Yesterday I was at the library for my now monthly #makingmath sessions for kids and their parents. The ages for this event seem to trend 8 and under, probably because parents with young children are often looking for something to do on Sunday afternoons. Our youngest participant yesterday was two and a little more, and this little story is about her.
I’ve written previously on this blog about what it looks like when children think and learn mathematically with their bodies. Yesterday my new friend was there with her mom and her brother. Her brother made this delightful “Dr. Seuss house with smoke coming out of the chimney” while she made a crown and earrings for herself out of the same materials.
Another activity we had going on was playing around with these cool hexagon building blocks that I found in a big box dollar bin a couple summers ago. A boy made an object that was just begging to be spun…
…after which my little two year old friend started rotating around in one spot exclaiming to me: “I’m spinning!”
This is just one more example of how children think and learn with their bodies. She was entranced by the toy and it’s gorgeousness. She spent a quick moment spinning exactly like the top and then went back to making earrings for her mother.
The body is where learning originates.Children use their bodies to show us every day what they know and think and wonder. This non-verbal, physical manifestation of cognition is present every day in some way. I invite you to put on your #movingmath glasses and, when you notice something tell us about it! Here’s a few places where you can share:
In the comments to this post
On Twitter with the hashtag #movingmath
-or-
On Facebook with privacy set to public with the hashtag #movingmath
I can’t wait to hear about (or see) what you notice!
Malke Rosenfeld is a dance teaching artist, author, editor, math explorer, and presenter whose interests focus on the learning that happens at the intersection of math and the moving body. She delights in creating rich environments in which children and adults can explore, make, play, and talk math based on their own questions and inclinations. Join Malke and other educators on Facebook as we build a growing community of practice around whole-body math learning.
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!
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…
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.
`When 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:
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?
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.
Lots has been written about how and why I brought math and dance together for young learners but it all started here: with the joy, the expression, the sounds, the rhythmic structure of traditional percussive dance.
This is a style of Canadian step dance. Small pieces of movement and rhythm combine to create a gorgeous improvisation. Children can use the elements of percussive dance to create their own gorgeous foot-based, math-based creations.
Malke Rosenfeld delights in creating rich environments in which children and their adults can explore, make, play, and talk math based on their own questions and inclinations. Her upcoming book, Math on the Move: Engaging Students in Whole Body Learning, will be published by Heinemann in Fall 2016.
When I showed up at my first 
Math on the Move 