supporting rigorous mathematics teaching and learning

© 2013 UNIVERSITY OF PITTSBURGH

Supporting Rigorous Mathematics Teaching and Learning

Tennessee Department of EducationElementary School Mathematics Grades 1 & 2

The Instructional Tasks Matter:Analyzing the Demand of Instructional Tasks

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, 2001, p. 335

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.

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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 Mathematical Practice addressed by each of the tasks.

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Comparing the Cognitive Demand of Mathematical TasksWhat are the similarities and differences between the two tasks?

• The Strings Task

• The Apples Task

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The Strings TaskSolve the set of addition expressions. Each time you solve a problem, try to use the previous equation to solve the problem.

7 + 3 = ___17 + 3 = ___27 + 3 = ___37 + 3 = ___47 + 3 = ___

Solve each problem two different ways. Make a drawing or show your work on a number line.

What pattern do you notice? If the pattern continues, what would the next three equations be?

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The Apples Task

One basket has 27 green apples and 3 fell out of the basket. How many green apples do we have?

Another basket has 37 red apples and 3 fell out of the basket. How many red apples do we have?

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The Common Core State Standards (CCSS)Examine the CCSS:

− for Mathematical Content

− for Mathematical Practice

• Will second grade students have opportunities to use the standards within the domain of Operations and Algebraic Thinking and Number Operations in Base Ten?

• What kind of student engagement will be possible with each task?

• Which Standards for Mathematical Practice will students have opportunities to use with each task?

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The CCSS for Mathematical Content: Grade 2

Common Core State Standards, 2010

Operations and Algebraic Thinking 2.OARepresent and solve problems involving addition and subtraction.

2.OA.A.1 Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Add and subtract within 20.

2.OA.B2 Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.

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The CCSS for Mathematical Content: Grade 2


Operations and Algebraic Thinking 2.OA

Work with equal groups of objects to gain foundations for multiplication.

2.OA.C.3 Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.

2.OA.C.4 Use addition to find the total number of objects arranged inrectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

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The CCSS for Mathematics: Grade 2

Number and Operations in Base Ten 2.NBTUnderstand place value.

2.NBT.A.1 Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

a. 100 can be thought of as a bundle of ten tens—called a “hundred.”

b. The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

Common Core State Standards, 2010, p. 19, NGA Center/CCSSO 10

The CCSS for Mathematics: Grade 2Number and Operations in Base Ten 2.NBTUnderstand place value.

2.NBT.A.2 Count within 1000; skip-count by 5s, 10s, and 100s.

2.NBT.A.3 Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.

2.NBT.A.4 Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.


The CCSS for Mathematics: Grade 2

Number and Operations in Base Ten 2.NBTUse place value understanding and properties of operations to add and subtract.

2.NBT.B.5 Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

2.NBT.B.6 Add up to four two-digit numbers using strategies based on place value and properties of operations.


The CCSS for Mathematics: Grade 2Number and Operations in Base Ten 2.NBTUse place value understanding and properties of operations to add and subtract.

2.NBT.B.7 Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

2.NBT.B.8 Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given number 100–900.

2.NBT.B.9 Explain why addition and subtraction strategies work, using place value and the properties of operations.


Table 1: Common Addition and Subtraction Situations


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.


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Comparing Two Mathematical Tasks

How do the differences between the Strings Task and the Apples Task impact students’ opportunity to learn the Standards for Mathematical Content and to use the Standards for Mathematical Practice?

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Linking to Research/Literature: The QUASAR Project

…Not all tasks are not created equal - different tasks

require different levels and kinds of student thinking.

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

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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

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Instructional Tasks: The Cognitive Demand of Tasks Matters

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The Mathematical Tasks Framework

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

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Linking to Research/Literature: The QUASAR Project (continued)

• Low-Level Tasks The Apples Task

• High-Level The Strings Task

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Linking to Research/Literature: The QUASAR Project (continued)

• Low-Level Tasks– Memorization– Procedures Without Connections (e.g., The

Apples Task)

• High-Level Tasks– Doing Mathematics (e.g., The Strings Task)– Procedures With Connections

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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 Strings Task relate to those on the TAG? Which characteristics describe the Apples Task?

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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 Mathematical Practice Standards.

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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?

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Relating the Cognitive Demand of Tasks to the CCSS for Mathematical Practice

What relationships do you notice between the cognitive demand of the written tasks and the Standards for Mathematical Practice?

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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.


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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.

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.

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Linking to Research/Literature

Tasks 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

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Gallery Walk Procedure

• Circulate and analyze the modified tasks of the other groups.

• On a yellow sticky-note, comment about the ways in which the task was modified to increase the cognitive demand of the task.

• On a pink sticky-note, write wonderings if you can think of other ways the demand of the task can be increased.

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References

Smith, M. S., Stein, M. K., Arbaugh, F., Brown, C. A., & Mossgrove, J. (2004). Characterizing the cognitive demands of mathematical tasks: A task-sorting activity. In G. W. Bright and R. N. Rubenstein (Eds.), Professional development guidebook for perspectives on the teaching of mathematics: Companion to the sixty-sixth yearbook (pp. 45 - 47). Reston, VA: National Council of Teachers of Mathematics.

Smith, M. S. & Stein, M. K. (1998). Selecting and creating mathematical tasks: From research to practice. Mathematics Teaching in the Middle School, 3 (5), 344 - 350.

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supporting rigorous mathematics teaching and learning

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