Tag Archives: Number Talks

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
  • Ask Yourself Questions
  • 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:

Teacher: “Add 1 more.”

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!

Decisions, Decisions

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

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

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

mr01

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

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

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

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

mr02

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

mr03

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

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

20160915_170313

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

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

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

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

mr04

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

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

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

20160915_170320

 

 

 

Purposeful Numberless Word Problems

[UPDATE – You can find all of my numberless word problem sets on this page.]

This year I read Sherry Parrish’s Number Talks from cover to cover as I prepared to deliver introductory PD sessions to K-2 and 3-5 teachers in November. She outlines five key components of number talks; you can read about them here. One of the components in particular came to the forefront of my thinking the past few days: purposeful computation problems. I’ll get back to that in a moment.

It all started when I got an email the other day asking whether I have a bank of numberless word problems I could share with a teacher. Sadly, I don’t have a bank to share, but it immediately got me thinking of putting one together. That led to me wondering what such a bank would look like: How would it be organized? By grade level? By problem type? By operation?

That brought to mind a resource I used last year when developing an extended PD program for our district interventionists: the Institute of Education Sciences practice guide Assisting Students Struggling with Mathematics: Response to Intervention (RtI) for Elementary and Middle Schools. The guide lays out 8 recommendations. I was reminded of this one:

Recommendation 4. Interventions should include instruction on solving word problems that is based on common underlying structures.

Students who have difficulties in mathematics typically experience severe difficulties in solving word problems related to the mathematics concepts and operations they are learning. This is a major impediment for future success in any math-related discipline.

Based on the importance of building proficiency and the convergent findings from a body of high-quality research, the panel recommends that interventions include systematic explicit instruction on solving word problems, using the problems’ underlying structure. Simple problems give meaning to mathematical operations such as subtraction or multiplication. When students are taught the underlying structure of a word problem, they not only have greater success in problem solving but can also gain insight into the deeper mathematical ideas in word problems.

(You can read the full recommendation here.)

And it was this recommendation that ultimately reminded me of the part of Sherry Parrish’s book where she talked about purposeful computation problems:

“Crafting problems that guide students to focus on mathematical relationships is an essential part of number talks that is used to build mathematical understanding and knowledge…a mixture of random problems…do not lend themselves to a common strategy. [They] may be used as practice for mental computation, but [they] do not initiate a common focus for a number talk discussion.”

All of this shaped my thoughts on how I should proceed if I were to create a bank of numberless word problems to share. Don’t get me wrong, the numberless word problem routine can be used at any time with any problem as needed. However, the purpose is to provide scaffolding, and we should provide scaffolding with a clear instructional end goal in mind. We’re not building ladders to nowhere!

The end goal, as I see it, is that we’re trying to support students so they can identify for themselves the structure of the problems they’re solving so they can successfully choose the operation or operations they need to use to determine the correct answer.

In order to reach that goal, we need to be very intentional in our work, in our selection of problems to pose to students. We need to differentiate practice for solving problems from purposefully selecting problems that initiate a common focus for problem solving.

What Sherry Parrish does to achieve this goal with regards to number talks is she creates problem strings and groups them by anticipated computation strategy. I didn’t create problem strings, per se, but what I did do was create small banks of word problems that all fit into the same problem type category. I’m utilizing the problem types shared in Children’s Mathematics: Cognitively Guided Instruction.

CGI

Here’s the document the image came from. It’s a quick read if you’re new to Cognitively Guided Instruction or if you want a quick brush up.

So far I’ve put together sets of 10 problems for all of the problem types related to joining situations. I plugged in numbers for the problems, but you can just as easily change them for your students. I did try to always select numbers that were as realistic as possible for the situation.

My goal is to make problem sets for all of the CGI problem types to help get teachers started if they want to do some focused work on helping students build understanding of the underlying structure of word problems.

I created these problems using the sample contexts provided by Howard County Public Schools. They’re simple, but what I like is that they help illustrate the operations in a wide variety of contexts. Addition can be found in situations about mice, insects, the dentist, the ocean, penguins, and space, to name a few.

As you read through the problems from a given problem type, it might seem blatantly obvious how all of the problems are related, but young students don’t always attend to the same features that adults do. Without sufficient experience, they may not realize what aspects of a problem make addition the operation of choice. We need to give them repeated, intentional opportunities to look for and make use of structure (SMP7).

Even though I’m creating sets of 10 problems for each problem type, I’m not recommending that a teacher should pick a problem type and run through all 10 problems in one go. I might only do 3-4 of the problems over a few days and then switch to a new problem type and do 3-4 of that problem type for a few days.

After students have worked on at least 2 problem types, then I would stop and do an activity that checks to see if students are beginning to be able to identify and differentiate the structure of the problems. Maybe give them three problems, 2 from one problem type and 1 from another. Ask, “Which two problems are of the same type?” or “Which one doesn’t belong?” The idea being that teachers should alternate between focused work on a particular problem type and opportunities for students to consolidate their understanding among multiple problem types.

On each slide in the problem banks, I suggest questions that the teacher could ask to help students make sense of the situation and the underlying structure. The rich discussion the class is able to have with the reveal of each new slide is just as essential as the slow reveal of information.

You may not need to ask all the questions on each slide. Also, you might come up with some of your own questions based on the discussion going on in your class. Do what makes sense to you and your goals for your students. I just wanted to provide some examples in case a teacher wasn’t quite sure how to facilitate a discussion of each slide for a given problem.

Creating these problem sets has prompted me to make a page on my blog dedicated to numberless word problems. You can find that here. I’ll post new problem sets there as they’re created. My current goal is to focus on creating problem sets for all of the CGI problem types. When that is complete, then I’d like to come back and tackle multi-step problems which are really just combinations of one or more problem types. After that I might tackle problems that incorporate irrelevant information provided in the problem itself or provided in a graph or table.

I’ve got quite a lot of work cut out for me!

Twelve Hours of Number Talks

November 3 and 4 were intense! Over the course of two days broken up into four half-day sessions, my colleague Regina and I introduced 150(!) K-5 teachers in our district to number talks. Whew! I still get tired thinking about it.

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The guy who wrote this doesn’t work in math education so “Number Talks” and “Numbers Talk” are all the same to him. I do wonder what a “Numbers Talk” PD would entail.

We offered two K-2 sessions and two 3-5 sessions. They were all called “Introduction to Number Talks” and that’s exactly what they were. We painted a big picture, got teachers excited and…ran out of time. I’m already formulating a follow up session for this summer that will dive more deeply into planning for number talks and building teacher confidence in the various computation strategies.

This post isn’t about looking ahead, however. Rather I’m going to look back and reflect on what we did accomplish. I’ll start by saying that for the most part the K-2 and 3-5 sessions were identical. There were some key points where we tailored the content to primary or intermediate grades, but the overall flow was the same.

Having led each session twice, I think that was a good call. In the places where it mattered, K-2 or 3-5 teachers experienced content that resonated with them, but as an introductory session, we were able to get more bang for our planning buck by keeping both sessions mostly the same. That said, I’m going to talk about the sessions as though they were all the same session. However, I’ll point out the places where they varied. Let’s get started!

Dot-Talk

How many dots do *you* see? Take a few moments and think about different ways you could prove your answer before reading on.

What better way to start than by diving in to our first number talk? I gave them the following directions to get them started:

  • Think quietly to yourself.
  • No pencils or paper.
  • Hold up one thumb when you have one way to find the answer. Hold up additional fingers as you find additional ways.

And then I left them to think.

You may be wondering why I started with this number talk rather than something more meaty like 25 × 32 or 198 + 136. And that is a good wondering! First of all, I didn’t want any numbers (symbols). Secondly, I wanted the quantities to be small. Given those constraints, how could anyone NOT figure out the answer?

Exactly!

When introducing number talks, you want to ensure the barrier to entry is low. You want to ensure that every single person can take part and succeed. While the two example problems I shared are cool and rich with possibilities, they can be intimidating to many students (and teachers!), especially when asked to solve them mentally.

The power of number talks is learning to recognize that there are multiple ways to approach problems. If the first problem I give you makes you anxious because you don’t consider yourself good at computation, or if the numbers seem too large and unwieldy to manipulate mentally, then you’re going to tune out quickly. Not exactly the way I want to start a 3 hour PD session, nor the way a teacher wants to start a promising new practice in her classroom.

After a minute or so, I saw everyone holding up multiple fingers. So far, so good! I asked someone what their answer was. The first person I called on in every session told me 10, but I still followed up with, “And does anyone else have a different answer?”

Crickets. (No surprise there.)

“Okay, who wants to defend this answer?” And we were off! Ahead of time I made copies of the image so that I could draw on them as needed as the teachers shared their thinking. Making extra copies like this requires some planning ahead, and I’m not sure it was necessary, but for this introduction, I did like having clean images available to dirty up with each person’s strategy.

By the way, did you actually stop and think about the image when I first shared it? If not, now would be a good time to do that because I’m going to share pictures I took of the board after each number talk. As you look at them, think about how the strategies our teachers shared compare to your own strategies. Then compare and contrast the strategies across the four sessions. It was fascinating to me to see how the same simple number talk played out with four different groups of people. By the way, if you want to know the order the strategies were shared in each session, go from right to left in each picture.

Our first number talk under our belts, Regina and I provided a brief rationale for number talks, including a reminder that fluency is more than speed and getting the correct answer. Procedural fluency is “skill in carrying out procedures flexibly, accurately, efficiently, and appropriately.” (From the Introduction of the Texas Math TEKS) We also talked about our district’s goals for K-12 mathematics, something we connect to every time we’re in front of teachers. We asked the teachers to compare our district’s goals with the goals of number talks. They noticed they fit together nicely!

After trying out a number talk together, I wanted the teachers to get a chance to see number talks in action with students. Sherry Parrish’s Number Talks book is a great resource because it includes a DVD chock full of 19 number talks from grades K, 2, 3, and 5. As much as I would have liked to show them all, there clearly wasn’t enough time. Not to mention, we didn’t want to bore teachers with video clips. We really wanted them to be incorporated intentionally into our sessions.

Chapter 9 includes some advice about providing teachers a schoolwide perspective of number talks. I thought this was crucial to build buy in. I want teachers to understand that number talks aren’t just for one grade level or grade band. They are a practice that can be used across all grade levels, and there is potential for great things to happen if students have the opportunity to do them year after year.

We showed the teachers a series of four number talks, one from each grade level – K, 2, 3, and 5. As they watched each video, their job was to observe how the following areas were exhibited:

  • Classroom community
  • Teacher’s role
  • Student’s role
  • Communication

I grouped the teachers so that during each video they only had to focus on one of the four areas. After the video was over, each group talked and recorded their observations on charts hanging around the room. Then they rotated to the next chart and watched the next video through a new lens.

Here are the sets of posters and some action shots from each session. I enjoyed listening in on their conversations and hearing how they were picking up on so many important features of number talks from watching the videos.

Session 1 (K-2):

Session 2 (3-5):

Session 3 (3-5):

Session 4 (K-2):

I heard a lot of good questions as they talked in groups. For example, many teachers observed that the classes in the videos seemed small, more like 15 students in a class rather than our usual 22-25. That made them a little skeptical about what number talks look like with larger groups of students. However, I also heard a lot of excited comments about what they were seeing, especially as they saw number talks moving up through the grades.

After the final video, I had each group stay at their final poster. Now their role shifted to analyzing all of the comments made through that one lens so they could share out to the whole group key similarities and differences. Their observations were on point. Listening to them share out each session showed me how powerful this activity had been.

We ended this activity by having a look through one final lens – the process standards. Each table group had 2-3 process standards, and they were tasked with identifying which of those process standards they observed in the four number talks.

Process-Standards

The teachers were already so excited to see all the great thinking and talking in the videos. When we talked and realized that pretty much all of these process standards appear in number talks in some form or fashion, that pretty much sealed the deal. “So you’re telling me that in 10-15 minutes my students can develop understanding of content while simultaneously incorporating 3, 4, 5, maybe even all 7 process standards? Count me in!”

That is what we call a great bang for your teaching buck.

Everything after this point was gravy, and boy did I take full advantage of it! Talking about the key components of number talks allowed me to sneakily embed some general teaching practices I feel passionately about.

Key-Components

Talking about component 1 allowed me to make my first plug of the session for Intentional Talk by Elham Kazemi and Allison Hintz. As the teachers watched the four videos, one of the recurring comments in each session was how obvious it was that the students felt safe taking intellectual risks in those classrooms. To me this is one of the most critical components of math instruction in general, not just number talks. We read and discussed a short excerpt about establishing norms from chapter 2 of Intentional Talk. My secret hope is that any work teachers do to establish and/or reinforce norms during number talks will inevitably bleed into other areas of math as well.

Component-2-1

Even though this component is about the discussions, I used it as a chance to reinforce the full routine:

  • Tell the students to solve the problem mentally, holding up a finger to show they have an answer.
  • Wait time is crucial. Wait until most of the students have an answer.
  • If you find that no one is answering, the problem may be too difficult. You are allowed to adjust! “I’m not seeing many thumbs on this one. Let’s pause on this one. I want to try a different problem first…”
  • Ask students to volunteer answers. Accept all answers – whether they are correct or incorrect – and do your best to keep a good poker face while writing them all on the board for students to consider.
  • Ask, “Who wants to defend one of these answers?”
  • Students share their strategies and justifications with their peers. This is powerful because it allows you to share authority with your students in determining whether an answer is correct.
  • Allow yourself and students to make mistakes. Use them as opportunities to learn.

This is where I got to make my second plug for Intentional Talk. While watching the four videos earlier, teachers were amazed at all of the mathematical ideas the students were sharing. We talked about how this is a learned skill for most students, which means they need our support in learning what and how to share. That’s where talk moves come in! We read another excerpt from chapter 2 of Intentional Talk, and then we watched a short video to see talk moves in action.

Video

I actually embedded talk moves throughout our PD sessions. I printed each one out on a piece of paper and posted them front and center so teachers would see them for the full 3 hours.

Talk-Moves-2

I told the teachers if they wanted to start using talk moves, they should post them in their classrooms. There are a lot of talk moves, and it can seem intimidating to start using them. Putting them up on the wall gives teachers a visual reminder to look at any time they want.

Having them posted is also a great way to get students to start using them. I shared one strategy I’ve seen where a teacher focuses on using one talk move throughout a lesson to help students learn and practice that particular talk move. (Check out this video, specifically around 1:13 to see an example.) Over time the students will have the opportunity to practice each talk move individually, and then they can start to use them as needed during number talks.

Or who knows? Maybe they’ll start being used in other areas of math or other parts of the school day?

My secret hope.

Next we talked about the role of mental math. We looked at this from two angles – efficiency and place value / quantity. This is one time where I provided different examples in the K-2 and 3-5 sessions. Well, I provided different examples when discussing efficiency, but I opted to keep the same example when discussing place value / quantity because I thought it would resonate with both groups.

Here’s what K-2 talked about for efficiency. The goal here was to illustrate that while the standard algorithm leads to the correct answer – that is never in question – it is much more cumbersome for this problem than a strategy such as using landmark numbers.

In the 3-5 session, we looked at the following multiplication example. Again, the standard algorithm will work – that is never in question – but other strategies will also lead to the correct answer and they may do so more efficiently.

Next we talked about place value / quantity. In both sessions, I shared the following problem which was posed to a group of 30 or so teachers in a PD session I attended back in my first or second year of teaching. Out of our entire group, only 2 teachers solved the problem by noticing 100 is 2 away from 98. The rest of us had used the standard algorithm. Until that moment at the age of 25, I had never in my life considered doing anything other than the standard algorithm for every single computation problem I solved. It was a life changing moment.

We wrapped up our discussion by revisiting the definition of procedural fluency and how number talks work to help students develop true procedural fluency.

Component-4-8-K-2

We finally come to the last number talks component, purposeful computation problems.

Component-5-1-K-2

Again, this is a time where we varied the examples for the K-2 and 3-5 sessions.

Here’s the K-2 problem string:

Component-5-2-K-2

And here’s the 3-5 problem string:

Component-5-2-3-5

The key understanding we wanted to convey here is that a mixture of random problems do not lend themselves to a common strategy. Sure, students will be doing mental computation practice, but the disconnectedness of the problems does not create a common focus for a number talk discussion.

We also emphasized that these problem strings may bait the hook for certain strategies, but there are no guarantees students will bite. There are good chances sure, and even better chances because of our purposeful planning, but never a guarantee. And that’s okay. We just need to be aware of that.

By this point we had spent a lot of time talking so we shifted gears and watched a couple more number talks videos. This time our goal was to look at vertical alignment. In the K-2 session we watched a video from Kinder and 2nd grade. In the 3-5 session we watched a video from 3rd grade and 5th grade.

Vertical-K-2Vertical-3-5

I knew we weren’t going to have time to dive deeply into all of the various computation strategies students might use. There are lengthy chapters devoted to them in Sherry Parrish’s Number Talks book, and they’re chock full of great information. This is a topic I will likely spend more time on during the session I’d like to plan for next summer. For now, I wanted to at least touch on the idea that math concepts build on each other and how this plays out during number talks.

Now that the teachers had seen 6 number talks, 7 if you include the one we did together at the start of the session, we wanted to stop and talk about the role of models and tools.

Models

It was impressive the variety that we saw – dot images, five and ten frames, rekenreks, hundred chart, number lines, and equations. There are a lot of great ways for students to use tools to think about computation and to show their thinking.

Because the videos from grades 3 and 5 had mostly shown symbolic representations, we did watch an extra video in the 3-5 session that showed how a 5th grade teacher used a number talk to help her students tackle misunderstandings about representing division using arrays.

Models-2

By this point in each session we were very much behind schedule. We had wanted to give teachers the chance to practice recording student thinking like they’ll have to do when their students share their strategies in a number talk. Unfortunately, we could tell there just wasn’t going to be enough time. In the first two sessions, I was annoyed that we didn’t get to it, but by the end, I think it worked out fine that it got cut. I don’t want to rush that activity, so it tells me that it is appropriate to wait and spend the right amount of time doing it in our summer session.

Even though we had to skip the activity, we did briefly talk about anticipating student thinking.

Nearing the end of our session, we wanted to close with some tangible steps for getting started implementing number talks once the teachers left and went back to their campuses.

We let them know where they can find pre-planned number talk strings in Sherry Parrish’s book. We showed them a document located in our curriculum guides where we collect various resources about number talks. It includes links to videos, articles, planning templates, etc. We closed by sharing final tips and advice for getting started.

And then we were done.

After the session was over, I sent out a link to a short survey to collect feedback from participants. We heard back from 20 folks. In case you’re considering leading a number talks PD, I thought I’d share their feedback so you can see what they liked and what they suggest we change for future sessions. Why not learn from my experiences?

1. What worked well in the Number Talks PD session?

  • Loved the videos – being able to see a number talk in action. I liked that we began the PD with a number talk.
  • There was lots of great information presented. It was nice to see number talks for various grade levels and to talk with teachers from other campuses.
  • Watching the videos and moving through the charts to debrief. Tying the number talk work to the Process Standards.
  • I liked reflecting on the video clips and observing them from 4 different perspectives. It really made me notice what was going on.
  • The clips and talking about each element of Number Talks reinforced what I am currently doing and answered some questions I had about certain parts.
  • You explained everything in great detail. I love the rotation on different lenses. The discussion was amazing and watching the videos from different grade levels really demonstrated the building blocks from K-5 and beyond.
  • Such a great PD! Loved all the videos and examples.
  • Videos and discussions with table groups were very helpful. I can’t wait to start using it.
  • Knowledgable instructor, great examples, great discussions facilitated nicely. Tied directly to NCTM standards, ARRC, TEKS, etc. Clearly showed vertical alignment and expectations.
  • Liked the jigsaw talk about the videos and examples.
  • Brian demonstrating a Number Talk. Sharing after the videos really helped us to put ourselves in the shoes of the teacher, student and classroom community. I liked doing each one individually so that we could focus on that area. It was also great to see the vertical alignment of Number Talks.
  • Videos were helpful; lots of dialogue/discussion among the group
  • I loved it all! I would like to do almost the exact same training for the staff at my school.
  • I knew the theory behind number talks and how they can helps students improve their numbers sense/computational fluency, but I like how this was practical for getting it started in the classroom.
  • Watching the classroom examples was helpful in visualizing and considering how this looks across all grade levels. It also helped in thinking about different ways in which the number talks could be presented.
  • Your pacing was excellent. It was great to see the videos at the different levels and with a different lens each time. The Talk Moves was also helpful to guide teachers.
  • Frequent movement, lots of discussion, the video slides, important information on the slides – not a lot of words/research/etc., lots of student examples, candy!!!
  • The videos helped to see examples of Number Talks, and using different lenses to view them was a great strategy.

2. What could we change to make this PD session more effective?

  • It was a well prepared and informative session. I am looking forward to doing number talks better in my classroom this year!
  • I think it would be nice to hear more from teachers who are already using it in the district just for some more direct input and ideas. It also would have been nice to be able to practice doing a few.
  • From classroom observation – encouraging teachers to understand the power of a number talk vs teaching a trick – especially one that will expire. Maybe a smidge of time to practice the talk moves – a few typed scenarios to read to get used to using the language, hearing it….
  • Just mentioning where to find the resources on the ARRC, instead of trying to explain what’s on the links.
  • The only thing I would possibly add is time to plan out a number talks for the current skill or every group plan a different one so we can have a bank of talk plans.
  • I thought it was great!
  • The only thing I could think to add would be to provide a week’s worth of number talks ready to implement in the classroom. (description, any displays or reproducibles already copied or on cardstock, etc) This would be assuming that the teacher needs to start at the beginning of the Number Talks continuum, or appropriate number talks for the current or upcoming math unit.
  • I would like a few starter ideas to take with me. Just some basic types of number talks like; dots, equations, and number line ideas.
  • This is the best PD I’ve ever been to with RRISD. I’m able to implement it asap.
  • I think this PD was perfect!
  • I thought it was great! The only thing might be to have grade levels discuss exactly what problems they would do for a number talk.
  • It would have been great to get with grade level teams within the PD to think about how these could work into current units. I have not used number talks quite like these before, but would love to start using them. It’s difficult to think of how this will fit in with our current fractions unit – perhaps decomposing fractions in different ways? It may be helpful to offer Number Talk planning sessions for grade levels – what we use and create could also be linked within units on the ARRC.
  • Nothing, it was great!
  • Print out the presentation slides as a notes handout so we can make notes next to the slides

Thoughts from Brian:

This definitely reinforces why I need a second session. It would be great in the summer if we could do a full day session so the first 3 hours could be an introduction, while the next 3 hours could dive into practice doing a number talk, recording student thinking, talking about computation strategies, and planning first number talks.

I have an idea for an activity where teachers work in trios to take turns leading a number talk. The other two people in the trio will act as students. It’s not the same as a class of 22 students, but at least the teachers would have the chance to practice recording someone else’s strategy who is telling it to them live and in person. They could also practice the talk moves a bit since there are three of them.

Knowing that teachers would want to hear from other teachers in our own district, I recently had everyone in the Math Rocks cohort I’m leading to write a blog post reflection about doing number talks in their classroom. These blog posts are all collected in a document that our teachers can access whenever they want to gain perspective from someone else in the district. If you start with Kari Maurer in the table, she and all of the folks below her are from my district.

3. What support do you need from the Teaching & Learning Department to help you succeed in implementing number talks?

  • It seems covered well! Thanks for putting Number Talks in the ARRC!
  • I know our coaches are very willing to help with the number talks. I’m excited to implement them in my classroom! Thanks!
  • Definitely summer PD Offering help to plan a number talk
  • I think just the book to help me have some numbers already to go. Perhaps coaching observations to help me know if I need to add anything or do anything differently.
  • Time 🙂 to plan for it
  • More PD, seeing Number Talks in action at one of our schools in RR. Or have a veteran Number Talk teacher run a talk with kids who have never been exposed to NT.
  • Resources are always great, but there’s already a lot available!
  • I won’t know for sure until I’ve had a chance to explore the tools available and look personally at the books referenced. So far I feel confident in my abilities to begin quickly and consistently. Thank you.
  • Number talk plans added to the ARRC to correspond with ARRC timeline would be amazing!
  • I know the skills covered during a Number Talk can vary, but putting samples in each unit under the Computational Fluency section might be helpful to give teachers ideas.
  • Specific number talk examples linked within units on the ARRC.
  • I know I can call or e-mail you as needed. Thank you!
  • The extra copies of the book on each campus is very helpful! Maybe a training/PD on that other book you discussed – Implementing Strategies that Work (not sure on title)?

In Closing

Whew! Writing up this blog post was almost as intense as leading the four PD sessions. If you made it this far, I tip my hat to you. I hope hearing about my experiences helps you in some way in your own work. If you have any questions, I’d love to discuss them in the comments.

And now I’m off to continue planning for 12 hours of fractions PD I’m leading this week based on the book Beyond Pizzas and Pies. The fun never ends!