (sin(x), x2, 1/x, and more) written below each figure. In Math on the Move, you suggest that this is an example of “less meaningful” math movement. There is clearly more to Math on the Move, then, than simply “moving a part of the body or being on one’s feet” (Rosenfeld, 2016, p. 10).
Could you expand on what you mean by “meaningful” and “less meaningful” math movement?
[perfectpullquote align=”right” bordertop=”false” cite=”” link=”” color=”” class=”” size=””]Whole-body math learning is not about re-representing math ideas we see on the page. Instead, we can learn new things from exploring those ideas with our whole bodies.[/perfectpullquote]Those stick figures are primarily focused on memorization, which is not necessarily the strongest or most meaningful use of the human body in a math investigation. I’ve always thought that a more meaningful body-based activity would be to turn those graph shapes into physical pathways and then investigate how different combinations of graphs create beauty or chaos! Even if you don’t want to to turn it into a dance activity, my overall point is that both math and dance utilize the ideas of space, time, and motion. Whole-body math learning is not about re-representing math ideas we see on the page. Instead, we can learn new things from exploring those ideas with our whole bodies. The math idea remains the same, but looking at it from a new angle can bring new perspectives to children, teens, and and adults alike. That’s where the meaning, and learning, is created.
Your work has focused on primary- and middle-grade-level mathematical concepts that can be expressed through whole body learning activities. Have you considered how Math on the Move might be brought into the secondary classroom? If so, what kinds of higher-level mathematical ideas might be explored in this way?
Well, different people see different mathematics in Math in Your Feet depending on their perspective and experience of “what math is.” For example, after reading Math on the Move David Butler (@DavidKButlerUoA), an Australian mathematician at the University of Adelaide wrote: “The dance moves within the tiny square spaces [used in Math in Your Feet] are abstract mathematical ideas that are 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 [the dance work in Math in Your Feet] to pure mathematical investigation is striking.”
So, if you want to investigate pure mathematics, give Math in Your Feet a try!
Barring that, in secondary and middle school classrooms we can also think about transformational geometry, which can be explored in meaningful ways at body-scale. Scott Steketee and Daniel Scher have written an article published in the NCTM journal Mathematics Teacher titled “Connecting Functions in Geometry and Algebra,” and I think their approach would translate very well to a body-based adaptation of that investigation.
Max Ray-Riek (@maxrayriek) of the Math Forum and Michael Pershan (@mpershan) have investigated the complex plane using an open, body-scale number line. Max has also advised me on middle and high school adaptations for the Rope Polygon activity from Math on the Move. For middle school classrooms, you might also consider Math in Your Feet pattern creation in a coding context.
Lastly: Where can mathematics teachers learn more about engaging their students in whole-body learning?
My book Math on the Move: Engaging Students in Whole Body Learning (Heinemann, 2016) is a thorough resource of information about the #movingmath approach, including 40 videos of the Math in Your Feet classroom action.
The Math on the Move Facebook book group is a resource for conversation with others starting out with a body-based approach to math learning and the potential for a book study and discussion. In addition, the Math on the Move book blog is an ongoing resource for learning more about whole-body math learning.
And, although I’m probably best known for my work with math and dance, the things we think about and do in a moving math context also translate to the art making modality of paper, glue, straws, color, Cuisenaire rods, and tape! To check out examples of my math art activities you can go to my blog Math in Unexpected Spaces (which has some examples of my work with scaled-up polyhedra) or visit the math art projects page on my website where you can read more about the projects I’ve created. Whatever you end up doing, I hope you have fun making math!
Interviewed by Ilona Vashchyshyn
References
Dienes, Z. P. (1973). Mathematics through the senses, games, dance and art. Windsor, England: The National Foundation for Educational Research.
Liljedahl, P. (2016). Building thinking classrooms: Conditions for problem solving. In P. Felmer, J. Kilpatrick , & E. Pekhonen (Eds.), Posing and solving mathematical problems: Advances and new perspectives. New York, NY: Springer.
Rosenfeld, M. (2011). Jump patterns: Percussive dance and the path to math. Teaching Artist Journal, 9(2), 78-89.
Rosenfeld, M. (2016). Math on the move: Engaging students in whole body learning. Portsmouth, NH: Heinemann.
