© 2013 UNIVERSITY OF PITTSBURGH Supporting Rigorous Mathematics Teaching and Learning Making Sense of the Number System Standards via a Set of Tasks Tennessee Department of Education Middle School Mathematics Grade 7

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

© 2013 UNIVERSITY OF PITTSBURGH Session Goals Participants will: make sense of the Number System Common Core State Standards (CCSS); determine the cognitive demand of tasks and make connections to the Standards for Mathematical Content and the Standards for Mathematical Practice; and differentiate between assessment items and instructional tasks. 3

© 2013 UNIVERSITY OF PITTSBURGH Overview of Activities Participants will: analyze a set of tasks as a means of making sense of the Number System Common Core State Standards (CCSS); determine the Mathematical 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

© 2013 UNIVERSITY OF PITTSBURGH The Research About Students’ Understanding of Rational Numbers 5

Linking to Research Virtually none of the making connections problems in the U.S. were discussed in a way that made the mathematical connections or relationships visible for students. Mostly, they turned into opportunities to apply procedures. Or, they became problems in which even less mathematical content was visible (i.e., only the answer was given). TIMSS Video Mathematics Research Group,

Linking to Research Once students have been introduced to the decimal computation procedure for a particular arithmetic operation, the type of errors they are most likely to make seems to remain nearly constant. There are no qualitative changes in the way students compute with decimals after they receive their first instructional lessons. Hiebert and Wearne,

Linking to Research/Literature Research has shown that children who have difficultly translating a concept from one representation to another are the same children who have difficulty solving problems and understanding computations. Strengthening the ability to move between and among these representations improves the growth of children’s concepts. Lesh, Post, & Behr,

© 2013 UNIVERSITY OF PITTSBURGH Analyzing Tasks as a Means of Making Sense of the CCSS The Number System 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

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

© 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

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. 14

© 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. 15

© 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

© 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

© 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? 18

The CCSS for Mathematical Content: Grade 7 Common Core State Standards, 2010, p. 48, NGA Center/CCSSO The Number System 7.NS Apply and extend previous understandings of operations and fractions to add, subtract, multiply, and divide rational numbers. 7.NS.A.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. 7.NS.A.1a Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged. 7.NS.A.1b Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts. 19

The CCSS for Mathematical Content: Grade 7 Common Core State Standards, 2010, p. 48, NGA Center/CCSSO The Number System 7.NS Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. 7.NS.A.1c Understand subtraction of rational numbers as adding the additive inverse, p – q = p +(-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. 7.NS.A.1d Apply properties of operations as strategies to add and subtract rational numbers. 7.NS.A.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. 7.NS.A.2a Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. 20

The CCSS for Mathematical Content: Grade 7 Common Core State Standards, 2010, p. 49, NGA Center/CCSSO The Number System 7.NS Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. 7.NS.A.2b Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then -(p/q) = (-p)/q = p/(-q). Interpret quotients of rational numbers by describing real-world contexts. 7.NS.A.2c Apply properties of operations as strategies to multiply and divide rational numbers. 7.NS.A.2d Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. 7.NS.A.3 Solve real-world and mathematical problems involving the four operations with rational numbers. 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

© 2013 UNIVERSITY OF PITTSBURGH A. Walking Task Mary Jane and her brother Paul each go on a walk starting from the same location. Mary Jane walks north 3 miles. Paul walks south 1.5 miles. 1.Use a number line to represent their starting and ending points. 2.Determine their distance from each other at the end of their walks using 2 different methods. 23

© 2013 UNIVERSITY OF PITTSBURGH B. Positive, Negative, or Neither Consider the following expressions. Use the number line above to determine whether each of the expressions has a value that is positive, negative, or equal to 0. Explain your reasoning. a.a + 1 b.b – b c.a + a d.a – b e.b – a f.–a g.ab +1 24

© 2013 UNIVERSITY OF PITTSBURGH C. Same or Different? 25

© 2013 UNIVERSITY OF PITTSBURGH D. Some Sum! Points A and B are the same distance from 0 on the number line. What is A + B? Explain how you determined your answer. 26

© 2013 UNIVERSITY OF PITTSBURGH E. Number Line Multiplication 1. Consider the product 2 x 5. a.Explain how the number line below models the product 2 x 5. b.Write a scenario that can be modeled by the expression 2 x Draw a number line model and write a scenario for each of the following products: a.2 x -5 b.-2 x 5 3. Is it possible to use a number line to model the product -2 x -5? Why or why not? 27

© 2013 UNIVERSITY OF PITTSBURGH F. Party Favors Destiny is making party favor bags for her birthday party next week. She has everything she needs to fill the bags except glow sticks. There are two stores near her house that sell glow sticks. Party Central sells packages of 6 glow sticks for $1.98. Party Time sells packages of 8 glow sticks for $2.56. a.Write number sentences using division to determine which store offers a better deal. Explain what each value in your number sentence represents in the problem context. b.Destiny has $10 to spend on glow sticks. Write number sentences using division to determine how many packages can she buy at each store. Explain what each value in your number sentence represents in the problem context. c.How did the operation of division help you think about this problem? d.Which store should Destiny buy glow sticks from? Justify your decision using mathematics. 28

© 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? 29

© 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? 30

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. 31

© 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? 32