Math on the Move: Part 1

I have a tendency to devour professional books. However, in my rush to read about all these new ideas, I rarely ever slow down and take the time to stop and reflect on what I’m reading. Don’t get me wrong, I do *a lot* of thinking about what I’m reading, but I’m not doing anything to make my thoughts permanent so I can easily engage with them later.

I’ve been meaning to change that, to clarify and capture my thoughts in my blog, and what better time to do that than with my colleague Malke Rosenfeld’s long-awaited book Math on the Move: Engaging Students in Whole Body LearningToday I’d like to write about my thoughts as I read the introduction and chapter 1. I’ll follow up with posts about the other chapters as I make my way through the book.

I’d like to start with my own introduction to how I first came to meet Malke and get to know her incredible work.

Back in the summer of 2014, I had the opportunity to attend my first Twitter Math Camp. Looking at the schedule of morning sessions, my curiosity was piqued by a session called “Embodied Mathematics: Tools, Manipulatives, and Meaningful Movement in Math Class” offered by Christopher Danielson and Malke Rosenfeld. Here’s the session description:

This workshop is for anyone who uses, or is considering using, physical objects in math instruction at any grade level. This three-part session asks participants to actively engage with the following questions:

  1. What role(s) do manipulatives play in learning mathematics?
  2. What role does the body play in learning mathematics?
  3. What does it mean to use manipulatives in a meaningful way? and
  4. “How can we tell whether we are doing so?”

In the first session, we will pose these questions and brainstorm some initial answers as a way to frame the work ahead. Participants will then experience a ‘disruption of scale’ moving away from the more familiar activity of small hand-based tasks and toward the use of the whole body in math learning. At the base of this inquiry are the core lessons of the Math in Your Feet program.

In the second and third sessions, participants will engage with more familiar tasks using traditional math manipulatives. Each task will be chosen to highlight useful similarities and contrasts with the Math in Your Feet work, and to raise important questions about the assumptions we hold when we do “hands on” work in math classes.

The products of these sessions will be a more mindful approach to selecting manipulatives, a new appreciation for the body’s role in math learning, clearer shared language regarding “hands-on” inquiry for use in our professional relationships and activities, and public displays to engage other TMC attendees in the conversation.

Sounds awesome, right? It was! I can’t tell you how many times I’ve brought up this experience in conversation with colleagues over the past couple years. It gave me a new perspective about how we construct knowledge with physical things, including manipulatives and the body. And how exciting is it that two years later I get to revisit and expand on these ideas as I read Malke’s new book.

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In pairs we created 8-beat dance patterns using movement variables.

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We analyzed each other’s dances and talked about the mathematics in the dance as well as the dance itself.

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Our work bled over into the evenings as we danced and talked math in the “Blue Tape Lounge.”

Now that my introduction is over, we can move on to Malke’s.

Malke is a percussive dancer and teaching artist. During her career she has explored the relationship between dancing and mathematics through a program she developed called Math In Your Feet. Check out this TEDx video to see her do a little dancing, but mostly to hear her talk about her vision and her work.

One thing Malke does early in her book is make it clear what she is and is not saying about teaching math and dance and what she is and is not saying about the role of the body in learning. I appreciate that she takes the time to do this because as humans we have a tendency to try to fit what we’re hearing into our pre-existing worldview. By sharing examples, and more importantly, nonexamples, Malke helps create some necessary disequilibrium before readers dive more deeply into the rest of the book. Here are a couple of examples:

The first is that this is not arts integration. According to Malke, arts integration is difficult to pull off well and often the core subjects, such as math and science, are truly the focus while art is brought in as a way to “liven” things up. Rather, Malke prefers to frame her work and the ideas in this book as interdisciplinary learning.

“Both math and dance are discrete disciplines that require students to gain content knowledge, develop skills, and cultivate thinking and reasoning fluency in order to create meaning within their respective systems.” (page xvii)

The goal is not to teach math with dance or to teach dance with math. Rather, students are able to engage with and learn concepts from both disciplines simultaneously. Reading about this reminded me about Annie Fetter’s Ignite talk where she talks about the intersection of art and mathematics in her mother’s weaving and quilting. It makes me wonder in what other disciplines mathematics intertwines where someone may not even be conscious of it.

A related and important point Malke makes is that not all math can be danced and not all dance is math. But where they overlap is a beautiful place to spend some time learning about both.

The second example is probably the most important before getting into the meat of her book. If someone is going to invest the time to dive deeper and explore her message, then she needs for the reader to understand what she does and does not mean about the role of the body in learning. She does not mean using our arms to represent types of graphs, bouncing on exercise balls as we recite multiplication facts, or having students create the sides of polygons with their bodies.

“Too often the moving body is used primarily as an object for literal interpretation, illustration, and memorization of math concepts. Conceptualizing the body in this way, as a drawing or mnemonic tool, severely limits its potential in a learning setting.” (page xvii)

In contrast, Malke wants us to consider how the body can be used as a thinking tool that puts the student at the center of the reasoning and doing within a particular context. From birth, we have used our bodies to explore and make sense of our world long before we had language skills or the ability to understand someone telling us what to do. Malke wants us consider how we can provide students opportunities to use their bodies in these same ways to explore math concepts in school. I’m not going to steal her thunder, but in chapter 1 she shares three lovely vignettes of this in action in kindergarten, second grade, and fifth grade. Be sure to read and think about those,  and then contrast them with the nonexamples she provides.

Then get ready to dance! Malke doesn’t let you off the hook as a reader. Chapter 1 has two Try It Yourself! boxes that encourage you to get some masking tape and make a square on the floor – I recommend blue painters tape. Then she poses questions and challenges that give you the opportunity to try using your body as a thinking tool. You might feel a bit silly, but you just might make some new insights as well. Give it a try!

With the groundwork laid, I look forward to diving in to chapter 2.

 

We Can Make Shapes!

In my previous post I shared one of two mathematical conversations I had with my daughter this morning. Here’s the second.

“Look I made a triangle.”

I look over and she’s sitting cross-legged on the floor. It takes me a moment, but I realize she’s talking about the square tile she’s sitting on and the triangle she can see in the corner. Here’s a re-creation of it since I didn’t take any photos.

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The third side of the triangle looked a lot cleaner with her crossed legs. This graphic of a child doesn’t quite work, but you get the idea.

“Oh! I see. How do you know it’s a triangle?”

As usual when I ask that question about a geometric shape – How do you know it’s a ___? –  she didn’t really say anything back. I turned around to put something in my lunchbox.

“Look! The triangle is smaller!”

I turned back around to look and she had scooted up on the tile. “So it is!”

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With pure delight she exclaimed, “We can make shapes!”

She started scooting back on the tile and stopped when she got here.

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“Is that a triangle, too?” I asked.

She looked down and thought for a moment. She slowly started scooting up until she got to the diagonal. Then she stopped and looked up at me.

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She doesn’t yet know how to articulate what a triangle is, but she is clearly grappling with and making judgments about the “triangleness” of her shapes. It’s fascinating.

Even better, her exclamation, “We can make shapes!” makes me so happy. It’s such a simple statement, but it felt so empowered. She came to the realization all on her own as she moved her body back and forth on our tile floor.

Counting Down to the Weekend

“Do I go to music class and swim class today?”

“No, today is Monday. Remember, I said you go to work for 5 days before you go to music class and swim class.” I hold up my fingers one by one as I call out, “Monday, Tuesday, Wednesday, Thursday, Friday.”

I put down all five fingers and continue, “So far we went to work on Monday and we’ll go today on Tuesday.” I put those two fingers back up as I talk.

Without skipping a beat she says, “Three more days! Today it will be 3, and then 2, and then 1.”

This was completely unexpected and so fascinating to hear! If only I hadn’t been in the middle of rushing to get dressed and ready to walk out the door to work. Looking back, I would have loved to ask, “How did you know there are three days left?”

In thinking about this conversation throughout the day, I’ve thought about all the play we’ve done with counting over the past several months. Fingers are a favorite of mine since they’re always close at hand.

In the car, one of the games we’ll play is that I hold up some number of fingers at my chest and ask, “Guess how many fingers I’m holding up.” She makes a guess and then I hold them up so she can see if she got it right. Nothing fancy, but it gives her a lot of opportunities to count and see quantities from 1 to 5.

Another game I like to play is, “Do you want me to show you 5 really fast?” She says, “Yes.” I put my hand behind my back and say, “Ready, set, go!” And then I whip out my hand with all my fingers out. She counts my fingers every time to prove there are 5 fingers, but I’m beginning to wonder if the counting is really necessary.

So I’m curious about how she knew it was 3 days until Saturday. The way I held my hand, she couldn’t see the three fingers that were down. Did she see them in her mind? Did she subitize? Did she count one by one super fast? There was hardly a heartbeat between what I said and her response. The counting back from 3 was really fast also.

Things to explore as we talk more.

I love being a parent and getting to have these kinds of conversations with my daughter. When she surprises me with a new understanding or insight, it’s like a wonderful gift. I treasure each and every one.

(Side note: Her Montessori school calls their learning time “work periods” so we’ve been calling it “going to work” since she started there a year ago. She likes the idea that she goes to work everyday like Daddy and Papa do. If I accidentally say something about going to school she’ll usually correct me, “No, I go to work!”)

[UPDATE 10/5/2016] This morning she asked a question she asks pretty much everyday without fail, “Is today a work day?”

“What did I say when you asked me last night?”

“It is a work day.”

I go back to eating my breakfast.

“We went on this day and this day, and this is today.” I look over and she’s holding up three fingers in front of her face. She’s grabbing the tip of her middle finger as she’s saying that this is today. She tells herself, “There’s two days left!”

Clearly our conversation yesterday wasn’t a fluke! She wasn’t even talking to me at the end. She was talking it out and making the observation all to herself. How cool!

A little later she’s in the kitchen and I ask her, “Can you show me how many workdays we’ve had on the Math Rack?” (By the way, we’ve had fun counting on the Math Rack, but I’ve never asked her to do anything like this before.)

She pulls over three beads, “One, two, three.” Then she holds up her thumb, touches it to the first bead and says, “One.” She holds up her pointer finger, touches it to the second bead and says, “Two.” Finally she holds up her middle finger, touches it to the third bead and says, “Three.”

“Can you show me how many days we have left down here?” I point to the bottom Math Rack.

She pulls over two beads, “One, two.” Then she puts her thumb, pointer, and middle fingers back up and moves her hand over to the two beads she just pulled over so that the two fingers that are still down are touching them.

I feel like she’s turned a corner developmentally and a whole new landscape has opened up. I’m so excited to explore it with her!

Revision

Free time. I wish I had more of it. Instead I have the amount I have and a wide variety of ways I’d like to fill it – going to the gym, paying bills, cleaning the house, spending time with my husband and daughter, blogging, reading comics. The list goes on. Lately I’ve been prioritizing time with my husband and daughter.

 

Except when that wasn’t an option. Back in August I went to Virginia for a few days to serve on a planning committee for the 2017 NCTM Innov8 conference. Our days were full of committee work, but my evenings were filled with hours of time to myself. It was a nice change of pace and the perfect opportunity to tackle a project I’ve been putting off time and again – revising our parent resource page on our district website.

 

The highlight of the revision work was creating curated collections of resources around the following topics:
* What Does It Mean to Teach and Learn Mathematics Today?
* Creating Positive Identities Toward Mathematics
* Talking Math With Your Kids
* Exploring Elementary Mathematics Topics
* Mathematics Games and Products
* Digital Mathematics Games and ProductsThe #MTBoS is a treasure trove of these kinds of resources, so I had a lot to pick from! I’m so happy to have the opportunity to share them with a wider audience. I’ve already had one of our instructional coaches share the link at her campus Back to School Night. She had over 75 parents ask for the link. Yay!

If you’d like to check out the resources, here’s a link to the page. And if you have ideas for other resources I should add to any of the resource collections, let me know in the comments.

Decisions, Decisions

This week our Math Rocks cohort met for the fourth time. We had two full days together in July, and we had our first after school session two weeks ago. One of our aims this year is to create a community of practice around an instructional routine, specifically the number talks routine. We spent a full day building a shared understanding of number talks back in July. You can read about that here. We also debriefed a bit about them during our session two weeks ago.

This week we put the spotlight on number talks again. We actually broke the group up by grade levels to focus our conversations. Regina led our K-2 teachers while I led our 3-5 teachers. The purpose of today’s session was to think about the decisions we have to make as teachers as we record students’ strategies. How do you accurately capture what a student is saying while at the same time creating a representation that everyone else in the class can analyze and potentially learn from?

We started the session with a little noticing and wondering about various representations of 65 – 32:

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Very quickly someone brought up exactly what I was hoping for which is that some of the representations show similar strategies but in different ways. For example, the number line in the top left corner shows a strategy of counting back and so do the equations closer to the bottom right corner.

This discussion also led into another discussion about the constant difference strategy – what it is and how it works. It wasn’t exactly in my plans to go into detail about it this afternoon, but since my secondary goal for the day was to focus specifically on recording subtraction strategies, it seemed a worthwhile time investment.

After our discussions I shared the following two slides that I recreated from an amazing session I attended by Pam Harris back in May. (For the record, every session I attend with her is amazing.)

The first slide differentiates strategies from models. Basically, if you have students telling you their strategy is, “I did a number line,” and you’re cool with that, then you should read this slide closely:

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The second slide differentiates tools for building relationships from tools for computation. This slide is crucial because it shows that while we want students to use tools like a hundred chart to learn about navigating numbers within 100, the goal is to eventually draw out worthwhile strategies, such as jumping forward and/or backward by 10s and then 1s.

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The strategy on the right that shows 32 + 30 followed by 62 + 3 is totally the type of strategy students should eventually do symbolically after building relationships with a tool like the hundred chart.

After blowing their minds with those two slides, I led them in a number talk of 52 – 37. During my recording of their strategies, I stopped a lot to talk about why I chose to do what I did, to solicit their feedback, and even to make some changes on the fly based on our discussion.

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For example, in the top right corner of the board I initially used equations to represent a compensation strategy. Someone asked if this could be modeled on a number line because she thought it might make more sense, so I did just that in the top left corner. By the time we were done they were like, “Oh, hey! That ends up looking like a strip diagram!”

It was amusing that the first strategies they shared involved constant difference. They were so excited about learning how the strategy worked that they wanted to give it a try. I didn’t want to quash their excitement by telling them that the strategy tends to work better, especially for students, when you adjust the second number to a multiple of ten. I wanted to stay focused on my goals for the day. We’ll discuss the strategy more in a future session.

(Unless you’re in Math Rocks and you’re reading this! In which case, see if you can figure out why that’s the case and share it at our next meeting.)

After some great discussion about recording a variety of strategies, we watched Kristin Gray in action leading a number talk of 61 – 27.

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We talked about how she recorded the students’ strategies. We also talked about some really lovely teacher moves that I made sure to draw attention to.

We wrapped up our time together talking about what new ideas they learned that they wanted to try out with their students. I had asked one of the teachers to lead us in another number talk, but we ran out of time so I think I’m going to have her do that at the start of our next session together. Hopefully everyone will have had some intentional experiences with recording strategies between now and then to draw on during that number talk.

Oh, another thing we talked about at various points during the session was how to lead students in the direction of certain strategies. This gets into problem strings, which may or may not happen in number talks depending on whom you talk to. Regardless, here are some we came up with. Can you figure out what strategies they might be leading students to notice and think about?

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Play With Me

On Wednesday I had the chance to visit my first classroom this school year. Sadly, in my role as curriculum coordinator, I don’t get to do this nearly enough. So I relish opportunities like this. Even better than visiting, the teacher allowed me to play a math game with her class.

I had so much fun!

I wanted something simple and quick to get the kids engaged before moving on to another activity. I also wanted it to involve adding 3-digit numbers because her class is in the middle of a unit on that very topic. I also wanted to bring in some place value understanding and reasoning, which are very much related to adding multi-digit numbers.

Basically I brought two decks of cards – one had Care Bears on the back and the other had Spider-Man on the back. I wanted different backs to the cards so it would be easier to tell which cards were mine and which were my opponent’s in case we needed to reference them during or after the game. I also pulled out all of the 10s and face cards, with the exception of the aces. I kept those and we decided to use them as zeroes. I tell you this because if you ever want to play a game that involves digit cards, here is a great way to get some without having to painstakingly cut out cards to make your own sets. Decks of cards are cheap enough. Just use those.

The game was me vs. the class. The goal is to make two 3-digit numbers. Whoever has the greater sum wins. On my turn, I drew a card, and I had a choice of putting it blank spots that I used to create two 3-digit numbers. Once a digit was placed it couldn’t be moved. On the class’ turn, I drew the card for them, but I let them tell me where to place the digit.

My favorite part of the game was at the end when the kids started shouting out that they’d won without even finding the sum. Take a look and see why they got excited: (Just pretend I hadn’t written the sums yet. I took the picture after the game was over.)

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“You have a 9 and a 4 in the hundreds place. We have a 5 and a 9.”

“Interesting, and how does that tell you you’ve won?”

“Because the 9s are the same. And we have a 5 which is greater than 4. You should have put your 5 in the hundreds place.”

“I was hedging my bets and I lost.”

Such wonderful thinking from a 3rd grader! How often do students rush to calculate and find an answer to a problem? How amazing that these students were paying attention to the place value that matters most in these numbers – the hundreds – and then comparing the digits to determine who had a greater sum?

Since I was just the lead-in to the day’s activities we only got to play once, but I would have loved to play again. I would have liked to change it up a bit. I would still construct my number on the board, but then I would have allowed everyone to create their own number at their desk using the cards that I drew on their turn. At the end we would discuss who thinks they have the greatest sum and talk about their placement of digits.

Even though I didn’t get to play again, I’ll take the time I did have. It was the highlight of my week!

And Now For Something A Little Different

Providing PD to teachers is tricky business. Our district offers two weeks of jam packed professional development every summer. The catch is that it happens while teachers are off contract, so there’s no requirement to be there. In addition, teachers have so many options of courses to attend – literacy, math, science, social studies, technology, TAG – that it can be hard to fill seats in some sessions.

During the school year, we periodically offer PD during the school day. We usually only do this when we have special funding that allows us to cover the costs of subs for teachers who attend the PD. Otherwise, you might not get many teachers to attend. However, when we do have sub funds, we usually can only afford to pay for one sub per campus so we’re only able to bring in 34 of our 1,200 or so elementary teachers. It’s a drop in the bucket.

For the past two school years, we’ve offered after school PD sessions called Just In Time. As the name implies, they were offered just in time for the start of the next nine weeks grading period. The purpose of these sessions was to give teachers a preview of the upcoming units. Now that our units are a few years old, attendance has dwindled because they’re no longer very timely.

So this year we decided to try something new.

We threw out the Just In Time sessions and created new mini-courses to bring some of the amazing topics from summer PD into the school year and to give teachers more choice in their professional development offerings. Instead of choosing from the Kindergarten, grade 1, grade 2, grade 3, grade 4, or grade 5 Just In Time sessions for math, teachers now have 7 course topics available to them. Here’s a link to a document that details each of our courses.

The sessions are still after school for an hour and a half, which is a turn off for some, but I’m hopeful that many more teachers will be drawn to a topic they want to explore this school year. I specifically designed our courses to be experiences over time because I believe that one-off PD experiences have little lasting impact on teaching practice. However, attending 4 sessions spread out over several months where teachers have the opportunity to try out what they’re learning in between sessions feels like a better recipe for success.

We had the very first session of our very first course yesterday. (Huge thanks to our amazing instructional coaches who will be leading all of these PD sessions!) The 16 teachers who attended were engaged and eager to learn about number talks. Here’s hoping this is a sign of even more great learning to come this school year!