The Donut Task

1. Dion chooses 3 chocolate donuts and 4 vanilla donuts. Draw a picture and write an equation to show Dion’s donuts.

2. Tamika has 4 vanilla donuts and 3 chocolate donuts. Draw a picture and write an equation to show Tamika’s donuts.

3. Tamika claims that she has more donuts than Dion. Who has more donuts, Dion or Tamika? Draw a picture and write an equation to show how you know who has more donuts.

Huinker, D. and Bill, V. Taking Action in Elementary School: Implementing Effective Mathematics Teaching Practices, NCTM, 2017

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© 2012 University of Pittsburgh

The Structures and Routines of a Lesson

The Explore Phase/

Private Work Time Generate Solutions

The Explore Phase/Small-Group Problem Solving

1. Generate and Compare Solutions

2. Assess and Advance Student Learning

Share Discuss and Analyze

Phase of the Lesson

1. Share and Model

2. Compare Solutions

3. Focus the Discussion on Key

Mathematical Ideas

4. Engage in a Quick Write

MONITOR: Teacher

Selects Examples for

the Share Discuss based on:

• Different solution

paths to the same

task

• Different representations

• Errors

• Misconceptions

SHARE: Students explain their methods,

repeat others’ ideas, put

ideas into their own

words, add on to ideas

and ask for clarification.

REPEAT THE CYCLE FOR

EACH SOLUTION PATH

COMPARE: Students

discuss similarities and differences between

solution paths.

FOCUS: Discuss the

meaning of

mathematical ideas in

each representation.

REFLECT: Engage

students in a Quick Write or a discussion of the

process.

Set Up the Task Set-Up of the Task

©2013, 2014 UNIVERSITY OF PITTSBURGH Clip ID 2390

The Donut Task Teacher: Amanda Smith District: Lebanon School District Grade: Kindergarten

Student: This, this is it. (Student shows 3 counters and 4 counters.) 1

Teacher: Do you agree with Jay Clayton? (Teacher puts up 3 fingers and 4 fingers and 2 engages students in counting all.) 3

Students: Yes…7… 4

Teacher: So what should we do? What should we do down here? 5

Student: What we have up here that’s what we write down here. 6

Teacher: Oh. Can you show us? 7

Student: Uh-huh. 8

Teacher: How many did Cooper say? He had 3 and 4 more. 9

Student: 7. 10

Teacher: How do you know? 11

Student: Because 3 + 4 = 7. (Student points to 3 counters and 4 counters.) 12

Teacher: 3 + 4 = 7. Do you agree with that? 13

Students: Yeah. 14

Teacher: Yea, alright. Good job, Alex. Thank you. 15

GRAPHIC SCREEN:

Are three chocolate donuts and four vanilla donuts more or less than four vanilla and three chocolate donuts?

Teacher: If I can think about my problem as 4 vanilla and 3 chocolate – can I think like 16 that? (Teacher moves set of 4 counters from the right to the left side and 3 17 counters to the left to the right.) 18

Student: Yes because you can…’cause it still makes 7. 19

©2013, 2014 UNIVERSITY OF PITTSBURGH 2

Teacher: Claire says that still makes 7. Do you agree with her? 20

Students: Yes. 21

Teacher: Oh, Claire, can you go show us? 22

Student: If 4 vanilla were over here and 3 chocolate were over here and we switched 23 them, it would still make 7 but it just got switched around. (Student points to the 24 counters.) 25

Teacher: She said, Yetzaira, she said it got what? 26

Student: 3 plus – 27

Teacher: Who heard what Claire said? It got – Will? 28

Students: Switched around. 29

Teacher: How would we write that? 30

Students: I know. 31

Teacher: Let’s see. How would we – 32

Student: If we wrote 4 and then we wrote a plus sign and then we put 3 then we would 33 put…we would put equal and then we would put 7 again. (Clair points to the 34 display on the overhead.) 35

Teacher: Oh. So Claire says that we would do it like this. 4 + 3 = 7. Evan, what are 36 different? What do you notice? (Teacher records 4 + 3 = 7.) 37

Student: This 3 is to the right and this one is to the left. 38

Teacher: Alright. So you’re telling me that it should look like this. It should look like that 3. 39 So is that what you’re thinking? So here we have how many? (Teacher writes 3 40 correctly.) 41

Students: 4. 42

Teacher: How many? 4, 5, 6, 7. So can we count on and get 7? (Teacher counts on from 4, 43 touching counters one by one and counting on three more touching her chin.) 44

Students: Yeah. 45

Teacher: Awesome. Great job, boys and girls. Ok now, Cooper. Let’s look back down here 46 at what you drew for us. Do you notice anything, Cooper, about what you drew 47 on this side and what you drew on this side in relation to our equations? Hmmm. 48

©2013, 2014 UNIVERSITY OF PITTSBURGH 3

Can you tell us? (Teacher points to Coopers drawing of circles showing 3 + 4 and 49 4 + 3.) 50

Student: There is more over there. 51

Teacher: You can just tell us, Renee. 52

Student: I had 3 down; that’s for the chocolate. And 4 down; that’s for the vanilla. Then… 53

Teacher: Then what did you draw here? 54

Student: The vanilla on the top and the chocolate on the bottom. 55

Teacher: Is that the same as our equations? 56

Student: Yes, ma’am. 57

Teacher: So, Cooper, were you already thinking that 3 and 4 and… 58

[End of Audio] 59

©2013, 2014 UNIVERSITY OF PITTSBURGH Clip ID 2394

The Donut Task Teacher: Amanda Smith District: Lebanon School District Grade: Kindergarten

Teacher: Tamika goes to the donut shop and she gets 3 chocolate, 2 vanilla, and 2 1

sprinkled donuts. Ooh, okay. So let’s think. Tamika gets 3 chocolate, 2 2 vanilla, 2 sprinkled. Alright, now I want you to write an equation for what 3 Tamika gets at her donut shop. Go ahead. Draw it. Show me. 4

Student: 7. 5

Teacher: Can you show me? How do you know it’s 7? 6

Student: Because. 3 + 2 and 2 = 7. 7

Teacher: Oh! So what kind of symbols can you put here to make an equation? 8

Student: I know. I get my 2… 9

Teacher: What did you discover when you read that equation to me? I heard you 10 say, what? 11

Student: 3 + 2 and 2 = 7. (Teacher points to the equation.) 12

Teacher: How can that be? 13

Student: 3 + 2 + 2 = 7 at the same way. 14

Teacher: It’s the same way. What do you mean by the same way? 15

Student: Because. Because if you count numbers… 16

Teacher: So our first equation was 3 + 4 = 7 and our second equation is 3 + 2 + 2 = 17 7. (Teacher points to the equations on the board.) So…We had 3, 2, and 2, 18 and our original problem was 3 and 4 = 7. What do you see about the 19 picture, Tyler? Tyler, what do you notice about our picture? How can the 20 4 and the 2 and the 2 be related? 21

Student: Because there’s two 2’s and two things. 22

Teacher: He said two 2’s make what? 23

Student: 4 24

Teacher: He said 2 and 2 make – (Points to the two sets of two on the overhead.) 25

©2013, 2014 UNIVERSITY OF PITTSBURGH 2

Student: 4. 26

Teacher: 4. Do you agree with that? 27

Student: Yes. 28

Teacher: So can that be the same? Can 3 + 4 be the same as 3 + 2 + 2? (Moves the 29 counters as she talks about each expression.) 30

Student: Yes. 31

Teacher: And they both equal what? 32

Students: 7. 33

[End of Audio] 34

Types of Questions in Mathematics Teaching

Question Type

Purpose Examples

Gathering information

These questions ask students to recall facts, definitions, or procedures.

• How many pieces of fruit did the caterpillar eat on Friday?

• Can you show me how you counted the fruit?

Probing thinking

These questions ask students to explain, elaborate, or clarify their thinking, including articulating the steps in solution methods or completion of a task.

• I see you wrote 10 + 5 on your paper. Where did the ten come from?

• Tell me about your picture. I see you wrote the days of the week and then drew squares.

Making the mathematics visible

These questions ask students to discuss mathematical structures and make connections among mathematical ideas and relationships.

• Marisa wrote 1+2+3+4+5=15. Is that okay to write an equation with all those plus signs?

• What pattern do you see in the equations 10 + 2 = 12, 10 + 3 = 13, 10 + 4 = 14, and 10 + 5 = 15?

Encouraging reflection and justification

These questions reveal deeper insight into student reasoning and actions, including asking students to argue for the validity of their work.

• I see you put a circle around the 1, 4, and 5. Why did you put these pieces of fruit together?

• What makes 10 + 6 equal to 9 + 5?

Engaging with the reasoning of others

These questions help students gain understanding of each other’s solution paths and thinking, and lead to the co-construction of mathematical ideas.

• Who understands Shyanne’s explanation and can say it back in your own words?

• Can you add on to what Nate’s said?

• Do you agree or disagree with Anne? Why?

Source: Huinker, D., & Bill, V. (2017). Taking Action: Implementing Effective Mathematics Teaching Practices in Grades K-5.

Reston, VA: National Council of Teachers of Mathematics.

Question Stems

Question Stems for Teachers Sentence Stems for Students

• That seems really important, who can say that again?

• Who can say that back in your own words … ?

• What does she mean when she says …? • Who can add on to that explanation …? • Do you agree or disagree with ____? Why? • Turn and talk with a partner about ….

Who can tell the class what your partner said?

• Let’s all try using ____’s approach on this new problem.

• Who has a similar way of looking at that? • Who has a different way? • Let’s look at these two approaches, how

are they similar? How are they different?

• I agree with ____ because … • I respectfully disagree with that because… • I still have questions about… • I’m confused by … • I have a different perspective because … • I connected with what ____ said because … • I chose this method because … • Can I add on to what ____ said about …? • I thought about it the same way because… • When you said …, that really helped me

understand it so much better. • I was wondering… • Could we try that strategy on a new

problem?

Source: Huinker, D., & Bill, V. (2017). Taking Action: Implementing Effective Mathematics Teaching Practices in Grades K-5.

Reston, VA: National Council of Teachers of Mathematics.

An Example of Revoicing 22 S: Because there’s two 2’s and two things. 23 T: He said two 2’s make what? 24 S: 4 ____________________________________ 42T: How many more? 43 S:4. 44T: How many? 4, 5, 6, 7. So can we count on and get 7? (Teacher counts on from 4, (touching counters one by one and counting on three more touching her chin.)

Revoicing #1

Teacher: How did you solve 3 + 4? Student says: I counted 4, 5, 6. You say:

Revoicing #2

Teacher: Tell me about 3 + 4 in your picture. Where can you see 3 + 2 + 2 in this picture? Student says: It is the same. You say:

Revoicing #3

Teacher: Tell me where you see 3 + 4 and 4 + 3 in your picture. Student says: 3 + 4 and 4 + 3 is the same. You say: