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!