Number Talk

Instructions

Note: Practice 3 of 5 in youcubed Mathematical Mindset practice collection.

To the Teacher

Number talks honor the fact that we all see math differently and that these differences are interesting and should be respected. Number talks also help students learn flexibility with numbers and how to calculate without paper and pencil.

With number talks, students have a chance to think through their understanding of numbers and explain their reasoning. In the number talks we did with our students, they had a chance to think about multiplication problems. The problems allowed for students to think flexibly about multiplication and develop number sense through their reasoning and the reasoning of their classmates.

We love when there is more than one answer because making and discussing mistakes lead to much more learning, and it also allows us space to give mindset messages about mistakes. When we did the 12×15 number talk, there was a student who got 168, and as she was explaining her thinking, she stopped and said, “Oh, wait! I made a mistake.” Jo’s response was, “That’s great! That means you have synapses firing in your brain because you made that mistake.” Jo then invited her to explain her thinking when she made the mistake so that the class could understand what she did (see video here). This was an important moment because the student who was sharing, and the rest of the class, saw that her thinking was respected and her mistake was celebrated.

Remember to value mistakes and say things like “This is great we have three different answers; we will have a really good discussion.”

Launch (2-5 minutes)

• Introduce students to the purpose of the number talk. Explain that students will see a number problem and be asked to determine the answer without paper or pencil. They will be asked to share their answers and to describe how they completed the calculation.

Explore (2-5 minutes)

• Show a number problem, e.g., 17 x 15. You may also choose to start with a two-digit number multiplied by a one-digit number like 21 x 3. (The first is appropriate for middle school or higher, the second might be used for elementary). Ask students to solve the problem using as many strategies as they can think of without using pencil and paper.
• Ask students to quietly place their thumb at chest level to signal when they have come up with at least one solution.
  • This is a better way of signaling to you than raising their hand. Hands shooting up can be intimidating and/or distracting to other students.
• Once you have given students enough time to think on their own about strategies, ask students to share their answers as you record them on the board.
  • If there are multiple answers, put all of them on the board and do not identify the correct answer or label any answer as incorrect. The purpose of the number talk discussion is for students to share and justify their answers. Students will oftentimes identify and correct mistakes on their own during the discussion.

Discuss (8+ minutes)

• Start by inviting students to share strategies by saying something like, “Who would like to justify their answer?” If there are different answers ask, “Who would like to defend one of these answers by sharing how you found it?”
• As students share, record their strategies and label their numerical representations with their name. Using the student’s name to label the representation of their thinking on the board allows students ownership over their particular strategy.
  • Capturing a student’s method can be challenging. Make sure to ask clarifying questions if you do not understand. Number talks are about communication. It is a good time to model how to interact when you do not understand what is being said. This is a great time for students to see you struggle and work to understand what is being shared. This is important modeling and it is also the reason we do not ask students to come to the board to share their thinking.
  • If students are challenged with finding or sharing methods, it is OK to introduce a method and share it with them. We recommend you do this by saying the method you are sharing is one you got from another student. This is an important message for them and a time when they should not see you as the expert in the classroom. By sharing a method created by another student you maintain the culture that you are a community of math learners.
  • Record students’ work horizontally rather than the traditional vertical method that students are taught. This will help students make sense of the numbers rather than working from the traditional algorithm. If a student says they used the traditional algorithm, ask the student to describe what they saw and record accurately. Then ask for other strategies.
  • It is also important to create visuals for student strategies. Choose a strategy and model how to draw a visual representation for the numerical calculations. Using color is helpful when creating visuals for student strategies. After creating a visual, ask students to choose a different strategy and create a visual that represents the calculation. (See PDF for examples of recorded student strategies to the problem 18 x 5.)
• To accurately represent students’ thinking with your numeric and visual representations, continually check in with the student who is sharing and ask them questions like:
  • Is it like this? (Referring to a part of your representation)
  • Is this what you saw?
  • Is it a little bit like this other one? What was different about it?
  • What did you do after that?
  • Maybe we could draw this one out because that would be helpful. Does this look like what you did?
  • Do you feel like this represents your thinking?
• A main goal during number talks is to get as many students sharing as many strategies as possible. One of the things we do to encourage more students to share is to invite more strategies by asking, “Did anyone do it differently?” “Did anyone see it differently?”

Extend

• Ask students to sketch the visual for a strategy.
• Ask students to solve a new problem using one of the strategies shared.

Look-Fors

• How are students engaging with mistakes? Multiple answers in number talks are an opportunity to honor and discuss mistakes. It is important to make sure that you keep your responses to right and wrong answers the same. When students recognize their teacher’s reactions and can connect them to right and wrong answers, they become fearful of their ideas being respected and will stop sharing.
• How are students taking numbers apart and putting them together? There are many ways to create equivalent expressions by composing and decomposing numbers. This is called number flexibility and it is very important for number sense and algebra. The more students experience number flexibility the more creative, confident, and fluent they will become. When you start number talks for the first time, you might notice students not sharing a variety of strategies and not breaking apart numbers beyond making 10s. This is likely connected to an absence of number flexibility. When students see different ways to compose and decompose numbers when doing a calculation, they often say, “We thought that wasn’t allowed.” Understanding that numbers represent quantities that can be redistributed and arranged is one of the reasons we love number talks.
• Which students are sharing strategies? While the number talk problem itself is often a simple looking arithmetic problem, there are many ways of solving the problem. Like most activities, some students participate vocally and others do not. This is not the kind of activity in which every student needs to share. There are many ways to think about engaging students who do not offer strategies. You might invite them to use one of the strategies shared on a new problem, create a visual for a strategy, or complete a reflection in their journal.

Reflect (5 minutes)

• Acknowledge the group for honoring all approaches to the problem as well as accepting and learning from mistakes.
• Ask students to reflect on their experience in their journal with a prompt like, “What is something you learned you could do with numbers that you did not know before?”

Source

This is a practice developed by prominent practitioners including Sherry Parrish, Ruth Parker, and Cathy Humphreys. It is recommended by Jo Boaler and featured on the website of youcubed, a center at Stanford University that she leads. In addition to classroom ideas and videos, youcubed offers a variety of resources for mathematics educators, including research summaries and professional development.

Evidence That It Works

Research has shown that students who learned about growth mindset with regards to mathematics reported more positive beliefs about math, were more engaged in math class, and did better on standardized math achievement tests. Mindset interventions in math benefit all students, but have demonstrated even more power for groups that may be more affected by myths about math learning, including girls, English language learners, and economically disadvantaged students.

In addition, a four-year study of high school students in different types of math classes showed that the students who learned math in mixed-ability classrooms that emphasized cooperative group work, open problem-solving, and the use of multiple strategies–compared to those in traditional math classrooms, which were often ability-grouped and focused on teacher lectures and individual work–demonstrated greater gains in math achievement and greater reductions in achievement gaps, enjoyed math more, and treated each other with more respect, support, and equity.

Why Does It Matter?

A substantial body of research has indicated that students who have a growth mindset about intelligence–who believe that, with effort, intelligence can be changed over time–are more likely to do well academically.

Importantly, evidence shows that growth mindset can be learned: in a nationally representative study, students who were taught about a growth mindset of intelligence went on to earn better grades (especially if they started out lower-achieving) and select more challenging classes. Grades improved even more in schools with more supportive learning climates, in which peer norms supported the growth mindset message.

Though much of the research on growth mindset has to do with beliefs about intelligence, other research suggests that social and emotional growth mindsets (e.g., believing that personality, emotions, etc., can grow and change) can reduce bias and promote well-being, social competence, and prosocial behavior.