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.

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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.

Slow-Paced Book Study of Math on the Move [June 1 – November 30, 2018]

 

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BOOK STUDY SECTION HASTAGS (Updated 6/9) 
#Foreword #movingmath
#Introduction #movingmath
#CH1pp1thru8 #movingmath
#CH1pp8thru15 #movingmath


OVERVIEW
Based on the tenet that learning takes time I am starting a slow-paced investigation and discussion of the ideas and activities in my book Math on the Move: Engaging Students in Whole Body Learning. The book study opens June 1, 2018 and wraps up November 30, 2018 on both Twitter and in our book group on Facebook. If you have a FB account you can join the group by clicking here.  When requesting to join (if you aren’t yet a member) please make sure to answer the question so I know you’re not a robot or whatever. And, just to be clear, You can progress through the book as quickly or slowly as you like.

This post will serve as an introduction and reminder of the processes by which we will be making meaning together around the topic of whole-body math learning and teaching. Our book study format is a combination of individual public reflection on the reading and conversations in community focused on the ideas and questions we have while reading.

MATERIALS NEEDED
If you do not yet have a copy of Math on the Move you can download the free sample chapter which includes the Foreword, Introduction and first chapter. This will begin your book study journey. Download this chapter and/or buy the book at the Heinemann website. If you are outside the U.S. please check a Book Depository website in your part of the world.

FORMAT
This book study includes small sections of reading followed by responses to four standard questions for each section. This format is adapted from the Reflective Review Protocol from the Artful Tools resource. Artful tools create a descriptive setting in which learners are supported in perceiving deeply, thinking critically, and making meaning, and asks:

  • How do we create a safe space for all voices to come forward?
  • How do we honor all perspectives and encourage critical thought and questions?
  • What is the value of deferring judgment in a learning setting?

KEEPING TRACK 
Each section of text will be denoted and searchable with the same hashtags used on both Facebook and Twitter. For example, we will start by reading the Foreword by Max Ray-Riek; the hashtags for this first section of reading will be #forward and #movingmath. From there we will use a #CHpp format (ex: #CH1pp1thru8) and #movingmath. As we move (ha!) through our reading You will be able to return to the discussions in any section using the specific hashtags, all of which will be updated and archived in this Google doc.

PROCESS FOR RESPONDING TO EACH SECTION OF TEXT
Book study participants respond to each section by answering the following four questions:

  1. What do you notice? Describe what you read without judgment. If judgment emerges, please provide evidence on which the judgment is based: What did you read that makes you say that? How did this section feel to you as a learner? As a teacher? Answer using descriptive terms, without making judgments about the quality of the work or offering personal preferences: “I notice that . . .”
  2. What questions does it raise? What  questions does the text trigger? Raise any questions about the work with “I wonder…”
  3. Speculate about what the text helps you understand: What do you think is the author’s intent? What do you think are the intended understandings? What is the author trying to help readers understand? Respond with what meaning you take away using the phrase: “I speculate that . . .”
  4. Respond/Open Dialogue/Reflect: Participate in an open dialogue with other book study participants about the section in question. This is time for participants to share new ideas for next steps and respond to one another directly about what they read, what they still wonder about, etc.

FINAL THOUGHTS
I am very much looking forward to learning and thinking with you! However, I can also imagine that once things get started there may be some hiccups or little things to be worked out in our process. If this happens I will communicate any changes/adjustments on Twitter and Facebook and record any changes I make to the process as edits to this post. Please don’t hesitate to get in touch with me if you have any questions or concerns along the way.

Let’s get started!


Malke Rosenfeld is a percussive 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.You can find out more about her work at malkerosenfeld.com,  on Twitter,  Instagram, or Facebook.

ISO Participants for the Moving Patterns Game Summer Pilot!

Do you have plans for summer programming? Do you want your participants to have an experience with math that is off the page, creative, and highly physical? If so, please consider joining the Moving Patterns Game by Math in Your Feet™ summer pilot! Help us further develop this interactive, moving, and mathematical game!

Overview

The first pilot of the Moving Patterns Game by Math in Your Feetwas with 3rd and 4th graders during school hours. It has also been played at family math nights and at last year’s Math on a Stick exhibit at the Minnesota State Fair. We know that it works but there are additional questions we want to explore.

The game combines the Math in Your Feet  school-based program with the best of informal, kid-initiated playground activity. The overall vision is to get kids moving and thinking mathematically at the same time, and as a creative free play option for recess (inside or out) and/or after-school enrichment.

About the Pilot

Pilot sites will be chosen based on a variety of factors including duration of summer programming, demographics of participating children, and level of commitment. If chosen you will receive all the materials necessary to run this activity and guidelines for getting started with the game including:

  • The original pattern cards AND a set of larger laminated cards that promote creative collaboration between dancers
  • A poster of Pattern Properties to help make sense of the pattern cards
  • Copies of the game when it has been professionally designed and released
  • An opportunity to contribute to the development of the game

If interested please fill out this survey no later than May 20, 2018. The survey includes more information about this pilot and a section for any unanswered questions you might have before committing. I’m looking forward to hearing from you!


Malke Rosenfeld is a percussive 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.You can find out more about her work at malkerosenfeld.com,  on Twitter,  Instagram, or Facebook.

Leaving Room for Question Asking

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NOTE: This post was originally published at my other blog  April 15, 2017.

How much of the math you do in your classroom is based on someone else’s questions? Someone far away from your classroom? Someone other than your students? Even with specter of standardized tests looming like dark, heavy clouds, how can we leave room for the most important part of learning: question asking?

“So many of the things that we do in math education — and maybe more generally in education — are giving students answers to questions that they would never think of asking. By definition, that’s what it is to be boring. If you’re sitting at a bar and someone’s telling you stuff that you’re not interested in and you would never think of asking about — what is more boring than that? That seems to be the model of our educational system: ‘Here’s the formula for the cosine of the double angle.’ ‘Well, I don’t care about that.’” —Steven Strogatz

The first time I experimented with building scaled-up geometric forms I was homeschooling my then-seven-year-old. At the time she was a “resistant” learner which basically meant she was happiest exploring her own questions, and, over a couple years of homeschooling I realized my best strategy was to influence the child by way of my own curiosities and the way I structured the environment, leaving out provocations to be discovered and, if of interest, investigated.  This was also a time when I was deep into finding answers for my own question “What is math?” and I was heavy into an investigation into Platonic Solids. I had never been a builder as a child, but I had a question and it needed to be answered.

Back in April, on Twitter, I had a conversation with Lana, a third grade teacher who is reading Math on the Move and who has been wondering about how how to “scale up” her students’ mathematical activity.  Specifically she’s been curious about my recent work with building body-scale polyhedra.  Body- or moving-scale means that the whole body/person is engaged in problem solving and mathematical thinking to investigate a mathematical challenge or project of some kind. I think Telanna’s question could be anybody’s question who is wondering how we can do math off the page.

My answer focused on how I structure a making activity and the learning environment in a way that motivates learners to collaborate and ask new questions in response to the activity in an intrinsic way.

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It’s in the process of making something with the freedom to try things out and see where it gets you that creates new questions.

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, like my daughter, 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.

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To bring kids to math we need to leave room for their own questions.

What happens next? There are more questions to be asked about this kind of approach. I have  my answers and am happy to share them. But I’d love to hear your questions first!


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.You can find out more about her work at malkerosenfeld.com,  on Twitter,  Instagram, or Facebook.

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.

 

 

Learning Math by Ear: The Role of Language in a Moving Classroom

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At first glance, this article about the value of reading aloud to older kids would not seem to connect to math learning. But, to me it does.  Here’s the piece that really stood out:

“The first reason to read aloud to older kids is to consider the fact that a child’s reading level doesn’t catch up to his listening level until about the eighth grade,” said Trelease [a Boston-based journalist, who turned his passion for reading aloud to his children into The Read-Aloud Handbook in 1979], referring to a 1984 study performed by Dr. Thomas G. Sticht showing that kids can understand books that are too hard to decode themselves if they are read aloud. “You have to hear it before you can speak it, and you have to speak it before you can read it. Reading at this level happens through the ear.”

Did you catch that? “You have to hear it before you can speak it, and you have to speak it before you can read it.”

I made a similar point while working with teachers and teaching artists in Minnesota in 2013 when participants noticed how the math language was woven naturally and seamlessly into our dance work. This vocabulary development, I said, was initially an attempt to help kids pay closer attention to the details of what they were doing while they created their dance patterns. I noticed that they became much better creators when they had the right words to help them identify their movement choices.

recent brain study focused on how the motor cortex contributes to language comprehension:
“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 [bolding emphasis mine]:

“An alternative is offered by an embodied or distributed view suggesting that the brain areas encoding the meaning of a word include both the areas specialised for representing linguistic information, such as the word’s acoustic form, but also those brain areas that are responsible for the control of the corresponding perception or action. On this account, in order to fully comprehend the meaning of the word ‘throw’, the brain needs to activate the cortical areas related to hand movement control. The representation of the word’s meaning is, therefore, ‘distributed’ across several brain areas, some of which reflect experiential or physical aspects of its meaning.”

 My take away from the study overview is this:

  1. Our whole bodies are just that: whole systems working in an fascinating and astoundingly connected ways.
  2. “Knowing” something, especially the ideas and concepts on the action side of math (transform, rotate, reflect, compose,  sequence, combine, etc) is strengthened by the partnership between mathematical language and physical experience.

In Math in Your Feet we start by moving to get a sense of the new (non-verbal) movement vocabulary in our bodies. At the same time we say together, as a group and out loud, the words that best match our movement. Sometimes we also pay attention to the words’  written forms on the board so all three modalities of the idea are clear to us.  When learners are more confident with their dancing they are asked to observe others’ work and choose the the specific  words that describe the attributes/properties of the moving patterns. There are over 40 video examples of this in action in Math on the Move.

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In addition to being able to parse our patterns, we use tons of other math terminology while we choreograph in conversations with our teammates and in whole group discussions.  This approach allows learners to fully grasp the real meaning and application of these ideas which, ultimately, allows them to write and talk confidently about their experiences making math and dance at the same time. Teachers consistently notice an increase of ‘math talk’ in their classrooms when children get up to explore math ideas with their whole bodies. As in, “I couldn’t believe how much math vocabulary they were using!”

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Math is a language but it’s not just about terminology, it’s about what those words MEAN.  To do this, learners need to play with mathematical ideas, notice and talk about patterns and structure, sort and compare, and share reasoning about and understanding of mathematical relationships.

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As such, language, in partnership with the body is our tool for thinking mathematically when we are up out of our seats and moving during math time.  Ideally, this language is facilitated by an adult through conversation, play and exploration, all before bringing it to the page to explore the ideas in a different modes and contexts.


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.

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.

Beautiful Objects, Redux

[Adapted from my post Beautiful Objects, from January 24, 2014]

I’ve thought a lot about the role of physical objects in math education.  Sometimes called manipulatives or, more generally, thinking tools, I’ve discovered conflicting opinions and strategies around the use of such objects. In her book Young Children Reinvent Arithmetic, Constance Kamii helpfully sums up some of the issues with which I’ve wrestled with [bolding emphasis mine]:

“Manipulatives are thus not useful or useless in themselves. Their utility depends on the relationships children can make…” p25

“Base-10 blocks and Unifix cubes are used on the assumption that they represent or embody the ‘ones,’ ‘tens,’ ‘hundreds,’ and so on. According to Piaget, however, objects, pictures and words do not represent. Representing is an action, and people can represent objects and ideas,but objects, pictures, and words cannot.” p31

So, it is not the object itself that holds the math, but rather the process in which the learner uses the tool that creates the meaning.  But, of course, when we use this kind of language we are talking abstractly about hypothetical objects and generalized characteristics of ‘the child,’ not any specific object or individual learner in particular.

Too much generality and abstraction drives me crazy so imagine how pleasantly surprised I was when this showed up in my mailbox one day:

triangle

What is it? Well…it’s an object. And a beautiful one, at that. An object that can be “manipulated” (the triangle comes out and can be turned). A thinking tool. It was designed and created by Christopher Danielson to investigate symmetry and group theory with his college students. Not only are parts of this tool moveable, but it also has the potential to help “facilitate [mathematical] conversations that might otherwise be impossible.” (Christopher on Twitter, Jan 17, 2014)

What was even better than getting a surprise package in my real life mailbox containing a real life manipulative (not a theoretical one) was my (real) then-eight year old’s interest in and reactions to said object. She spotted the envelope and said, “Hey! What’s that?!” I told her that a math teacher friend of mine had sent me something he made for his students to use. I took it out of the envelope for her to look at.

First thing she noticed was the smell — lovely, smokey wood smell which we both loved.  She investigated the burned edges, tried to draw with them (sort of like charcoal). This led to a discussion about laser cutters (heat, precision) and the fact Christopher had designed it. I pointed out the labeled vertices on the triangle, showed her how you can turn it, and mentioned that the labels help us keep track of how far the shape has turned. She immediately took over this process.

She repeatedly asked if she could take it to school! I asked her, “What would you do with it?”  She said, matter-of-factly: “Play around with the triangle…and discover new galaxies.” Then, she turned the triangle 60° and said, “And make a Jewish star…” Then she put the triangle behind the the opening so it (sort of) made a hexagon.  I asked, “What did you make there?” She said, “A diaper.” Ha!

I hope Christopher’s students were just as curious about and enthralled with the “object-ness” of this gorgeous thing as they were with the idea that it helped them talk and think about things that might otherwise be impossible to grasp.  I know that the objects themselves hold no mathematical meaning but watching how intrigued my daughter was with Christopher’s gift, I am left thinking about what we miss out on if we consider a tool simply a bridge to the ‘real’ goal of mental abstraction.  

Beautiful and intriguing objects, I think, have a role in inspiring the whole of us, all our senses, kinetics, and curiosities, not just our minds, to engage in the process of math learning.  An object doesn’t necessarily have to be tangible; narrative contexts are highly motivating ‘tools’ when working with children. As I blend math, dance and basic art making I see over and over again how presenting the object (idea) first pulls my learners in — they are curious about what this dance is, how they might weave their own wonderful designs using math, what does she mean “growing triangles” and why are these pennies on the table?

Learning is hard work, but my experience is that students will gladly work hard if they have even a small sense of the direction in which they’re headed. The whole, moving body is one of those beautiful objects which can create other beautiful objects (in this case a dance pattern) using the elements of time, space, and kinetic energy. This first video is from a session I did with undergraduate math majors at the University of Michigan:

And these two videos are of me and Max Ray-Riek last summer playing around a little while setting up the after-hours Blue Tape Lounge at Twitter Math Camp. The first video shows some interesting inverse and symmetry action, and the second one…can you tell what kind of symmetry is happening 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.