Notes from a #movingmath Summer Classroom

I am so excited to share the work of Lisa Ormsbee at Fairhill School in Dallas, Texas  who has spent the past two weeks running a math and movement summer enrichment camp using resources from Math on the Move, the Move with Math in May lesson plans, a rhythm-based exercise program called Drumfit, and a lot of other great ideas she pulled together to meet the needs of her students through rhythm and movement. She has ten students with most of the students “learning different” (e.g. dyslexia, dysgraphia, ADHD, mild autism, and selective mutism)  not all of them fans of math, what she described as a “general math reluctance.”

Fairhill MiYF

I was thrilled to get her wonderful email updates on the first and second week of programming which showed just how much of an impact a #movingmath approach can have for all learners. I especially love the progression Lisa created to gently lead reluctant movers (and math-ers) into what has become enthusiastic engagement! Here’s some of what Lisa shared with me:

Monday:  I did a couple of ice breaker activities which involved moving around and were non-threatening (meaning no one HAD to talk in front of the  group).  I started by challenging them to put THEMSELVES into patterns during this warm up time – it was totally spontaneous but it was fun for them. We also got oriented to our class space.  I had removed all the desks and chairs and had the [Math in Your Feet] squares taped on the floor.  They had to adjust to the idea that we weren’t going to sit in desks. I also introduced Drumfit on this day and used that activity time to introduce “follow me” patterns with the drumming rhythms. These kids are fairly reluctant to move around and have pretty low physical literacy and body confidence so I wanted to be sure to take the introduction of the program slowly.  They did extremely well with the movement during the icebreakers!  The drumming is growing on them but took several days for them to feel confident and, some still do not, but I’m not pushing them in that area as it’s a “fun” time. It’s such a good fit with patterns and using your body to make them though! 

Tuesday: We did the pattern game sitting in a circle that you outline in one of your lessons [Clap Hands: A Body-Rhythm Pattern Game].  This was HARD for some of them!  They were all engaged in it though.  We could certainly do this again!  Then we went to our gym space and used the ladders to prove the center [Proving Center lesson] in teams and also to create patterns as a team using bodies and any other items they wanted to use. They were told to be as creative as they wanted with their repeatable pattern. We discussed symmetry here too.  I used my purple circle discs to have them create a game using their ladders also. The game had to have some “math” in it. It was so very, very interesting to watch them do all of this!! We discussed a lot after that and talked about what they had done and how they had thought of their games and patterns.

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After one more day of getting kids used to moving and thinking about math at the same Just turns postertime Lisa introduced the first step in the  Math in Your Feet “pattern/partner/dance process.”  Lisa wrote:

It was slow and I didn’t hurry them.  It took a while to orient them to the squares, talk about sameness (congruence), and review the movement variables. We also took a LONG time talking about the turns. That’s all we got done but I told them we’d be making a pattern with our partner the next day and we’d be concerned with precision and sameness.

On Friday they started working with their partners on creating their 4-beat patterns.

The kids were ALL so engaged in this activity!  I couldn’t believe it.  They had some trouble with cooperation and with identifying sameness. 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 right!  AWESOME!!  I felt like it was a successful day and I can’t wait to do more. 

Next week, I want to have them write a little bit about their patterns and make a drawing etc. like you do in the book.  I also want to let them do this part again then work on combining and transforming.  When we get to the mirroring piece we will have to go pretty slowly I’m guessing. 

During the second week of summer school Lisa did the mirroring/reflection lessons and was also able to extend and connect the physical work by having them having them map their patterns and then read/decode each other’s pattern maps.

Once I added music to the activity they had a blast!  I feel we were all inspired by the Math in Your Feet program to be open to new ways to learn through movement. I was so caught up in our activities I didn’t get any pictures!

But she did eventually get some videos! Here are a couple showing the children’s awesome physical thinking around reflection. One person is keeping their rights and lefts the same as they originally designed the pattern, and the other person is dancing the pattern with opposite lefts and rights. And this is all on top of some tricky rotations. A mighty feat!

Lisa says: I hope [this account] helps others dive into the program because my kids really engaged with it and I am 100% sure that they would not have been so engaged had I chosen a more traditional program for the summer enrichment. I really hope this will help them with their understanding of math and also with their movement confidence and honestly, their joy of moving! I’ll be the P.E. teacher here next year – although I must say this might actually make me a fan of math too. Yay!

Thank you, Lisa, for sharing your work with us!


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.

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.”

whiteboards5

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

WELCOME!

This is a special nearing-the-end-of-the-school-year event featuring four math-and-movement lesson plans to chose from. The goal? An opportunity to try out whole-body math in a low-key way to get a sense of what it’s all about. Here’s how to play:

A LITTLE OR A LOT:
The lessons are presented in full but you are free to do as little or as much of the plan as you like. The most important thing is that the moving is the math as much as anything else so just getting your learners moving in a mathematical setting is a GREAT start.

SHARE YOUR EXPERIENCE BY MAY 31, 2017 to enter the random drawing to win a copy of Math on the Move: Engaging Students in Whole Body Learning. Share your experience via blog post or video on Twitter or Facebook with #movingmath and, just to make sure, share the link with me

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!!

Header Proving Center 1 FINAL

In 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


Header Rope Polygons 2 FINAL

In 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


Header Clap Hands 3 FINAL

Clapping 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


 

Header MiYFeet 4 FINAL

Have 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!”

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.

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.

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.

Math off the Page: Relational Understanding & Scale

`me-and-the-boysWhen I showed up at my first Twitter Math Camp in 2014 I was a ball of nerves. It was my first chance to meet, in real life, the math teachers of the #MTBoS from whom I had learned so much. I’m probably not the only person who feels nervous at their first in-real-life meeting of online friends or colleagues. What I noticed was that, while I could anticipate certain things about a person from our online interactions, having a chance to interact with them in real time and space enriched and deepened our interactions. By the end of our three days learning together, making math after hours, and chatting over a variety of meal times, I found myself with a much more nuanced understanding of my friends and colleagues.

I think it’s the same kind of situation for math learning. 

Richard Skemp defined the difference between Instrumental and Relational Understanding in math. Here’s a visual overview (via David Wees) of the difference between the two kinds of understanding:

instrumental

Math teacher Mark Chubb explains in his blog post Focusing on Relational Understanding:

Students who are taught instrumentally come to see mathematics as isolated pieces of knowledge. They are expected to remember procedures for each and every concept/skill.  Each new skill requires a new set of procedures.  However, those who are taught relationally make connections between and within concepts and skills.  Those with a relational understanding can learn new concepts easier, retain previous concepts, and are able to deviate from formulas/rules given different problems easier because of the connections they have made.

My perspective on relational understanding focuses getting to know a math idea in multiple contexts. Zoltan Dienes (creator of the base 10 blocks you use in your classroom) thought so too (bolding mine):

According to Dienes … mathematical abstractions occur when students recognize structural similarities shared by several related models. For example, when base-ten blocks are used to teach arithmetic regrouping operations, Dienes claimed that it is not enough for students to work with a single model; they must also investigate “mappings” to other models, such as bundling sticks or an abacus … a primary goal is to help students recognize how patterns of relationships in one model correspond to patterns of relationships in another model.

Because math is frequently presented in a static way, whether in textbooks or on worksheets, the dynamic action represented by those symbols and figures are often lost in the shuffle. The experience of math in this single mode and a series of fixed images, ideas, and answers might leave us to wonder:

How can we  learn math out of our seats? How can we learn math if its not written down? 

As part of my new First Steps series for bringing #movingmath into the classroom in a low-stress way we’re gonna’ have a TON of fun exploring math off the page in the next few months!

To kick things off let’s start by finding the math idea of scale as it exists off the page. Scale is a ratio that compares the size of one thing to another. It is what we are thinking about when we ask “how much bigger/smaller, taller/shorter, or faster/slower.” For example: In this picture of the Louisville Slugger Museum and Factory, how much bigger is the bat to the building? How much smaller is my kid compared to the giant bat?

bat-and-building  kid-and-big-bat

Another example of scale off the page (which also does double duty as a great example of whole-body #movingmath) are the videos from OK GO, below.  To create the video for the song I Won’t Let You Down the music was slowed down 50% to record the complex movements at half the speed. It clocks in at about 10 minutes. The song and moving images were sped up for the final video which clocks in a little more than 5 minutes .

Overall, it’s not about whether one mode of math thinking and doing is better than another. It’s about providing opportunities for our students to really get to know a math idea in all its forms. We do this when we provide opportunities for learners to reflect on the process by which they arrived at an answer, by recognizing that watching an OK GO video during unit about scale might provide students with new insights, or by creating a lesson where students use their own bodies as measuring tools.

Whole-body math learning is one part of a whole variety of experiences that, taken together, help build a personal relationship to math so that we can recognize and rely on our new friend … on the page … and off.


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. Her new book Math on the Move: Engaging Students in Whole Body Learning was recently published by Heinemann (2016). Join Malke and other educators on Facebook as we build a growing community of practice around whole-body math learning.

New Year, New Tools for Making Sense of Math!

happy-hoppy-new-year

Happy 2017!! This year harness the original “thinking tool” to help your learners make sense of math! What is this tool, you ask? Why, your students’ own bodies and creative spirits of course!

Math on the Move: Engaging Students in Whole Body Learning is now available from Heinemann. Included in the book are specific, actionable ideas for including your students’ moving bodies in the math you are already doing in your classroom!

Here is your first tip in the New Year for a simple first step in bringing Math in Your Feet and other #movingmath activities into your classroom in a low key way.  All the best to you for a new year filled with enthusiastic math making!

new-year-new-tools-tip-1

You can download the Movement Variables from the Classroom Materials page.

Interested in having First Steps show up in your inbox in a semi-regular and non-irritating fashion? Join my mailing list! 

Will you tell us about your #movingmath adventures with us? I’d love to hear your stories. Share with us on Twitter or at the book group on Facebook.


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. Her new book Math on the Move: Engaging Students in Whole Body Learning was recently published by Heinemann (2016). Join Malke and other educators on Facebook as we build a growing community of practice around whole-body math learning.