© 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER Supporting Rigorous Mathematics Teaching and Learning Tennessee Department of

© 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER

Supporting Rigorous Mathematics Teaching and Learning

Tennessee Department of Education

Elementary School Mathematics

Grade 1

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, 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.

3© 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER

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 TaskMary, 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 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? ____________________________________________

Mary

Nick

Jean


Nine Plus a Number TaskSolve 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 1Operations and Algebraic Thinking 1.OA

Represent and solve problems involving addition and subtraction.

1.OA.A.1 Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

1.OA.A.2 Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Common Core State Standards, 2010, p. 15, NGA Center/CCSSO


Understand and apply properties of operations and the relationship between addition and subtraction.

1.OA.B.3 Apply properties of operations as strategies to add and subtract. Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)

1.OA.B.4 Understand subtraction as an unknown-addend problem. For example, subtract 10 – 8 by finding the number that makes 10 when added to 8.



Add and subtract within 20.

1.OA.C.5 Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

1.OA.C.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).


Common Core State Standards for Mathematics: Grade 1

Operations and Algebraic Thinking 1.OA

Work with addition and subtraction equations.

1.OA.D.7 Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.

1.OA.D.8 Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = ? – 3, 6 + 6 = ?.


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


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

…Not all tasks are created equal - different tasks will

provoke 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

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


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


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

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.

4 + 5 + 9 4 + (5 + 9)

9 + 8 10 + 8

8 + 6 4 + 4 + 6


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 DManipulatives/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 EManipulatives/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.

1. Jim has 23 red pencils and 8 pencils are not sharpened. How many pencils are sharpened?

2. Jamie has 48 cookies and some of them are chocolate and some are vanilla. 26 cookies are chocolate. How many cookies are vanilla?

3. 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


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 JManipulatives/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


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


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

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


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.

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

Documents

© 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER Supporting Rigorous Mathematics Teaching and Learning Tennessee Department of