Supporting Rigorous Mathematics Teaching and Learning

1. Supporting Rigorous Mathematics Teaching and Learning Making Sense of the Number and Operations – Fractions Standards via a Set of Tasks Tennessee Department of Education Elementary School Mathematics Grade 3

2. Rationale Tasks form the basis for students’opportunities to learn what mathematics is and how one does it, yet not all tasks afford the same levels and opportunities for student thinking. [They] are central to students’learning, shaping not only their opportunity to learn but also their view of the subject matter. Adding It Up, National Research Council, 2001, p. 335 By analyzing instructional and assessment tasks that are for the same domain of mathematics, teachers will begin to identify the characteristics of high-level tasks and differentiate between those that require problem-solving and those that assess for specific mathematical reasoning.

3. Session Goals Participants will: • make sense of the Number and Operations – Fractions Common Core State Standards (CCSS); • determine the cognitive demand of tasks and make connections to the Mathematical Content Standards and the Standards for Mathematical Practice; and • differentiate between assessment items and instructional tasks.

4. Overview of Activities Participants will: • analyze a set of tasks as a means of making sense of the Number and Operations – Fractions Common Core State Standards (CCSS); • determine the Content Standards and the Mathematical Practice Standards aligned with the tasks; • relate the characteristics of high-level tasks to the CCSS for Mathematical Content and Practice; and • discuss the difference between assessment items and instructional tasks.

5. The Data About Students’ Understanding of Fractions

6. The Data About Fractions Siegler, Carpenter, Fennell, Geary, Lewis, Okamoto, Thompson, & Wray (2010). IES, U.S. Department of Education Only a small percentage of U.S. students possess the mathematics knowledge needed to pursue careers in science, technology, engineering, and mathematics (STEM) fields. Moreover, large gaps in mathematics knowledge exist among students from different socioeconomic backgrounds and racial and ethnic groups within the U.S. Poor understanding of fractions is a critical aspect of this inadequate mathematics knowledge. In a recent national poll, U.S. algebra teachers ranked poor understanding about fractions as one of the two most important weaknesses in students’ preparation for their course.

7. The Data About Fractions: Conceptual Understanding A high percentage of U.S. students lack conceptual understanding of fractions, even after studying fractions for several years; this, in turn, limits students’ ability to solve problems with fractions and to learn and apply computational procedures involving fractions. • 50% of 8th graders could not order 3 fractions from least to greatest. • 27% of 8th graders could not correctly shade of a rectangle. • 45% of 8th graders could not solve a word problem that required dividing fractions (NAEP, 2004). • Fewer than 30% of 17-year-olds correctly translated 0.029 as (Kloosterman, 2010).

8. The Data About Fractions: Conceptual Understanding Siegler, Carpenter, Fennell, et al; U.S. Dept. of Education, IES Practice Guide: Developing Effective Fractions Instruction for Kindergarten through 8th Grade. A lack of conceptual understanding of fractions has several facets, including…students’ focusing on numerators and denominators as separate numbers rather than thinking of the fraction as a single number. Errors such as believing that > arise from comparing the two denominators and ignoring the essential relationship between each fraction’s numerator and denominator.

9. Making Sense of the CCSSNumber and Operations Fractions

10. Making Sense of the Number and Operations – Fractions Standards Work in groups. Each group will study and make sense of one or more of the Standards for Mathematical Content. Your goal is to provide examples, counter-examples, visuals, or contexts that will help others understand your assigned standard. Each group will have five minutes to help us make sense of their standard(s). Group 1 – 3.NF.A.1 Group 2 – 3NF.A.2a, 3.NF.A.2b Group 3 – 3.NF.A.3 Group 4 - 3.NF.A.3a, 3.NF.A.3b Group 5 – 3.NF.A.3c, 3.NF.A.3d Group 6 - 3.NF.A.3d

11. The CCSS for Mathematical Content: Grade 3 Common Core State Standards, 2010, p. 24, NGA Center/CCSSO

12. The CCSS for Mathematical Content: Grade 3 Common Core State Standards, 2010, p. 24, NGA Center/CCSSO

13. The CCSS for Mathematical Content: Grade 4 Common Core State Standards, 2010, p. 30, NGA Center/CCSSO

14. The CCSS for Mathematical Content: Grade 4 Common Core State Standards, 2010, p. 30, NGA Center/CCSSO

15. The CCSS for Mathematical Practice Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO • Make sense of problems and persevere in solving them. • Reason abstractly and quantitatively. • Construct viable arguments and critique the reasoning of others. • Model with mathematics. • Use appropriate tools strategically. • Attend to precision. • Look for and make use of structure. • Look for and express regularity in repeated reasoning.

16. Analyzing Tasks as a Means of Making Sense of the CCSSNumber and Operations Fractions

17. Linking to Research/Literature: The QUASAR Project The Mathematical Tasks Framework TASKS as set up by the teachers TASKS as implemented by students TASKS as they appear in curricular/ instructional materials Student Learning Stein, Smith, Henningsen, & Silver, 2000

18. Linking to Research/Literature: The QUASAR Project The Mathematical Tasks Framework TASKS as set up by the teachers TASKS as implemented by students TASKS as they appear in curricular/ instructional materials Student Learning Stein, Smith, Henningsen, & Silver, 2000 Setting Goals Selecting Tasks Anticipating Student Responses • Orchestrating Productive Discussion • Monitoring students as they work • Asking assessing and advancing questions • Selecting solution paths • Sequencing student responses • Connecting student responses via Accountable Talk®discussions Accountable Talk®is a registered trademark of the University of Pittsburgh

19. Linking to Research/Literature: The QUASAR Project • Low-level tasks • High-level tasks

20. Linking to Research/Literature: The QUASAR Project • Low-level tasks • Memorization • Procedures without Connections • High-level tasks • Doing Mathematics • Procedures with Connections

21. The Cognitive Demand of Tasks(Small Group Discussion) Analyze each task. Determine if the task is a high-level task. Identify the characteristics of the task that make it a high-level task. After you have identified the characteristics of the task, then use the Mathematical Task Analysis Guide to determine the type of high-level task. Use the recording sheet in the participant handout to keep track of your ideas.

22. The Mathematical Task Analysis Guide Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000) Implementing standards-based mathematics instruction: A casebook for professional development, p. 16. New York: Teachers College Press.

23. The Cognitive Demand of Tasks(Whole Group Discussion) What did you notice about the cognitive demand of the tasks? According to the Mathematical Task Analysis Guide, which tasks would be classified as • Doing Mathematics Tasks? • Procedures with Connections? • Procedures without Connections?

24. Analyzing Tasks: Aligning with the CCSS(Small Group Discussion) Determine which Content Standards students would have opportunities to make sense of when working on the task. Determine which Mathematical Practice Standards students would need to make use of when solving the task. Use the recording sheet in the participant handout to keep track of your ideas.

25. A. Half of a Whole Task Identify all of the figures that have one-half shaded. Be prepared to show and explain how you know that one-half of a figure is shaded. If a figure does not show one-half shaded, explain why. Make math statements about what is true about a half of a whole. Adapted from Watanabe, 1996

26. B. Equal Parts on the Geoboard John claims that all of the sections on the geoboard are equal. Do you agree or disagree with John? If you agree, how can you prove that he is correct?

27. C. Making Quilts Tell what fraction of the whole rectangle that each numbered section represents. Explain how you know that the fraction represents the amount of the figure. Adapted from Connected Mathematics, Prentice Hall, 2007

28. D. Pieces of Ribbon 1 0 Here are 3 fractions. Each fraction is greater than 0 and less than 1: Joan’s Ribbon Frank’s Ribbon Cheryl’s Ribbon ft. ft. ft. Locate all 3 students’ pieces of ribbon on the number line below. Locate them as accurately as you can. Cheryl claims that she has the longest piece of ribbon. Do you agree or disagree and why?

29. E. Fractions on a Number Line Explain how and are related. How are these two fractions different from each other? 0 1 3 2 Place a point on the number line for each fraction.

30. F. Eating Pizza Harold and Jed each bought a small pizza for lunch. Harold cut his pizza into 8 equal pieces and ate 3 of them. Jed cut his pizza into 4 equal pieces and ate 2 of them. Which boy ate more pizza? Use diagrams, words, and numbers to show you are right.

31. G. Locating Points on a Number Line Put these fractions on the number line below.

32. H. Shaded and Not Shaded What fraction of the figure is shaded gray? Write 2 different fractions that describe the shaded portion of the figure. Write 2 different fractions that describe the shaded portion of the figure. Explain how your fraction represents the shaded area.

33. I. Sharing Pizza Jenny, Gina, and Petra shared a pizza. Jenny ate of the pizza.Gina ate of it.Petra ate the rest.Did Petra eat more or less pizza than each of the other girls?Show with a diagram each person’s share of the pizza and explain how you know Petra’s share of the pizza.

34. J. A Fraction of the Whole In each case below, the area of the whole rectangle is 1. Shade an area equal to the fraction underneath each rectangle. Compare the 2 amounts. • Which is more and how do you know? Use the greater than or less than sign to compare the amounts. • Give another fraction that describes the shaded portion.

35. Analyzing Tasks: Aligning with the CCSS(Whole Group Discussion) How do the tasks differ from each other with respect to the content that students will have opportunities to learn? Do some tasks require that students use Mathematical Practice Standards that other tasks don’t require students to use?

36. Reflecting and Making Connections • Are all of the CCSS for Mathematical Content in this cluster addressed by one or more of these tasks? • Are all of the CCSS for Mathematical Practice addressed by one or more of these tasks? • What is the connection between the cognitive demand of the written task and the alignment of the task to the Standards for Mathematical Content and Practice?

37. Differentiating Between Instructional Tasks and Assessment Tasks Are some tasks more likely to be assessment tasks than instructional tasks? If so, which and why are you calling them assessment tasks?

38. Instructional Tasks Versus Assessment Tasks

39. Reflection So, what is the point? What have you learned about assessment tasks and instructional tasks that you will use to select tasks to use in your classroom next school year?