supporting rigorous mathematics teaching and learning

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH

Supporting Rigorous Mathematics Teaching and Learning

Tennessee Department of Education

Elementary School Mathematics, Grade 1

December 7, 2012

Illuminating Student Thinking: Assessing and Advancing Questions

Rationale

Effective teaching requires being able to support students as they work on challenging tasks without taking over the process of thinking for them (NCTM, 2000). Asking questions that assess student understanding of mathematical ideas, strategies, or representations provides teachers with insights into what students know and can do. The insights gained from these questions prepare teachers to then ask questions that advance student understanding of mathematical concepts, strategies, or connections between representations.

By analyzing students’ written responses, teachers will have the opportunity to develop questions to both assess and advance student understanding of Mathematical Concepts and Mathematical Practice.

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The Mathematical Task Framework

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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

Stein, Smith, Henningsen, & Silver, 2000


Overview of Activities

• Discuss solutions to the Marble Tasks.

• Analyze student work to determine what students know and can do.

• Develop assessing and advancing questions and generalize the characteristics of each.

• Discuss the benefits of engaging in this process.

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Session Goals

• Learn to ask assessing and advancing questions

based on student responses to what is learned

about student thinking from an assessing question.

• Develop characteristics of assessing and advancing

questions and be able to distinguish the purpose of

each type.

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The Structures and Routines of a Lesson

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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 Lesson

1. Share and Model

2. Compare Solutions

3. 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 PATH

COMPARE: Students discuss similarities and difference between solution paths.FOCUS: Discuss the meaning of mathematical ideas in each representationREFLECT by engaging students in a quick write or a discussion of the process.

Set Up the TaskSet Up of the Task

Marbles Tasks: One-Digit Addition and Subtraction Situations (First Grade)

1. Connie had 5 marbles. Juan gave her 8 more marbles. How many marbles does Connie have altogether?

2. Connie has 5 marbles. How many more marbles does she need to have 13 marbles altogether?

Carpenter, Fennema, Franke, Levi, & Empson, 1999, p. 12

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The Common Core State Standards (CCSS) for Mathematical Content

Which of the CCSS for Mathematical Content can be

addressed when solving and discussing the tasks?

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Table 1: Common Addition and Subtraction Situations

9Common Core State Standards, 2010, p. 88, NGA Center/CCSSO

Common Core State Standards for Mathematics: Grade 1


Operations and Algebraic Thinking 1.OA

Represent and solve problems involving addition and subtraction.

1.OA.1 Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

1.OA.2 Solve word problems that call for addition of three whole numbers

whose sum is less than or equal to 20, e.g., by using objects,

drawings, and equations with a symbol for the unknown number to

represent the problem.




Understand and apply properties of operations and the relationship between addition and subtraction.

1.OA.3 Apply properties of operations as strategies to add and subtract. Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)

1.OA.4 Understand subtraction as an unknown-addend problem. For example, subtract 10 – 8 by finding the number that makes 10 when added to 8.




Add and subtract within 20.

1.OA.5 Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

1.OA.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).


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


Work with addition and subtraction equations.

1.OA.7 Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.

1.OA.8 Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = ? - 3, 6 + 6 = ?.

Common Core Standards for Mathematical Practice

What would have to happen in order for students to have opportunities to make use of 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

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What Does Each Student Know?

Individually examine the 3 pieces of student work A, B,

and C for the Marbles Tasks in your participant

handout.

What does each student know?

Be prepared to share and justify your conclusions.

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What Does Each Student Know?

Why is it important to make evidence-based comments

and to not make inferences when identifying what

students know and can do?

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Using Questioning During the Exploration PhaseImagine that you are walking around the room as your groups of students work on the Marbles Tasks, observing what they are doing.

Consider what you would say to the groups who produced responses A , B, and C in order to assess and advance their thinking about key mathematical ideas, problem-solving strategies, or use of and connection between representations.

Specifically, for each response, indicate what questions you would ask:

– to determine what the student knows and understands

(ASSESSING QUESTIONS).

– to move the student towards the target mathematical goals

(ADVANCING QUESTIONS).

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LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH 18

TargetMathematical

Goal

StudentsStudents’’ Mathematical Mathematical UnderstandingsUnderstandings

Assess

Assess


TargetTargetMathematical Mathematical

GoalGoal

A StudentA Student’’s Current s Current UnderstandingUnderstanding

AdvanceAdvance

MathematicalMathematicalTrajectoryTrajectory


TargetTargetMathematical Mathematical

GoalGoal

StudentsStudents’’ Mathematical Mathematical UnderstandingsUnderstandings


Three Goals of Assessing and Advancing Questions

• a mathematical understanding;

• a problem-solving strategy; and/or

• the connections between representations.

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Assessing and advancing questions prompt students to advance in their understanding of:

Linking to Research/LiteratureConnections Between Representations

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Pictures

WrittenSymbols

ManipulativeModels

Real-worldSituations

Oral Language

Adapted from Lesh, Post, & Behr, 1987


Asking Assessing and Advancing QuestionsStudent A

Connie had 5 marbles. Juan gave her 8 more marbles. How many marbles does Connie have altogether?

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Asking Assessing and Advancing QuestionsStudent B

Connie has 5 marbles. How many more marbles does she need to have 13 marbles altogether?

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Connie has 5 marbles. How many more marbles does she need to have 13 marbles altogether?

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Asking Assessing and Advancing QuestionsStudent C


Discussing Assessing Questions

• Listen as several assessing questions are read aloud.

• Consider how the assessing questions are similar to or different from each other.

• Are there any questions that you do not believe belong in this category and why?

• What are some general characteristics of the assessing questions?

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Looking for Patterns

• Why are some students’ assessing questions other students’ advancing questions?

• Why do all students need to be asked both an assessing and an advancing question?

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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.

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Reflection

• Why is it important to ask students both assessing

and advancing questions? What message do you

send to students if you ask ONLY assessing

questions?

• Look across the set of both assessing and

advancing questions. Do we ask more questions

related to Mathematical Content or Practice?

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Reflection

• All tasks are not created equal.

• Assessing and advancing questions can be asked of some tasks but not others. What are the characteristics of tasks in which it is worthwhile to ask assessing and advancing questions?

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Preparing to Ask Assessing and Advancing Questions

How does a teacher prepare to ask assessing and advancing questions?

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Supporting Student Thinking and Learning

In planning a lesson, what do you think can be gained

by considering how students are likely to respond to a

task and by developing questions in advance that can

assess and advance their learning, depending on the

solution path they choose?

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Reflection

What have you learned about assessing and

advancing questions that you can use in your

classroom tomorrow?

Turn and Talk

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Bridge to Practice

• Choose a high-level task. Plan a lesson with colleagues.

• Anticipate student responses, errors, and misconceptions.

• Write assessing and advancing questions related to the student responses. Keep copies of your planning notes.

• Teach the lesson. When you are in the Explore Phase of the lesson, tape your questions or ask a colleague to scribe your questions and the student responses.

• Following the lesson, reflect on the kinds of assessing and advancing questions you asked and consider the benefit to student learning.

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