I have used the Number Talks from your original book, Number Talks: Helping Children Build Mental Math and Computation Strategies, with my students for a number of years now, and I’ve seen tremendous growth in their ability to work flexibly with numbers. For those who might not be familiar with Number Talks, can you briefly explain what they are and why they are important for our students?
A Number Talk is a five to fifteen minute classroom conversation around purposefully crafted problems that are solved mentally. During a Number Talk teachers ask students to mentally solve problems to help students focus on number relationships, encourage and elicit students’ individual strategies, and help students construct important mathematical ideas. As students share and defend their solutions and strategies, they have opportunities to collectively reason about numbers while building their mathematical understanding.
Number Talks are rooted in Jean Piaget’s theory of learning which centers on the idea that there are actually three different types of knowledge. (We discuss this more in Number Talks: Fractions, Decimals, and Percentages.) Knowledge that has its ultimate source in other people, and changes from society to society, like the names for things or how something is written, is something we have to tell students. It’s called social knowledge. Things like, “This is called the distributive property,” or “You can write the multiplication sign as an ‘x’ but you can also write it as a dot,” are social conventions and can change from society to society or over time.
We ask instead of tell, however, when we are dealing with logico-mathematical knowledge. Logico-mathematical knowledge has its ultimate source in the mental relationships that students make for themselves. There are different ways to solve the problem 16 x ¾ because there is logic involved. When we ask students, “How could you solve this? How do you know?” it’s because we know we are dealing with logico-mathematical knowledge and that the source of this knowledge is students’ own mental relationships.
Understanding the difference between different types of knowledge changed our classroom practice from giving students procedures to asking students to create or “invent” strategies based upon mental relationships they were making.
Teachers operate under a wide variety of instructional routines and utilize many different math programs. How can teachers fit Number Talks into what they are currently doing?
Teachers often wonder about:
- how often to do a Number Talk;
- time constraints;
- the best time to do a Number Talk; and
- connecting the Number Talk to their lesson.
We recommend doing whole-group Number Talks at least three times a week to develop a community of learners that are focused on accuracy, flexibility, and efficiency. A whole-group setting offers an opportunity to exchange and consider a greater range of diverse ideas. Some teachers also incorporate small group Number Talks into their weekly routine to allow them to focus on students with similar needs.
Our instructional time is precious, so we encourage teachers to limit their Number Talk to somewhere between 5 and 15 minutes. We have even set a timer to help keep us mindful of the time! If you only have 5 minutes, consider using an individual problem or using the problems from a Number Talk string over several days. For example, on Monday you might pose 1/2 + 1/2; on Tuesday introduce the second problem in the string, 1/2 + 3/4, and on Wednesday share the final problem, 1/2 + 5/8. Keeping the previous problems and their solutions posted, provides students support as they think about the new problem.
A Number Talk can be used at any time during the day; however, our preference is to begin our math time with a Number Talk. There is something about coming together and exchanging different perspectives in a Number Talk that sets the tone for the rest of our lesson. When we begin with Number Talks, we notice students’ mathematical dispositions shift toward sense making and justification of ideas.
A Number Talk does not always have to connect to your lesson content. For example, if your lesson is focused on geometry, your Number Talk might incorporate problems to address multiplication strategies or “soft spots” you notice from a particular operation.
As an instructional coach who was trying to help my teachers implement Number Talks, I found that some teachers were intimidated because they didn’t understand the strategies themselves and they weren’t sure they would be able to scribe the strategies the students used. What advice can you give teachers who are hesitant to try Number Talks for those reasons? What supports does the book provide to help teachers feel more comfortable as they get started?
We completely understand. It can be uncomfortable to ask students to share their strategies and then not understand their way of solving the problem. The good news is that the more you do Number Talks, the more strategies you hear and the less often you find yourself in that uncomfortable position.
When we first started doing Number Talks, we found ourselves asking co-workers, friends and family members how they would solve the problems we were going to give our students. The more strategies we heard, the more we were able to anticipate what our students might do and the more we were able to plan how we would record those strategies.
We also found that depending on the problem, as we asked several people how they would solve it, there were usually only 3 or 4 common solutions. Anticipating students’ solutions helped build our confidence to open the classroom discussion with Number Talks.
When a student does share a strategy that we haven’t considered before, we usually handle the situation in one of two ways.
- We ask another student if they can explain the first students’ strategy. Sometimes a peer can understand another students’ thinking when we can’t.
- We let the student know that we are having trouble following their solution right now, and we record and put that solution to the side until we can have a private discussion to try to understand their thinking. Then if it’s appropriate, we come back to that discussion the next day with a fresh understanding and acknowledge that student’s thinking.
I find that in classrooms that are the most successful with Number Talks, the teacher has fostered a culture of respect and helped students develop strong mathematical communication skills. How do number talks support mathematical discourse and why is that important?
Number Talks are based on the ability to exchange ideas and consider other people’s points of view. A classroom culture of respect is critical to successful Number Talks. As soon as you ask students questions such as, “What do you think?” or “How do you know?” or “Who would like to defend their thinking?” you are asking students to be vulnerable. That can be frightening unless our classroom is a place where it’s okay to be wrong, where mistakes are viewed as opportunities to learn, and where the class a whole is trying to learn from each other.
Turn and Talk is an important instructional strategy that can help build students’ confidence in sharing their ideas. When you pose a problem and ask students to turn and talk about what they are thinking, everyone gets a chance to share their ideas. Then when you ask students to share, they are not only sharing their own ideas, but ones they have collaborated on and had a chance to solidify by talking them through with a partner. It takes some of the pressure off of sharing your own ideas.
When students are new to sharing their ideas and discussing the reasoning of others, it is often helpful to post a list of open-ended prompts to support students as they learn to engage in mathematical discourse. We have found the following to be useful in our classroom discussions.
I noticed ________________________________________.
Can you explain why you __________________________?
Will it always work if we __________________________?
I understand ________but am confused about _________.
I am confused about ______________________________.
What are some of the reasons that students have such a poor understanding of fractions and decimals?
There are three primary reasons students have difficulty with understanding fractions and decimals.
- A fraction can have multiple representations and interpretations; For example, 3/4 can be represented as a fraction, a decimal, or a percentage. Three-fourths can also be interpreted as a part-whole relationship, a measure, a ratio, a quotient, or an operator. Navigating between these different representations and interpretations can be confusing and problematic for students.
- When a student’s knowledge of fractions rests solely on memorized procedures, their foundation for making sense of solutions when operating with fractions is fragile. Tricks such as “yours is not to wonder why, just invert and multiply” or “keep, change, flip,” may produce quick results but often result in errors when students forget steps to procedures they do not own. By intentionally focusing on building conceptual understanding first, student use fractional reasoning and number sense to decide whether a solution is reasonable.
- One of the biggest barriers to understanding fractions is when students think about fractions as two whole numbers stacked on top of each other instead of as distinct quantities. Examples of students using inappropriate whole-number reasoning are when they add or subtract across numerators and denominators or when they order fractions by focusing solely on the denominator (such as ½, 1/3, 1/4). These kinds of mistakes are not careless errors, but indications of students trying to use what they know about whole numbers to understand fractions. These student misconceptions do not have to be inevitable. Number Talks can be an important vehicle for helping students to make sense of fractions.
The floor is yours! What do you feel teachers ought to know about this new book?
We hope teachers will begin with Number Talks that focus on building fractional reasoning before moving into computation. Students’ ability to reason and make sense of the mathematics is what supports them when they begin operating with fractions.
While the book is organized by chapters, please do not feel you should go page by page and use the Number Talks in the order in which they are presented. Listen to your students and let them guide you as you select individual problems or Number Talk strings. The Number Talk strings are provided as a way to support students as they look for and make numerical relationships; however, it is certainly fine to select an individual problem.
Finally, enjoy the journey! Enjoy learning with and from your students. We hope you will be comfortable becoming a part of the learning community and that your own fractional reasoning and strategies will grow through this process. The strategies in the book were ones we learned from students, so we urge you to resist teaching the strategies by telling.
What a thrill for me to have the opportunity to interview Sherry and share her words of wisdom with you!