(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
- Use-Ten Strategy for Multiplication
- Doubling Strategy for Multiplication
- Build-Up Strategy for Multiplication
- Build-Down Strategy for Multiplication
- Use a Rule Strategy for Multiplication – There isn’t a video for this one. This thinking strategy involves multiplying with 0 or 1.
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.
- Sample Number Talks – Use-Ten Strategy for Multiplication
- Sample Number Talks – Doubling Strategy for Multiplication
- Sample Number Talks – Build-Down Strategy for Multiplication
- Sample Number Talks – Build-Up Strategy for Multiplication
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:
- Ask Yourself Questions
- 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.
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.
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.