Supporting Rigorous Mathematics Teaching and Learning

Supporting Rigorous Mathematics Teaching and Learning The Instructional Tasks Matter: Analyzing the Demand of Instructional Tasks Tennessee Department of Education Elementary School Mathematics Grade 1

Rationale –Comparing Two Mathematical Tasks 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, p. 335, 2001 By analyzing two tasks that are mathematically similar, teachers will begin to differentiate between tasks that require thinking and reasoning and those that require the application of previously learned rules and procedures.

Learning Goals and Activities Participants will: compare mathematical tasks to determine the demand of the tasks; and identify the Common Core State Standards (CCSS) for Mathematical Content and the Standards for Mathematical Practice addressed by each of the tasks.

Comparing the Cognitive Demand of Two Mathematical Tasks What are the similarities and differences between the two tasks? • Counting Houses Task • Nine Plus a Number Task

Counting Houses Task Mary, Nick, and Jean are collecting donations to support homeless people. Each student starts on a different path. The houses are side-by-side. Which student will visit the most houses and how do you know? Write an equation that describes each part of the students’ paths and explain which student visited the most houses and how you know. Mary Nick Jean • Mary claims she sees a pattern in the Counting Houses Task that she can use to solve the tasks below. • 9 + 8 = ___ 9 + 7 = ___ 9 + 6 = ___ • 8 + 9 = ___ 7 + 9 = ___ 6 + 9 = ___ • 10 + 7 = ___ 10 + __ = 16 10 + __ = 15 • What pattern do you see?____________________________________________

Nine Plus a Number Task Solve each addition problem. Use the blocks when solving each problem. 9 + 5 = ___ 5 + 9 = ____ 10 + 4 = ___ Solve the problems below. 9 + 8 = ___ 7 + 7 = ___ 6 + 6 = ___ 8 + 8 = ___ 7 + 9 = ___ 6 + 9 = ___ 9 + 7 = ___ 8 + 6 = ___ 5 + 8 = ___ 6 + 7 = ___ 8 + 5 = ___ 8 + 5 = ___

The Common Core State Standards Examine the CCSS • for Mathematical Content • for Mathematical Practice • Will first grade students have opportunities to use the standards within the domain of Operations and Algebraic Thinking? • What kind of student engagement will be possible with each task? • Which Standards for Mathematical Practice will students have opportunities to use?

Common Core State Standards for Mathematics: Grade 1 Common Core State Standards, 2010, p. 15, NGA Center/CCSSO

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.

Comparing Two Mathematical Tasks How do the differences between the Counting Houses Task and the Nine Plus a Number Task impact students’ opportunities to learn the Standards for Mathematical Content and to use the Standards for Mathematical Practice?

Linking to Research/Literature: The QUASAR Project Stein M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standards-based mathematics instruction: A casebook for professional development, p. 3. New York: Teachers College Press …Not all tasks are created equal - different tasks will provoke different levels and kinds of student thinking.

Linking to Research/Literature There is no decision that teachers make that has a greater impact on students’ opportunities to learn and on their perceptions about what mathematics is than the selection or creation of the tasks with which the teacher engages students in studying mathematics. Lappan & Briars, 1995

Instructional Tasks: The Cognitive Demand of Tasks Matters

Linking to Research/Literature: The QUASAR Project TASKS as they appear in curricular/ instructional materials TASKS as set up by the teachers TASKS as implemented by students Student Learning Stein, Smith, Henningsen, & Silver, 2000, p. 4 The Mathematical Tasks Framework

Linking to Research/Literature: The QUASAR Project (continued) • Low-Level Tasks • Nine Plus a Number Task • High-Level Tasks • Counting Houses Task

Linking to Research/Literature: The QUASAR Project (continued) • Low-Level Tasks • Memorization • Procedures Without Connections (e.g., Nine Plus a Number Task) • High-Level Tasks • Doing Mathematics (e.g., Counting Houses Task) • Procedures With Connections

The Mathematical Task Analysis Guide Research has identified characteristics related to each of the categories on the Mathematical Task Analysis Guide (TAG). How do the characteristics that we identified when discussing the Counting Houses Task relate to those on the TAG? Which characteristics describe the Nine Plus a Number Task?

The Cognitive Demand of Tasks(Small Group Work) • Working individually, use the TAG to determine if tasks A – L are high- or low-level tasks. • Identify and record the characteristics on the TAG that best describe the cognitive demand of each task. • Identify the CCSS for Mathematical Practice that the written task requires students to use. • Share your categorization in pairs or trios. Be prepared to justify your conclusions using the TAG and the Standards for Mathematical Practice.

Identifying High-level Tasks(Whole Group Discussion) Compare and contrast the four tasks. Which of the four tasks are considered to have a high level of cognitive demand and why?

Relating the Cognitive Demand of Tasks to the Standards for Mathematical Practice What relationships do you notice between the cognitive demand of the written tasks and the Standards for Mathematical Practice?

Addition Task A Determine the sum of each addition problem. 5 + 6 = ___ 6 + 4 = ____ 7 + 9 = ___ 5 + 5 = ____ 8 + 9 = ___ 7 + 6 = ____ 8 + 9 = ___ 8 + 5 = ____ 6 + 8 = ___ 8 + 4 = ____ 7 + 7 = ___ 6 + 5 = ____ 8 + 8 = ___ 9 + 9 = ____

Addition Task B 4 + (5 + 9) 4 + 5 + 9 10 + 8 9 + 8 4 + 4 + 6 8 + 6 Tell if the scale will balance or tilt. If the scale does not balance, write which side will tilt down and why and indicate what would have to change to make the scale balance.

Addition Task C Use your solution to one problem to solve the second problem. The first problem is given as an example. 6 + 6 = 12 6 + 7 = ___ Solve each set of problems by using the first problem to solve the second problem. 7 + 7 = ____ 8 + 8 = ____ 5 + 5 = ___ 7 + 8 = ____ 8 + 9 = ____ 5 + 6 = ___ The problems below work in the opposite way as the ones above. How can you use the first problem to solve the second problem in each set of problems? 7 + 7 = ____ 8 + 8 = ____ 5 + 5 = ___ 7 + 6 = ____ 8 + 7 = ____ 5 + 4 = ___

Addition Task D Manipulatives/Tools available: Counters, cubes, grid paper, base ten blocks Write a word problem for the number sentence. 8 + 6 = ___ Ask a question with your story problem so we know what we are supposed to figure out. Write a word problem for the number sentence. 14 – 5 = ___ Ask a question with your story problem so we know what we are supposed to figure out. Compare the two word problems. How do they differ from each other?

Subtraction Task E Manipulatives/Tools available: base ten blocks Solve this problem in two different ways: 32 - 17 After each way, write about how you did it. Be sure to include: what materials, if any, you used to solve this problem; how you solved it; and an explanation of your thinking as you solved it. First Way: Second Way: Adapted from Investigations in Number, Data, and Space, Dale Seymour, Menlo Park, CA, 1998.

Subtraction Task F Manipulatives/Tools available: base ten blocks Use base ten blocks to model the situations below. Write a number sentence for each problem. Jim has 23 red pencils and 8 pencils are not sharpened. How many pencils are sharpened? Jamie has 48 cookies and some of them are chocolate and some are vanilla. 26 cookies are chocolate. How many cookies are vanilla? 32 cookies are in the box and you ate some of them. Now there are 26 cookies left. How many cookies did you eat? Explain how the problems are similar to each other. Explain how the problems differ from each other.

Subtraction Task G Manipulatives/Tools available: none Solve the subtraction number sentences. 14 – 7 = ____ 16 – 8 = ___ 12 – 8 = ____ 18 – 9 = ___ 14 – 7 = ____ 16 – 8 = ___ 12 – 8 = ____ 18 – 9 = ___ 14 – 7 = ____ 16 – 8 = ___ 12 – 8 = ____ 18 – 9 = ___

Subtraction Task H Manipulatives/Tools available: none Study the strategy of rounding the subtrahend in order to subtract all of the ones available and doing mental subtraction. 65 – 26 = ___ 83 – 37 = ___ 65 – 25 = 40 83 – 33 = 50 40 – 1 = 39 50 – 4 = 46 45 – 26 = ____ 62 – 28 = ____

Place Value Task I Manipulatives/Tools available: base ten blocks Order the amounts from smallest to largest. 234 243 284 254 233 Order the amounts from smallest to largest. 348 349 345 384 336

Place Value Task J Manipulatives/Tools available: base ten blocks Identify the number of tens possible in each of the amounts. 236__________________ 368__________________ 589__________________ 2389_________________ 3458_________________

Place Value Task K Manipulatives/Tools available: base ten blocks Study each set of addition problems and write about what keeps changing with each sum. 345 + 10 = ______________ 355 + 10 = ______________ 365 + 10 = ______________ 375 + 10 = ______________ Which numbers stayed the same? Which numbers changed? Explain why only one number kept changing.

Place Value Task L Manipulatives/Tools available: base ten blocks Circle the number in the ones place in each of the numbers below. 45 56 67 78 89 Circle the number in the tens place in each of the numbers below. 45 345 567 678 689 Circle the number in the hundreds place in each of the numbers below. 3,459 459 5,679 3,457 2,349

The CCSS for Mathematical Practice Common Core State Standards, 2010 • 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.

Linking to Research/Literature: The QUASAR Project Stein, M. K. & Lane, S. (1996). Instructional tasks and the development of student capacity to think and reason: An analysis of the relationship between teaching and learning in a reform mathematics project. Educational Research and Evaluation, 2 (4), 50-80. If we want students to develop the capacity to think, reason, and problem-solve then we need to start with high-level, cognitively complex tasks.

Linking to Research/Literature Adding It Up, National Research Council, p. 335, 2001 Tasks are central to students’ learning, shaping not only their opportunity to learn but also their view of the subject matter.

Supporting Rigorous Mathematics Teaching and Learning