Supporting Rigorous Mathematics Teaching and Learning

© 2013 University Of Pittsburgh

Supporting Rigorous Mathematics Teaching and Learning

Using Assessing and Advancing Questions to Target Essential Understandings

Tennessee Department of EducationElementary School MathematicsGrade 4

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.

3© 2013 University Of Pittsburgh

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.


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.

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


Solving and Discussing Solutions to the Thirds and Sixths Task


The Structure 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


Thirds and Sixths: Task Analysis

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

• Anticipate possible student responses to the task.


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.


Thirds and Sixths: Task Analysis

• Study the Grade 4 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 Mathematical Content − Grade 4

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

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.

The CCSS for Mathematical Content − Grade 4


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


Common Core Content Standards and Mathematical

Practice StandardsEssential Understandings

The Common Core State Standards

Mathematical Essential Understanding(Fractional Equivalence)

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.

Objective Essential UnderstandingStudents will explain with words and diagrams why two fractions are equivalent.

When you multiply a fraction by a fraction equivalent to one, the denominator is partitioned into a new designated number of pieces that are smaller in size, but larger in the number of pieces than the original and as a result of the partitioning, all of the pieces referenced by the numerator end up being partitioned, too.


Mathematical Essential Understanding(Adding Iterations of a Unit Fraction)

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

Objective Essential UnderstandingStudents will recognize that adding unit fractions gives you a non-unit fraction.


Mathematical Essential Understanding(Multiplying Iterations of a Unit Fraction)

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).

Objective Essential Understanding

Students use multiplication of a unit fraction to derive a non-unit fraction.


Mathematical Essential Understanding(Equivalent Unit Fraction Expressions)

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.)

Objective Essential Understanding

Students will write equivalent multiplication expressions to represent the same value.


Essential UnderstandingsEssential Understanding CCSS

When creating equivalent fractions, all of the pieces in a whole are subdivided or partitioned, thus the amount of pieces named in the numerator are automatically partitioned in the same way. What is created is an equivalent fraction.

4.NF.A.1

When you multiply a fraction by a fraction equivalent to one, the denominator is partitioned into a new designated number of pieces that are smaller in size, but larger in the number of pieces than the original and as a result of the partitioning of the denominator, all of the pieces referenced by the numerator end up being partitioned, too.

4.NF.A.2

Iterating a unit fraction is an interpretation of a fraction that illustrates the equal parts that the fraction is composed of ( = + + or 3 x , = 5 x ).

4.NF.B.3

When decomposing a fraction into iterations of the unit fraction, the number of iterations is the same as the value of the numerator.

4.NF.B.4

When “a” identical things are divided into “b” equal parts, each of “a” things contributes 1/b. So, a x 1/b = a/b. (NCTM Essential Understandings, 2011)

4.NF.B.4b


Asking Advancing Questions that Target the Essential

Understanding

© 2013 University Of Pittsburgh 22

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

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

Essential UnderstandingsEssential Understanding CCSS

When creating equivalent fractions, all of the pieces in a whole are subdivided or partitioned, thus the amount of pieces named in the numerator are automatically partitioned in the same way. What is created is an equivalent fraction.

4.NF.A.1

When you multiply a fraction by a fraction equivalent to one, the denominator is partitioned into a new designated number of pieces that are smaller in size, but larger in the number of pieces than the original and as a result of the partitioning of the denominator, all of the pieces referenced by the numerator end up being partitioned, too.

4.NF.A.2

Iterating a unit fraction is an interpretation of a fraction that illustrates the equal parts that the fraction is composed of ( = + + or 3 x , = 5 x ).

4.NF.B.3

When decomposing a fraction into iterations of the unit fraction, the number of iterations is the same as the value of the numerator.

4.NF.B.4

When “a” identical things are divided into “b” equal parts, each of “a” things contributes 1/b. So, a x 1/b = a/b. (NCTM Essential Understandings, 2011)

4.NF.B.4b


Group A


Group B


Group C


Group D


Group E


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

Documents

student thinking

students mathematical

practice standards

content standards

instructional goals

mathematical practice

mathematical underpinnings

mathematical tasks