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 Paterson Public Schools, New Jersey, 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 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 Victoria Bill and the Bubble Gum Task Grade 3 Principles to Actions Effective Mathematics Teaching Practices The Case of Victoria Bill and the Bubble Gum Task Grade 3 INSTITUTE for LEARNING

2. Overview of the Session Watch a video clip of a third grade class engaged in a whole class discussion of a fraction task. Discuss what the teacher does to support the students’ learning of mathematics. Relate teacher actions in the video to the effective mathematics teaching practices.

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. Principles to Actions (NCTM, 2014, p. 7)

4. Ms. Bill Third Grade Classroom “Bubble Gum Task” Ms. Bill Third Grade Classroom “Bubble Gum Task” INSTITUTE for LEARNING

5. Ms. Bill (Visiting Teacher) Mathematics Learning Goals Students will understand that: fractions which refer to the same whole and have the same-size pieces, or common denominators, can be easily compared with each other because the size of the pieces is the same. comparison to a known benchmark quantity can help determine the relative size of a fraction because the benchmark quantity can be seen as greater than, less than, or the same as the fraction. fractions are equivalent if they represent the same portion of area of a figure or they name the same point on the number line. when you multiply by a fraction equivalent to one, n/n (n > 1), the whole is partitioned into a new number of pieces that are smaller in size but larger in total number of pieces than the original; pieces referenced by the numerator are partitioned in the same way. INSTITUTE for LEARNING

6. The Bubble Gum Task (Student Version) 1. Use the number line below to illustrate:  Which friend chewed the most gum?  Which friend chewed the smallest piece?  Which 2 friends chewed the same sized piece? 2.Explain how you can use 3/4 to help you determine which value is greater, 5/6 or 3/5, if each fraction refers to the same whole. Four friends each bought a roll of bubble gum tape. Carlos chewed of his gum. Helen chewed of her gum. Jamal chewed of his gum. Lizbeth chewed of her gum. Adapted from: Schifter, D., Bastable, V., & Russell, S. J. (1999). Developing mathematical ideas: Making meaning for operations. Parsippany, NJ: Dale Seymour.

7. The Bubble Gum Task (Adult Version) 1. Use the number line below to illustrate:  Which friend chewed the most gum?  Which friend chewed the smallest piece?  Which 2 friends chewed the same-sized piece? 2.Explain how you can use 3/4 to help you determine which value is greater, 5/6 or 3/5, if each fraction refers to the same whole. 3.Explain why a/b = (a/b) x (n/n) and connect this equation to a visual diagram. Four friends each bought a roll of bubble gum tape. Carlos chewed of his gum. Helen chewed of her gum. Jamal chewed of his gum. Lizbeth chewed of her gum.

8. Reflecting on the Demands of the Task Why is it important that the whole remain the same for each strip of bubble gum? What are the advantages of not specifying that the same whole must be used for each strip of bubble gum when the task is given to students initially?

9. Connections to the CCSSM Standards for Mathematical 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 National Governors Association Center (NGA) & Council of Chief State School Officers (CCSSO). (2010). Common core state standards for mathematics. Washington DC: Author.

10. Number and Operations — Fractions Develop understanding of fractions as numbers. 3.NF.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b. 3.NF.2 Understand a fraction as a number on the number line; represent fractions on a number line diagram. 3.NF.2a Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line. 3.NF.2b Represent a fraction a/b on a number line diagram by marking off a lengths of 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line. 3.NF.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. 3.NF.3a Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line. Connections to the CCSSM Standards for Mathematical Content: Grade 3 NGA/CCSSO). (2010). Common core state standards for mathematics. Washington DC: Author.

11. The Bubble Gum Lesson Context Visiting Teacher:Victoria Bill Grade level:3 School:School #26 District:Paterson Public Schools, New Jersey The classroom teacher, Millie Brooks, and Ms. Bill, a visiting teacher, want students to use a linear representation of fractions and to understand how a benchmark fraction can help them think about and compare fractions. Although students are not expected to be proficient with benchmark fractions until fourth grade the teachers know it is important to begin developing this understanding now. The students work privately for five minutes and then discuss solution paths in small groups. While students work, the teacher circulates asking assessing and advancing questions and notices students are struggling to use the number line representation. The teacher decides to co-construct a solution path with the students. INSTITUTE for LEARNING

12. Lens for Watching the Video As you watch the video, Identify the mathematical insights that surfaced for students. Make note of what the teacher does to support student learning of mathematics.

13. Watch the Video “Bubble Gum Lesson”

14. Discussing Student Learning of Mathematics In small groups, discuss the following: 1.What mathematical insights surfaced for students? 2.In what ways did the teacher advanced student understanding toward the math learning goals and content standards?

15. A Closer Look at the Effective Mathematics Teaching Practices A Closer Look at the Effective Mathematics Teaching Practices

16. 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.

17. Effective Mathematics Teaching Practice Pose Purposeful Questions Effective Mathematics Teaching Practice Pose Purposeful Questions Teachers’ questions are crucial in helping students make connections and learn important mathematics and science 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). Weiss, I. R., & Pasley, J. D. (2004). What Is High-Quality Instruction? Educational Leadership, 61(5), 24–28.

18. Pose Purposeful Questions Effective questions: 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. Principles to Actions (NCTM, 2014, p. 35-41)

19. Effective Mathematics Teaching Practice Use and Connect Mathematical Representations Effective Mathematics Teaching Practice Use and Connect Mathematical Representations

20. Strengthening the ability to move between and among these representations improves the growth of children’s concepts ( Lesh, Post, & Behr, 1987 ). Use and Connect Mathematical Representations Lesh, R., Post, T., & Behr, M. (1987). Representations and translations among representations in mathematics learning and problem solving. In C. Janvier, (Ed.), Problems of representations in the teaching and learning of mathematics (pp. 33-40). Hillsdale, NJ: Lawrence Erlbaum.

21. 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. Principles to Actions (NCTM, 2014, p. 24-29)

22. Teacher and Student Actions Count off by fours. Posing Purposeful Questions: – Ones study the teacher actions. – Twos study the student actions. Use and Connect Mathematical Representations: – Threes study the teacher actions. – Fours study the student actions.

23. 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. (NCTM, 2014, p. 41)

24. 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 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. (NCTM, 2014, p. 29)

25. Watch the Video “Bubble Gum Lesson” using your assigned lens of teacher or student actions for the focus mathematics teaching practices.

26. Discussing the Video “Pose Purposeful Questions” Which of the teacher and student actions were attended to in the lesson and how did each support student learning of fractions? What were some key questions posed by the teacher and what purpose did each serve to support student learning?

27. 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. (NCTM, 2014, p. 41)

28. Which of the teacher and student actions were attended to in the lesson and how did each support student learning of fractions? How were representations used to promote students’ ability to make sense of mathematical ideas and reason mathematically? Discussing the Video “Use and Connect Representations”

29. 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 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. (NCTM, 2014, p. 29)

30. Reflecting on the... Effective Mathematics Teaching Practices Reflecting on the... Effective Mathematics Teaching Practices

31. As you reflect on the effective mathematics teaching practices examined in this session, summarize one or two ideas or insights you might apply to your own classroom instruction.