## Mathematical Discourse

In Principles to Actions (2014), NCTM proposes 8 principles for teachers to implement in their teaching of mathematics. The fourth principle is “Facilitate Meaningful Mathematical Discourse”, while principle number five is “Post Purposeful Questions.” One of the best ways I have discovered this year to implement these practices it to introduce “Number Talks” to the teachers I work with. Although there are several things that may be referred to as Number Talks, I’m going to describe a specific protocol that we have found very successful.

These Number Talks are based on the books by Sherry Parrish. I was able to acquire Number Talks: Whole Number Computation Grades K – 5 for my Elementary math teachers, and Number Talks: Fractions, Decimals, and Percentages for my Middle school math teachers. These books are a great resource with some carefully scripted sets of problems for the number talks.

## So what is a Number Talk?

A Number Talk is about a 15 minute conversation with students about numbers. When I do a number talk with elementary students, I have them sit on the floor near a board or some chart paper, so that they can see the things I am writing. These problems are designed mostly for mental computation and thinking.

Students are instructed that they cannot raise their hands. The first time I say this, I get some very puzzled looks from them! They are always taught to raise their hands! Instead, I explain that I want them to put their fist on their chest, and when they have an answer and strategy, simply put a thumb up. However, they then need to think about a second strategy, and if they are able to find a second strategy put a finger up with their thumb.

This protocol is fabulous for a variety of reasons. Every class has “Susie” who is the first one with her hand raised with an answer. Once Susie puts her hand up, many students shut down, because they “are always too slow”, or “Susie already has the answer”. In addition, Susie shuts down – the thought going through her mind is “I got the answer! I got the answer!” By having them simply show a thumbs up, the rest of the class is not distracted by hands waving, and those who have an answer still have a task to work on.

Once there has been sufficient time, I will then start asking for answers. Recently I added a second piece – once a student gives an answer, I’ll ask for those that agree to signal that with a particular motion – putting up the thumb and little finger and moving the hand back and forth is a sign for “Me Too”. That way, at a quick look I can see how many agree.

But the answer is not all we are after – I want to have them explain their thinking. There are often multiple strategies that students can use to arrive at an answer, and I want students to share a whole variety of ways they thought about the problem. I have been amazed when I hear students give a strategy that I myself had not thought about! It’s a wonderful experience to hear students giving all these different strategies that lead to the same answer.

## Example

Recently I did a Number Talk in 3 different third grade classes. I did the same sequence of problems in each class. I kept each problem/answer combination on the board so they had it as a reference. Here is what it looked like:

### 2 x 125 =

Students would give various strategies, such as:

“I did 2 times 100, and then 2 times 20, and then 2 times 5. When I added up I got 250.”

“I knew that 2 times 25 was 50, and 2 times 100 was 200, so I add them and get 250.”

The next problem was:

### 6 x 100 =

Most students talked about “counting by 100, 6 times”. Some talked about “adding up 100, 6 times.” I even had one  young boy said that he counted by 6, 100 times. I thought that a little unusual, so I asked “Did you really count by 6, or did you just know that you COULD count?” He said he knew that it would give 600.

The third problem:

### 6 x 20 =

I saw some similar strategies, such as counting by 20 (20, 40, 60, 80, 100, 120). Another student said “I knew that 6 x 2 was 12, so I added a zero to get 120.” (That one I had to talk a little about. I told the student that every time I added 0 to 12 I only got 12, making the point that we are not “adding zero”, and that we had to work on another way to express it.)

The fourth problem:

### 6 x 5 =

This was kind of fun, because some counted by 5, a total of 6 times, while others counted by 6 a total of 5 times. Others talked about just adding up all the fives, while still others said they “just knew it.”

The final problem in the set is really the culminating piece. Skipping this last problem defeats the careful construction of questions that the students have just done.

### 6 x 125 =

A whole variety of answers come up with this, and some of them are incorrect. A student who gives an incorrect answer will then hear other students talking about THEIR strategy. They are then given the option to “revise” their answer if they wish. For this problem, some students talked about 6 x 100, 6 x 20, and 6 x 5. This is much like the first problem. Others found different ways to explain. The two explanations that I was thrilled to hear were:

“I looked at the 2 x 125 = 250, and knew that to get 6 x 125 I needed to do the 250 three times. That makes it 750.”

“I saw 6 x 100 = 600, 6 x 20 = 120, and 6 x 5 = 30 from what we just did, so I added 600 + 120 + 30 = 750.”

These two explanations showed that the students were processing facts that they had just determined, and were applying them to knew problems.

## Final thoughts

As I spend more time doing number talks with students I am more and more impressed with their ability to explain their thinking. The teachers who observe this are also very impressed. Watching me do a number talk with their students allows them to watch carefully their own students thinking and to make notes about it, to better know where they are struggling.

One concern teachers have is this: “I am already struggling to get everything done – how can I add more things in?” In a recent series of grade level meetings that I hosted we talked about this. I think an important consideration is “how much learning happens in a 15 minute period of time in my regular instruction?” I then challenged them to think about whether these number talks would increase the amount of learning happening, still within the 15 minute period. If that happens, then we end up further ahead than before doing the number talks.

The links above to the two books that I have acquired for my teachers will take you to those books on Amazon. It’s just one of the easiest places I have to provide links to these materials, and I hope you give them a try and see how they work.