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
The Structures and Routines of a Lesson Set Up the Task Set-Up of the Task 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. 2012 University of Pittsburgh
The Donut Task Amanda Smith District: Lebanon School District Grade: Kindergarten 1 2 3 4 5 6 7 8 9 This, this is it. (Student shows 3 counters and 4 counters.) Do you agree with Jay Clayton? (Teacher puts up 3 fingers and 4 fingers and engages students in counting all.) Yes 7 So what should we do? What should we do down here? What we have up here that s what we write down here. Oh. Can you show us? Uh-huh. How many did Cooper say? He had 3 and 4 more. 10 7. 11 12 13 14 15 How do you know? Because 3 + 4 = 7. (Student points to 3 counters and 4 counters.) 3 + 4 = 7. Do you agree with that? Yeah. Yea, alright. Good job, Alex. Thank you. GRAPHIC SCREEN: Are three chocolate donuts and four vanilla donuts more or less than four vanilla and three chocolate donuts? 16 17 18 If I can think about my problem as 4 vanilla and 3 chocolate can I think like that? (Teacher moves set of 4 counters from the right to the left side and 3 counters to the left to the right.) 19 Yes because you can cause it still makes 7. 2013, 2014 UNIVERSITY OF PITTSBURGH Clip ID 2390
20 21 22 23 24 25 26 Claire says that still makes 7. Do you agree with her? Yes. Oh, Claire, can you go show us? If 4 vanilla were over here and 3 chocolate were over here and we switched them, it would still make 7 but it just got switched around. (Student points to the counters.) She said, Yetzaira, she said it got what? 27 3 plus 28 29 30 31 Who heard what Claire said? It got Will? Switched around. How would we write that? I know. 32 Let s see. How would we 33 34 35 36 37 38 If we wrote 4 and then we wrote a plus sign and then we put 3 then we would put we would put equal and then we would put 7 again. (Clair points to the display on the overhead.) Oh. So Claire says that we would do it like this. 4 + 3 = 7. Evan, what are different? What do you notice? (Teacher records 4 + 3 = 7.) This 3 is to the right and this one is to the left. 39 40 41 42 43 44 Alright. So you re telling me that it should look like this. It should look like that 3. So is that what you re thinking? So here we have how many? (Teacher writes 3 correctly.) 4. 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.) 45 46 47 48 Yeah. Awesome. Great job, boys and girls. Ok now, Cooper. Let s look back down here at what you drew for us. Do you notice anything, Cooper, about what you drew on this side and what you drew on this side in relation to our equations? Hmmm. 2013, 2014 UNIVERSITY OF PITTSBURGH 2
49 50 51 52 53 54 55 56 57 58 59 [End of Audio] Can you tell us? (Teacher points to Coopers drawing of circles showing 3 + 4 and 4 + 3.) There is more over there. You can just tell us, Renee. I had 3 down; that s for the chocolate. And 4 down; that s for the vanilla. Then Then what did you draw here? The vanilla on the top and the chocolate on the bottom. Is that the same as our equations? Yes, ma am. So, Cooper, were you already thinking that 3 and 4 and 2013, 2014 UNIVERSITY OF PITTSBURGH 3
The Donut Task Amanda Smith District: Lebanon School District Grade: Kindergarten 1 2 3 4 5 6 7 Tamika goes to the donut shop and she gets 3 chocolate, 2 vanilla, and 2 sprinkled donuts. Ooh, okay. So let s think. Tamika gets 3 chocolate, 2 vanilla, 2 sprinkled. Alright, now I want you to write an equation for what Tamika gets at her donut shop. Go ahead. Draw it. Show me. 7. Can you show me? How do you know it s 7? Because. 3 + 2 and 2 = 7. 8 Oh! So what kind of symbols can you put here to make an equation? 9 I know. I get my 2 10 11 12 13 14 15 16 What did you discover when you read that equation to me? I heard you say, what? 3 + 2 and 2 = 7. (Teacher points to the equation.) How can that be? 3 + 2 + 2 = 7 at the same way. It s the same way. What do you mean by the same way? Because. Because if you count numbers 17 18 19 20 21 So our first equation was 3 + 4 = 7 and our second equation is 3 + 2 + 2 = 7. (Teacher points to the equations on the board.) So We had 3, 2, and 2, and our original problem was 3 and 4 = 7. What do you see about the picture, Tyler? Tyler, what do you notice about our picture? How can the 4 and the 2 and the 2 be related? 22 23 Because there s two 2 s and two things. He said two 2 s make what? 24 4 25 He said 2 and 2 make (Points to the two sets of two on the overhead.) 2013, 2014 UNIVERSITY OF PITTSBURGH Clip ID 2394
26 4. 27 28 29 30 31 32 4. Do you agree with that? Yes. So can that be the same? Can 3 + 4 be the same as 3 + 2 + 2? (Moves the counters as she talks about each expression.) Yes. And they both equal what? 33 34 7. [End of Audio] 2013, 2014 UNIVERSITY OF PITTSBURGH 2
Types of Questions in Mathematics Teaching Question Type Gathering information Probing thinking Making the mathematics visible Encouraging reflection and justification Engaging with the reasoning of others Purpose These questions ask students to recall facts, definitions, or procedures. These questions ask students to explain, elaborate, or clarify their thinking, including articulating the steps in solution methods or completion of a task. These questions ask students to discuss mathematical structures and make connections among mathematical ideas and relationships. These questions reveal deeper insight into student reasoning and actions, including asking students to argue for the validity of their work. These questions help students gain understanding of each other s solution paths and thinking, and lead to the co-construction of mathematical ideas. Examples How many pieces of fruit did the caterpillar eat on Friday? Can you show me how you counted the fruit? 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. 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? 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? 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 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? Sentence Stems for Students 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 How did you solve 3 + 4? Student says: I counted 4, 5, 6. You say:
Revoicing #2 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 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: