Supporting Rigorous Mathematics Teaching and Learning

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 EducationElementary School MathematicsGrade 3

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 and differentiate between those that require problem-solving and those that assess for specific mathematical reasoning.

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 3 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 8th Grade.


Making Sense of the CCSS

Number and Operations Fractions


Making Sense of the Number and Operations – Fractions Standards

• Work in groups.• Each group will study and make sense of one or more of

the Standards for Mathematical Content. Your goal is to provide examples, counter-examples, visuals, or contexts that will help others understand your assigned standard.

• Each group will have five minutes to help us make sense of their standard(s).

Group 1 – 3.NF.A.1Group 2 – 3NF.A.2a, 3.NF.A.2bGroup 3 – 3.NF.A.3 Group 4 - 3.NF.A.3a, 3.NF.A.3bGroup 5 – 3.NF.A.3c, 3.NF.A.3dGroup 6 - 3.NF.A.3d

The CCSS for Mathematical Content: Grade 3

Common Core State Standards, 2010, p. 24, NGA Center/CCSSO

Number and Operations—Fractions 3.NFDevelop understanding of fractions as numbers.

3.NF.A.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

3.NF.A.2 Understand a fraction as a number on the number line; represent fractions on a number line diagram.

3.NF.A.2a Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.

3.NF.A.2b Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.



Number and Operations—Fractions 3.NFDevelop understanding of fractions as numbers.

3.NF.A.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.

3.NF.A.3a Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.

3.NF.A.3b Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction model.

3.NF.A.3c Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.

3.NF.A.3d Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.



Number and Operations – Fractions 4.NFExtend 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.



Number and Operations – Fractions 4.NFBuild 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 this understanding 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


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® discussions Accountable 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.

A. Half of a Whole Task Identify all of the figures that have one-half shaded. Be prepared to show and explain how you know that one-half of a figure is shaded. If a figure does not show one-half shaded, explain why. Make math statements about what is true about a half of a whole. Adapted from Watanabe, 1996


B. Equal Parts on the Geoboard

John claims that all of the sections on the geoboard are equal. Do you agree or disagree with John? If you agree, how can you prove that he is correct?

C. Making QuiltsTell what fraction of the whole rectangle that each numbered section represents. Explain how you know that the fraction represents the amount of the figure.

Adapted from Connected Mathematics, Prentice Hall, 2007


D. Pieces of Ribbon

0 1

Here are 3 fractions. Each fraction is greater than 0 and less than 1: Joan’s Ribbon Frank’s Ribbon Cheryl’s Ribbon ft. ft. ft.

Locate all 3 students’ pieces of ribbon on the number line below. Locate them as accurately as you can. Cheryl claims that she has the longest piece of ribbon. Do you agree or disagree and why?


E. Fractions on a Number Line

0 1 2 3

Place a point on the number line for each fraction.

Explain how and are related. How are these two fractions different from each other?


F. Eating Pizza

Harold and Jed each bought a small pizza for lunch.

Harold cut his pizza into 8 equal pieces and ate 3 of them.

Jed cut his pizza into 4 equal pieces and ate 2 of them. Which boy ate more pizza? Use diagrams, words, and numbers to show you are right.


G. Locating Points on a Number Line

Put these fractions on the number line below.


H. Shaded and Not Shaded

What fraction of the figure is shaded gray? Write 2 different fractions that describe the shaded portion of the figure.

Write 2 different fractions that describe the shaded portion of the figure. Explain how your fraction represents the shaded area.


Jenny, Gina, and Petra shared a pizza.

Jenny ate of the pizza.

Gina ate of it.

Petra ate the rest.

Did Petra eat more or less pizza than each of the other girls?

Show with a diagram each person’s share of the pizza and explain how you know Petra’s share of the pizza.

I. Sharing Pizza


J. A Fraction of the Whole

In each case below, the area of the whole rectangle is 1. Shade an area equal to the fraction underneath each rectangle. Compare the 2 amounts. a. Which is more and how do you know? Use the greater than

or less than sign to compare the amounts.b. Give another fraction that describes the shaded portion.


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?


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?


Instructional Tasks Versus Assessment Tasks

Instructional Tasks Assessment TasksAssist 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 school year?

Supporting Rigorous Mathematics Teaching and Learning

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