1. This module was developed by Victoria Bill, University of Pittsburgh Institute for Learning; DeAnn Huinker, University of Wisconsin-Milwaukee; and Amy Hillen, Kennesaw State University. Video courtesy of New York City Public Schools and the University of Pittsburgh Institute for Learning. These materials are part of the Principles to Actions Professional Learning Toolkit: Teaching and Learning created by the NCTM project team that includes: Margaret Smith (chair), Victoria Bill (co-chair), Melissa Boston, Fredrick Dillon, Amy Hillen, DeAnn Huinker, Stephen Miller, Lynn Raith, and Michael Steele. Principles to Actions Effective Mathematics Teaching Practices The Case of Amanda Smith and the Donuts Task Kindergarten Principles to Actions Effective Mathematics Teaching Practices The Case of Amanda Smith and the Donuts Task Kindergarten

2. Overview of the Session Solve and discuss the Donuts Task. Watch video clips of a Kindergarten class working on the Donuts Task. Relate teacher and student actions in the video to the effective mathematics teaching practices. Consider how you might apply specific mathematics teaching practices to your work.

3. Teaching and Learning Principle An excellent mathematics program requires effective teaching that engages students in meaningful learning through individual and collaborative experiences that promote their ability to make sense of mathematical ideas and reason mathematically. National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author. (p. 7)

4. “The Donuts Task” and Connections to the Common Core State Standards “The Donuts Task” and Connections to the Common Core State Standards

5. The Donuts 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. Extensions Tamika changes her mind and she gets 3 chocolate, 2 vanilla, and 2 sprinkle donuts. Draw a picture and write an equation to show Tamika’s donuts. Tamika claims that she has more donuts than Dion because she has three kinds of donuts. What do you think about Tamika’s claim? Who has more donuts and how do you know?

6. Connections to the CCSSM Kindergarten Standards for Mathematical Content Operations and Algebraic Thinking (OA) Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from. K.OA.A. 1 Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations. K.OA.A. 2 Solve addition and subtraction word problems*, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. Kindergarten students work with “add to” and “take from” result unknown situations and “put together/take apart” total unknown and both addends unknown situations Common Core Standards Writing Team. (2011). Progressions for the common core state standards in mathematics (draft): Counting and cardinality, operations and algebraic thinking. http://ime.math.arizona.edu/progressions. National Governors Association Center for Best Practices (NGA Center) and Council of Chief State School Officers (CCSSO). (2014). Common core state standards for mathematics. http://www.corestandards.org/Math/Content/1/NBThttp://www.corestandards.org/Math/Content/1/NBT

7. Connections to the CCSSM Standards for Mathematical Practice National Governors Association Center for Best Practices (NGA Center) and Council of Chief State School Officers (CCSSO). (2014). Mathematics. Common core state standards for mathematics. Retrieved from http://www.corestandards.org/Math/Practicehttp://www.corestandards.org/Math/Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.

8. Connections to the CCSSM Standards for Mathematical Practice National Governors Association Center for Best Practices (NGA Center) and Council of Chief State School Officers (CCSSO). (2014). Mathematics. Common core state standards for mathematics. Retrieved from http://www.corestandards.org/Math/Practicehttp://www.corestandards.org/Math/Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.

9. Ms. Smith’s Mathematics Learning Goals Ms. Smith’s Mathematics Learning Goals

10. Establish Mathematics Goals to Focus Learning Learning Goals should: clearly state what it is students are to learn and understand about mathematics as the result of instruction; be situated within learning progressions; and frame the decisions teachers make during a lesson. “Formulating clear, explicit learning goals sets the stage for everything else.” (Hiebert, Morris, Berk, & Janssen, 2007, p.57)

11. Ms. Smith’s Mathematics Learning Goals Students will understand that: the sum of two or more sets can be combined in many different ways (counting, counting on, or the use of known facts); two addition expressions can have the same sum although the addends appear in a different order (the commutative property of addition); and more sets of smaller amounts can still add up to the same amount as fewer sets with more items.

12. Effective Mathematics Teaching Practice Effective Mathematics Teaching Practice

13. Effective Mathematics Teaching Practices 1.Establish mathematics goals to focus learning. 2.Implement tasks that promote reasoning and problem solving. 3.Use and connect mathematical representations. 4.Facilitate meaningful mathematical discourse. 5.Pose purposeful questions. 6.Build procedural fluency from conceptual understanding. 7.Support productive struggle in learning mathematics. 8.Elicit and use evidence of student thinking. National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author.

14. Effective Mathematics Teaching Practices 1.Establish mathematics goals to focus learning. 2.Implement tasks that promote reasoning and problem solving. 3.Use and connect mathematical representations. 4.Facilitate meaningful mathematical discourse. 5.Pose purposeful questions. 6.Build procedural fluency from conceptual understanding. 7.Support productive struggle in learning mathematics. 8.Elicit and use evidence of student thinking. National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author.

15. Pose Purposeful Questions Effective Questions should: Reveal students’ current understandings; Encourage students to explain, elaborate, or clarify their thinking; and Make the mathematics more visible and accessible for student examination and discussion. Teachers’ questions are crucial in helping students make connections and learn important mathematics concepts. Teachers need to know how students typically think about particular concepts, how to determine what a particular student or group of students thinks about those ideas, and how to help students deepen their understanding. Weiss & Pasley, 2004

16. Pose Purposeful Questions Teacher and Student Actions What are teachers doing?What are students doing? Advancing student understanding by asking questions that build on, but do not take over or funnel, student thinking. Making certain to ask questions that go beyond gathering information to probing thinking and requiring explanation and justification. Asking intentional questions that make the mathematics more visible and accessible for student examination and discussion. Allowing sufficient wait time so that more students can formulate and offer responses. Expecting to be asked to explain, clarify, and elaborate on their thinking. Thinking carefully about how to present their responses to questions clearly, without rushing to respond quickly. Reflecting on and justifying their reasoning, not simply providing answers. Listening to, commenting on, and questioning the contributions of their classmates.

17. Use and Connect Mathematical Representations Different Representations should: Be introduced, discussed, and connected; Focus students’ attention on the structure or essential features of mathematical ideas; and Support students’ ability to justify and explain their reasoning. Strengthening the ability to move between and among these representations improves the growth of children’s understanding of mathematical concepts. Lesh, Post, & Behr, 1987

18. Use and Connect Mathematical Representations What is the distinction between physical and visual representations? Describe some physical representations that might support students’ thinking in the Donuts Task. Describe are some visual representations you would expect students to produce. In the diagram, why do the arrows go both ways?

19. Use and Connect Mathematical Representations Teacher and Student Actions What are teachers doing?What are students doing? Selecting tasks that allow students to decide which representations to use in making sense of the problems. Allocating substantial instructional time for students to use, discuss, and make connections among representations. Introducing forms of representations that can be useful to students. Asking students to make math drawings or use other visual supports to explain and justify their reasoning. Focusing students’ attention on the structure structure or essential features of mathematical ideas that appear, regardless of the representation. Designing ways to elicit and assess students’ abilities to use representations meaningfully to solve problems. Using multiple forms of representations to make sense of and understand mathematics. Describing and justifying their mathematical understanding and reasoning with drawings, diagrams, and other representations. Making choices about which forms of representations to use as tools for solving problems. Sketching diagrams to make sense of problem situations. Contextualizing mathematical ideas by connecting them to real-world situations. Considering the advantages or suitability of using various representations when solving problems.

20. Ms. Smith’s Kindergarten Classroom Ms. Smith’s Kindergarten Classroom

21. Teacher:Amanda Smith Grade:Kindergarten School:Sam Houston Elementary School District:Lebanon School District, Tennessee Date:April 10, 2013 The teacher poses a problem to the students. Students work with manipulatives to represent and solve the problem. The teacher circulates, asking students what they know and pressing students to tell her the number of chocolate and vanilla donuts. The teacher also works to help students understand the goals for their learning. The Donuts Task The Context of the Video Segment

22. Lens for Watching Video Clip 1 As you watch the first video clip, pay attention to the teacher and student indicators associated with these two Effective Mathematics Teaching Practices. Pose Purposeful Questioning Use and Connect Mathematical Representations

23. Teacher and Student Actions Review your assigned list of teacher or student actions from Principles to Actions. Identify actions from the list that you observed in the video. Identify the line numbers from the transcript that indicate or point to evidence of specific actions. Be prepared to summarize the teacher or student actions related to the two teaching practices.

24. Use and Connect Mathematical Representations

25. Video Clip 2 The Donuts Task Extension Tamika changes her mind and she gets 3 chocolate, 2 vanilla, and 2 sprinkle donuts. Draw a picture and write an equation to show Tamika’s donuts. Tamika claims that she has more donuts than Dion because she has three kinds of donuts. What do you think about Tamika’s claim? Who has more donuts and how do you know?

26. Lens for Watching Video Clip 2 As you watch the second video clip of the students engaged with the extension problems, pay attention to the teacher and student indicators related to: Pose Purposeful Questioning Use and Connect Mathematical Representations

27. Revisiting Ms. Smith’s Mathematics Learning Goals Students will understand that: the sum of two or more sets can be combined in many different ways (counting, counting on, or the use of known facts); two addition expressions can have the same sum although the addends appear in a different order (the commutative property of addition); and more sets of smaller amounts can still add up to the same amount as fewer sets with more items.

28. Applying Ideas to Your Own Classrooms

29. As you reflect on the Effective Mathematics Teaching Practices examined in this session, how might these ideas apply to your own classroom instruction?

30. Effective Mathematics Teaching Practices 1.Establish mathematics goals to focus learning. 2.Implement tasks that promote reasoning and problem solving. 3.Use and connect mathematical representations. 4.Facilitate meaningful mathematical discourse. 5.Pose purposeful questions. 6.Build procedural fluency from conceptual understanding. 7.Support productive struggle in learning mathematics. 8.Elicit and use evidence of student thinking. National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author.

31. Principles to Actions: www.nctm.org/pta