# Multiplication Number Talks Using Models

(This post is re-blogged from my other math blog.)

In my previous post I discussed the importance of planning number talks with the four stages of using models. I used a 1st grade example in that post, and almost immediately my colleague Heidi Fessenden shared this wondering.

Great minds think alike, because this is exactly what I’ve been thinking about lately!

In Round Rock ISD, we want our students to learn thinking strategies for multiplication, rather than attempting to memorize facts in isolation. Thinking strategies have the following benefits for our students:

• There’s less to memorize because there are 5 thinking strategies to learn instead of 121 isolated facts.
• They create consistent language across grade levels.
• They afford a strategic mindset around how we think about computation facts.
• Their utility extends beyond basic facts to computation with larger numbers.

The thinking strategies we want our students to learn are from ORIGO’s Book of Facts series. (Each strategy is linked to a one-minute video if you’d like to learn more.)

#### One-Minute Overview Videos

In our curriculum, students learn about these thinking strategies in their core instruction. We have two units in 3rd grade that focus on building conceptual understanding of multiplication and division across a total of 51 instructional days. In between those units, students practice these thinking strategies during daily numeracy time so they can build procedural fluency from their conceptual understanding. My hope is that planning number talks with the four stages of using models will facilitate this rigorous work.

I also hope it supports students in maintaining their fluency at the start of both 4th and 5th grade. Our daily numeracy time at the beginning of both of those grade levels focuses on multiplication and division. Even if every 3rd grade student ended the year fluent, it’s naive to think that fluency will continue into perpetuity without any sort of maintenance.

To help teachers envision what a number talk might look like at different stages of using models, I’ve designed a bank of sample number talks for each thinking strategy.

Each bank includes a variety of examples from the different stages of using models:

• Stage 2 Referring to a complete model (Number Talks 1-4)
• Stage 3 Referring to a partial model (Number Talks 5-8)
• Stage 4 Solving the problem mentally (Number Talks 9-10)

You’ll notice some “Ask Yourself” questions on many slides. You’re welcome to delete them if you don’t want them visible to students. Ever since reading Routines for Reasoning by Grace Kelemanik and Amy Lucenta, I’ve been utilizing the same pedagogical strategies they baked into their routines to support emergent bilingual students and students with learning disabilities:

• Think-Pair-Share
• Annotation
• Sentence Stems and Sentence Starters
• The 4Rs: Repeat, Rephrase, Reword, Record

Since not all of the teachers in our district might be aware of “Ask Yourself” questions, I embedded them on the slides to increase the likelihood they’ll be used by any given teacher utilizing these slides.

#### Caveat

These sample banks are not designed to be followed in order from Number Talk 1 through Number Talk 10. Student thinking should guide the planning of your number talks. As Kathy Richardson shared in a tweet responding to my previous post, the four stages of using models are about levels of student thinking, not levels of instruction.

What these number talks afford is different ways of thinking about computation. A traditional number talk that presents a symbolic expression allows students to think and share about the quantities and operations the symbols represent. The teacher supports the students by representing their thinking using pictures, objects, language, and/or symbols.

A number talk that presents models, on the other hand, allows students to think and share about the the quantities shown and the operation(s) implied. The teacher supports the students by representing their thinking with language and/or symbols.

#### Trying It Out in the Classroom

For example, I led a number talk in a 5th grade class today, and I started with this image:

A student said she saw 10 boxes with 3 dots in each box. I wrote that language down verbatim, and then asked her how we could represent what she said with symbols. She responded with 10 × 3.

I asked the 5th graders to turn and talk about why we can use multiplication to represent this model. This was challenging for them! They’ve been multiplying since 3rd grade, but they haven’t necessarily revisited the meaning of multiplication in a while.

They were able to use the model to anchor their understanding. They said it’s because the number 3 repeats. This led us into talking about how there are 10 groups of 3 and how multiplication is a way that we can represent counting equal groups of things.

The number talk continued with this second image:

The first student I called on to defend their answer said, “I know 10 times 3 is 30, so I just took away 3.”

I recorded (10 × 3) – 3 = 27, but I didn’t let the students get away with that. I reminded them that multiplication is about equal groups. If we had 10 groups of 3, then we didn’t just take 3 away, we took away something else.

One of the students responded, “You took away a group.”

We continued talking which led to me recording (10 groups of 3) – (1 group of 3) = 9 groups of 3 under the original equation and then (10 × 3) – (1 × 3) = 27 under that.

I have to admit I screwed up in that last equation because I should have written 9 × 3 instead of 27. Thankfully number talks are an ongoing conversation. Students’ number sense is not dependent on any given day’s number talk, which means they’re forgiving of the occasional mistake.

What we did today is hopefully the start of a series of number talks to get students thinking about how taking away groups is one thinking strategy to help them derive facts they don’t know. Students don’t own that strategy right now, but our conversation today using the model was an excellent start.

#### Final Thoughts

I’m hoping these samples might inspire you to create number talks of your own based on the kinds of conversations you’re having with your students. Here is a document with dot images you can copy and paste from to create your own number talk images.

If you try out these number talks in your classroom, I’d love to hear how it went. Either tag me in a tweet (@EMathRRISD) or share your experience in the comments.

# Planning Number Talks with the Four Stages of Using Models

(This post is re-blogged from my other math blog.)

A year ago I attended the Math Perspectives Leadership Institute led by Kathy Richardson. One idea she shared that really resonated with me was the four stages of using models.

#### Four Stages of Using Models

Stage 1 Moving the model. Students need to actually touch and move the model.

Stage 2 Referring to a complete model. Students can look at models that represent all the numbers in the problem.

Stage 3 Referring to a partial model. Students can look at a model and think about what would happen if a number was added or taken away or the model was reorganized.

Stage 4 Solving the problem mentally. The student can solve the problem mentally without a model but can also use the model to demonstrate their thinking or prove their answer.

Kathy Richardson went on to share the following points about the importance of models:

• Models are used so the quantities become meaningful to the students
• Models allow the students to look for structure, parts of numbers, and relationships between them
• Every child has a way to work in the problem
• Everyone can participate because they solve the problem in ways they understand

This got me thinking about number talks. Do we capitalize on the value of models when planning number talks? Or do we have a tendency to gravitate to stage 4 without considering whether each and every student is actually ready for it? If we spend the bulk of our time in stage 4, are we considering issues of access? Whose knowledge do we privilege when we consistently present problems symbolically and assume that students are thinking flexibly about how to manipulate the numbers mentally?

Don’t get me wrong, I want students to reach stage 4, but I wonder how we can ensure we’re taking the necessary steps to build each and every student up to this kind of thinking. If you revisit the four stages of using models, what it looks like to me is a progression of transferring the actions of computation from physical, hands-on actions to increasingly mental actions. If we want students to mentally compose and decompose numbers, then we can use these stages to build a bridge from physically performing the action to mentally performing the action, and each stage creates a pathway for this to occur.

Let’s look at this progression in 1st grade as students are learning and practicing the Count-On Strategy for Addition.

#### Stage 1

Moving the model. In this number talk, students build the count-on addition facts on a ten frame.

Teacher: “In today’s number talk we’re going to think like mathematician as we use what we know about addition to solve some problems.”

Teacher: “Show 5.”

Students:

Students:

Teacher: “What is 5 and 1 more?”

Students: “5 and 1 more is 6.”

Teacher: “How can we record what we did using an equation?”

Students: “5 + 1 = 6”

Follow up questions the teacher should ask to help students make connections between the two representations:

• Where is the 5 in your model?
• Where is this 1?”
• Then what does the 6 mean?”
• What do these two symbols mean, + and =?

Repeat to solve and discuss more problems as time permits.

#### Stage 2

Referring to a complete model. In this number talk, the teacher shows both addends using a visual model.

Teacher: “In today’s number talk we’re going to think like mathematician as we use what we know about addition to solve some problems.”

Teacher: “How many on top?”

Students: “5.”

Teacher: “How many on the bottom?”

Students: “2.”

Teacher: “What is 5 and 2 more?”

Give students think time and then collect answers like you would in a regular number talk before having students share their strategies.

While counting all is a valid strategy, I purposefully set our mathematical goal for this number talk for students to use what they know about addition to solve problems. I would accept counting strategies, but I would emphasize strategies involving addition, such as counting on 2 from 5.

Like the previous example from stage 1, I would also be sure we create and analyze an addition equation that this model represents.

Repeat to solve and discuss more problems as time permits.

#### Stage 3

Referring to a partial model. In this number talk, the teacher shows both addends using a visual model.

Teacher: “In today’s number talk we’re going to think like mathematician as we use what we know about addition to solve some problems.”

Teacher: “How many dots are there?”

Students: “4.”

Teacher: “What if I added two more? How many would we have altogether?”

Give students think time and then collect answers like you would in a regular number talk before having students share their strategies.

Like the previous example from stage 2, I would accept all strategies, but I would emphasize strategies that relate to our mathematical goal, which is using what we know about addition to solve problems. I would also be sure we create and analyze an addition equation that this problem represents.

Notice that throughout all three of these stages, the action still exists, “add 1 more” or “add 2 more.” The difference is that while students can physically perform the action in stage 1, they have to mentally perform the action in stages 2 and 3. In stage 2, they can see both quantities so they can refer to both and they can even mentally try to manipulate them, if necessary.

In stage 3, students are anchored with the first quantity, but now they not only have to imagine the second quantity, but they have to imagine the action as well. In the example above, while they cannot physically add two more counters to the ten frame, their repeated experiences with the physical action means they have a greater chance of “seeing” the action happening in their mind. The work through these three stages prepares students for the heavy lifting they have to do in stage 4.

#### Stage 4

Solving the problem mentally. In this number talk, the teacher shows a symbolic expression.

Teacher: “In today’s number talk we’re going to think like mathematician as we use what we know about addition to solve some problems.”

Teacher writes the problem on the board and asks students to solve it mentally:

Give students think time and then collect answers like you would in a regular number talk before having students share their strategies.

This stage flips the script with regard to creating and connecting representations. Now the teacher can select from a variety of models she can draw to illustrate a student’s strategy. She might draw a ten frame, draw a math rack, draw hands and label the fingers with the numbers a student said, draw a number line, or even write equations.

As Pam Harris says, the goal of creating a model to represent a student’s thinking during a number talk is to make that student’s thinking more “take-up-able” by the rest of the class. Just because we’ve reached the point where students can solve problems represented symbolically doesn’t mean we stop making connections to models. We don’t want to unintentionally send the message that the symbols “5 + 2” somehow mean addition more than all of the other representations students have created and used.

The advantage of moving to symbols is just that they allow us to communicate in more efficient ways. While the efficiency is less obvious in the case of 5 + 2 – recording 3 symbols vs drawing 7 dots – it is much more obvious with a 25 + 12 – recording 3 symbols vs drawing 37 dots.

#### Final Thoughts

As you continue to plan number talks this year, consider the four stages of using models, particularly how these stages can help create access to the critical mathematical ideas at your grade level for a wider range of learners in your classroom. The final goal may be solving symbolic problems mentally, but it doesn’t mean that’s where we have to start or even where we have to spend the majority of our time.

#### Not So Final Thoughts

After sharing this post, Kathy Richardson responded with the following tweet and I wanted to be sure to share it here since this post is heavily influenced by her work.

I also heard that her new book Number Talks in the Primary Grades is going to be released in January. I look forward to checking it out!

# 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.