Trick or Treat!

Now that I’ve completed sets of numberless word problems for all of the addition and subtraction CGI problem types, I wanted to do something fun.

This school year, my co-worker Regina Payne and I have been visiting the teachers in our Math Rocks cohort. One of the things they’ve been graciously letting us do is model how to facilitate a numberless word problems. In addition to making this a learning experience for the teachers, we’ve made it a learning experience for ourselves by putting a twist on the numberless word problem format.

Instead of your usual wordy word problem, we’ve been trying out problems that include visuals, specifically graphs. Instead of revealing numbers one at a time, we’ve been revealing parts of the graph. Let me walk you through an example I made tonight.

Here’s the graph I started with. I created it with some data I found on the Internet.

If I threw this graph at a 4th or 5th grader along with a word problem, they would probably ignore what the graph is all about and just focus on getting the numbers they need for doing whatever computations they’ve decided to do. They would probably also ignore a vital piece of information – the scale that says “In Millions” – which means their answer is going to be about 1,000,000 times too small.

But what if we could change that by starting with something a little more accessible like this?

What do you notice? What do you wonder?

I’m guessing at least one student in the class would comment that it looks like a bar graph. Interesting. What do you think this bar graph could represent?

Oh, and you think a bar is missing in the middle. Interesting. What makes you say that?

What new information was added to the graph? How does it change your thinking?

Oh, so there is a bar between Hershey’s and M&M’s. How tall do you think the bar for Snickers might be? Why do you say that?

Now we know how tall the bar for Snickers is. How does that compare to our predictions?

Considering everything we know so far, what do you think this bar graph is about? What other information do we need in order to get the full story of this graph?

What new information was added to the graph? How does it change your thinking about what this graph is about?

What are Sales? How do they relate to candy?

What does “In Millions” mean? How does that relate to Sales?

I know we don’t have any numbers yet, but what relationships do you see in the graph? What comparisons can you make?

What new information was added? How does it change your thinking?

Hmm, how many dollars in sales do you think each bar represents? How did you decide?

How do the actual numbers compare to your estimates?

What were the total sales for Reese’s in 2013? (I’d sneak in this question if I felt like the students needed a reminder about the scale being in millions.)

What are some other questions you could use answer using the data in this bar graph?

What is this question asking?

How can you use the information in the graph to help you answer this question?

*****

I may or may not actually show that last slide. After reading this blog post by one of our instructional coaches Leilani Losli, I like the idea of letting the students generate and answer their own questions. In addition to being motivating for the students, it makes my time creating the graph well spent. I don’t want to spend a lot of time digging up data, making a graph, and then asking my students a whopping one question about it! That doesn’t motivate me to make more graphs. I  also want students to recognize that we can ask lots of different questions to make sense of data to better understand the story its telling.

Some thoughts before I close. This takes longer than your typical numberless word problem. There are a lot more reveals. Don’t be surprised if this takes you at least 15-20 minutes when you take into account all of the discussion. When I first do a graphing problem like this with a class, it’s worth the time. I like the extra scaffolding. Kids without a lot of sense making practice tend to be pretty terrible about paying attention to details in graphs, especially if their focus is on solving an accompanying word problem.

If I were to use this type of problem more frequently with a group of students, I could probably start to get away with fewer and fewer reveals. Remember, the numberless word problem routine is a scaffold not a crutch. My hope is that over time the students will develop good habits for attending to features and data in graphs on their own. If you’re looking for a transition to scaffold away from numberless and toward independence, you might start by showing the full graph and then have students notice and wonder about it before revealing the accompanying word problem.

If you’d like to try out this problem, here’s a link to a slideshow with all of the graph reveals. You’ll notice blank slides interspersed throughout. I’ve found that if you have a clicker or mouse that has a tendency to jump ahead a slide or two, the blank slide can prevent accidental reveals. It also helps with graphs because when I snip the pictures in they aren’t always exactly the same size. If the blank slides weren’t there, you’d probably notice the slight differences immediately, but clearing the screen between reveals mitigates that problem.

Happy Halloween!

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.

Math with Bad Drawings

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.

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.

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!”

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

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

“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.

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!