# Represent! Part 1

This week at #ElemMathChat I had the pleasure to lead the chat. I used the opportunity to talk about using and connecting mathematical representations, a topic that has been on my mind a lot this school year.

I kicked off the chat with this quote:

“Because of the abstract nature of mathematics, people have access to mathematical ideas only through the representations of those ideas.” –National Research Council, 2001, p. 94

and this question:

What does it mean that people only have access to mathematical ideas through representations?

I wanted this to be our guiding question throughout the rest of the chat.

I immediately followed up with this question:

As expected, the folks in the chat remarked that the symbolic form of this number does not convey anything about the number seven. Even if someone told you this is the number seven, what that means to you will vary depending on what you already understand about that number. Just being able to see this symbol and say the word, “Seven,” does not necessarily mean a person understands anything about the number seven or the quantity it represents.

But what if I show you this?

So what do these representations convey to you about the meaning of the number 7? Before reading on, take a moment to analyze the different representations. Do they all represent the same thing about the number seven? Do some representations give you different understandings than others? How many different things can you learn about the number seven from these representations?

Here are some of the things these representations convey to me:

• 7 can be made with combinations of smaller numbers: 1 and 6, 2 and 5, 3 and 4.
• At first I usually see a specific combination within a representation, like 4 and 3 in the domino or 5 and 2 in the math rack.
• After spending time looking at them, I start to notice multiple combinations within some representations. The teddy bears show me 4 and 3 if I look at the rows. However, I also see 6 and 1 if I look at the group of 6 with 1 teddy bear hanging off the end.
• I also see that 7 can be made with combinations of more than two numbers: 3, 3, and 1 for example as shown in the matches and the teddy bears.
• The number track shows me where 7 is in relation to other numbers. I can see that 6 is just before 7 and 8 is just after 7.
• I also see how 7 is related to 10. The math rack, number path, and fingers all show me that 7 is 3 less than 10.

This is hardly an exhaustive list of all the ways the meaning of 7 is conveyed, but hopefully it serves to demonstrate the point that the more representations of 7 I have access to, the more robust my understanding of the number 7 may become. The same applies for any number.

I followed up with this quote:

“There is no inherent meaning in symbols. Symbols always stand for something else. The meaning a symbol has for a child depends on what the child knows and understands about the concepts the symbol represents.” — Kathy Richardson, How Children Learn Number Concepts, p. 20

and this question:

Sometimes it’s hard to put ourselves in the shoes of our students, but doing so can help us better understand our students’ struggles and frustrations. We have been seeing numeric symbols for years and years. We see 7 and immediately have access to meaning. When in our adult lives might we encounter symbols we don’t understand?

For me it’s any time I encounter writing that doesn’t use the Roman alphabet. Even if I can’t speak Spanish or German, I can at least read the words I see (despite any horrible pronunciation problems):

• Buenos días.
• Por favor hable más despacio.
• Entschuldigen Sie bitte.
• Lange nicht gesehen!

And if there are any cognates involved, I just might be able to make some sense of what I’m reading.

But when I encounter writing in Hebrew or Chinese?

• בוקר טוב
• נעים מאוד
• 你好嗎?
• 我很高興跟你見面

These symbols have absolutely no meaning to me. They are inaccessible. Visiting Israel several times for work, it was always disconcerting to be bombarded by street signs, advertisements, and menus and have no way to even map any sounds to the text I was seeing.

Now am I saying that teachers are not currently providing students access to multiple representations of numbers like 7? No.

But that doesn’t mean it isn’t worth reflecting on our practices to ensure we are providing students access to these concepts via multiple and varied representations and that we aren’t rushing to the use of a symbol because that’s our “goal.” There is nothing inherently more mathematical about a symbol like 7 than a collection of dots on a domino or seven fingers on my hands. What numeric symbols do allow for is efficiency of representing quantity, especially once the place value system comes into play. But that efficiency is lost on students, especially those who struggle, if they do not have a solid foundation in the concepts the symbols represent.

# Math with More Bad Drawings

In my last post, I shared some abominable strip diagrams. Last night, my friend messaged me again about some different models. Also pretty terrible.

“Sorry to hit you up for math help but I can’t find any like this on the internet.”

There are two reasons for this, the second of which I’ll get to later in this post. The first is because this model is too bloated and trying to show competing ideas.

Here’s a cleaned up version of the model.

Any (good) area model should simultaneously represent multiplication and division. They’re inverses of each other. If you understand the components of the model, you should be able to write equations related to the model using both operations.

If I look at this model in terms of multiplication, I know I can multiply the length (7) times the width (13) to find the area (91). This area model represents 7 × 13 using the partial products of 7 × 10 and 7 × 3.

If I look at this model in terms of division, I know I can divide the area (91) by the width (7) to find the length (13). This area model represents 91 ÷ 7 using the partial quotients of 70 ÷ 7 and 21 ÷ 7.

All that from this one model. I don’t need all the “noise” included in the original model. For example, what is the purpose of writing the dimensions along the top as “10|70” and “3|21”? Knowing how an area model works, the only place 70 and 21 appropriately appear are inside the rectangle to show they represent area. Putting them along the top edge creates confusion about their meaning. Our students don’t need more confusion in their lives.

The repeated subtraction underneath isn’t terrible, but it’s unnecessary if you just want to know what multiplication or division sentence this model represents. Now, if a student were building the area model while using the partial quotients strategy, then the subtraction might be a useful recording strategy, but that’s not the same as being part of the model itself. I think it’s important to distinguish between those two things: features of the model itself and recording strategies a person might use as they build the model.

So the first problem my friend shared wasn’t great, but of course there was a second problem.

And it’s worse.

Holy cow! Bring on the tears.

I get that a student solving 46 ÷ 2 might think about and possibly even jot down potential options for partial quotients, but there is no reason this needs to be shown to children on their homework. And there’s still the problem of there being two numbers side-by-side along the length. Does someone think interpreting bad models is a sign of rigorous math instruction? I don’t.

Here’s the cleaned up version.

While the original model was terrible, the question wasn’t bad at all. I’d probably revise it slightly though. I might say, “Gina found partial quotients to solve 46 ÷ 2. She recorded her work in the area model shown. Circle the number(s) in Gina’s model that shows the quotient of 46 ÷ 2. Convince me you circled the right numbers in the model.”

So earlier in the post I mentioned there are two reasons my friend couldn’t find anything like this on the internet. The first is because these were bad drawings. I tried looking for videos of someone solving a division problem using partial quotients and an area model which led me to the second problem. So many videos out there of varying quality. And by varying, I mean it’s easy to find videos that aren’t all that great. Many demonstrate either a limited view of partial quotients or a limited understanding of the area model.

One of the great things about using partial quotients to divide is the flexibility in how you can choose to decompose the dividend. In the first problem in this post, for example, the dividend (91) was decomposed into 70 and 21, which are both easy to divide by 7. It could just as easily been decomposed in to

• 90 and 1
• 35, 35, and 21
• 63 and 28

While looking for videos to share with my friend, I found these (Video 1 | Video 2). What I noticed is that the partial quotients method is carried out in a rigid way that maps closely to the long division algorithm. In one of the videos, the presenter even connects the area model to long division notation.

The emphasis on place value is appreciated, but students deserve to know that they do have choice in how they decompose the dividend. Place value isn’t the only way.

These were the good videos. They might have missed out on sharing the power of this strategy, but at least the math is good. (I still didn’t share them with my friend.)

Sadly, there were also the bad videos. My major beef with these is that if you aren’t familiar or comfortable with partial quotients, you could just as easily watch a bad video and think you’re getting good information. These videos are so bad because, intentionally or not, they demonstrate big misunderstandings about the area model.

Example 1

In this example, the students are writing the numbers in the wrong place on the model. The partial areas (800, 370, and 23) should all be inside the rectangle while the lengths (100, 70, and 4) should all be outside along the top. I’m not blaming the kids. From what I can tell, they invented this strategy in their class (Cool!) but their teacher helped them make this video to share their strategy far and wide on the internet (Not cool!). Rather, as a teacher, I would have noted the students’ misunderstandings, helped them develop a better understanding of the area model, and then helped them create a video to show off their strategy.

Example 2

This one doesn’t even try to represent the values of the numbers. For whatever reason, the long division algorithm is carried out in boxes. Which, by the way, I don’t care if your video calls this the “box method” or “rectangle method.” It does not excuse you from misrepresenting the area model, because that’s what you’re doing. So many people believe math is confusing enough. Don’t add fuel to the fire.

Considering the time and effort that goes into building an understanding of area as a model for multiplication and division, we shouldn’t be making or showing these bad models to our students. We shouldn’t be showing them to our parents either. Seriously, if you share YouTube videos with your parents, please preview them and make sure the mathematics is good. Make sure they model the kinds of thinking, reasoning, and representing we want our own students to be developing.

Remember, the only people who should be making bad drawings are our students because they’re still figuring all of this out. Our job is to help them so that over time they get better.

Before you say anything, yes, the title of this blog post also happens to be the title of Ben Orlin’s amazing blog. I don’t care. I want it to be the title of this post, too.

Math should make sense. Or at least, you should be able to make sense of math. And any drawings you create along the way should aid in that sense making. And any drawings you encounter drawn by someone else should similarly aid in your sense making.

But what happens when they don’t? What happens when kids are forced to do math with – literally – bad drawings?

A few days ago a friend of mine sent me the following message:

“I am so lost trying to help my 4th grader. Do you have a secret website where I can find a strip diagram cheat sheet? I have never seen anything like this before.”

Yeah, me neither. Because this drawing is crap.

I mean, seriously, where do I begin? The three boxes with 32 in them actually make sense. Everything underneath? Not so much, especially to a 9 or 10 year old.

• I tend to prefer curly braces to bracket off clearly defined amounts, like, say, the total. This looks like someone just dragged it over partway to the right and then went, “Eh, good enough.”
• Then there’s a random gap which technically should represent a quantity of its own since this whole thing is built as a linear model.
• And finally we have that little scratch at the end with a 4 under it. Why is that not a curly brace? Are children supposed to know the difference between quantities represented by curly braces and those by line segments that have a slight curve at the end?

Here, let me fix this.

Can I guarantee that the meaning of this particular strip diagram will jump off the page and make sense to anyone who views it? Of course not. But at least now we have some consistency to the stuff on the bottom and the random gap is removed. At least now a child might be able to notice, “Oh the two numbers on the bottom (m and 4) should add up to the total of the numbers in the three boxes above.”

By the way, I should probably stop here and say: Strip diagrams are a TOOL, not a math skill unto themselves! They are meant to be used as a way to represent relationships so that you have an easier time determining which operation(s) to use. So rather than giving a strip diagram and asking students to write an equation and solve it, why not ask, “You set up the following strip diagram to solve a problem. How could you use the diagram to help you find the unknown value? Describe the steps you would take.” Honestly, I care less about students’ computation accuracy with this particular question than I do their ability to tell me that they would do something like multiply 32 times 3 and then subtract 4.

Unfortunately, this wasn’t the only example my friend sent.

Do we hate children? Do we want to make them cry? Because they have every right to as they try to make sense of these horrendous models.

My loathing is not because I can’t figure these out. I have figured them out. And I hate them. They’re just so cumbersome and confusing. Any mathematical meaning they’re trying to convey is muddled by inconsistencies and disproportionate boxes.

Let me make some attempt to fix this. No promises.

I just couldn’t with choice D. That was just a bad model all around. Sure it’s a wrong answer, but there’s no reason it has be a bad model on top of being the wrong answer.

No wonder parents take to Facebook to vent about math these days. If you’re required to use these materials, and I hope you aren’t, then please, please, please keep them at school. For the love of god, don’t send them home.

By the way, all of the drawings I made for this post can be made fairly easily in Google Drawing or the newest version of Powerpoint. They both include automatic features that help you line up and center your boxes and curly braces. Play around and practice. It is well worth your time, not to mention it’s pretty empowering to be able to create the exact strip diagram, number line, or other image you want to use in math class.

Our students deserve to make sense of math with drawings that make sense. Please do everything you can to ensure the only bad drawings are ones students are making themselves because they’re still figuring all this out. With practice and your help, over time they’ll get better.

# 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.

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

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

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.

# 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:

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:

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.

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.

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.

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?