Illuminating Student Thinking: Assessing and
Advancing Questions
Tennessee Department of Education
Elementary School Mathematics, Grade 1
December 7, 2012
LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH

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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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
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Discuss solutions to the Marble Tasks.
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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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• 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 Explore Phase/Private Work Time
Generate Solutions
The Explore Phase/Small-Group Problem
Solving
1. Generate and Compare Solutions
2. 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 representation
REFLECT by engaging students in a quick write or a discussion of the process.
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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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Which of the CCSS for Mathematical Content can be addressed when solving and discussing the tasks?
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Common Core State Standards, 2010, p. 88, NGA Center/CCSSO 9

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.
Common Core State Standards, 2010, p. 15, NGA Center/CCSSO
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Operations and Algebraic Thinking 1.OA
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.
Common Core State Standards, 2010, p. 15, NGA Center/CCSSO
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Operations and Algebraic Thinking 1.OA
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).
Common Core State Standards, 2010, p. 15, NGA Center/CCSSO
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Operations and Algebraic Thinking 1.OA
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 State Standards, 2010, p. 15, NGA Center/CCSSO
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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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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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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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Imagine 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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Students ’ Mathematical
Understandings
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Target
Mathematical
Goal
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Mathematical
Trajectory
A Student ’ s Current Understanding
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Target
Mathematical Goal
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Students ’ Mathematical
Understandings
LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH
Target
Mathematical Goal
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Assessing and advancing questions prompt students to advance in their understanding of:
• a mathematical understanding;
• a problem-solving strategy; and/or
• the connections between representations.
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Pictures
Manipulative
Models
Written
Symbols
Real-world
Situations
Oral
Language
Adapted from Lesh, Post, & Behr, 1987 22

Asking Assessing and Advancing Questions
Student 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 Questions
Student B
Connie has 5 marbles. How many more marbles does she need to have 13 marbles altogether?
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Asking Assessing and Advancing Questions
Student C
Connie has 5 marbles. How many more marbles does she need to have 13 marbles altogether?
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• 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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• 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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Assessing Questions
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Based closely on the work the student has produced.
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Clarify what the student has done and what the student understands about what s/he has done.
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Provide information to the teacher about what the student understands.
Advancing Questions
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Use what students have produced as a basis for making progress toward the target goal.
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Move students beyond their current thinking by pressing students to extend what they know to a new situation.
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Press students to think about something they are not currently thinking about.
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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?
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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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All tasks are not created equal.
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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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How does a teacher prepare to ask assessing and advancing questions?
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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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What have you learned about assessing and advancing questions that you can use in your classroom tomorrow?
Turn and Talk
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Choose a high-level task. Plan a lesson with colleagues.
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Anticipate student responses, errors, and misconceptions.
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Write assessing and advancing questions related to the student responses. Keep copies of your planning notes.
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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.
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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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