Tag Archives: Kathy Richardson

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!

Represent! Part 1

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

I kicked off the chat with this quote:

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

and this question:

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

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

I immediately followed up with this question:

04

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?

05

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.