Transcript Mathematical Tasks: The Study of Equivalence November 18

Mathematics Instruction:
Planning, Teaching, and Reflecting
Modifying Tasks to Increase the Cognitive Demand
Tennessee Department of Education
Elementary School Mathematics
Grade 4
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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.
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Rationale
There is no decision that teachers make that has a
greater impact on students’ opportunities to learn and on
their perceptions about what mathematics is than the
selection or creation of the tasks with which the teacher
engages students in studying mathematics.
Lappan & Briars, 1995
By determining the cognitive demands of tasks and being
cognizant of the features of tasks that make them highlevel or low-level tasks, teachers will be prepared to
select or modify tasks that create opportunities for
students to engage with more tasks that are high-level
tasks.
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Session Goals
Participants will:
• deepen understanding of the cognitive demand of
a task;
• analyze a set of original and modified tasks to
learn strategies for increasing the cognitive
demand of a task; and
• recognize how increasing the cognitive demand of
a task gives students access to the Common Core
State Standards (CCSS) for Mathematical
Practice.
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Overview of Activities
Participants will:
• discuss and compare the cognitive demand of
mathematical tasks;
• identify strategies for modifying tasks; and
• modify tasks to increase the cognitive demand of
the tasks.
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Mathematical Tasks:
A Critical Starting Point for Instruction
All tasks are not created equal−different tasks require
different levels and kinds of student thinking.
Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standardsbased mathematics instruction: A casebook for professional development, p. 3.
New York: Teachers College Press.
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Mathematical Tasks:
A Critical Starting Point for Instruction
The level and kind of thinking in which students engage
determines what they will learn.
Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Olivier, & Human, 1997
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Mathematical Tasks:
A Critical Starting Point for Instruction
If we want students to develop the capacity to think,
reason, and problem-solve, then we need to start with
high-level, cognitively complex tasks.
Stein & Lane, 1996
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Revisiting the Characteristics of
Cognitively Demanding
Tasks
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Comparing the Cognitive Demand of
Two Tasks
• Compare the two tasks.
• How are the tasks similar? How are the tasks
different?
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Task #1: A Place Value Task
Identify the place value for each of the underlined digits.
a. 351
b. 76
c. 4,789
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Task #2: What is Changing?
Solve each equation.
234 + 10 = ___
379 + 10 = ___
389 + 10 = ___
399 + 10 = ___
489 + 10 = ___
499 + 10 = ___
When ten is added to each of the numbers above, how is the sum
changing from one equation to the next?
Sometimes the tens place changes and sometimes the hundreds
place change when ten is added to the number. Why does this
happen and when does it happen?
Look at the number 2,399. Which numbers will change when ten is
added to this number and why?
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Linking to Research/Literature:
The QUASAR Project
• Low-Level tasks
– Memorization
– Procedures without Connections
• High-Level tasks
– Procedures with Connections
– Doing Mathematics
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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:
© 2013 UNIVERSITY OF PITTSBURGH
A casebook for professional development, p. 16. New York: Teachers College Press
.
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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
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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
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
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Talk discussions
Identify Goals for Instruction
and Select an Appropriate Task
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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.
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The CCSS for Mathematics: Grade 4
Number and Operations – Fractions
4.NF
Extend understanding of fraction equivalence and ordering.
4.NF.A.1
4.NF.A.2
Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b) by
using visual fraction models, with attention to how the number and size
of the parts differ even though the two fractions themselves are the
same size. Use this principle to recognize and generate equivalent
fractions.
Compare two fractions with different numerators and different
denominators, e.g., by creating common denominators or numerators,
or by comparing to a benchmark fraction such as 1/2. Recognize that
comparisons are valid only when the two fractions refer to the same
whole. Record the results of comparisons with symbols >, =, or <, and
justify the conclusions, e.g., by using a visual fraction model.
Common Core State Standards, 2010, p. 30, NGA Center/CCSSO
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The CCSS for Mathematics: Grade 4
Number and Operations – Fractions
4.NF
Understand decimal notation for fractions, and compare decimal
fractions.
4.NF.C.5 Express a fraction with denominator 10 as an equivalent fraction with
denominator 100, and use this technique to add two fractions with
respective denominators 10 and 100. For example, express 3/10 as
30/100, and add 3/10 + 4/100 = 34/100.
4.NF.C.6
Use decimal notation for fractions with denominators 10 or 100. For
example, rewrite 0.62 as 62/100; describe a length as 0.62 meters;
locate 0.62 on a number line diagram.
4.NF.C.7
Compare two decimals to hundredths by reasoning about their size.
Recognize that comparisons are valid only when the two decimals
refer to the same whole. Record the results of comparisons with the
symbols >, =, or <, and justify the conclusions, e.g., by using a visual
model.
Common Core State Standards, 2010, p. 31, NGA Center/CCSSO
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Mathematical Practice Standards 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
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Identify Goals: Solving the Task
(Small Group Discussion)
Revisit the Pizza Task.
Solve the task.
Discuss the possible solution paths to the task.
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The Pizza Task
1
4
Jolla has of a pizza.
Sarah has
Maria has
30
of a pizza.
100
3
of a pizza.
100
Tim’s pizza is shaded on the pizza. How much pizza is Tim’s
share?
Jake has
Juan has
3
of a pizza.
10
1
of a pizza.
5
1. Show each of the student’s amount of pizza.
2. Compare the students’ amounts of pizza. Explain with
words and use the >, <, or = symbols to show who has the
most pizza.
3. Explain with words and use the >, <, or = symbols to show
who has the least amount of pizza.
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The Pizza Task (continued)
Jolla’s Pizza
Tim’s Pizza
Juan’s Pizza
Sarah’s Pizza
Maria’s Pizza
Jake’s Pizza
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Identify Goals Related to the Task
(Whole Group Discussion)
Does the task provide opportunities for students to
access the Mathematical Content Standards and
Practice Standards that we have identified for student
learning?
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Giving it a Go:
Modifying Textbook Tasks to
Increase the Cognitive Demand
of Tasks
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Your Turn to Modify Tasks
• Form groups of no more than three people.
• Discuss briefly important NEW mathematical
concepts, processes, or relationships you will want
students to uncover by the textbook page. Consult
the CCSS.
• Determine the current demand of the task.
• Modify the textbook task by using one or more of the
Textbook Modification Strategies.
• You will be posting your modified task for others to
analyze and offer comments.
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Strategies for Modifying Textbook Tasks
Increasing the cognitive demands of tasks:
• Ask students to create real-world stories for “naked
number” problems.
• Include a prompt that asks students to represent
the information another way (with a picture, in a
table, a graph, an equation, with a context).
• Include a prompt that requires students to make a
generalization.
• Use a task “out of sequence” before students have
memorized a rule or have practiced a procedure
that can be routinely applied.
• Eliminate components of the task that provide too
much scaffolding.
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Strategies for Modifying Textbook Tasks
(continued)
Increasing the cognitive demands of tasks:
• Adapt a task so as to provide more opportunities for students
to think and reason—let students figure things out for
themselves.
• Create a prompt that asks students to write about the meaning
of the mathematics concept.
• Add a prompt that asks students to make note of a pattern or to
make a mathematical conjecture and to test their conjecture.
• Include a prompt that requires students to compare solution
paths or mathematical relationships and write about the
relationship between strategies or concepts.
• Select numbers carefully so students are more inclined to note
relationships between quantities (e.g., two tables can be used
to think about the solutions to the four, six, or eight tables).
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Gallery Walk
• Post the modified tasks.
• Circulate, analyzing the modified tasks. On a “Post-It”
Note,” describe ways in which the tasks were
modified and the benefit to students.
• If the task was not modified to increase the cognitive
demand of the task, then ask a wondering about a
way the task might be modified.
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The Cognitive Demand of Tasks
• Does the demand of the task matter?
• What are you now wondering about with respect to
the task demands?
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The CCSS for Mathematics: Grade 4
Number and Operations – Fractions
4.NF
Extend understanding of fraction equivalence and ordering.
4.NF.A.1
Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b) by
using visual fraction models, with attention to how the number and size
of the parts differ even though the two fractions themselves are the same
size. Use this principle to recognize and generate equivalent fractions.
4.NF.A.2
Compare two fractions with different numerators and different
denominators, e.g., by creating common denominators or numerators, or
by comparing to a benchmark fraction such as 1/2. Recognize that
comparisons are valid only when the two fractions refer to the same
whole. Record the results of comparisons with symbols >, =, or <, and
justify the conclusions, e.g., by using a visual fraction model.
Common Core State Standards, 2010, p. 30, NGA Center/CCSSO
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The CCSS for Mathematics: Grade 4
Number and Operations – Fractions
4.NF
Understand decimal notation for fractions, and compare decimal
fractions.
4.NF.C.5 Express a fraction with denominator 10 as an equivalent fraction with
denominator 100, and use this technique to add two fractions with
respective denominators 10 and 100. For example, express 3/10 as
30/100, and add 3/10 + 4/100 = 34/100.
4.NF.C.6
Use decimal notation for fractions with denominators 10 or 100. For
example, rewrite 0.62 as 62/100; describe a length as 0.62 meters;
locate 0.62 on a number line diagram.
4.NF.C.7
Compare two decimals to hundredths by reasoning about their size.
Recognize that comparisons are valid only when the two decimals
refer to the same whole. Record the results of comparisons with the
symbols >, =, or <, and justify the conclusions, e.g., by using a visual
model.
Common Core State Standards, 2010, p. 31, NGA Center/CCSSO
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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.
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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.
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Accountable Talk Moves
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?
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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?
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An Example: Accountable Talk Discussion
The Focus Essential Understanding
Creating Equivalent Fractions
When the denominator is multiplied or divided then the numerator is
automatically divided into the same number of pieces because it is a
subcomponent of the denominator.
Group A
Group B
•
•
•
•
•
•
Explain your set of equivalencies.
Who understood what he said about the 100 and the 10? (Community)
3
Can you say back what he said how the model shows 10? (Community)
Who can add on and talk about the 3 and the 30? (Community)
The denominator tells the number of equal parts in the whole. (Marking)
30
Do we see 100 in both pieces of work? (Rigor)
•
Tell us how you found 10 in your picture (Group A). (Rigor)
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An Example: Accountable Talk Discussion
The Focus Essential Understanding
Creating Equivalent Fractions
When the denominator is multiplied or divided then the numerator is
automatically divided into the same number of pieces because it is a
subcomponent of the denominator.
Group A
Group B
•
•
•
•
•
𝟑
𝟑𝟎
Both groups say that 𝟏𝟎 is equal to 𝟏𝟎𝟎. How can this be when the
fractions use different numbers? (Hook)
30
3
3
Can Group B explain why 100 = 100 = 10?
Who understood what they said about the denominators? (Community)
Can you say back what they said about the numerator changing?
(Community)
Each group made statements about equivalency. How does the visual
model differ from/support the symbolic model? (Rigor)
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Reflecting: 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.”
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Step Back and Application to Our
Work
What have you learned today that you will apply when
planning or teaching in your classroom?
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