Supporting Rigorous Mathematics Teaching and Learning Using Assessing and Advancing

© 2013 UNIVERSITY OF PITTSBURGH 1

Supporting Rigorous Mathematics Teaching and Learning

Using Assessing and Advancing Questions to Target Essential Understandings

Tennessee Department of EducationElementary School MathematicsGrade 3

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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© 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.

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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

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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

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® discussion Accountable Talk ® is a registered trademark of the University of Pittsburgh

7© 2013 UNIVERSITY OF PITTSBURGH

Solving and Discussing Solutions to the Half of a Whole Task

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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


Half of a Whole: Task Analysis

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

• Anticipate possible student responses to the task.

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


Half of a Whole: Task Analysis

• Study the Grade 3 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 3

12Common 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.

The CCSS for Mathematical Content: Grade 3

13Common Core State Standards, 2010, p. 24, NGA Center/CCSSO

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.

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.

14Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO


Common Core State Standards for Mathematical Content and Mathematical

Practice

Essential Understandings

The Common Core State Standards

Mathematical Essential UnderstandingNot All Halves Are Created Equal

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What is mathematically true about the figures shown above?

ObjectiveStudents will discover, via the use of the fractional pieces, that not all halves are equal.

Essential UnderstandingIf the wholes differ, then a fractional piece from each of the wholes will not be equal because their initial whole was not the same (e.g., of a large pizza is not the same as of a small pizza).

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.

Mathematical Essential UnderstandingA Whole Can Be Represented as a Fraction

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ObjectiveStudents will recognize that fractions in the form , , , , etc., are fractional names for a whole, or 1.

Essential Understanding

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.

Mathematical Essential UnderstandingHalf of the Denominator Is Half of the Whole

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ObjectiveStudents recognize that two equal parts of the whole represent half of the figure.


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.

Mathematical Essential UnderstandingHalves Take Up the Same Amount of Space

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ObjectiveStudents recognize half of a figure as two spaces of equal 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.

Mathematical Essential UnderstandingContinuous and Discrete Figures Represent a Whole

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ObjectiveStudents recognize equivalent fractions as those that have the same amount of space of a figure.


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.

Essential UnderstandingsEssential Understanding CCSS

Equal Size PiecesA fraction describes the division of a whole or unit (region, set, segment) into equal parts. A fraction is relative to the size of the whole or unit.

3.NF.A.3d

Half of the Denominator is Half of the WholeThe denominator (bottom number) tells how many equal parts into which the whole or unit is divided. The numerator (top number) tells how many equal parts of that subdivided whole are indicated. 3 equal pieces is half of 6 pieces because 3 + 3 makes 6. 2 equal pieces is half of 4 pieces because 2 + 2 makes 4.

3.NF.A.1

Halves Take Up the Same Amount of SpaceA fraction can be continuous (linear model), or a measurable quantity (area model), or a group of discrete/countable things (set model) but regardless of the model what remains true about all of the models is that they represent equal parts of a whole.

3.NF.A.3a

Continuous and Discrete Figures Can Both Represent HalvesFigures show halves if the same amount of area exists for each half, regardless of the location of the areas in the figures.

3.NF.A.3b

Not all Halves are Created EqualA fraction describes the division of a whole or unit (region, set, segment) into equal parts. A fraction is relative to the size of the whole or unit.

3.NF.A.3d

A Whole Can be Represented as a FractionIf all of the equal pieces in the denominator are represented in the numerator, then the whole figure is represented. = 1 = 1 = 1

3.NF.A.3c


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.

Essential UnderstandingsEssential Understanding CCSS

Equal Size PiecesA fraction describes the division of a whole or unit (region, set, segment) into equal parts. A fraction is relative to the size of the whole or unit.

3.NF.A.3d

Half of the Denominator is Half of the WholeThe denominator (bottom number) tells how many equal parts into which the whole or unit is divided. The numerator (top number) tells how many equal parts of that subdivided whole are indicated. 3 equal pieces is half of 6 pieces because 3 + 3 makes 6. 2 equal pieces is half of 4 pieces because 2 + 2 makes 4.

3.NF.A.1

Halves Take Up the Same Amount of SpaceA fraction can be continuous (linear model), or a measurable quantity (area model), or a group of discrete/countable things (set model) but regardless of the model what remains true about all of the models is that they represent equal parts of a whole.

3.NF.A.3a

Continuous and Discrete Figures Can Both Represent HalvesFigures show halves if the same amount of area exists for each half, regardless of the location of the areas in the figures.

3.NF.A.3b

Not all Halves are Created EqualA fraction describes the division of a whole or unit (region, set, segment) into equal parts. A fraction is relative to the size of the whole or unit.

3.NF.A.3d

A Whole Can be Represented as a FractionIf all of the equal pieces in the denominator are represented in the numerator, then the whole figure is represented. = 1 = 1 = 1

3.NF.A.3c


Supporting Students’ ExplorationAnalyzing Student Work

Analyze the students’ responses.

Analyze the 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?


Group A: Lauren and Austin


Group B: Jacquelyn, Alex, and Ethan

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Group C: Tylor, Jessica, and Tim


Group D: Frank, Juan, and Kimberly


Group E: JT, Fiona, and Keisha

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Reflecting on the Use of Essential Understandings

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

Documents

Supporting Rigorous Mathematics Teaching and Learning Using Assessing and Advancing