We briefly touched on this work in our session at the 2017 NCTM annual conference, and it’s been exciting to see my #MTBoS colleagues taking the idea and running with it in their schools! In case you don’t follow them – which will hopefully change after reading this post! – I want to share their work so you don’t miss out on all the great stuff they’re doing.

Kassia has written two wonderful blog posts about how she took our ideas and tinkered with them to create a data routine called Notice and Wonder Graphs. I like this name because it’s more inclusive than numberless graphs. When it comes to graphs, you might hide the numbers, but you could just as easily hide other parts of the graph first. It all depends on your goals and how you want the conversation to unfold. In Kassia’s first post, she shares this graph with students. Notice it has numbers, and little else.

Curious what it’s about? Then check out Kassia’s post. I’m betting you’ll be quite surprised when you reach the final reveal.

I will share this snippet from her post:

I love this routine for many of the reasons that I love Brian’s numberless word problems–it slows the thinking down and focuses on sense-making rather than answer-getting.

But I also love it because it brings out the storytelling aspect of data. So often in school (especially elementary school!) we analyze fake data. Or, perhaps worse, we create the same “What is your favorite ice cream flavor?” graph year after year after year for no apparent purpose.

I’ve decided to make it a goal to think more about data as storytelling, data as a way to investigate the world, and data as a tool for action. In my next two posts (YES, people! I’m firing the ole blog back up again!) I’m going to delve into the idea that we can use data to discuss social justice ideas and critical literacy at the elementary level. I’m just dipping my toe into this waters, but I’m really excited about it!

And Kassia did just that! So far she’s followed up with one post where her students noticed and wondered about a graph showing the percent of drivers pulled over by police in 2011, by race. I love how the graph sparked a curiosity that got her students to dive more deeply into the data. How often does a graph about favorite desserts or our birthday months spark much of any curiosity?

Jenna shared a numberless graph that immediately got me curious! This is one she created to use with 6th grade students.

I can’t help but notice a bunch of dots grouped up at the beginning with a just few outliers streeetttcchhiiing across almost to the very end.

Once she included some numbers, my first instinct was that this graph is about ages. Apparently I wasn’t alone in that assumption!

And then there’s the final reveal.

So why did Jenna create and share this graph? What was her mathematical goal?

I especially loved this observation about how her students treated the dot at 55 before they had the full context about what the graph is really about.

Chase wrote a detailed post about how he worked with 2nd grade teachers to do a lesson study about interpreting graphs.

…there’s so many rich opportunities for meaningful student discourse about data. That is, if it’s done right. Most textbooks suck all the life out of the content. Students need to understand that data tells a story; it has contextual meaning that is both cohesive and incomplete. Students need to learn how to ask questions about data and to learn to identify information gaps. In other words, students need to learn to be active mathematical agents rather than passive mathematical consumers.

Chase walks you through the lesson he and the teachers created and tried out in three different classrooms. I love how he details all of the steps and even shares the slides they used in case you want to use them in your own classroom.

He closes the post with a great list of noticings and wonderings about continuing this work going forward. Here are a couple of them about numberless graphs specifically:

- We need to give students more choice and voice about how they make meaning of problems and which problems they choose to solve. Numberless Data problems like these can be be a tool for that.
- The missing information in the graph created more engagement.

A huge thank you to Kassia, Jenna, and Chase for trying out numberless graphs and sharing their experiences so we can all be inspired and learn from them. I can’t wait to see how this work continues to grow and develop next school year!

If you’re interested in reading more first-hand accounts of teachers using numberless word problems and graphs, be sure to check out the ever-growing blog post collection on my Numberless Word Problems page. I recently added a post by Kristen Acosta that I really like. I’m especially intrigued by a graphic organizer she created to help students record their thinking at various points during the numberless problem. Check it out!

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**short** – which makes sharing them and using them in PD very easy – and they’re **to the point** – which means a focused message to get others thinking and talking.

Here’s the catch. To ensure each talk is short and to the point there’s a format each one follows:

- Each talk is 5 minutes long.
- You have exactly 20 slides.
- The slides auto-advance every 15 seconds.

Sounds stressful, right? And up until a couple of months ago I was always happy someone else was in the hot seat. That is until I got an email from Suzanne Alejandre from The Math Forum inviting me to be one of 10 speakers at the NCSM 2017 Ignite event in San Antonio in April. My emotions upon reading her email were a mix of feeling honored and terrified.

The event has come and gone, so now I get to look back on it with relief and a sense of accomplishment that I took the challenge and saw it through. I’m proud of the final result. I chose to talk about **doing** **less** of 3 key things in our math classrooms, which sounds counter-intuitive, but it actually results in **getting more** of a few things we want. Check it out:

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You might notice a theme to all three posts. Maybe.

Here’s links to the three posts:

I’ve been on a bit of a geometry kick lately. In my work, I’ve focused a lot on computation generally and number talks specifically the past couple of years. Geometry is an area I haven’t explored much recently so I’ve been making a more focused effort to do so.

The blog also has a great resources page with lots of stuff for teachers and parents. Take a peek at that if you go by for a visit.

I’ll still be writing here. I just wanted to make sure people knew I was writing there, too.

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Then I asked them to answer the question. I purposefully set up the survey so participants could select more than one answer, though I gave no encouragement to do this. All the question said was, “Your answer.” Finally, I left a section for comments.

When I tweeted out the survey, I didn’t provide any details about the problem or my intent. I was trying really hard to see if other people saw the same thing I did without leading them to it.

Turns out many people did. I feel validated.

So here’s the story. This is question #28 from the 2016 Grade 3 Texas state math test, called STAAR, that students took last spring. Back in February, one of our district interventionists emailed me to say that he thought both choice G and J are correct answers. I opened up the test, analyzed the question, and realized he was right. I immediately drafted an email to the Texas Education Agency to ask about it.

Good morning,

I have a question about item 28 on the grade 3 STAAR from spring 2016.The correct answer is listed as J. This makes sense because the number line directly models a starting amount of 25 people and then taking some away to end at 13, the number of people still in the library.However, the question isn’t asking for the model that most closely represents the story. Rather, it asks which model can be used to determine the number of people who left the library.In that case, answer choice G is also correct. Our students understand that addition and subtraction are inverse operations. Rather than thinking about this as 25 – __ = 13, answer choice G represents it as 13 + __ = 25, which is a completely valid way of determining the number of people still in the library.I look forward to hearing TEA’s thoughts about this question. You can reach me at this email address or by phone.Have a great day!

About a month later I still hadn’t received a response so I emailed again and got a call the next day. It turns out I wasn’t the only person who had submitted feedback about this question. Unfortunately, according to the person on the phone, after internal review TEA has decided not to take any action. However, they do acknowledge that the wording of this question could be better so they will do their best to ensure this doesn’t happen again.

I told her I wasn’t happy with that answer and that I would like to protest that decision. She didn’t think that’s possible, but she offered to pass my email along to her supervisor or ask the supervisor to call me. I asked for her supervisor to call me.

Surprisingly, my phone rang about two minutes later.

The supervisor asked me to go over my concern with her so I explained pretty much what I said in my email. She said she understood, but if we looked at G that way then all of the answer choices could potentially be right answers. This was confusing to me because I don’t think F would help you determine the answer at all. If anything it shows 25 + 13, which will not give you an answer of 12.

I stressed that my concern is that answer choice G is **mathematically correct*** with regards to answering the question asked*. I get that J is a closer match to representing the situation, and if the question had asked, “Which number line best represents the situation?” then I probably wouldn’t be emailing and calling.

But it doesn’t.

The question asks, ‘Which number line represents **one way** to determine the number of people who left the library?” If you know how to use addition to solve a subtraction problem, then answer choice G is totally **a** **way** to find the number of people who left the library.

She said that is a strategy, not a way of representing the problem.

“That’s exactly what a **way** is. How you would do something, your strategy,” I replied.

She decided to redirect the conversation, “Let’s look at the data on this question. 68% of students chose J. 9% chose F, 12% chose G, and 10% chose H. The data shows students weren’t drawn to choice G. It’s not a distractor that drew them from choosing J.”

“I don’t care about that. The number of students who selected G doesn’t change the fact that it’s mathematically correct. If anything we should give those students the benefit of the doubt because we don’t know why they picked it.”

“Exactly,” she replied. “We don’t know why they picked it, so we can’t assume they were adding.”

“That’s not okay. Since we don’t know why they picked it, we’re potentially punishing students who chose to use a perfectly appropriate strategy of addition to solve this problem. There are a lot of 3rd graders in Texas, and 12% of them is a large number of kids. Who knows if this is the one question they missed that could have raised their score to passing?”

From this point she steered the conversation back to the question and how J is still the best choice because this is a subtraction problem.

“But you aren’t required to subtract to solve it! We work really hard in our district to ensure our students have the depth of understanding necessary of addition and subtraction to know that they can add to find the answer to a subtraction problem. We want them to be flexible in how they choose to solve problems. And again, the question isn’t asking students which number line best matches the situation. It just asks for **one way** to find the number of people who left, and both G and J do that.”

She went back to her original argument that if I’m correct then all of the answer choices could be used to find the answer to the question. She talked about how choice F shows both parts of the problem, 25 and 13, so you could technically find the answer. I disagreed because you end up with a total distance of 38. There’s nothing that makes me see or think of the number 12.

We went round and round a few more times. She wasn’t budging, and I was having a hard time listening to her justifications. She assured me they were going to be much more diligent about how number lines are used in future questions, but this question was going to remain as-is because she believes J is the best answer.

The whole exchange left me livid. In some small way, TEA is acknowledging that this question is flawed, but they aren’t willing to do the right thing by either throwing it out or making it so either G or J could be counted as correct.

They’re just going to do better next time.

But we’re talking about a **high stakes test**! Our students, teachers, principals, and schools don’t get to just “do better next time.” They are held accountable for their scores now. They can be punished for their scores. People can be moved out of their jobs because of students’ scores. So much is at stake that if a question is this flawed, TEA should show compassion to our students, not stubbornness. They should admit that both answers are mathematically correct and update each students’ score.

Because we’re not talking about a small handful of kids.

12% may not sound like much, but when 327,905 students took this test, that means nearly **40,000(!)** of them chose answer choice G. That’s 40,000 students who are being punished because of a poorly worded item that has two answers.

That’s not correct.

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- No time pressure
- Conceptual basis for the operations
- Mistakes must be handled properly

Tracy goes on to share two apps she does recommend, one of which is DreamBox Learning. After Tracy’s enthusiastic review, I wanted to get my hands on it and try it out. I invited a sales rep to our district for a demo, but that was underwhelming as always. For whatever reason, edtech companies tend to reveal only the briefest of glimpses of the actual student experience of their products. This is so frustrating to me!

Back in the spring I was part of a request for proposal (RFP) process that looked at various computer-based math programs. One of my biggest questions when reviewing programs is always, “What is it like for a kid using this program?” The reps will show you a few screens, but generally not enough to get a sense of what kids are really experiencing. Rather, the bulk of the time is spent talking about things like adaptive pre-assessments, teacher dashboards, and the plethora of reports that can be generated and dissected. Since the companies aren’t marketing to children, they focus their time and energy on the features that the adults will use. However, students are the ones using these products the most to (hopefully!) learn more about math. Their user experience is the one I care most about understanding and evaluating.

I did get some sample DreamBox accounts to play with, but I really wanted to see how it works in the hands of a kid, especially considering the adaptive nature of the program.

Enter my daughter, @SplashSpeaks. We’ll call her Splash for short. Splash is going to be 5 years old in March. She’s on the young side to be using the program – it says it’s designed for grades K-8 – but we have been doing so much counting and talking about numbers in our day-to-day lives that I thought it would be worth giving it a shot. Since the program is adaptive, I figured it would ensure she was in appropriate content.

Over winter break, I decided to create a personal account and start a two-week free trial. This post is about how, at least for now, I’m not going to subscribe now that the trial is over.

When Splash and I sat down at my iPad Mini to play DreamBox for the first time, she was excited to try a new app. The first few activities were a piece of cake for her. All they asked her to do was determine either “Which has more?” or “Which has less?” from two images of dots. The only challenge was paying attention enough to know which was being asked for. Her default was to assume that it was going to ask her to find the one with more. All in all, she did well enough and new activities started opening up for her.

I will say I’m impressed with the variety of representations she encountered in DreamBox. These included ten frames, dot images, math racks, and number tracks. It was interesting to see which ones resonated more with her. The math rack is definitely her favorite!

Sometimes the interactivity to complete one screen was a bit cumbersome for her. In the above example, she had to count the beads in the static image, create a representation of the same number of beads in the interactive math rack, count the number of beads again to make sure she remembered the number, count along the number track until she found the number she was looking for, click it, and then click the green arrow to indicate she’s done.

Whew!

Thankfully not every screen was this involved, but when they were, she would often skip a step. For example, she would build the number on the math rack and then jump down to the green arrow, forgetting to also select the number on the number track.

The first red flag for me that this may not be a good choice for her was her reaction whenever her answers were checked by the system. If she got the answer right, she would turn to me and smile, but if she got it wrong, she had a physical reaction of frustration. Rather than knowing it for herself, she started putting her faith in the system to tell her whether she had counted correctly. I didn’t feel comfortable with that shift in authority. I want *her* to trust that she counted correctly or built the number correctly, not wait for a computer to tell her. And I didn’t like how that subtle shift so dramatically changed her reactions to being wrong.

I will recommend that if your children use DreamBox, young ones especially, you should sit with them. There are some activities that ask for things Splash didn’t understand at first. For example, after building some numbers with the math rack, it started asking her to do it in the fewest number of moves possible. She had no idea what that meant.

Perhaps I should have said nothing and let her fail at the task. Since the system is adaptive, it might have shifted her back to other activities. However, considering how quickly the system brought her to this point in the first place, my guess is that after another activity or two she would have been prompted with these same directions.

I opted to explain to her what the phrase meant and she was able to start doing it on her own with the math rack. It was definitely more confusing with the ten frame, but even then I started seeing her grab larger chunks of dots rather than just counting out one at a time.

Here’s where another red flag came up. If you make a mistake on a screen that asked for the fewest clicks possible and then correct your mistake, the system will chide you for not getting the answer in the fewest number of moves and make you do it again. For example, let’s say you were supposed to drag 7 beads on the math rack but you mis-click and drag 6. If you drag all the beads back and then click 7, which is what my daughter did, your answer is still wrong because it counted all the clicks you made on that turn. It doesn’t matter that your last click was the efficient one.

This caused my daughter a lot of distress because she felt pressure to make sure she was completing the task perfectly, but the mix of her 5 year old hand-eye coordination and my small iPad Mini screen meant this happened somewhat frequently. She had a similar issue with the number track where she’d be counting and pointing at the numbers on the track to find the number she wanted and accidentally click one of the numbers she was counting. In certain activities there is no green check mark. If you click the number track that’s it; the system thinks that’s your answer. It was frustrating to watch her getting discouraged at being told her answer was wrong even though it was a user interface issue.

Her frustration reached a breaking point when DreamBox started introducing Quick Images activities. If you’re not familiar, an image is flashed for about 2 seconds and then covered. The user has to select the number of beads/dots that were in the image. This just blew Splash’s mind! She can identify 1, 2, and 3 on sight, but if it’s 4, 5, or greater, she relies on counting one by one. This activity made her so annoyed the first time she did it. That is, until she had an idea. She hopped up and said, “I’ll be right back!” She came back with her personal math rack:

Suddenly the activity became much more do-able for her. By building the images herself, she started to notice that some images only had red beads and others had red or white. If the image had red or white then she learned she only had to count the white beads. Clever girl! She still hasn’t had the “a-ha” moment that all of the red beads are 5, but it’ll happen at some point down the road. I’m not worried.

Bringing in a math tool was a lifesaver for her. She had a renewed interest in the program and felt empowered using her tool to support her thinking. That is until she started getting Quick Images with dot images. This is where I’m curious how DreamBox gauges student ability with regards to numbers to 10. I already know my daughter is super comfortable with 1, 2, and 3. She clearly needs more work on 4 and 5. Numbers 6-10 I’m less concerned about though I know she can count them accurately.

The Quick Images activity is all over the map. It would show an image of 3 dots. Cool, no problem there. But then it would follow up with an image like this:

She took one look at this and was defeated. She had no idea how many dots there were. We haven’t played a lot of dice games yet, so she doesn’t know that arrangement of 5. And she doesn’t understand counting on yet so even though she can see two orange dots, that’s not useful for finding the total.

I let her take her best guess and get it wrong. I kept telling her it’s okay. If she gets it wrong the system knows she’s not ready for that problem and will give her a different one. This is where I ran into two big problems with DreamBox. First, the way it decides what numbers to give her seems random. After getting large quantities wrong, I figured it would adjust and only give her small quantities, but it kept ping ponging back and forth showing 3, then 9, then 2, then 8, then 10. In my head I was like, “Clearly she can’t figure out the big numbers, stop giving them to her!”

The other issue has to do with the length of the activities. Normally it seems like she answers 6-8 questions and the activity is over. There’s even a visual on the screen to help show progress. For example, a long dinosaur neck is inching along the bottom of the screen towards some leaves. In this same Quick Images activity, I saw that she was close to the point where the activity normally ends, so I encouraged her to do what felt like must be the last problem. And the one after that. And the one after that. And the one after that. It never ended! The dinosaur neck just kept inching and inching and inching toward that leaf. I felt like we were trapped in Zeno’s paradox. Each time, Splash got more and more upset and frustrated until she finally broke down in tears, and that’s when I ended it. If I had known the system was capable of extending an activity that long I would have backed out of it much, much sooner. As it is, I felt terrible! I love talking about and doing math with my daughter. The last thing I want is to bring her up to and well beyond the point of frustration.

We took a break from DreamBox for a day or two. When I asked her to try it again she said, “I don’t want to do it. I don’t like it.” That made me sad. I didn’t want to stop using DreamBox on the negative note of her last activity. I wanted to help remind her about all the amazing thinking she had been doing while using the program. I encouraged her to try again, but this time we would ignore those Quick Images activities. She was hesitant, but she agreed and we ended up having a good session. The next day we played again, but this time she chose an activity that I thought was something different but it turned out to be that dreaded Quick Images activity. Aargh!

Rather than give up, I took a quick look around the dining room and saw a tub of beads. Splash wanted to get right out of the activity, but I stopped her and said, “Why don’t you try using these to build the picture like you did with your math rack?” Building with beads sounded fun so she agreed. I also prompted her to look for small groups of dots in the pictures to help her. What a difference that made! She blew me away with her subitizing skills.

I was so proud of her! She managed to build every single image thrown at her. It wasn’t until the activity ended and said, “That’s okay, we’ll try Quick Images again another time,” that I realized the system was not as impressed with her performance. Apparently she was being timed. It took her a while to build and count each image. Even though she got every single answer correct, DreamBox considered it a failure and didn’t count the activity as complete.

A day or so later our 14-day trial ended and I was left with the decision about whether I should pay for a subscription. Splash clearly demonstrated some wonderful strategizing and thinking while using DreamBox, and I was tempted to see where it would take her, but I had a feeling in my gut that it wasn’t the right decision for her.

I couldn’t quite put it into words why until a week or so later when I read chapter 2 of Tracy Zager’s new book Becoming the Math Teacher You Wish You’d Had. The chapter is titled “What Do Mathematician’s Do?” In it, she shares the story of a primary classroom where students are asked what it means to do math. Initially their answers have to do with worksheets and giving answers. The teacher and Tracy work together to develop a mini-unit to open students’ eyes to what mathematicians really do. By the end of the unit, students are beginning to understand that math is about some wonderful verbs including *noticing*, *wondering*, *asking*, *investigating*, *figuring*, *reasoning*, *connecting*, and *proving*. They’re learning that math is all around them. Reading about the experiences of these students made me want to be in that classroom, experiencing that joy of discovery with them.

And then I thought of my daughter and all of the experiences we have daily with math. I realized that DreamBox might be better than nearly every other edtech program for practicing specific skills and working through a coherent progression of ideas, but it’s not the kind of math I want my 5 year old daughter to experience. I don’t want her worrying about whether a computer is telling her her answers are correct or whether she’s taking too long to come up with them or whether she’s finding them in the most efficient way possible.

In just two weeks I already saw that path leading to frustration and negative feelings toward mathematics. No thank you.

I want to continue down the joyful, meandering path we are already on where she investigates making shapes using her body and our tile floor:

Where she wonders about the biggest shape we can possibly make with plastic strips called Exploragons:

Where we figure out important things in our daily life, such as, “How many more days until the weekend?” and where we notice and play with math:

Down the road I might revisit DreamBox for my daughter, but not anytime soon. Lest you think I’m just being a harsh critic, I will still happily recommend it for parents and teachers who have older kids. When a child has more math under their belt and you want a system to be able to flexibly move backward and forward to meet their needs, then this is a great choice. It’s not perfect, but it’s far better than other programs I’ve seen. Kent Haines said it best:

But for a child just starting out and just beginning to develop her identity and relationship with mathematics, I’ll pass.

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Let’s start with this question from the chat:

Before reading on, pick one of the models yourself and analyze it.

- How does it represent 2/3?
- Where is the numerator represented in the model?
- Where is the denominator represented in the model?
- Can you answer these questions with all three models?

First it might help to differentiate the three models. The top left corner is an area model, the top right corner is a set model, and the bottom middle is a number line.

If you look at the **area model**, you’ll see that the whole rectangle – all of its area – has been partitioned into three equal parts, each with the same area. When we divide a shape or region into three parts with equal area, we actually have a name for each of those parts: thirds. Those thirds are countable. If I count all of the thirds in my area model, I count, “1 third, 2 thirds, 3 thirds.”

Two of them have been shaded orange. So if I count only the parts that are orange, “1 third, 2 thirds,” I can say that 2 thirds, or 2/3, of the whole rectangle is shaded orange.

If you look at the **set model**,** **you might think at first that this model is the same as the area model, but this representation actually has some different features from the area model. In the set model, the focus is on the number of objects in the set rather than a specific area. I used circles in the above image, which are 2D and might make you think of area, but I could have just as easily used two yellow pencils and one orange sharpener to represent the fraction 2/3.

I can divide the whole set into three equal groups. Each group contains the same number of objects. When we divide a set of objects into three groups with the same number of objects in each group, we actually have a name for each of those groups: thirds. Those thirds are countable. If I count all of the thirds in my set model, I count, “1 third, 2 thirds, 3 thirds.”

Two of the groups contain only pencils. So if I count only those groups, “1 third, 2 thirds,” I can say that pencils make up 2 thirds, or 2/3, of the objects in this set.

Finally, we have the **number line** model which several people in the chat said is the most difficult for them to make sense of. While we have a wide amount of flexibility in how we show 2/3 using an area model or set model, the number line is limited by the fact that 2/3 can only be located at one precise location on the number line. It is always located at the same point between 0 and 1.

In this case, our whole is not an area or a set of objects. Rather, the whole is the interval from 0 to 1. That interval can be partitioned into three intervals of equal length. When we divide a unit interval into three intervals of equal length, we actually have a name for each of those intervals: thirds. Those thirds are countable. If you start at 0, you can count the intervals, “1 third, 2 thirds, 3 thirds.”

However, what’s unique about the number line is that we label each of these intervals at the end right before the next interval begins. This is where you’ll see tick marks.

- So 1/3 is located at the tick mark at the end of the first interval after 0.
- 2/3 is located at the tick mark at the end of the second interval after 0, and
- 3/3 is located at the tick mark at the end of the third interval that completes the unit interval. We know we have completed the unit interval because this is the location of the number 1.

This quote sums up what I was aiming for with this discussion of representations of 2/3:

“Helping students understand the meaning of fractions in different contexts builds their understanding of the

relevant featuresof different fraction representations and therelationships between them.” – Julie McNamara and Meghan Shaughnessy, Beyond Pizzas and Pies, p. 117

The bold words are very important to consider when working with students. What is obvious to adults, who presumably learned all of these math concepts years and years ago, is not necessarily obvious to children encountering them for the first time. What children attend to might be correct or it might be way off base. One common problem is that children tend to overgeneralize. A classic example is shared in *Beyond Pizzas and Pies*. Students were shown a model like this:

They overwhelmingly said 1/3 is shaded. The relevant features to the students were shaded parts (1) and total parts (3). They weren’t attending to the critical feature of equal parts (equal areas).

I’ll close this post with a Which One Doesn’t Belong? challenge that I shared during the #ElemMathChat. (Note: I revised the image of the set model from what was presented during the chat.) As you analyze the four images, think about the relevant features of the area model, set model, and number line; look for relationships between them; and then look for critical differences that prove why one of the models doesn’t belong with the other three.

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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:

Have you ever encountered symbols in your adult life that had no inherent meaning for you?

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.

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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?

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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!

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*“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.

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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.

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