Transcript Mathematical Tasks: The Study of Equivalence November 18

Supporting Rigorous Mathematics
Teaching and Learning
Selecting and Sequencing Based on Essential
Understandings
Tennessee Department of Education
Middle School Mathematics
Grade 6
© 2013 UNIVERSITY OF PITTSBURGH
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.
© 2013 UNIVERSITY OF PITTSBURGH
2
Session Goals
Participants will learn about:
• goal-setting and the relationship of goals to the
CCSS and essential understandings;
• essential understandings as they relate to selecting
and sequencing student work;
• Accountable Talk® moves related to essential
understandings; and
• prompts that problematize or “hook” students during
the Share, Discuss, and Analyze phase of the
lesson.
Accountable Talk is a registered trademark of the University of Pittsburgh.
© 2013 UNIVERSITY OF PITTSBURGH
3
“The effectiveness of a lesson depends
significantly on the care with which the
lesson plan is prepared.”
Brahier, 2000
4
“During the planning phase, teachers make
decisions that affect instruction dramatically.
They decide what to teach, how they are going
to teach, how to organize the classroom, what
routines to use, and how to adapt instruction for
individuals.”
Fennema & Franke, 1992, p. 156
5
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
6
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
Setting Goals
Selecting Tasks
Anticipating 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 discussions
7
Identify Goals for Instruction
and Select an Appropriate Task
© 2013 UNIVERSITY OF PITTSBURGH
8
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
© 2013 UNIVERSITY OF PITTSBURGH
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.
9
Contextualizing Our Work Together
Imagine that you are working with a group of students who have the
following understanding of the concepts:
• 70% of the students need to make sense of what it means to
represent rational numbers on a number line. (6.NS.C.6,
C.6a, C.6c)
• 20% of the students need additional work understanding
when values in context should be represented with negative
numbers (6.NS.C.5). These students also need opportunities
to struggle with and make sense of the problem. (MP1)
• 5% of the students are consistently able to represent positive
and negative rational numbers as points on the number line
and are working on understanding absolute value as distance
from 0. (6.NS.C.7c)
• 5% of the students struggle to pay attention and their
understanding of numbers and operations is two grade levels
below sixth grade.
© 2013 UNIVERSITY OF PITTSBURGH
10
The CCSS for Mathematics: Grade 6
The Number System
6.NS
Apply and extend previous understandings of numbers to the system of
rational numbers.
6.NS.C.5
Understand that positive and negative numbers are used together
to describe quantities having opposite directions or values (e.g.,
temperature above/below zero, elevation above/below sea level,
credits/debits, positive/negative electric charge); use positive and
negative numbers to represent quantities in real-world contexts,
explaining the meaning of 0 in each situation.
6.NS.C.6
Understand a rational number as a point on the number line.
Extend number line diagrams and coordinate axes familiar from
previous grades to represent points on the line and in the plane
with negative number coordinates.
6.NS.C.6a
Recognize opposite signs of numbers as indicating locations on
opposite sides of 0 on the number line; recognize that the opposite
of the opposite of a number is the number itself, e.g., -(-3) = 3, and
that 0 is its own opposite.
Common Core State Standards, 2010, p. 43, NGA Center/CCSSO
11
The CCSS for Mathematics: Grade 6
The Number System
6.NS
Apply and extend previous understandings of numbers to the system of
rational numbers.
6.NS.C.6c
Find and position integers and other rational numbers on a
horizontal or vertical number line diagram; find and position pairs of
integers and other rational numbers on a coordinate plane.
6.NS.C.7c
Understand the absolute value of a rational number as its distance
from 0 on the number line; interpret absolute value as magnitude
for a positive or negative quantity in a real-world situation. For
example, for an account balance of -30 dollars, write |-30| = 30 to
describe the size of the debt in dollars.
Common Core State Standards, 2010, p. 43, NGA Center/CCSSO
12
Standards for Mathematical Practice
Related to the Task
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
13
Identify Goals: Solving the Task
(Small Group Discussion)
Solve the task.
Discuss the possible solution paths to the task.
© 2013 UNIVERSITY OF PITTSBURGH
14
Hiking Task
Dia’Monique and Yanely picnicked together. Then
Dia’Monique hiked 17 miles. Yanely hiked 14 miles in
the opposite direction. What is their distance from each
other? Draw a picture to show their distance from each
other.
© 2013 UNIVERSITY OF PITTSBURGH
15
Identify Goals Related to the Task
(Whole Group Discussion)
Does the task provide opportunities for students to
access the Standards for Mathematical Content and
Standards for Mathematical Practice that we have
identified for student learning?
© 2013 UNIVERSITY OF PITTSBURGH
16
Identify Goals: Essential Understandings
(Whole Group Discussion)
Study the essential understandings associated with the
Number System Common Core Standards.
Which of the essential understandings are the goals of
the Hiking Task?
© 2013 UNIVERSITY OF PITTSBURGH
17
Essential Understandings
(Small Group Discussion)
Essential Understanding
Positive and Negative Numbers Can be Used to Represent Real-World
Quantities
Positive numbers represent values greater than 0 and negative numbers represent
values less than 0. Many real-world situations can be modeled with both positive
and negative values because it is possible to measure above and below a baseline
value (often 0).
Rational Numbers Can be Located on a Number Line
Any rational number can be modeled using a point on the number line because the
real number line extends infinitely in the positive and negative directions. The sign
and the magnitude of the number determine the location of the point.
Absolute Value is a Measure of a Number’s Distance From 0
The absolute value of a number is the number’s magnitude or distance from 0. If
two rational numbers differ only by their signs, they have the same absolute value
because they are the same distance from zero.
Distance Between Two Values on a Number Line Can be Determined Using
Arithmetic
The distance between a positive and negative value on a number line is equal to the
sum of their absolute values because they are located on opposite sides of zero.
© 2013 UNIVERSITY OF PITTSBURGH
18
Selecting and Sequencing
Student Work for the
Share, Discuss, and Analyze
Phase of the Lesson
© 2013 UNIVERSITY OF PITTSBURGH
19
Analyzing Student Work
(Private Think Time)
• Analyze the student work.
• Identify what each group knows related to the
essential understandings.
• Consider the questions that you have about each
group’s work as it relates to the essential
understandings.
© 2013 UNIVERSITY OF PITTSBURGH
20
Prepare for the Share, Discuss, and Analyze Phase:
Selecting and Sequencing Student Work
(Small Group Discussion)
Assume that you have circulated and asked students
assessing and advancing questions.
Study the student work samples.
1. Which pieces of student work will allow you to
address the essential understanding?
2. How will you sequence the student’s work that you
have selected? Be prepared to share your rationale.
© 2013 UNIVERSITY OF PITTSBURGH
21
The Share, Discuss, and Analyze Phase:
Selecting and Sequencing Student Work
(Small Group Discussion)
In your small group, come to consensus on the work
that you select, and share your rationale. Be prepared
to justify your selection and sequence of student work.
Essential Understandings
Group(s)
Order Rationale
Positive and Negative Numbers Can
be Used to Represent Real-World
Quantities
Rational Numbers Can be Located
on a Number Line
Absolute Value is a Measure of a
Number’s Distance From 0
Distance Between Two Values on a
Number Line Can be Determined
Using Arithmetic
© 2013 UNIVERSITY OF PITTSBURGH
22
The Share, Discuss, and Analyze Phase:
Selecting and Sequencing Student Work
(Whole Group Discussion)
What order did you identify for the EUs and student work?
What is your rationale for each selection?
Essential Understandings
#1 via
Gr.
#2 via
Gr.
#3 via
Gr.
#4 Via
Gr.
Positive and Negative Numbers Can be
Used to Represent Real-World
Quantities
Positive numbers represent…
Rational Numbers Can be Located on a
Number Line
Any rational number…
Absolute Value is a Measure of a
Number’s Distance From 0
The absolute value…
Distance Between Two Values on a
Number Line Can be Determined Using
Arithmetic
The distance between…
© 2013 UNIVERSITY OF PITTSBURGH
23
Group A
© 2013 University Of Pittsburgh
24
Group B
© 2013 University Of Pittsburgh
25
Group C
© 2013 University Of Pittsburgh
26
Group D
© 2013 UNIVERSITY OF PITTSBURGH
27
Group E
© 2013 UNIVERSITY OF PITTSBURGH
28
Group F
© 2013 UNIVERSITY OF PITTSBURGH
29
Group G
© 2013 UNIVERSITY OF PITTSBURGH
30
The Share, Discuss, and Analyze Phase:
Selecting and Sequencing Student Work
(Whole Group Discussion)
What order did you identify for the EUs and student work?
What is your rationale for each selection?
Essential Understandings
#1 via
Gr.
#2 via
Gr.
#3 via
Gr.
#4 Via
Gr.
Positive and Negative Numbers Can be
Used to Represent Real-World
Quantities
Positive numbers represent…
Rational Numbers Can be Located on a
Number Line
Any rational number…
Absolute Value is a Measure of a
Number’s Distance From 0
The absolute value…
Distance Between Two Values on a
Number Line Can be Determined Using
Arithmetic
The distance between…
© 2013 UNIVERSITY OF PITTSBURGH
31
Academic Rigor in a Thinking
Curriculum
The Share, Discuss, and Analyze
Phase of the Lesson
© 2013 UNIVERSITY OF PITTSBURGH
32
Academic Rigor In a Thinking
Curriculum
A teacher must always be assessing and advancing
student learning.
A lesson is academically rigorous if student learning
related to the essential understanding is advanced in
the lesson.
Accountable Talk discussion is the means by which
teachers can find out what students know or do not
know and advance them to the goals of the lesson.
© 2013 UNIVERSITY OF PITTSBURGH
33
Accountable Talk Discussions
Recall what you know about the Accountable Talk
features and indicators. In order to recall what you
know:
• Study the chart with the Accountable Talk moves.
You are already familiar with the Accountable Talk
moves that can be used to Ensure Purposeful,
Coherent, and Productive Group Discussion.
• Study the Accountable Talk moves associated
with creating accountability to:
 the learning community;
 knowledge; and
 rigorous thinking.
© 2013 UNIVERSITY OF PITTSBURGH
34
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
35
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.
That’s an important point. One factor tells
use the number of groups and the other
factor tells us how many items in the group.
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.
S: 4 + 4 + 4.
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?
Recapping
You said three groups of four.
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?
36
Accountable Talk Moves (continued)
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
37
The Share, Discuss, and Analyze Phase of
the Lesson: Planning a Discussion
(Small Group Discussion)
• From the list of potential EUs and its related student
work, each group will select an essential
understanding to focus their discussion.
• Identify a teacher in the group who will be in charge
of leading a discussion with the group after the
Accountable Talk moves related to the EU have been
written.
Write a set of Accountable Talk moves on chart paper
so it is public to your group for the next stage in the
process.
© 2013 UNIVERSITY OF PITTSBURGH
38
An Example: Accountable Talk Discussion
The Focus Essential Understanding
Positive and Negative Numbers Can be Used to Represent Real-World
Quantities
Positive numbers represent values greater than 0 and negative numbers represent
values less than 0. Many real-world situations can be modeled with both positive
and negative values because it is possible to measure above and below a baseline
value (often 0).
Group B
Group G
•
•
•
•
•
•
Group B, how did your group determine where to place the point for each girl?
Who understood what she said about the opposite directions? (Community)
Can you say back what he said about the sign of the numbers? (Community)
Signed numbers can be used to represent positions in opposite directions from
a set point. (Revoicing)
Group G didn’t use negative numbers. How does your model represent the
problem situation? (Knowledge)
What do the 14 and 17 in Group G’s model represent? Is it possible to walk a
negative distance? What does the -14 represent? (Rigor)
© 2013 UNIVERSITY OF PITTSBURGH
39
Problematize the Accountable Talk Discussion
(Whole Group Discussion)
Using the list of essential understandings identified earlier, write Accountable
Talk discussion questions to elicit from students a discussion of the
mathematics.
Begin the discussion with a “hook” to get student attention focused on an
aspect of the mathematics.
Type of Hook
Example of a Hook
Compare and
Contrast
Compare the half that has two equal pieces with the
figure that has three pieces.
Three equal pieces of the six that are on one side of
Insert a Claim and Ask the figure show half of the figure. If I move the three
if it is True
pieces to different places in the whole, is half of the
figure still shaded?
Challenge
You said two pieces are needed to create halves. How
can this be half; it has three pieces?
A Counter-Example
If this figure shows halves (a figure showing three
sixths), tell me about this figure (a figure showing three
sixths but the sixths are not equal pieces).
© 2013 UNIVERSITY OF PITTSBURGH
40
An Example: Accountable Talk Discussion
The Focus Essential Understanding
Positive and Negative Numbers Can be Used to Represent Real-World Quantities
Positive numbers represent values greater than 0 and negative numbers represent
values less than 0. Many real-world situations can be modeled with both positive and
negative values because it is possible to measure above and below a baseline value
(often 0).
Group B
Group G
•
•
•
•
•
•
You can’t walk negative 14 miles, can you? Is it okay to use a negative number
to represent Yanely? Do you have to use negative numbers? (Hook)
Group B, how did your group determine where to place the point for each girl?
Who understood what she said about the opposite directions? (Community)
Can you say back what he said about the sign of the numbers? (Community)
Signed numbers can be used to represent positions in opposite directions from a set
point. (Revoicing)
Group G didn’t use negative numbers. How does your model represent the problem
situation?
© 2013 UNIVERSITY OF PITTSBURGH
41
Revisiting Your Accountable Talk Prompts
with an Eye Toward Problematizing
Revisit your Accountable Talk prompts.
Have you problematized the mathematics so as to draw
students’ attention to the mathematical goal of the
lesson?
• If you have already problematized the work, then
underline the prompt in red.
• If you have not problematized the lesson, do so
now. Write your problematizing prompt in red at
the bottom and indicate where you would insert it
in the set of prompts.
We will be doing a Gallery Walk after we role-play.
© 2013 UNIVERSITY OF PITTSBURGH
42
Role-Play Our Accountable Talk Discussion
• You will have 15 minutes to role-play the discussion of
one essential understanding.
• Identify one observer in the group. The observer will
keep track of the discussion moves used in the lesson.
• The teacher will engage you in a discussion. (Note:
You are well-behaved students.)
The goals for the lesson are:
 to engage all students in the group in developing
an understanding of the EU; and
 to gather evidence of student understanding
based on what the student shares during the
discussion.
© 2013 UNIVERSITY OF PITTSBURGH
43
Reflecting on the Role-Play: The
Accountable Talk Discussion
• The observer has 2 minutes to share observations
related to the lessons. The observations should be
shared as “noticings.”
• Others in the group have 1 minute to share their
“noticings.”
© 2013 UNIVERSITY OF PITTSBURGH
44
Reflecting on the Role-Play: The
Accountable Talk Discussion
(Whole Group Discussion)
Now that you have engaged in role-playing, what are
you now thinking about regarding Accountable Talk
discussions?
© 2013 UNIVERSITY OF PITTSBURGH
45
Zooming In on Problematizing
(Whole Group Discussion)
Do a Gallery Walk. Read each others’ problematizing
“hook.”
What do you notice about the use of hooks? What role
do “hooks” play in the lesson?
© 2013 UNIVERSITY OF PITTSBURGH
46
Step Back and Application to Our
Work
What have you learned today that you will apply when
planning or teaching in your classroom?
© 2013 UNIVERSITY OF PITTSBURGH
47
Summary of Our Planning Process
Participants:
• identify goals for instruction;
– Align Standards for Mathematical Content and
Standards for Mathematical Practice with a task.
– Select essential understandings that relate to the
Standards for Mathematical Content and Standards
for Mathematical Practice.
• prepare for the Share, Discuss, and Analyze Phase of
the lesson.
– Analyze and select student work that can be used to
discuss essential understandings of mathematics.
– Learn methods of problematizing the mathematics in
the lesson.
© 2013 UNIVERSITY OF PITTSBURGH
48