Category: language

How Moving Patterns was made better by colleagues

On By MalkeIn first steps, language, making, math ed, Math in Your Feet, moving math, scale up, UncategorizedLeave a comment

[ Go Straight to the Kickstarter https://bit.ly/MPKickstarter

MP.2 (2)Sooo…I have been bringing percussive dance and math  together into elementary classrooms for over 16 years. During that time I’ve shared my approach to whole-body math learning with my book Math on the Move: Engaging Children in whole body learning. book Published by  Heinnemann. learning in 2016. 

The last few years I’ve focused on re-imagining my flagship program Math in Your Feet™ to make this school day program more accessible to teachers, parents, and caregivers than I could ever reach one classroom at a time…and the Moving Patterns Game was born!

My advisers in this game making endeavor include:

  • Max Ray-Riek (@maxrayriek) who did some awesome brainstorming with me around the mathematical content.
  • John Golden (@mathhombre) who helped  me with making it a real game! 
  • Finally, I am indebted to Christopher Danielson (@Trianglemancsd) and the awesome volunteers at the  Math on a Stick exhibit at the Minnesota State Fair over the past few years. 

Here’s how the game works in a nutshell…

The Moving Patterns Game is an active, self-directed game featuring patterns, footwork, friends, and math. Dancing makes life fun, and math makes the dancing more interesting! This playful and creative body-based game challenges players to collaboratively decode and dance a series of footwork-based “maps” (called Pattern Cards.)

Blue Cards

The blue Pattern cards function as little footwork “maps” that show the player how and when to move their feet.

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The orange Challenge cards provide mathematical prompts for making the footwork maps more interesting and changing each pattern in some way, leading to the creation of new footwork based dance patterns. Paired together they become potent choreographic prompts where players can literally play around with both math and dance at the same time.

About the Game:

The Moving Patterns Game is based on a style of dance called “percussive dance” where you make rhythm and patterns with your feet at the same time. Percussive dance includes tap dance, step dance, clogging, and many other foot based styles.

I am acutely aware that I bring the two most anxiety producing subjects together but don’t let that get in the way of a playing a really fun game!  If you are not sure about how it works, please know there will be a number of supports provided along the way will be a variety of supports to help you learn how to play the game, especially an online instructional video and the Facebook Group Moving Patterns Game Support. |You can join the group any time. Check it out now!! Hope to see you there!

Malke 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.

Whole-Body Planar Graph Investigation

On By MalkeIn Higher Ed, language, math ed, moving math, scale upLeave a comment

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Earlier this year I was contacted by Eric Stucky who was lecturing a course for math majors at the University of Minnesota. Although I am generally in the realm of elementary mathematics, over the years I have had the pleasure of interacting with professional mathematicians and others who think expansively about both the mathematical subject matter AND the pedagogy of their instruction.

These experiences and conversations continue to show me that whole-body math learning is for everyone. This particular story is an email conversation I had with Eric that shows just how powerful whole-body math investigations can be for fourth graders AND math majors. This is also a story about coming to understand how to negotiate a whole-body math lesson. I especially appreciate Eric’s reflections later on in this post on how his lesson went.

[ERIC]

 Hello Malke,

I am a graduate student at UMN, looking for advice on an activity. I am currently a lecturer for an upper-division course for math majors; this is my first lecturing position. Class meets for two hours straight, so I’m always on the lookout for activities to break up the time.

On Wednesday, the class begins a unit on planar graphs. I had the idea to build some large physical models of graphs out of index cards [vertices] and yarn [edges], and then have them play around trying to see if they can get the edges to not cross each other.

Figuring I should know what the research is, I found your blog. Your post “A Framework for Whole-Body Math Teaching & Learning” made me consider integration of movement may not be substantial enough: they might as well lay the cards on the floor. The models should be big enough that they would have to walk/stretch to move the vertices around, but at that point, perhaps the movement becomes less of a reasoning tool and more just a nuisance? At least they would still have the experience of moving the edges physically instead of metaphorically (i.e. as they would when redrawing the graph on a worksheet), but this feels weak to me.

Unfortunately, there’s a logistical difficulty that doesn’t arise in the Rope Polygons exercise: students only have two hands, so it’s not as straightforward to make *them* the vertices of the graph because they can’t just “hold on to their edges” in a straightforward way. I really feel that if this could be done somehow, it would provide that extra something to make this activity be really special. 

If you have any thoughts, or any references to point me to, I’d love to hear them. Thank you for your hard work!

-eric

[MALKE]

Hi Eric! Thanks for getting in touch. I’m excited that you are wanting to try a whole-body math activity with your students.  My first thought after reading about your plans for the activity is that it’s not weak at all. Movement is important but in this case I think it’s more about the change of scale and the collaborative effort to meet a series of challenges you provide (I’m assuming you might already have a specific graph or graphs in mind?)  I wonder if you could find some sturdy elastic instead of rope to make the activity more dynamic? The change of scale cannot be underestimated as a learning tool.

[ERIC]

Sorry that I left you hanging… the activity ended up getting delayed for a few weeks because of some pressing issues that we discovered from the homework, but it did end up happening! Your kind words gave me the push I needed to do it, and you were spot-on: the collaborative aspect was definitely reinforced by the change of scale [emphasis added] and I think it worked out well all around.

I ended up using the yarn thinking that I could just use multiple strands per edge, but when I was doing some test runs I realized that this was a terrible idea because it was too much work to get the strands to stay together so they ended up being single-stranded yarn strongly taped to hard-plastic plates; definitely would do that differently next time.

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I decided to give them one non-planar graph (pictured left) to see what they would do with it. After that group worked on it for a while, and I told other people to go work on it with them, I asked the class what they thought. Half of them decided that it wasn’t possible, and half of them were not so convinced. I wonder how related those results were to how much interaction the students had with that graph. (Unfortunately, the idea was IMO undercut by the fact that one of the planar graphs I gave was a bit too complicated, and that group wasn’t able to untangle it. I didn’t tell them that it was planar— except one student who asked me about it after class. But I think everyone could see that the two groups were getting stuck in qualitatively different ways. I wonder how much the results were influenced by that graph).

The graph on the bottom right was the planar graph that the class wasn’t able to figure out.

malke-np (1)A further note from Eric: It was important to me that the graph to the left  would be fairly hard to identify for the more advanced students. In particular, if they already knew Euler’s formula and the q≤3v-6 inequality this would be useless, because it has the same number of vertices and edges as a cube, which is planar. Moreover, although it has K_{3,3} as a minor, it does not have K_{3,3} as a subgraph, so some passing familiarity with the classic 3-utilities puzzle would not be enough to immediately detect nonplanarity. In practice, although I had a few very advanced students in my class, including one who worked on this graph from the beginning, this wasn’t an issue— or perhaps it was just a success :P)

Malke 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.

The Cognitive and Intellectual Aspects of Dance (and math)

On By MalkeIn first steps, language, making, moving mathLeave a comment

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 it—we 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 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.

Prepare to be Inspired! New Math & Dance Resources from a Canadian School Board to Help Guide Your Way

On By MalkeIn book study, language, making, math ed, Math in Your Feet, moving math, scale up, spatial reasoningLeave a comment
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During the fall and winter of the 2017-18 school year teachers and students in the Kawartha Pine Ridge District School Board (KPRDSB), Ontario, Canada took the plunge. Using Math on the Move: Engaging Students in Whole Body Learning they bravely began the process of bringing math and dance together into the same learning context. Mary Walker Hope, who spearheaded the process, invited me to observe and celebrate the final presentations of children in grades one through eight. During my video chat observations I was incredibly inspired to see how the process laid out in Chapters 4 & 5 of Math on the Move had supported both children and teachers alike.

At the very end of their math and dance project Mary created three individual e-books recounting their work, with a special emphasis on the process. She writes:

Through integrating math and the arts, we engaged our students as inquirers, collaborators, creators, problem solvers, artists, dancers and mathematicians.

We began our journey from a creatively curious stance and with humility. We inquired, persevered, and solved. We learned how to teach math through dance and dance through math. We discovered through our collaborative inquiry that math, dance, language, music, and art are as interconnected as the processes we use to understand, solve, and create.

These three e-books are divided by grade band and FULL of documentation of their math/dance making process from start to finish including:

  • Introductory activities
  • Insights and encouragement for teachers around negotiating math and dance in the classroom at the same time
  • Details about what each step of the process looks like in each grade band
  • Lots of videos illustrating a variety of student work
  • Step-by-step examples of the making process
  • Examples of what they did to apply, extend, reflect, and assess the math/dance work
  • Finally, these e-books provide an overall positive and encouraging message for teachers who might be ready to jump in to #movingmath!

These are real kids and real teachers making gorgeous math and dance.  YOU CAN TOO!

The books are linked below. You might also be interested in another post on this blog inspired by the Canadian crew called “Why Math in your FEET?” which provides an explanation of percussive dance and the different kinds of sounds you can make with your feet while dancing.

Teaching Math AND Dance! Primary edition Gr. 1-3
Teaching Math AND Dance! Junior edition Gr. 4-6
Teaching Math AND Dance! Intermediate edition Gr. 7-8
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Photos courtesy of Mary Walker Hope

Malke 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.

When “The Movement IS the Reasoning Tool”

On By MalkeIn first steps, language, math ed, moving math, scale up, spatial reasoningLeave a comment

 

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.

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

On By MalkeIn language, making, math ed, Math in Your Feet, moving math2 Comments

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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.

A 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!”

BIG PICTURE #1

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.

BIG PICTURE #2

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.