Supporting Rigorous Mathematics Teaching and Learning Using Assessing and Advancing Questions to

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH

Supporting Rigorous Mathematics Teaching and Learning

Using Assessing and Advancing Questions to Target Essential Understandings

Tennessee Department of EducationElementary School MathematicsGrade 5

Rationale

There is wide agreement regarding the value of teachers attending to and basing their instructional decisions on the mathematical thinking of their students (Warfield, 2001).

By engaging in an analysis of a lesson planning process, teachers will have the opportunity to consider the ways in which the process can be used to help them plan and reflect, both individually and collectively, on instructional activities that are based on student thinking and understanding.

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Session Goals

Participants will:

• learn to set clear goals for a lesson;

• learn to write essential understandings and consider the relationship to the CCSS; and

• learn the importance of essential understandings (EUs) in writing focused advancing questions.

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Overview of Activities

Participants will:

• engage in a lesson and identify the mathematical goals of the lesson;

• write essential understandings (EUs) to further articulate a standard;

• analyze student work to determine where there is evidence of student understanding; and

• write advancing questions to further student understanding of EUs.

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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

Linking to Research/Literature: The QUASAR Project

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

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH 7

Solving and Discussing Solutions to the Multiplying and Dividing

FractionsTask


The Structures and Routines of a Lesson

The Explore Phase/Private Work TimeGenerate Solutions

The Explore Phase/Small Group Problem Solving

1. Generate and Compare Solutions2. Assess and Advance Student Learning

Share, Discuss, and Analyze Phase of the Lesson1. Share and Model2. Compare Solutions3. Focus the Discussion on Key

Mathematical Ideas 4. Engage in a Quick Write

MONITOR: Teacher selects examples for the Share, Discuss,and Analyze Phase based on:• Different solution paths to the same task• Different representations• Errors • Misconceptions

SHARE: Students explain their methods, repeat others’ ideas, put ideas into their own words, add on to ideas and ask for clarification.REPEAT THE CYCLE FOR EACH

SOLUTION PATHCOMPARE: Students discuss similarities and difference between solution paths.FOCUS: Discuss the meaning of mathematical ideas in each representationREFLECT: Engage students in a Quick Write or a discussion of the process.

Set Up the TaskSet Up of the Task


Multiplying and Dividing Fractions: Task Analysis

• Solve the task. Write sentences to describe the mathematical relationships that you notice.

• Anticipate possible student responses to the task.


Multiplying and Dividing Fractions

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7 x =

Use either a number line or an area model to show the correct answer.

What division equations can also describe the SAME model you created for 7 x ? Explain how they relate to the model.


Multiplying and Dividing Fractions: Task Analysis

• Study the Grade 5 CCSS for Mathematical Content within the Number and Operations-Fractions domain.

Which standards are students expected to demonstrate when solving the fraction task?

• Identify the CCSS for Mathematical Practice required by the written task.

The CCSS for Mathematics: Grade 5

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Number and Operations – Fractions 5.NF

Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

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

5.NF.B.4a Interpret the product (a/b) × q as a parts of a partition of qinto b equal parts; equivalently, as the result of a sequence ofoperations a × q ÷ b. For example, use a visual fraction model to show (2/3) x 4 = 8/3, and create a story context for this equation. Do the same with (2/3) x (4/5) = 8/15. (In general, (a/b) x (c/d) = ac/bd.)

5.NF.B.4b Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.

Common Core State Standards, 2010, p. 36, NGA Center/CCSSO

CCSS for Mathematics: Grade 5

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Number and Operations – Fractions 5.NF

Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

5.NF.B.5 Interpret multiplication as scaling (resizing), by:

5.NF.B.5a Comparing the size of a product to the size of one factor onthe basis of the size of the other factor, without performing theindicated multiplication.

5.NF.B.5b Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n × a)/(n × b) to the effect of multiplying a/b by 1.

5.NF.B.6 Solve real-world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.

Common Core State Standards, 2010, p. 36, NGA Center/CCSSO

CCSS for Mathematics: Grade 5

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Number and Operations – Fractions 5.NFApply and extend previous understandings of multiplication and division to multiply and divide fractions.

5.NF.B.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.

5.NF.B.7a Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) x 4 = 1/3.

5.NF.B.7b Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 x (1/5) = 4.

5.NF.B.7c Solve real-world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?

Common Core State Standards, 2010, p. 36-37, NGA Center/CCSSO

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.

15Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO


Common Core Content Standards and Mathematical

Practice Standards

Essential Understandings

The Common Core State Standards

Mathematical Essential UnderstandingMultiplying Fractions

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Objective

Students will show multiplication of a fraction by a whole number with a model.

Essential Understanding

When you multiply a fraction by whole number, the partitioning of the whole is based on the denominator. The number of partitions taken is the product of the whole number and numerator. This can be show by repeated addition of the fraction the number of times indicated by the whole number.

5.NF.B.4a Interpret the product (a/b) × q as a parts of a partition of qinto b equal parts; equivalently, as the result of a sequence ofoperations a × q ÷ b. For example, use a visual fraction model to show (2/3) x 4 = 8/3, and create a story context for this equation. Do the same with (2/3) x (4/5) = 8/15. (In general, (a/b) x (c/d) = ac/bd.)

Common Core State Standards, 2010

Mathematical Essential UnderstandingMultiplying Fractions

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Objective

Students will compute multiplication of fractions.


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


Mathematical Essential UnderstandingDividing Fractions by Whole Numbers

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ObjectiveStudents will divide unit fractions by whole numbers.


5.NF.B.7a Interpret division of a unit fraction by a non-zero whole number, and compute such quotients


Mathematical Essential UnderstandingDividing by a Unit Fraction

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ObjectiveStudents will whole numbers by unit fractions.


5.NF.B.7b Interpret division of a whole number by a unit fraction, and compute such quotients.



Essential Understanding CCSS

When multiplying a fraction by whole number, the partitioning of the whole is based on the denominator. The number of partitions taken is the product of the whole number and numerator. This can be show by repeated addition of the fraction the number of times indicated by the whole number..

5.NF.B.4a

When multiplying by a fraction, there is less than 1 whole group/less than 1 in a group therefore, the product will be less than one or both factors. 5.NF.B.4

When dividing a fraction by a whole number, every iteration of the unit fraction needs to be divided by the whole number.

5.NF.B.7a

When dividing a whole number by a unit fraction, the number of times that the unit fraction fits inside the whole number is determined by the denominator. 5.NF.B.7b



Asking Advancing Questions that Target the Essential

Understanding


TargetMathematical

Goal

Students’ Mathematical Understandings

Assess


TargetMathematical

Goal

A Student’s Current Understanding

Advance

MathematicalTrajectory


Target Mathematical Understanding

Illuminating Students’ Mathematical Understandings


Characteristics of Questions that Support Students’ Exploration

Assessing Questions• Based closely on the

work the student has produced.

• Clarify what the student has done and what the student understands about what s/he has done.

• Provide information to the teacher about what the student understands.

Advancing Questions• Use what students have

produced as a basis for making progress toward the target goal.

• Move students beyond their current thinking by pressing students to extend what they know to a new situation.

• Press students to think about something they are not currently thinking about.


Supporting Students’ Exploration(Analyzing Student Work)

Analyze the students’ responses.

Analyze the student’s group work to determine where there is evidence of student understanding.

What advancing question would you ask the students to further their understanding of an EU?


Essential Understanding CCSS

When multiplying a fraction by whole number, the partitioning of the whole is based on the denominator. The number of partitions taken is the product of the whole number and numerator. This can be show by repeated addition of the fraction the number of times indicated by the whole number..

5.NF.B.4a

When multiplying by a fraction, there is less than 1 whole group/less than 1 in a group therefore, the product will be less than one or both factors. 5.NF.B.4

When dividing a fraction by a whole number, every iteration of the unit fraction needs to be divided by the whole number.

5.NF.B.7a

When dividing a whole number by a unit fraction, the number of times that the unit fraction fits inside the whole number is determined by the denominator. 5.NF.B.7b


Group A


Group B

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Group C


Group D


Group E

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Reflecting on the Use of Essential Understandings

How does knowing the essential understandings help you in writing advancing questions?

Supporting Rigorous Mathematics Teaching and Learning Using Assessing and Advancing Questions to

Documents

development center

goals teachers

students mathematical

content standards

instructional goals

mathematical practice

mathematical underpinnings

mathematical tasks