LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH 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

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. 2

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH 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. 3

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH 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. 4

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH The Data About Students’ Understanding of Fractions 5

The Data About Fractions 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. Siegler, Carpenter, Fennell, Geary, Lewis, Okamoto, Thompson, & Wray (2010). IES, U.S. Department of Education 6

The Data About Fractions: Conceptual Understanding 7

Siegler, Carpenter, Fennell, et al; U.S. Dept. of Education, IES Practice Guide: Developing Effective Fractions Instruction for Kindergarten through 8 th Grade. 8

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH Making Sense of the CCSS Number and Operations Fractions 9

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH 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 NF.A.3a, 3.NF.A.3b Group 5 – 3.NF.A.3c, 3.NF.A.3d Group NF.A.3d 10

The CCSS for Mathematical Content: Grade 3 Common Core State Standards, 2010, p. 24, NGA Center/CCSSO Number and Operations—Fractions 3.NF Develop understanding of fractions as numbers. 3.NF.A.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.A.2 Understand a fraction as a number on the number line; represent fractions on a number line diagram. 3.NF.A.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.A.2b Represent a fraction a/b on a number line diagram by marking off a lengths 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. 11

The CCSS for Mathematical Content: Grade 3 Common Core State Standards, 2010, p. 24, NGA Center/CCSSO Number and Operations—Fractions 3.NF Develop understanding of fractions as numbers. 3.NF.A.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. 3.NF.A.3a Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line. 3.NF.A.3b Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction model. 3.NF.A.3c Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram. 3.NF.A.3d Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. 12

The CCSS for Mathematical Content: Grade 4 Common Core State Standards, 2010, p. 30, NGA Center/CCSSO Number and Operations – Fractions 4.NF Extend understanding of fraction equivalence and ordering. 4.NF.A.1 Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. 4.NF.A.2 Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. 4.NF.B.3 Understand a fraction a/b with a > 1 as a sum of fractions 1/b. 4.NF.B.3a Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. 13

The CCSS for Mathematical Content: Grade 4 Common Core State Standards, 2010, p. 30, NGA Center/CCSSO Number and Operations – Fractions 4.NF Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. 4.NF.B.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. 4.NF.B.4a Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4). 4.NF.B.4b Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.) 4.NF.B.4c Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie? 14

The CCSS 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. Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO 15

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH Analyzing Tasks as a Means of Making Sense of the CCSS Number and Operations Fractions 16

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

TASKS as they appear in curricular/ instructional materials TASKS as set up by the teachers TASKS as implemented by students Student Learning The Mathematical Tasks Framework Stein, Smith, Henningsen, & Silver, 2000 Linking to Research/Literature: The QUASAR Project 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 18

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH Linking to Research/Literature: The QUASAR Project Low-level tasks High-level tasks 19

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH Linking to Research/Literature: The QUASAR Project Low-level tasks – Memorization – Procedures without Connections High-level tasks – Doing Mathematics – Procedures with Connections 20

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH 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. 21

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. 22

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH 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? 23

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH 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. 24

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,

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH 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? 26

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,

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH D. Pieces of Ribbon

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH E. Fractions on a Number Line

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH 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. 30

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH G. Locating Points on a Number Line 31

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH 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. 32

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH I. Sharing Pizza 33

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH 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. a.Which is more and how do you know? Use the greater than or less than sign to compare the amounts. b.Give another fraction that describes the shaded portion. 34

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH 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? 35

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH 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? 36

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH 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? 37

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH Instructional Tasks Versus Assessment Tasks Instructional TasksAssessment Tasks Assist learners to learn the CCSS for Mathematical Content and the CCSS for Mathematical Practice. Assess fairly the CCSS for Mathematical Content and the CCSS for Mathematical Practice of the taught curriculum. Assist learners to accomplish, often with others, an activity, project, or to solve a mathematics task. Assess individually completed work on a mathematics task. Assist learners to “do” the subject matter under study, usually with others, in ways authentic to the discipline of mathematics. Assess individual performance of content within the scope of studied mathematics content. Include different levels of scaffolding depending on learners’ needs. The scaffolding does NOT take away thinking from the students. The students are still required to problem-solve and reason mathematically. Include tasks that assess both developing understanding and mastery of concepts and skills. Include high-level mathematics prompts. (The tasks have many of the characteristics listed on the Mathematical Task Analysis Guide.) Include open-ended mathematics prompts as well as prompts that connect to procedures with meaning. 38

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH 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? 39