LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH Supporting Rigorous Mathematics Teaching and Learning Making Sense of the Number

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 4


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.

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LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH 3

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.


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.


The Data About Students’ Understanding of Fractions

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.

Siegler, Carpenter, Fennell, Geary, Lewis, Okamoto, Thompson, & Wray (2010).

IES, U.S. Department of Education

The Data about Fractions:Conceptual Understanding

A high percentage of U.S. students lack conceptual understanding of

fractions, even after studying fractions for several years; this, in turn,

limits students’ ability to solve problems with fractions and to learn and

apply computational procedures involving fractions.

• 50% of 8th graders could not order three fractions from least to greatest;

• 27% of 8th graders could not correctly shade of a rectangle;

• 45% of 8th graders could not solve a word problem that required dividing fractions (NAEP, 2004).

• Fewer than 30% of 17-year-olds correctly translated 0.029 as (Kloosterman, 2010).

The Data about Fractions:Conceptual Understanding

A lack of conceptual understanding of fractions has

several facets, including…students’ focusing on

numerators and denominators as separate numbers

rather than thinking of the fraction as a single number.

Errors such as believing that > arise from comparing the

two denominators and ignoring the essential relationship

between each fraction’s numerator and denominator.

Siegler, Carpenter, Fennell, et al; U.S. Dept. of Education, IES Practice Guide:

Developing Effective Fractions Instruction for Kindergarten through 8 th Grade.


Analyzing Tasks as a Means of Making Sense of the CCSS

Number and Operations – Fractions

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

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


Setting Goals Selecting TasksAnticipating 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® discussionsAccountable Talk® is a registered trademark of the University of Pittsburgh



• Low-level tasks

• High-level tasks



• Low-level tasks

– Memorization

– Procedures without Connections

• High-level tasks

– Doing Mathematics

– Procedures with Connections


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.

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.


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?


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.


Analyzing Tasks: Aligning with the CCSS(Whole Group Discussion)

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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 4

Common Core State Standards, 2010, p. 30, NGA Center/CCSSO

Number and Operations – Fractions 4.NF

Extend understanding of fraction equivalence and ordering.

4.NF.A.1 Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

4.NF.A.2 Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.

4.NF.B.3 Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

4.NF.B.3a Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.

The CCSS for Mathematical Content − Grade 4

Common Core State Standards, 2010, p. 30, NGA Center/CCSSO

Number and Operations – Fractions 4.NF

Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.

4.NF.B.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.

4.NF.B.4a Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).

4.NF.B.4b Understand a multiple of a/b as a multiple of 1/b, and use thisunderstanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)

4.NF.B.4c Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?

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


A. Writing a Rule for Comparing

Isabelle is comparing fractions. She says that she can

see, without doing any calculations, which one is greater

in each of the pairs below:

and

and

and

What rule can be written for comparing the fractions in each pair without finding a common denominator?

Does the rule you have written work with all fractions?


B. Leftover Pizza

Frankie orders a pizza. He eats of the pizza. His little sister eats of the pizza. How much is left?


C. Thirds and Sixths

Joel looks at the picture below and says, “I see of the picture is shaded.”

Sammy says, “No, of the picture is shaded.”

Who is correct? Write addition and multiplication equations to prove your answer.


D. Four-Fifths of His Homework

Jesse has been working on homework. He looks at the

number of problems he has completed and figures out

that he has finished of his homework. If he has 20

problems for homework, how many does he still have

to complete?


E. Thirds of Sandwiches

Tara invited friends over to work on homework. She is

ordering submarine sandwiches for dinner. They are

large sandwiches so she plans on giving of a sandwich

to each person. If she wants to feed 7 friends and

herself, how many sandwiches does she need to order?


F. Three Cakes

Ashlee brought 3 cakes to school to share with

classmates. There are 30 students in the class. How

much cake does each student get?


G. Decorating Gifts

Sarah bought 5 feet of ribbon. She needs to wrap 3 gifts

and wants to decorate each gift with an equal amount of

ribbon. How many feet of ribbon will be used per gift if

she wants to use all 5 feet?


H. Eating Cereal

Sam buys a box of his favorite cereal. He eats of the

box per day. How much of the box has he eaten by the

5th day? Show how you know you are correct.

Sam’s sister likes a different cereal. By the 5th day, she

has eaten of her box. Who has eaten more cereal by

day 5 if the boxes are the same size?


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?


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?


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.


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.

Instructional Tasks Versus Assessment TasksInstructional Tasks Assessment Tasks

Assist learners to learn the CCSS for Mathematical Content and the CCSS for Mathematical Practice.

Assess 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.


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 year?

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

Documents

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH Supporting Rigorous Mathematics Teaching and Learning Making Sense of the Number