Transcript (ASSESSING QUESTIONS).
Supporting Rigorous Mathematics Teaching and Learning
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
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
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 3
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. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH 5
The Structures and Routines of a Lesson
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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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 7
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? LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH 8
Table 1: Common Addition and Subtraction Situations
Common Core State Standards, 2010, p. 88, NGA Center/CCSSO 9
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. Common Core State Standards, 2010, p. 15, NGA Center/CCSSO 10
Common Core State Standards for Mathematics: Grade 1
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 11
Common Core State Standards for Mathematics: Grade 1
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 12
Common Core State Standards for Mathematics: Grade 1
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 13
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 14
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 Phase
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 LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Target Mathematical Goal 18
Mathematical Trajectory A Student ’ s Current Understanding LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Target Mathematical Goal 19
Students ’ Mathematical Understandings LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Target Mathematical Goal 20
Three Goals of Assessing and Advancing Questions
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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Linking to Research/Literature Connections Between Representations
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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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? LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH 26
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? LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH 29
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? LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH 30
Preparing to Ask Assessing and Advancing Questions
How does a teacher prepare to ask assessing and advancing questions? LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH 31
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. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH 34