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 5

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH 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, 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

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH 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. 6 Siegler, Carpenter, Fennell, Geary, Lewis, Okamoto, Thompson, & Wray (2010). IES, U.S. Department of Education

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 Analyzing Tasks as a Means of Making Sense of the CCSS Number and Operations – Fractions 9

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 10

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 11

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH Linking to Research/Literature: The QUASAR Project Low-level tasks High-level tasks 12

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 13

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. 14

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. 15

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? 16

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. 17

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH Analyzing Tasks: Aligning with the CCSS (Whole group discussion) 18 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?

The CCSS for Mathematical Content: Grade 5 Common Core State Standards, 2010, p. 36, NGA Center/CCSSO Number and Operations—Fractions 5.NF Apply and extend previous understandings of multiplication and division to multiply and divide fractions. 5.NF.B.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. 5.NF.B.4a Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) x 4 = 8/3, and create a story context for this equation. Do the same with (2/3) x (4/5) = 8/15. (In general, (a/b) x (c/d) = ac/bd.) 5.NF.B.4b Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas. 19

The CCSS for Mathematical Content: Grade 5 Common Core State Standards, 2010, p. 36, NGA Center/CCSSO Number and Operations – Fractions 5.NF Apply and extend previous understandings of multiplication and division to multiply and divide fractions. 5.NF.B.5 Interpret multiplication as scaling (resizing), by: 5.NF.B.5a Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication. 5.NF.B.5b Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n × a)/(n × b) to the effect of multiplying a/b by 1. 5.NF.B.6 Solve real-world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem. 20

The CCSS for Mathematical Content: Grade 5 Common Core State Standards, 2010, p , NGA Center/CCSSO Number and Operations – Fractions 5.NF Apply and extend previous understandings of multiplication and division to multiply and divide fractions. 5.NF.B.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. 5.NF.B.7a Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) x 4 = 1/3. 5.NF.B.7b Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 x (1/5) = 4. 5.NF.B.7c Solve real-world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins? 21

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 22

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH A. Stretching a Rubber Band 23

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

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH C. Dividing Fractions on a Number Line 25

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH D. Time to do Jobs 26

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH E. Dividing Fractions Relationship 27

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH F. Multiplying and Dividing Fractions 28

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH G. Servings of Mashed Potatoes 29

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

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? 31

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? 32

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? 33

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH Characteristics of Performance-Based Assessments Each task is cognitively demanding. (TAG: require math reasoning, an explanation for why formulas or procedures work; analysis of patterns, formulation of a generalization, prompt connection making between representations, strategies, or mathematical concepts and procedures.) The task addresses several of the CCSS for Mathematical Content. The task may require students to use more than one strategy or representation when solving the task. The expectations in the task are clear and explicit regarding the extent of the work expected. The task may ask students to (1) explain their reasoning in words or to (2) use mathematical reasoning to justify their response. 34

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH Characteristics of Instructional Tasks A variety of tasks at the different levels of cognitive demand are used. (TAG: require math reasoning, an explanation for why formulas or procedures work; analysis of patterns, formulation of a generalization, prompt connection making between representations, strategies, or mathematical concepts and procedures.) The task addresses one or more of the CCSS for Mathematical Content. The task may require students to use more than one strategy or representation when solving the task. The task may be open-ended because the teacher can guide the instruction. 35

Instructional Tasks Versus Assessment Tasks Instructional TasksAssessment Tasks Assist learners to learn the CCSS for Mathematical Content and the CCSS for Mathematical Practice. Assesses 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. 36

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 week? How do we give students the opportunities during instructional time to learn math so that they are successful on the next generation assessment items? 37