Transcript Mathematical Tasks: The Study of Equivalence November 18

Supporting Rigorous Mathematics
Teaching and Learning
Strategies for Scaffolding Student
Understanding: Academically Productive Talk
and the Use of Representations
Tennessee Department of Education
High School Mathematics
Geometry
© 2013 UNIVERSITY OF PITTSBURGH
Rationale
Teachers provoke students’ reasoning about
mathematics through the tasks they provide and the
questions they ask. (NCTM, 1991) Asking questions
that reveal students’ knowledge about mathematics
allows teachers to design instruction that responds to
and builds on this knowledge. (NCTM, 2000) Questions
are one of the only tools teachers have for finding out
what students are thinking. (Michaels, 2005)
Today, by analyzing a classroom discussion, teachers
will study and reflect on ways in which Accountable
Talk® (AT) moves and the use of representations
support student learning and help teachers to maintain
the cognitive demand of a task.
Accountable talk® is a registered trademark of the University of Pittsburgh.
Session Goals
Participants will learn about:
• Accountable Talk moves to support the
development of community, knowledge, and
rigorous thinking;
• Accountable Talk moves that ensure a productive
and coherent discussion and consider why moves
in this category are critical; and
• representations as a means of scaffolding student
learning.
© 2013 UNIVERSITY OF PITTSBURGH
Overview of Activities
Participants will:
• analyze and discuss Accountable Talk moves;
• engage in and reflect on a lesson in relationship to
the CCSS;
• analyze classroom discourse to determine the
Accountable Talk moves used by the teacher and
the benefit to student learning;
• design and enact a lesson, making use of the
Accountable Talk moves; and
• learn and apply a set of scaffolding strategies that
make use of the representations.
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Review the
Accountable Talk Features
and Indicators
Learn Moves Associated With
the Accountable Talk Features
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Linking to Research/Literature:
The QUASAR Project
The Mathematical Tasks Framework
TASKS
TASKS
TASKS
as they
appear in
curricular/
instructional
materials
as set up by
the teachers
as
implemented
by students
Student
Learning
Stein, Smith, Henningsen, & Silver, 2000
The Structure and Routines of a Lesson
Set Up
Up the
of the
Task
Set
Task
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
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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: Engage students
in a Quick Write or a discussion
of the process.
Accountable Talk Discussion
• Review the Accountable Talk features and indicators.
• Turn and Talk with your partner about what you recall
about each of the Accountable Talk features.
- Accountability to the learning community
- Accountability to accurate, relevant knowledge
- Accountability to discipline-specific standards
of rigorous thinking
© 2013 UNIVERSITY OF PITTSBURGH
Accountable Talk Features and Indicators
Accountability to the Learning Community
• Active participation in classroom talk.
• Listen attentively.
• Elaborate and build on each others’ ideas.
• Work to clarify or expand a proposition.
Accountability to Knowledge
• Specific and accurate knowledge.
• Appropriate evidence for claims and arguments.
• Commitment to getting it right.
Accountability to Rigorous Thinking
• Synthesize several sources of information.
• Construct explanations and test understanding of concepts.
• Formulate conjectures and hypotheses.
• Employ generally accepted standards of reasoning.
• Challenge the quality of evidence and reasoning.
© 2013 UNIVERSITY OF PITTSBURGH
Accountable Talk Moves
Consider:
• In what ways are the Accountable Talk moves
different within each of the categories?
− Support Accountability to Community
− Support Accountability to Knowledge
− Support Accountability to Rigorous Thinking
• There is a fourth category called, “To Ensure
Purposeful, Coherent, and Productive Group
Discussion.” Why do you think we need the set of
moves in this category?
© 2013 UNIVERSITY OF PITTSBURGH
Accountable Talk Moves
Talk Move
Function
Example
To Ensure Purposeful, Coherent,
and Productive Group Discussion
Marking
Direct attention to the value and
importance of a student’s
contribution.
It is important to say describe to
compare the size of the pieces and
then to look at how many pieces of
that size.
Challenging
Redirect a question back to the
students or use students’
contributions as a source for further
challenge or query.
Let me challenge you: Is that
always true?
Revoicing
Align a student’s explanation with
content or connect two or more
contributions with the goal of
advancing the discussion of the
content.
You said 3; yes, there are three
columns and each column is of the
whole.
Recapping
Make public in a concise, coherent
form the group’s achievement at
creating a shared understanding of
the phenomenon under discussion.
Let me put these ideas all together.
What have we discovered?
© 2013 UNIVERSITY OF PITTSBURGH
Accountable Talk Moves (continued)
Talk Move
Function
Example
To Support Accountability to
Community
Keeping the
Channels Open
Ensure that students can hear
each other, and remind them that
they must hear what others have
said.
Say that again and louder.
Can someone repeat what was just
said?
Keeping
Everyone
Together
Ensure that everyone not only
heard, but also understood, what
a speaker said.
Can someone add on to what was
said?
Did everyone hear that?
Linking
Contributions
Make explicit the relationship
between a new contribution and
what has gone before.
Does anyone have a similar idea?
Do you agree or disagree with
what was said?
Your idea sounds similar to his
idea.
Verifying and
Clarifying
Revoice a student’s contribution,
thereby helping both speakers
and listeners to engage more
profitably in the conversation.
So are you saying...?
Can you say more?
Who understood what was said?
© 2013 UNIVERSITY OF PITTSBURGH
Accountable Talk Moves (continued)
Talk Move
Function
Example
To Support Accountability to
Knowledge
Pressing for
Accuracy
Hold students accountable for the
accuracy, credibility, and clarity of
their contributions.
Why does that happen?
Someone give me the term for
that.
Building on
Prior
Knowledge
Tie a current contribution back to
knowledge accumulated by the
class at a previous time.
What have we learned in the
past that links with this?
To Support Accountability to
Rigorous Thinking
Pressing for
Reasoning
Elicit evidence to establish what
contribution a student’s utterance
is intended to make within the
group’s larger enterprise.
Say why this works.
What does this mean?
Who can make a claim and then
tell us what their claim means?
Expanding
Reasoning
Open up extra time and space in
the conversation for student
reasoning.
Does the idea work if I change
the context? Use bigger
numbers?
© 2013 UNIVERSITY OF PITTSBURGH
Five Representations of Mathematical Ideas
What role do the representations play in a discussion?
Pictures
Written
Manipulative
Models
Symbols
Real-world
Situations
Oral & Written
Language
Modified from Van De Walle, 2004, p. 30
Five Different Representations of a Function
What role do the representations play in a discussion?
Language
Context
Table
Graph
Van De Walle, 2004, p. 440
Equation
Engage In and Reflect On a Lesson
Building a New Playground Task
© 2013 UNIVERSITY OF PITTSBURGH
Building a New Playground Task
The City Planning Commission is considering building a
new playground. They would like the playground to be
equidistant from the two elementary schools,
represented by points A and B in the coordinate grid
that is shown.
© 2013 UNIVERSITY OF PITTSBURGH
Building a New Playground
PART A
1. Determine at least three possible locations for the park
that are equidistant from points A and B. Explain how you
know that all three possible locations are equidistant from
the elementary schools.
2. Make a conjecture about the location of all points that are
equidistant from A and B. Prove this conjecture.
PART B
3. The City Planning Commission is planning to build a third
elementary school located at (8, -6) on the coordinate
grid. Determine a location for the park that is equidistant
from all three schools. Explain how you know that all
three schools are equidistant from the park.
4. Describe a strategy for determining a point equidistant
from any three points.
© 2013 UNIVERSITY OF PITTSBURGH
The Cognitive Demand of the Task
Why is this considered to be a cognitively demanding
task?
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The Mathematical Task Analysis Guide
Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000) Implementing standards-based mathematics instruction:
A casebook for professional development, p. 16. New York: Teachers College Press.
The Common Core State Standards
(CCSS)
Solve the task.
Examine the CCSS for Mathematics.
• Which CCSS for Mathematical Content will
students discuss when solving the task?
• Which CCSS for Mathematical Practice will
students use when solving and discussing the
task?
© 2013 UNIVERSITY OF PITTSBURGH
The CCSS for Mathematical Content
CCSS Conceptual Category – Geometry
Congruence
(G-CO)
Understand congruence in terms of rigid motions.
G-CO.B.6 Use geometric descriptions of rigid motions to transform
figures and to predict the effect of a given rigid motion
on a given figure; given two figures, use the definition of
congruence in terms of rigid motions to decide if they
are congruent.
G-CO.B.7 Use the definition of congruence in terms of rigid
motions to show that two triangles are congruent if and
only if corresponding pairs of sides and corresponding
pairs of angles are congruent.
G-CO.B.8 Explain how the criteria for triangle congruence (ASA,
SAS, and SSS) follow from the definition of congruence
in terms of rigid motions.
Common Core State Standards, 2010, p. 76, NGA Center/CCSSO
The CCSS for Mathematical Content
CCSS Conceptual Category – Geometry
Congruence
(G-CO)
Prove geometric theorems.
G-CO.C.9
Prove theorems about lines and angles. Theorems include:
vertical angles are congruent; when a transversal crosses
parallel lines, alternate interior angles are congruent and
corresponding angles are congruent; points on a
perpendicular bisector of a line segment are exactly those
equidistant from the segment’s endpoints.
G-CO.C.10 Prove theorems about triangles. Theorems include: measures
of interior angles of a triangle sum to 180°; base angles of
isosceles triangles are congruent; the segment joining
midpoints of two sides of a triangle is parallel to the third side
and half the length; the medians of a triangle meet at a point.
G-CO.C.11
Prove theorems about parallelograms. Theorems include:
opposite sides are congruent, opposite angles are congruent,
the diagonals of a parallelogram bisect each other, and
conversely, rectangles are parallelograms with congruent
diagonals.
Common Core State Standards, 2010, p. 76, NGA Center/CCSSO
The CCSS for Mathematical Content
CCSS Conceptual Category – Geometry
Similarity, Right Triangles, and Trigonometry
(G-SRT)
Define trigonometric ratios and solve problems involving right
triangles.
G-SRT.C.6
Understand that by similarity, side ratios in right triangles are
properties of the angles in the triangle, leading to definitions
of trigonometric ratios for acute angles.
G-SRT.C.7
Explain and use the relationship between the sine and
cosine of complementary angles.
G-SRT.C.8
Use trigonometric ratios and the Pythagorean Theorem to
solve right triangles in applied problems.★
★ Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling
standards appear throughout the high school standards indicated with a star (★). Where an entire domain is marked with a
star, each standard in that domain is a modeling standard.
Common Core State Standards, 2010, p. 77, NGA Center/CCSSO
The CCSS for Mathematical Content
CCSS Conceptual Category – Geometry
Expressing Geometric Properties with Equations
(G-GPE)
Use coordinates to prove simple geometric theorems algebraically.
G-GPE.B.4
G-GPE.B.5
Use coordinates to prove simple geometric theorems algebraically.
For example, prove or disprove that a figure defined by four given
points in the coordinate plane is a rectangle; prove or disprove that
the point (1, √3) lies on the circle centered at the origin and containing
the point (0, 2).
Prove the slope criteria for parallel and perpendicular lines and use
them to solve geometric problems (e.g., find the equation of a line
parallel or perpendicular to a given line that passes through a given
point).
G-GPE.B.6
Find the point on a directed line segment between two given points
that partitions the segment in a given ratio.
G-GPE.B.7
Use coordinates to compute perimeters of polygons and areas of
triangles and rectangles, e.g., using the distance formula.★
★Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling
standards appear throughout the high school standards indicated with a star (★). Where an entire domain is marked with a
star, each standard in that domain is a modeling standard.
Common Core State Standards, 2010, p. 78, NGA Center/CCSSO
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
Analyzing a Lesson: Lesson Context
Teacher:
Grade Level:
Debbee Campbell
Geometry
School:
School District:
Tyner Academy
Hamilton County School District
The students and the teacher in this school have been
working to make sense of the Common Core State
Standards for the past two years.
The teacher is working on using the Accountable Talk
moves and making sure she targets the mathematics
standards in very deliberate ways during the lesson.
© 2013 UNIVERSITY OF PITTSBURGH
Building a New Playground Task
The City Planning Commission is considering building a
new playground. They would like the playground to be
equidistant from the two elementary schools,
represented by points A and B in the coordinate grid
that is shown.
© 2013 UNIVERSITY OF PITTSBURGH
Building a New Playground
PART A
1. Determine at least three possible locations for the park
that are equidistant from points A and B. Explain how you
know that all three possible locations are equidistant from
the elementary schools.
2. Make a conjecture about the location of all points that are
equidistant from A and B. Prove this conjecture.
PART B
3. The City Planning Commission is planning to build a third
elementary school located at (8, -6) on the coordinate
grid. Determine a location for the park that is equidistant
from all three schools. Explain how you know that all
three schools are equidistant from the park.
4. Describe a strategy for determining a point equidistant
from any three points.
© 2013 UNIVERSITY OF PITTSBURGH
Instructional Goals
Debbee’s instructional goals for the lesson are:
• students will determine a set of points that are
equidistant from two points, A and B;
• students will recognize and conjecture that all such
points fall on the perpendicular bisector of 𝐴𝐵 ; and
• students will prove their conjecture.
© 2013 UNIVERSITY OF PITTSBURGH
Reflection Question
(Small Group Discussion)
As you watch the video segment, consider what
students are learning about mathematics.
Name the moves used by the teacher and the purpose
that the moves served.
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Reflecting on the Accountable Talk
Discussion
(Whole Group Discussion)
• Step back from the discussion. What are some
patterns that you notice?
• What mathematical ideas does the teacher want
students to discover and discuss?
• How does talk scaffold student learning?
© 2013 UNIVERSITY OF PITTSBURGH
Five Representations of Mathematical Ideas
What role did tools or representations play in scaffolding student
learning?
Pictures
Manipulative
Written
Models
Symbols
Real-world
Situations
Oral & Written
Language
Modified from Van De Walle, 2004, p. 30
Five Different Representations of a Function
What role did tools or representations play in scaffolding
student learning?
Language
Context
Table
Graph
Van De Walle, 2004, p. 440
Equation
Giving it a Go: Planning for an
Accountable Talk Discussion of a
Mathematical Idea
• Identify a person who will be teaching the lesson to
others in your small group.
• Plan the lesson together. Anticipate student
responses.
• Write Accountable Talk questions/moves that the
teacher will ask students to advance their
understanding of a mathematical idea.
© 2013 UNIVERSITY OF PITTSBURGH
Building a New Playground Task
The City Planning Commission is considering building a
new playground. They would like the playground to be
equidistant from the two elementary schools,
represented by points A and B in the coordinate grid
that is shown.
© 2013 UNIVERSITY OF PITTSBURGH
Building a New Playground
PART A
1. Determine at least three possible locations for the park
that are equidistant from points A and B. Explain how you
know that all three possible locations are equidistant from
the elementary schools.
2. Make a conjecture about the location of all points that are
equidistant from A and B. Prove this conjecture.
PART B
3. The City Planning Commission is planning to build a third
elementary school located at (8, -6) on the coordinate
grid. Determine a location for the park that is equidistant
from all three schools. Explain how you know that all
three schools are equidistant from the park.
4. Describe a strategy for determining a point equidistant
from any three points.
© 2013 UNIVERSITY OF PITTSBURGH
Focus of the Discussion
Recognize that, to fully answer the question, students
must prove points are equidistant from A and B iff they
fall on the perpendicular bisector of 𝐴𝐵 .
© 2013 UNIVERSITY OF PITTSBURGH
Reflection: The Use of Accountable
Talk Moves and Tools to Scaffold
Student Learning
What have you learned?
© 2013 UNIVERSITY OF PITTSBURGH