In this first post of our 2023 Stenhouse Summer Series, math educator Pierre Tranchemontagne writes about building mathematical understanding and community through Number Talk Images.
“Is there more than one layer?”
I was not expecting this answer from the student.
I mean, I asked a simple question to the class—“How many, and how do you know?”. In the many, many times I had looked at the image above I had never considered the possibility that there might be more than one layer of peaches (Or are they nectarines? Maybe plums?). In my moment of slight panic, I did what a lot of teachers do when they are not sure what to answer.
I fired a question right back.
“Well, what do you think?”
The above is a snippet of a number talk I facilitated in a grade 8 class years ago and, in case you are wondering, the image is from Number Talk Images, a website dedicated to gathering interesting images to be used as launching points for number talks. As the person who created the website, you can imagine I have spent quite a bit of time looking at images and thinking about their use in math class. I have seen firsthand the positive effects images have in number talks. I’ve often wondered why that is…
What is a Number Talk?
We should probably first explore a little about number talks in case they are not a familiar routine for you. Here is a definition that I quite like:
“Number talks are short, 5–15-minute teaching and learning practices that involve students mentally solving computational problems and talking about the strategies they have used. Number talks are powerful tools for developing computational fluency and conceptual understanding at the same time. They focus on sense making as students use number relationships and structures of numbers to mentally add, subtract, multiply and divide. Teachers then facilitate class discussion in which students explain their reasoning and reflect on the reasoning of others. During this time, teachers visually represent student thinking to reflect the relationships and structures a student has used and make their thinking clear to all students. Students with different approaches to a problem are invited to present their thinking to highlight that there are multiple ways to arrive at a solution, promoting flexibility and confidence in working with numbers.” (Number Talks Factsheet from the Queensland Curriculum and Assessment Authority)
Though this is a simplified definition, number talks can be nuanced in how the routine is launched, how the teacher uses questions to engage students in others’ thinking, how the conversation is facilitated, among many other variations and subtleties that are guided by the goals of the teacher.
The Power of Images
In a traditional number talk, the teacher displays a computational problem (or a series of problems one by one, generally called a number string). Often a number talk is chosen intentionally to surface the use of a particular strategy or mathematical concept. A number talk with images may share some of these same goals, but the use of images rather than numbers has many advantages when compared to “naked numbers.” So this brings me back to my wondering—What is it about number talks with images?
Interest and Engagement
Take a look at some of my favorite examples from the Photos section of the website.
Did you engage with these images? Do you find them inviting in some way? How or why? I find it difficult to pinpoint exactly why certain images work well for number talks, but often the colors and organization of the objects themselves are such that students, and adults, are immediately engaged. They are like beautiful invitations to think, talk and wonder about.
For students, aside from the aesthetics of the image, I think part of the appeal is also that the pictures are of objects that often can be found in everyday life, not just at school. They may have seen, held and even eaten them at home, in the grocery store or in their communities. Students have connections to the images and this is powerful.
Making Mathematical Connections
I have come to see number talks as a microcosm of mathematics teaching—like a time-compressed version of what teachers do when they lead longer activities or lessons. Below, you will find a list of the Mathematics Teaching Practices from NCTM. Can you think of moments or examples within a number talk when a teacher might use some of these practices?
Mathematics Teaching Practices
- Establish mathematics goals to focus learning
- Implement tasks that promote reasoning and problem solving
- Use and connect mathematical representations
- Facilitate meaningful mathematical discourse
- Pose purposeful questions
- Build procedural fluency from conceptual understanding
- Support productive struggle in learning mathematics
- Elicit and use evidence of student thinking
(Mathematics Teaching Practices; Principles to Actions- Executive Summary, NCTM 2014)
Together we could probably identify each of these teaching practices within an effective number talk. However, let’s dive into one in particular: Use and connect mathematical representations. Take a look at the following teacher annotations of student thinking around the image of the box of chocolates.
Notice how the teacher represented the wide variety of student thinking while using different mathematical models (visual drawing, number line, ten frame, symbolic representations). In doing so, the teacher created many opportunities to ask questions and have interesting conversations about how representations show student thinking. Here are a few follow-up questions a teacher might ask to prompt connections:
- (re. Cody’s thinking) “Why did I start the jumps at 2 on the number line? Why didn’t I start at another number?”
- “How does the ten frame show the equation 5+5=10?”
- “Could we use the number line to represent any of the strategies? How come?”
These sorts of questions help students establish links between physical representations and symbolic/abstract models. The image of the box of chocolates gives a visual connection to the many equations and representations drawn on the whiteboard.
Here is another example. It’s from a number talk I facilitated with a grade 4 class. I asked students, “How many cupcakes are there? Think about how you know and see if you can come up with different ways of knowing”.
Take a look at the image on the right of the annotations I made of student thinking. What questions could you ask to push mathematical reasoning forward?
What jumped out to me during the number talk was the opportunity to talk about properties and strategies for multiplication. After students shared ways they figured out that there were 30 cupcakes, I asked some questions to help nudge them towards the generalization of the commutative property of multiplication.
- “I see 6x5 and 5x6. We know both are 30 because we didn’t remove or add any cupcakes. Isn’t that weird? How can that be? Why are these two equations the same?”
- “Can we always switch the position of the numbers when multiplying 2 numbers? Why is that?”
I also wanted to get students thinking about the associative property and the “doubling and halving” strategy for multiplication:
- (pointing to the visual representations) “Here is 3x10 and here is 6x5. How did we transform 6x5 into 3x10? The numbers in the equations are different but we know they are equivalent because, again, there are 30 cupcakes for each.”
- “How about 6x5 and 2x15? Do you see them represented? How did we transform one equation to the other?”
In this example, the use of an image in the number talk helped students in sense making. It gave them a visual reference point as to why certain things can be done in multiplication. Properties of operations and numerical manipulations are abstract concepts. Images can help students make connections between thinking strategies and symbolic representations.
Accessibility
“Is there more than one layer? ” This question the student asked about the box of peaches has stayed with me for a long while now. It’s been about eight years since I facilitated the number talk that I described at the start of this blog post. In reflecting on this question, I’ve figured out one of the most important reasons as to why images are so powerful for number talks: accessibility.
Just by their nature, an image invites the observer to see it in their own way. And when an idea is shared out, the image allows others to see it in different ways. Ways that are often personal and sometimes unique.
Images are often less intimidating at the start of a number talk when compared to equations. If you count one-by-one or if you can multiply in different ways, you still have a place in the number talk. I have seen students disengage at the start of a number talk because they quickly realize that they do not have a strategy to compute a given equation. However, with an image, these same students perk up because they have a way in.
They have access and they have an invitation to the math party. (Shout out to Kaneka Turner for this notion.)
So I invite you to try out an image number talk. Notice student engagement. See if students who sometimes disconnect with math are a little more engaged. Take note of the thinking the image prompts. Maybe you’ll find evidence of the power of images that I haven’t described in this blog post. Maybe your students will surprise you with the unique ways they see the image and the connections they make. Heck, if you listen to students closely enough, you might even find other hidden layers of Number Talk Images.
About the Author
Pierre Tranchemontagne is the creator and curator of the Number Talk Images website. A classroom teacher for ten years and a pedagogical mathematics lead at the school board level for five years, he is currently teaching grade 7 and 8 science, math and physical education at Carrefour Jeunesse, a French public elementary school in Rockland, Ontario, Canada. Pierre is incredibly grateful to the many, many people who have shared images and contributed in different ways to the Number Talk Images website. Aside from Number Talk Images, his current thinking revolves around Thinking Classrooms and teacher education pedagogies/mathematical knowledge for leadership. You can connect with Pierre on Twitter at @Pierre_Tranche.
