The Cognitive and Intellectual Aspects of Dance (and math)

Dance Magazine recently published an interesting article titled: Don’t get It Twisted: Dance Is An Intellectual Pursuit and parts of the article really resonated with my approach to combining math and dance. Below are some excerpts from that article interspersed with related #movingmath posts from this blog (bolding mine).

People have a tendency to think of dance as purely physical and not intellectual. But when we separate movement from intellect, we are limit what dance can do for the world. It’s not hard to see that dance is thought of as less than other so-called “intellectual pursuits.” How many dancers have been told they should pursue something “more serious”? How many college dance departments don’t receive funding on par with theater or music departments, much less science departments? Perhaps that’s because dance only leaves behind traces. The words and decisions that go into making dances have a hard time being accounted for, and choreographic notes and videos cannot fully capture a dance work.

Dance depends on the presence of the body. Unfortunately, it’s difficult to explain to non-dancers how corporal movement is a means of thinking and engaging with complex ideas. That’s why it’s so important that dancers can talk or write about their work, translating the corporal knowledge into language.

When we acknowledge that our bodies think, move, translate, react—often in conjunction with linguistic thought or prior to itwe can use dance as a tool. 

Related: When the Movement IS the Reasoning Tool | What does it look and sound like when kids use their whole bodies during a math lesson? What happens prior to, during, and after the activity? Luckily Deb Torrance and Lana Pavolova have provided us with some stellar documentation so we can get a closer look at what happens when we give kids a mathematical challenge to explore with their whole bodies.

Linguistic Intelligence Has Its Place in Dance, Too
That’s not to say that language isn’t part of dance. Choreographers craft dancers’ intentions and movements with words, images and metaphors. Even in improvisation, a director dictates a score, and dancers translate the imagery into corporal form. When choreographers layer dance and words, it engages the audience in new ways. As Bill T. Jones explains, “You see one thing and you hear another thing, and then the audience puts together what they mean.”

Related: Learning Math by Ear: The Role of Language in a #movingmath classroom  | “Comprehension of a word’s meaning involves not only the ‘classic’ language brain centres but also the cortical regions responsible for the control of body muscles, such as hand movements.” To me this study explains part of why a “moving math” approach that includes a focus on math language used in context can open up new pathways for our learners.”

Dance Can Help Us Better Understand Our World
Many choreographers use dance to shed light on today’s most pressing topics. Some use dance in conjunction with social activism, like Ananya Dance Theatre’s Ananya Chatterjea, who recently created Shyamali as a tribute to women across the world who have stood up against oppression. Others explore the nuances of science: Michelle Dorrance’s Myelination, for example, translates the biological process of a myelin sheath forming around a nerve into tap dance. Not to mention artists who use their dance practice as research, focusing on the process of dance making to explore a question or subject…

Related: Leaving Room for Question Asking | “It’s these questions, arising in the moments when they’re needed, born of collaboration, that help learners notice structure and pattern and purpose in what they’re doing. From there we can move to the more formal learning. But… I think kids in general are most motivated when they are provided agency by the adults in their lives. Their work may not be technically perfect, but they are in the best part of learning (to me, anyhow): inside the flow of an investigation filled with their wonderings.”

Dancers Connect Multiple Parts of Ourselves
Dance intertwines the cerebral, physical and emotional; science tries to unravel the connections between these. Dance uses these inherent connections to delve deeper into our humanity, and create new ways of reflecting on the world. In that way, dance is a crucial tool in intellectual pursuits.

Related: Learning Without a Body |”The body is not simply a vehicle toward realizing the perceived pinnacle of abstracted knowledge housed in the mind.  The body is where learning originates. Living in a body is also the way children learn personal agency as they make decisions about how their bodies will move and act and how that power can influence and shape their world. And, in the process, learning that there are obvious consequences and responses in relation to their actions. This is literally and viscerally democracy in action.”


Malke Rosenfeld is a percussive dance teaching artist, Heinemann author, editor, math Malke Rosenfeld Croppedexplorer, and presenter whose interests focus on the learning that happens at the intersection of math and the moving body. She delights in creating rich environments in which children and adults can explore, make, play, and talk math based on their own questions and inclinations.You can find out more about her work at malkerosenfeld.com,  on Twitter,  Instagram, or Facebook.

“What I’ve Learned (so far) About Teaching and Learning (Mathematics) in Dance Class”

“It’s hard to put your finger on the cause of a slow-burning problem. One day, you feel lucky to be paid to do something you love and the next, you’ve forgotten why you loved it in the first place. It can be even harder to put your finger on a solution. I turned to dance.”
–Ilona Vashchyshyn in her post “This one’s about dance (and burn out)”

Among many other things, Ilona is a secondary teacher in Saskatoon, Saskatchewan, and co-editor of the Saskatchewan Mathematics Teachers’ Society’s monthly periodical, The Variable. I have had the honor being interviewed by Ilona in which she heard all about my work. She posted a gorgeous, forthright story on her blog recently where I got to know more about her. In particular, I want to share her reflection on what she learned about teaching, learning, and mathematics during dance class (bolding mine)but I also encourage you to read her entire post.

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What I’ve learned (so far) about teaching and learning (mathematics) in dance class, by Ilona Vashchyshyn:

Don’t underestimate the importance of routine.

Almost without fail, dance instructors begin class with a warm-up routine that engage dancers’ bodies and minds in preparation for the work ahead. At HDC (Harbour Dance Centre), most instructors kept the warm-up routine more-or-less similar from one day to the next, although new movements might be added (and some left behind) as the class gained proficiency. E.g., in ballet, a barre exercise might be repeated but with new arm movements, or with a relevé balance to finish. Of course, warm-ups are particularly important in dance because they help to prevent injury, but I found that having a common thread from class to class was valuable in other ways: notably, it gave me repeated opportunities to learn and improve the movements / pathways / exercises which, in turn, gave me the opportunity to develop my confidence as a dancer. Seeing my improvement from day to day encouraged me to continue, and experiencing small successes at the start of class offset the occasional frustration of learning new movements and combinations later on in the lesson. Not to mention, slowing down to practice movements in isolation meant that I could perform them with more ease and fluidity when it came to using them in choreography or in improvisation.

Dance is not about [learning] the choreography.

Early in the week, one of my instructors at HDC cut the music just before we began our routine to say: “Stop. Why are you here? If music is playing in dance class, you dance.” And he insisted that we did, even if it felt awkward at first. Later, referring to a complicated move: “Listen, I’m not gonna teach you that shit. You can learn it on the internet. This is dance class.”

More than any dance instructor I’ve ever had, he emphasized that dance isn’t about the choreography. Choreography, which is a sequence of memorized movements, will be quickly forgotten. A more valuable outcome of dance class is increased confidence and joy in responding with movement to music. Of course, an additional benefit of dance lessons is picking up new steps or pathways, but the value of these is not that you can perform them in a specific sequence, but rather that you can go on to combine and remix them in endless ways, in addition to creating your own. In fact, most of the dance instructors regularly set aside time during class for improvisation – even in ballet, which is generally regarded as the most rigid of the dance styles.

The connection to math class, I think, almost goes without saying: formulas, procedures, algorithms are our version of choreography. Of course, it can be fun to learn and execute the moves. There’s a great sense of accomplishment in accurately performing a complicated routine. But the main outcome of math class shouldn’t be a memorized sequence of steps; more important, I think, is increased confidence and persistence when facing new problems (the analogue of improvisation, or developing your own routine), maybe with a few more tools to tackle them.

Note that I don’t feel that there is a clash between routine (c.f. above), choreography, and improvisation – all have their place and their value. All parts of the elephant, as they say. Lest I stretch the metaphor past its usefulness, I will stop here, but the idea of dancing as problem solving (and problem solving as dancing) is something I’d really like to keep exploring.

A teacher learns.

A dance studio can be a prime location to study how great teachers differentiate and adapt instruction to meet their students’ needs. Especially in adult beginner dance classes, students tend to be very diverse in their abilities and previous experiences. While some really are starting from scratch, many are dancers returning after a long break, and still others are trying a new style but have extensive experience in another. At HDC it was instructive to watch how teachers adapted their plans for the week as they learned more about their students, as well as how they provided options in the moment for dancers with different levels of experience (“for now, just focus on the feet,” or “try lifting your hands off the bar if you’d like a challenge”).

On the flip side, the experience of being a student again was also invaluable. As teachers, we know that the struggle of learning something for the first time can be quickly forgotten, replaced by the illusion that it was easy all along. Forgetting is all the more likely when you teach the same course semester after semester, year after year. And maybe the quickest way to dismantle this illusion is to put on a pair of ballet slippers and a leotard and step into a mirrored room full of strangers who all seem to know what they’re doing while you’re still struggling to remember the difference between a frappé and a fondu and to balance with both feet on the ground. Even with the most encouraging of teachers, I sometimes found myself afraid to ask for help, hiding in the back when I felt lost, and even fighting the urge to escape to the bathroom when I started to get overwhelmed. Pushing through the frustration of just not getting it and the fear that I just never will, especially when it feels like everyone else does, can take tremendous effort. And a teacher who – in addition to providing focused feedback – takes time to remind the class that it’s okay to feel lost sometimes, that being on the verge of this is too hard is just where you’re supposed to be, can make all the difference between choosing to stay home tomorrow and showing up to try again.

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Malke here again. More than anything I’d like to encourage you to think about whole- body #movingmath (whether dance or non-dance based) as a process of coming to understand not just the math but the process by which we come to know math, a topic that to many people, seems foreign and inaccessible.  Using the whole body in math class teaches us to persevere. Engaging our learners’ bodies in math class can help this process and as well as help build a community of learners. There are lots of resources on this blog for getting started. Please don’t hesitate to get in touch if you have questions!


Malke Rosenfeld is a percussive dance teaching artist, Heinemann author, editor, math Malke Rosenfeld Croppedexplorer, and presenter whose interests focus on the learning that happens at the intersection of math and the moving body. She delights in creating rich environments in which children and adults can explore, make, play, and talk math based on their own questions and inclinations.You can find out more about her work at malkerosenfeld.com,  on Twitter,  Instagram, or Facebook.

Why Math in Your FEET?

In Chapter One of Math on the Move: Engaging Students in Whole Body Learning I share a little of the history of how I developed Math in Your Feet and also describe how, in the years before making the connection between my dance style of percussive dance and mathematics, I worked to support children in becoming dance makers using the elements of this not-quite mainstream dance form. So what is this percussive dance and how does it connect to math and, most puzzling, why do we need to use our feet?

This question about feet came up a lot during almost weekly Skype sessions with the amazing Mary Walker Hope in Ontario, Canada. In January 2018 I was lucky enough to get a peek at the awesome Math in Your Feet patterns and footwork resulting from a 2nd through 8th grade exploration in her school board. I figure that  others may ask this “Why are we using our feet?” question from time to time so in this post I’d like to elucidate the “feet” aspect of the math and dance that we make while creating patterns with our feet in Math in Your Feet.

First of all, there many, many (MANY!) different dance styles in this world. Some of them are percussive (rhythm based) and some of them are not. Sometimes they’re a combination of both. Percussive dance is probably best known through the international performance phenomenon called Riverdance which features Irish step dancing. You might also be familiar with percussive dance through all those tap classes either you or your kid took at some point.

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The main focus of percussive dance styles is to be a rhythm maker using different parts of your feet. The Pattern Properties I created lay out the basic elements of this kind of dance, similar to what creative movement is to modern dance. To answer “why feet” look at the Movement category of the Math in Your Feet Pattern Properties chart on the left. Each movement has a clear sound associated with the movement. Making sure you produce that sound is one aspect that makes this kind of dance so satisfying. In the chart below I’ve provided some information about how these sounds are made.

Pattern Properties description FINAL
So, there you have it! Making patterns and sounds with your feet is fun and satisfying, not to mention mathematical!


Malke Rosenfeld Cropped.jpgMalke Rosenfeld is a percussive dance teaching artist, Heinemann author, editor, math explorer, and presenter whose interests focus on the learning that happens at the intersection of math and the moving body. She delights in creating rich environments in which children and adults can explore, make, play, and talk math based on their own questions and inclinations.You can find out more about her work at malkerosenfeld.com,  on Twitter,  Instagram, or Facebook.

A Powerful Tool for Both Learners & Teachers

 

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Larry Ferlazzo invited me to answer this question on his Ed Week Teacher blog: “What is an instructional strategy and/or teaching concept that you think is under-used/under-appreciated in the classroom that you think should be practiced more widely?”  I am sharing my response here but do check out the other answers!


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Every day students of all ages come to school with a powerful tool for mathematical reasoning but rarely get the opportunity to harness its full potential. When this tool, our students’ own bodies, are used, the activity is typically relegated to acts of memorization that lead no further than the next test. In contrast, taking math off the page and into the spatial, embodied realm of the whole, moving body has great potential to open up new avenues for understanding. Here are some examples of how a whole body, #movingmath approach can open up new opportunities for learning in a variety of grades and settings:

  1. Changing the scale:  When you change the scale of the math you are already exploring in your classroom you provide learners with the opportunity to get to know math from a completely new and novel perspective. Whether it’s exploring  number patterns on a scaled-up hundred chart, physically experiencing magnitude, scale, distance, and direction on an open body-scale number line, or noticing new things about polygons using lengths of knotted rope, learners collaborate, discuss, evaluate, reflect upon, record their activity, and start to connect it to other experiences in which they encounter and use these ideas. Seeing connections develops intuition,” Dan McQuillan at the University of Norwich tweeted recently. “Proofs are great; just like climbing trees, but the ability to swing from tree to tree is also great.”CH3P26
  2. Reasoning in action: During a Proving Center lesson Kindergarten students were asked to work in teams of four or five to find the center of an 11- cell structure, which looks a bit like a ladder.  Children were able to find the “center” of the object with their bodies rather quickly but their biggest challenge was to justify their physical reasoning. Lana Pavlova, an elementary teacher from Calgary, Canada told me that some of her students’ reasoning included “Because five is the same as five”, “Because these two sides are equal”, “Because it is exactly the half”.  Another student said that “not all numbers have the middle, six doesn’t. One has the middle and it’s one.” Lana told me, “I was very impressed by the kids’ reasoning. I also want to highlight how important the initial ‘explore’ stage is [and that] the movement IS the reasoning tool.”Fairhill ladder
  3. A reason to persevere: Lisa Ormsbee, a P.E. teacher at Fairhill School in Dallas,TX spent three weeks this past June running an enrichment program using movement and rhythm to explore and deepen enjoyment and understanding of math with intermediate students, many of whom exhibited what she called “math reluctance.” One of her main activities was Math in Your Feet  which requires precise physical/spatial reasoning around rotations,  categories of pattern properties, unitzing, complex patterning, equivalence, and perseverance to create original foot-based patterns. Lisa told me, “The kids were ALL so engaged in this activity! It was extremely hard for a couple of students, but because they were working with a partner they were more interested in “sticking in there” where it was uncomfortable until they got it!”rb-5a
  4. Cognition is embodied: “Conceptualising the body, in mathematics, as a dynamic cognitive system enables students and teachers’ physical, visual, verbal, written, mental, and (in)formal activity to be taken not simply as representations  of abstract spatial concepts but…as corporeal and contextually grounded forms of cognition.” [Spatial Reasoning in the Early Years, Davis et al. 2015]

Overall, no math concept can be understood completely in one representation or modality.  Similarly, not all math can be explored with the body. Whole-body math may be a novel approach for many but it’s also clear that it can be a powerful tool for both learners and teachers.


Malke Rosenfeld is a dance teaching artist, author, editor, math explorer, and presenter coverwhose interests focus on the learning that happens at the intersection of math and the moving body. She also delights in creating rich environments in which children and adults can explore, make, play, and talk math based on their own questions and inclinations. Her book Math on the Move: Engaging Students in Whole Body Learning,  was published by Heinemann in 2016.

When “The Movement IS the Reasoning Tool”

 

ch3p26What does it look and sound like when kids use their whole bodies during a math lesson? What happens prior to, during, and after the activity? Luckily Deb Torrance and Lana Pavolova have provided us with some stellar documentation so we can get a closer look at what happens when we give kids a mathematical challenge to explore with their whole bodies.

Let’s start with this video in which children work collaboratively to explore a body scale 25-cell ladder-like structure in pursuit of proving how they know they’ve found its center. Although teams may solve the initial challenge rather quickly, the core mathematical experience is in using space, structure, and their bodies as tools for making sense of the challenge as they work to prove that they have found the right location.

These are first graders; Deb Torrance, their wellness teacher, and their classroom teacher have teamed up to run the lesson.

What do you notice about what the children are doing?



Here is the first key aspect of a moving math activity — it moves, but in a very focused manner and it also inspires on-topic conversations. Deb reported that “During four minutes of ‘free explore’ time with the ladders I was amazed at the different ways children were attempting to cross the structure! As the wellness teacher, it always excites me to see students moving and they were certainly doing that; hopping patterns, cartwheels, keeping hands in boxes, crawling… Students were then pulled to the center of the gym to discuss their thoughts and ideas about the ladders. The math vocabulary that was already being discussed [with peers in the context of the physical exploration] was amazing (symmetrical, middle. center, odd/even…).”

The role of the adults during the exploration phase of a moving math lesson is to keep tabs on the activity and check in occasionally with the learners about what they’re thinking or wondering. Teachers also play a role when it’s time for teams share out to the whole class; in this lesson the sharing would be focused on the strategies teams used and how they knew they had found the center of the space.

Lana Pavlova did the same Proving Center lesson with a group of kindergarten students and an 11-cell ladder. She reports that “proving was where the fun started. Many students could find the middle and count five squares on each side but weren’t sure how to explain why five and five was the middle but four and six squares was not. So, a lot of conversations revolved around trying to prove it and showing with their bodies what’s going on.”

Although the kindergarten kids were in groups, they mostly worked individually. Some of their reasoning included “because five is the same as five”, “because these two sides are equal”, “because it is exactly the half”. Some students were convinced that the middle was on the line, so they counted both lines and squares; if you stand in the middle “there will be six lines on each side”. One student said that “not all numbers have the middle, six doesn’t. One has the middle and it’s one.”

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When the kindergarten students went back to their classroom they used whiteboards to explain what they did during the moving portion of the lesson. Lana says, “The physical activity helped [most of] them to remember that there were five squares on each side. One student drew a “9 frame” and wrote the number five on each side. As he was explaining it to me, he noticed he had counted it incorrectly and went back to change his number to four on each side. He shared how he was in the middle because there was the “same on both sides.” 


Lana’s final thoughts after running this lesson get right to the core of what what #movingmath is and can do. “I was very impressed by the kids’ reasoning. I also want to highlight how important the initial ‘explore’ stage is; the movement IS the reasoning tool.”

No math concept can be understood completely in one representation or modality. Similarly, not all math can be explored with the body. Whole-body math may be a novel approach but it’s also clear that it can be a powerful tool for both learners and teachers.

You can find the Proving Center lesson plan as well as three other moving math lessons for K-12 learners here.  When you try it out please consider sharing  a picture, video, or blog post to Twitter or Facebook with the hashtag #movingmath.


Malke Rosenfeld is a dance teaching artist, author, editor, math explorer, and presenter whose interests focus on the learning that happens at the intersection of math and the moving body. She also delights in creating rich environments for math art making in which children and adults can explore, make, play, and talk math based on their own questions and inclinations.

Move with Math in May: Four #movingmath Lessons

UPDATE, September 2017: This post was originally celebrating a special nearing-the-end-of-the-school-year event titled “Move with Math in May. The event featured four math-and-movement lesson plans to chose from. The goal was an opportunity to try out whole-body math in a low-key way to get a sense of what it’s all about…but you can use these lesson plans any time you want! Below you’ll find overviews of and links to each lesson. If you have any questions, feel free to get in touch on Twitter or via the contact form. Most importantly, HAVE FUN!!

 

MOTM Proving Center Lesson 1 HeaderIn this activity, children work collaboratively in teams of three to five (four being an optimal number) to determine the center of a taped ladder-like structure on the floor. Although teams may solve the initial challenge rather quickly, the core mathematical experience is in using space and their bodies as tools for making sense of the challenge as they work to prove that they have found the right location. GO TO THE LESSON


MOTM Rope Polygons Lesson 2 HeaderIn this activity, created in collaboration with Max Ray-Riek from the Math Forum at NCTM, students work collaboratively in teams of three to five to investigate and construct polygons with their bodies and a twelve-foot knotted rope. Although this lesson attends to regular polygons, the activity has been extended to address learning goals for middle and high school students.  GO TO THE LESSON


MOTM Clap Hands Lesson 3 HeaderClapping games are a part of the natural mathematics of childhood; they are also filled with pattern, spatial reasoning, and rhythm.  This activity, which can be different every time you play, was developed by John Golden (@mathhombre) with a class of preservice teachers.  GO TO THE LESSON


MOTM MIYF Lesson 4 HeaderHave you ever wondered what Math in Your Feet would look and sound like in your classroom? Here is a game-based version of this work, developed in collaboration with wellness teacher Deb Torrance (@Mrs_Torrance), as a way for you to see what math and dance can look like when both are happening at the same time. GO TO THE LESSON.

I’m looking forward to seeing and hearing how things go!


Malke Rosenfeld is a dance teaching artist, author, editor, math explorer, and presenter whose interests focus on the learning that happens at the intersection of math and the moving body. She delights in creating rich environments in which children and adults can explore, make, play, and talk math based on their own questions and inclinations.

 

 

Getting Started with Whole-Body Math Learning: Scale Up!

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The goal of a moving math classroom is to harness students’ whole bodies and energy in a way that also focuses their attention on the mathematics in question. Learning to facilitate this kind of thinking/learning/moving activity doesn’t happen overnight but there are some specific small steps to help you and your students, get used to this new mode of math investigation.

First of all, children, need to experience what it means to learn math off the page. After all, movement during the school day is usually the bailiwick of the playground and P.E. class. Being explicit about expectations for a dual focus on both body agency and on a mathematical task, whether inside or outside the classroom, will also support the development of executive function and self-regulation skills, both of which can have a positive impact on their learning overall.

The key to learning self-regulation skills … is not to avoid situations that are difficult for kids to handle, but to coach kids through them and provide a supportive framework — clinicians call it “scaffolding” the behavior you want to encourage — until they can handle these challenges on their own. [Child Mind Institute]

Here are two related ways to help children “learn to learn” with their bodies while learning math at the same time.

1. Change the scale

We can provide opportunities for “learning to learn” with your whole body by “changing the scale” of a familiar math idea from what is normally the size of a piece of paper (hand-scale) to “body-scale.” Here are some examples of familiar math investigations that have been “scaled-up.”

One of my favorite off-the-page math investigation is a scavenger hunt, often as a photo challenge,  like this school showed in their tweet, below:

Also, don’t miss this account from MathsExplorers, based in England, who blogged recently about the creation of “an impromptu large-scale dice game” and how changing the scale motivated children during a challenging time of day.

2. Create a non-permanent, body-scale problem solving context that encourages math talk and conversation

The familiar hundred chart scaled up to body-scale (sometimes called moving-scale) is big enough to walk in/on during an investigation. Allowing students’ bodies to interact with this tool in a new way can deepen their understanding of its structure and inspire new insights about the relationship between the numbers within. As in any #movingmath activity, these insights are created by the scale of the activity as well as collaboration and conversation.

A paper hundred chart is a useful collaborative tool between, at most, two children. A body-scale hundred chart allows for many more people to think and talk together. It’s also a wonderful example of what a whole-body non-permanent problem solving context looks like. Scaling up a math activity  that is focused on making sense of math instead memorization can create a flexible problem solving context that allows the learner to adjust their answers and reasoning as their thinking progresses.

In the video below, Jenn Kranenburg, whose work with body-scale math is featured in the first half of Chapter 3 of Math on the Move, shows us how this looks and sounds in her classroom.

If you’re interested in learning more about how and why a moving math classroom is beneficial to both math learning and our students’ overall growth check out the post 5 Articles that Answer: “How can they learn math if they’re moving?”

And, if you’ve scaled up a math activity I’d love to hear about it!


Malke Rosenfeld is a dance teaching artist, author, editor, math explorer, and presenter whose interests focus on the learning that happens at the intersection of math and the moving body. She delights in creating rich environments in which children and adults can explore, make, play, and talk math based on their own questions and inclinations. Join Malke and other educators on Facebook as we build a growing community of practice around whole-body math learning.

5 Articles that Answer: “How can they learn math if they’re moving?”

Or, more succinctly, “How is this math?” There is an entire chapter in Math on the Move that answers this question in great detail, but here are some research-based articles, as well as bonus perspectives from mathematicians, that I hope will provide a strong rationale for you when explaining to others the benefits of whole-body math learning.

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1. A recent study in Denmark has concluded “Math is learned best when children move…and it  helps to use the whole body.”

Participation in math lessons focusing on integrating gross motor activity can positively contribute to mathematical achievements in preadolescent children. In normal math performers, gross motor enrichment led to larger improvements than fine motor enrichment and conventional teaching. Across all children gross motor enrichment resulted in greater mathematical achievement compared to fine motor enrichment. From a practical perspective, teachers and related personnel should consider integrating gross motor activity in learning activities relevant to the academic curriculum as a promising way to engage children and improve academic achievement.

This is great news but we need to keep our eye on what it means to do this in a meaningful way in the classroom!

2. Spatial Reasoning IS mathematics: “It is almost as if they are one and the same thing.”

Even though spatial reasoning includes the body (see information in #3, below), there has been little research on whole-body-based spatial reasoning. Nevertheless, spatial reasoning is a foundational skill for learning math and Math on the Move is, in part, about illustrating in great detail how we can harness and develop whole-body spatial reasoning during math time.

“The relation between spatial ability and mathematics is so well established that it no longer makes sense to ask whether they are related” (p. 206). Researchers have underlined that the link between spatial reasoning and math is so strong that it is “almost as if they are one and the same thing” (Dehaene, 1997, p. 125). Reflecting on the strength of this relationship, others have noted that “spatial instruction will have a two-for-one effect” that yields benefits in mathematics as well as the spatial domain…”

 3. Paying Attention to Spatial Reasoning

A succinct document targeted to educators that explains the importance of spatial reasoning in mathematics and what it looks like when it’s integrated into math class in grades K-8.

Students need to be explicitly taught and given opportunities to practice using executive functions to organize, prioritize, compare, contrast, connect to prior knowledge, give new examples of a concept, participate in open-ended discussions, synthesize new learning into concise summaries, and symbolize new learning into new mental constructs, such as through the arts or writing across the curriculum.

4. Developing Executive Function

Math is more than facts and being in control of your own body while focusing on a specific body-based task is an opportunity for students to develop Executive Function as well as apply and deepen their learning.

Creative opportunities — the arts, debate, general P.E., collaborative work, and inquiry — are sacrificed at the altar of more predigested facts to be passively memorized. These students have fewer opportunities to discover the connections between isolated facts and to build neural networks of concepts that are needed to transfer learning to applications beyond the contexts in which the information is learned and practiced … When you provide students with opportunities to apply learning, especially through authentic, personally meaningful activities with formative assessments and corrective feedback throughout a unit, facts move from rote memory to become consolidated into related memory bank, instead of being pruned away from disuse.

5. Children think and learn through their bodies

We conclude that children think and learn through their bodies. Our study suggests to educators that conventional images of knowledge as being static and abstract in nature need to be rethought so that it not only takes into account verbal and written languages and text but also recognizes the necessary ways in which children’s knowledge is embodied in and expressed through their bodies.

BONUS: Mathematicians can recognize the whole-body activity as “doing math”

“Its [the second part of[Math on the Move] 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 is striking.”

 David Butler, University of Adelaide, Australia [Read full review]

“The movement activities described [by Malke] naturally link to the notions of transformational geometry and the subtle questions of sameness and difference that are explored. Enabling people to find the links between that physical understanding and the mathematical abstractions is a wonderful way to make mathematics open up. Overall this is a wonderful book on the power and importance of mathematical thinking to explore all sorts of surprising topics, and conversely the importance of physical movement and dance to explore mathematics.”

—Edmund Harriss, Clinical Assistant Professor, Department of Mathematical Sciences, University of Arkansas [Read full review]


Malke Rosenfeld is a dance teaching artist, author, editor, math explorer, and presenter whose interests focus on the learning that happens at the intersection of math and the moving body. She delights in creating rich environments in which children and adults can explore, make, play, and talk math based on their own questions and inclinations. Join Malke and other educators on Facebook as we build a growing community of practice around whole-body math learning.

A Quick Example of Body-Based Cognition: Spinning

library-poster

Yesterday I was at the library for my now monthly #makingmath sessions for kids and their parents. The ages for this event seem to trend 8 and under, probably because parents with young children are often looking for something to do on Sunday afternoons. Our youngest participant yesterday was two and a little more, and this little story is about her.

I’ve written previously on this blog about what it looks like when children think and learn mathematically with their bodies. Yesterday my new friend was there with her mom and her brother. Her brother made this delightful “Dr. Seuss house with smoke coming out of the chimney” while she made a crown and earrings for herself out of the same materials.

Another activity we had going on was playing around with these cool hexagon building blocks that I found in a big box dollar bin a couple summers ago.  A boy made an object that was just begging to be spun…

…after which my little two year old friend started rotating around in one spot exclaiming to me: “I’m spinning!”

This is just one more example of how children think and learn with their bodies. She was entranced by the toy and it’s gorgeousness.  She spent a quick moment spinning exactly like the top and then went back to making earrings for her mother.

The body is where learning originates. Children use their bodies to show us every day what they know and think and wonder. This non-verbal, physical manifestation of cognition is present every day in some way. I invite you to put on your #movingmath glasses and, when you notice something tell us about it! Here’s a few places where you can share:

In the comments to this post
On Twitter with the hashtag #movingmath
-or-
On Facebook with privacy set to public with the hashtag #movingmath

I can’t wait to hear about (or see) what you notice!


Malke Rosenfeld is a dance teaching artist, author, editor, math explorer, and presenter whose interests focus on the learning that happens at the intersection of math and the moving body. She delights in creating rich environments in which children and adults can explore, make, play, and talk math based on their own questions and inclinations. Join Malke and other educators on Facebook as we build a growing community of practice around whole-body math learning.

 

Math off the Page: Units & Part-Whole Relationships

This week, as part of series of posts on “First Steps” for bringing math off the page and into our students’ bodies, we’ll continue investigating what familiar math concepts look like in the wild. In this post I’ll be looking at the idea of units and part-whole relationships as they present themselves in daily life.

One of the  places we can find units and other examples of parts and wholes off the page is in classic children’s pattern- and rhythm-based play like jump rope or clapping rhymes, like in this video of spontaneous game play at a summer program I did a while back. One thing I know for certain: when there is tape on the floor where there once was none, interesting things always happen!

The body can be harnessed as informal thinking tool  on the playground and also more formally in math class, and is well suited to investigations of part-whole relationships and is at the core of our math-dance making in Math in Your Feet.

Types of Units: Breakfast Edition

A unit is a single quantity regarded as a whole.

Composed units begin with a single thing which we assemble with others of these single things to make a larger unit: the assemblage of units becomes a single whole. For example, in your refrigerator you likely have a carton of eggs. The original unit is an egg. The composed unit is 12 of these: a dozen eggs.

A loaf of bread however, is not a composed unit because we don’t make the loaf out of slices. Instead, we start with a loaf and partition it into smaller units…and then toast it up to go with our egg.

Also consider a natural unit which refers to a composed unit that has to be the size that it is, like a pair of shoes or a pair of mittens.

Here are a couple quick videos of original Math in Your Feet patterns created by the dancers themselves! The base unit is four beats, and the two teams combined their patterns to create a longer pattern composed of two four-beat patterns.

Here’s another fun 8-beat pattern which, I’m pretty sure, Max created. We were at Twitter Math Camp this Summer and we were setting up for some after-hours math-dancing in the Blue Tape Lounge. You can read more about our evening here.

Building a flexible understanding of part-whole includes understanding the myriad ways this idea presents itself in a variety of contexts. This includes the familiar operations of addition/subtraction, multiplication/division and measurement (which you can experience both on and off the page) but Sarama and Clements (2009) also include, among other things, unitizing, grouping, partitioning, and composing as operations as well, leaving the door wide open to pretty much everything we do while we are thinking mathematically.

The idea also shows up in some unexpected places, like the sidewalk…or the sky…or during breakfast…

bricks-part-whole

Here is my all-time favorite piece of math art, probably because it’s math that moves! The video starts by partitioning a humble equilateral triangle. Math off the page sometimes moves quickly, but I bet you can follow the different relationships that develop as different forms are composed or partitioned.

What every-day examples of units or part/whole relationships can you find off the page this week? Share your answers with us at the Math on the Move book group or, if you’re on Twitter check in and/or post to the #unitchat hashtag. Hope to see you there!


Malke Rosenfeld is a dance teaching artist, author, editor, math explorer, and presenter whose interests focus on the learning that happens at the intersection of math and the moving body. She delights in creating rich environments in which children and adults can explore, make, play, and talk math based on their own questions and inclinations. Join Malke and other educators on Facebook as we build a growing community of practice around whole-body math learning.