Supporting Rigorous Mathematics
Teaching and Learning
Enacting Instructional Tasks:
Maintaining the Demands of the Tasks
PARTICIPANT PACKET
Tennessee Department of Education
High School Mathematics
© 2013 UNIVERSITY OF PITTSBURGH
-1-
Using the Assessment to Think About Instruction
In order for students to perform well on the Constructed Response Assessments (CRAs), what are
the implications for instruction?
•
What kinds of instructional tasks will need to be used in the classroom?
•
What will teaching and learning look like and sound like in the classroom?
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). By analyzing the classroom actions and
interactions of five teachers enacting the same high-level task, teachers will begin to identify
classroom-based factors that are associated with supporting or inhibiting students’ high-level
engagement during instruction.
© 2013 UNIVERSITY OF PITTSBURGH
-2-
Session Goals
Participants will:
•
learn about characteristics of the written tasks that impact students’ opportunities to
think and reason about mathematics; and
•
learn about the factors of implementation that contribute to the maintenance and
decline of thinking and reasoning.
Overview of Activities
Participants will:
•
discuss the differences between two written tasks and their relationship to the
CCSSM;
•
discuss how tasks are implemented in classrooms and the impact on students’
opportunities to learn; and
•
make connections to what research says about task implementation.
© 2013 UNIVERSITY OF PITTSBURGH
-3-
Comparing Two Mathematical Tasks
Task A: The Hexagon Pattern Task
Trains 1, 2, 3, and 4 (shown below) are the first 4 trains in the hexagon pattern. The first
train in this pattern consists of one regular hexagon. For each subsequent train, one
additional hexagon is added.
1. Compute the perimeter for each of the first four trains. .
2. Make some observations that help you describe the perimeter of larger trains.
3. Determine a perimeter of the 25th train without constructing it.
4. Write the function that can be used to compute the perimeter of any train in the
pattern. Explain how you know it will always work.
Extension
How can you find the perimeter of a train that consisted of triangles? Squares? Pentagons?
Can you write a general description that can be used to find the perimeter of a train of any
regular polygons?
(Adapted from Visual Mathematics Course I, Lessons 16-30 published by The Math Learning Center.
© 1995 by The Math Learning Center, Salem, Oregon.)
Task B: The Square Tiles Task
Using the side of a square pattern tile as a measure, find the perimeter (i.e., distance
around) of each train in the pattern block figure shown below.
train 1
train 2
train 3
(Adapted from Visual Mathematics Course I, Lessons 16-30 published by The Math Learning Center.
© 1995 by The Math Learning Center, Salem, Oregon.)
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Comparing Two Mathematical Tasks
Compare the two tasks. How are they similar and how are they different?
•
•
The Hexagon Pattern Task
The Square Tiles Task
Task A
Task B
Characteristics of the
Task
Math Content
Standards
Math Practice
Standards
© 2013 UNIVERSITY OF PITTSBURGH
-5-
The CCSS for Mathematical Content
CCSS Functions Conceptual Category
Building Functions
F-BF
Build a function that models a relationship between two quantities.
F-BF.A.1
Write a function that describes a relationship between two quantities.
F-BF.A.1a
Determine an explicit expression, a recursive process, or steps for calculation
from a context.
Interpreting Functions
F-IF
Understand the concept of a function and use function notation.
F-IF.A.2
Use function notation, evaluate functions for inputs in their domains, and
interpret statements that use function notation in terms of a context.
F-IF.A.3
Recognize that sequences are functions, sometimes defined recursively,
whose domain is a subset of the integers. For example, the Fibonacci
sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥
1.
Interpret functions that arise in applications in terms of the context.
F-IF.B.5
Relate the domain of a function to its graph and, where applicable, to the
quantitative relationship it describes. For example, if the function h(n) gives
the number of person-hours it takes to assemble n engines in a factory, then
the positive integers would be an appropriate domain for the function.
Analyze functions using different representations.
F-IF.C.8
Write a function defined by an expression in different but equivalent forms to
reveal and explain different properties of the function.
F-IF.C.8a
Use the process of factoring and completing the square in a quadratic function
to show zeros, extreme values, and symmetry of the graph, and interpret
these in terms of a context.
-6-
The CCSS for Mathematical Content
CCSS Algebra Conceptual Category
Seeing Structure in Expressions
A-SSE
Interpret the structure of expressions.
A-SSE.A.1
Interpret expressions that represent a quantity in terms of its context.
A-SSE.A.1a
Interpret parts of an expression, such as terms, factors, and coefficients.
A-SSE.A.1b
Interpret complicated expressions by viewing one or more of their parts as
a single entity. For example, interpret P(1+r)n as the product of P and a
factor not depending on P.
A-SSE.A.2
Use the structure of an expression to identify ways to rewrite it.
For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference
of squares that can be factored as (x2 – y2)(x2 + y2).
Write expressions in equivalent forms to solve problems.
A-SSE.B.3
Choose and produce an equivalent form of an expression to reveal and
explain properties of the quantity represented by an expression.
Creating Equations
A-CED
Create equations that describe numbers or relationships.
A-CED.A.2
Create equations in two or more variables to represent relationships
between quantities; graph equations on coordinate axes with labels and
scales.
The CCSS for Mathematical Practices
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
-7-
The Enactment of the Task
(Private Think Time)
•
Read the vignettes.
•
Consider the following question:
What are students learning in each classroom?
The Enactment of the Task
(Small Group Discussion)
Discuss the following question and cite evidence from the cases:
What are students learning in each classroom?
The Enactment of the Task
(Whole Group Discussion)
What opportunities did students have to think and reason in each of the classes?
© 2013 UNIVERSITY OF PITTSBURGH
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Vignettes: The Hexagon Task
Scenarios of Six Common Patterns of Task Set-Up and Implementation
Scenario A
Mrs. Fox has posted the Hexagon problem on the blackboard. As students arrive in the classroom,
they begin to work on the problem immediately with their partners.
For the pattern shown below, compute the perimeter for the first four trains, determine the
perimeter for the tenth train without constructing it, and then write a description that can be
used to compute the perimeter of any train in the pattern. Find as many ways as you can to
compute (and justify) the perimeter.
train 1
train 2
train 3
Mrs. Fox walks around the room as students begin their work, stopping at different groups to listen
in on their conversations and to provide support as needed. She notes that many students start out
by describing the number of units on the top and bottom and on the sides of the hexagon train. As
they keep adding a hexagon, they realize that there are two on the top and two on the bottom
added but the original two on the sides remain. The students are keeping track of their work in a
table. During this time, Mrs. Fox circulates among the groups, asking such questions as, “How do
you know what will happen if you add one more hexagon?,” “Can you describe the 20th train without
constructing it?,” and “Do you see a pattern?” These questions lead students to see the need to
organize their data, make conjectures, and test them out. As the period draws to a close, none of
the groups have completed the task, but most are well on their way to discovering that the pattern
grows at a constant rate of four with each additional train. For homework, Mrs. Fox asks students to
summarize what they have learned so far from their exploration and what they want to continue to
work on in the next class.
© 2013 UNIVERSITY OF PITTSBURGH
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Scenario B
Mr. Chambers has posted the following problem on the blackboard. As they arrive in the classroom,
his students begin to work on the problem immediately with their partners.
For the pattern shown below, compute the perimeter for the first four trains, determine the
perimeter for the tenth train without constructing it, and then write a description that can be
used to compute the perimeter of any train in the pattern. Find as many ways as you can to
compute (and justify) the perimeter.
train 1
train 2
train 3
As students begin to work on the problem, Mr. Chambers circulates around the room, noticing
approvingly that students are taking the task seriously. There are plenty of interesting ideas being
discussed although some of them, he has to admit, will not lead students toward a strategy for
solving the core problem. For example, he hears question such as, “How tall would the train be?,”
“What does the front of a train look like?,” and “Should we put wheels on our drawing?” Mr.
Chambers decides not to intervene and tell the students how to solve the problem; rather he keeps
circulating and observing, hoping that the students will make progress on their own.
With 10 minutes remaining, a few pairs have reached the correct answer for the first question.
However, none of the pairs has made progress toward discovering the big mathematical idea: that
for every additional train added the perimeter grows by four. Mr. Chambers decides to have the
students continue to work on the problem for homework and to revisit it again the next day.
© 2013 UNIVERSITY OF PITTSBURGH
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Scenario C
Ms. Fagan has posted the following problem on the blackboard. As they arrive in the classroom, her
students begin to work on the problem immediately with their partners.
For the pattern shown below, compute the perimeter for the first four trains, determine the
perimeter for the tenth train without constructing it, and then write a description that can be
used to compute the perimeter of any train in the pattern. Find as many ways as you can to
compute (and justify) the perimeter.
train 1
train 3
train 2
One-third of the way into a 40-minute period, Ms. Fagan notices that students have made little or no
progress. Although many have begun testing out different perimeters for trains of various lengths,
most are unsystematic in how they keep track of their work. At this point, Ms. Fagan determines
that students will never get to the answer by the end of the period. She constructs the following
table on the board and tells students to complete it.
Train Number
1
2
3
4
Perimeter
6
10
22
6
7
Students busily complete the table, relieved that they now know “what to do.” With 5 minutes left in
the period, Ms. Fagan asks a pair of students to come to the front of the room to complete the table
and identify the pattern of four in the table. For homework, Ms. Fagan asks the students to figure
out the perimeter for the 20th train.
© 2013 UNIVERSITY OF PITTSBURGH
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Scenario D
Ms. Jackson has posted the following problem on the blackboard. As they arrive in the classroom,
her students begin to work on the problem immediately with their groups.
For the pattern shown below, compute the perimeter for the first four trains, determine the
perimeter for the tenth train without constructing it, and then write a description that can be
used to compute the perimeter of any train in the pattern. Find as many ways as you can to
compute (and justify) the perimeter.
train 1
train 2
train 3
As she walks around the room, Ms. Jackson gives each group of students a large sheet of chart
paper, explaining that she wants each group to produce a poster showing their work in an
organized way. She notices that as soon as they get the chart paper and markers, the students’
attention immediately turns to the creation of posters as works of art rather than as the result of
mathematical thinking and activity. The students produce elaborate drawings of trains and use
carefully drawn calligraphy to produce a title. Although some students try to turn the discussion to
figuring out the problem, the students who are not engaged in the artistic work are beginning to lose
interest and to talk about other things. Ms. Jackson successfully pulls students’ attention back to the
task when she stops at a group, but the group’s attention is not sustained once she leaves.
As the bell rings at the end of the period, Ms. Jackson looks up from the group she currently is
talking to and tells the students to drop their posters off at the front of the room. They will discuss
the posters in class the following day, she says, as students file out the door.
© 2013 UNIVERSITY OF PITTSBURGH
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Scenario E
Mr. Cooper gives his students a worksheet with the hexagon task.
Using the side of a hexagon pattern tile as a measure, find the perimeter (i.e., distance
around) of each train in the pattern block figure shown below.
train 1
train 2
train 3
Mr. Cooper begins the lesson telling students that the distance around a figure is the perimeter of
figure. He shows students the first train and asks for the perimeter of the hexagon if a unit is the
side of the hexagon. A student responds 6. Mr. Cooper adds another hexagon to the first hexagon
and asks for the perimeter of the train of the two hexagons together. One student says 12 because
he has counted all of the units on each hexagon. Mr. Cooper corrects the child, saying “No, we only
count the units around the two trains, not the units inside.” He asks the child to try again. The child
says 10. Mr. Cooper says, “Nice job.” He adds a third hexagon to the train and asks for the
perimeter. He tells students to continue to find the perimeter of the perimeter of the train with four,
five and then six hexagons. He emphasizes the importance of using the blocks and counting
carefully to determine the perimeter. He reminds the students to keep track of how they are thinking
through the problems. As he moves around the room, he watches and listens and reminds students
to work with their partners. He also pairs students so he has one that listens and usually knows
what to do with one that does not listen well. After some time, a pair of students claims that they
have found the perimeter of the 20th train. Mr. Cooper has the students read their answers. The
students say 6, 10, 14, 18, 22, 26, 30, 34, 38, 42 ….. 82. Mr. Cooper asks other students in the
class to finish finding the perimeter of the 20th train for homework. Mr. Cooper gives students the
triangle pattern. He asks students for the perimeter of the triangle. A student says, “3.” He puts two
together and asks for the perimeter, as he touches each side. A student says, “4.”
© 2013 UNIVERSITY OF PITTSBURGH
-13-
Scenario F
Ms. Gorman gives her students a worksheet with the hexagon task.
For the pattern shown below, compute the perimeter for the first four trains, determine the
perimeter for the tenth train without constructing it, and then write a description that can be
used to compute the perimeter of any train in the pattern. Find as many ways as you can to
compute (and justify) the perimeter.
train 1
train 2
train 3
Ms. Gorman starts by asking a successful student to show how to use the hexagon blocks to find
the perimeter of the perimeter of the train. The student shows one hexagon and says the perimeter
is six. He puts two hexagons together and says that the perimeter is 10. He points to the sides of
the two hexagons and counts each unit. The student makes a table that shows 1 hexagon with 6
sides, 2 hexagons with 10 sides. He even tells the students to be careful to not count the units in
the middle. Ms. Gorman asks students if they have any questions. The students do not have any
questions after the presentation, so Ms. Gorman lets the students start working on the problem with
a partner. Ms. Gorman visits each pair to insure they are “doing it correctly,” sometimes helping
them count the units around the train. Ms. Gorman tries to make sure the students learn the right
steps, first count the sides, then record the number in the table, then add another hexagon, count,
and then record. As she circulates, she keeps repeating the steps. She is sure that if they do
several examples by following this set of procedures, they will know how to do other problems.
Some students can get through the procedure if guided directly by Ms. Gorman, but she is not sure
they really understand that every time they add another hexagon, they are only adding four sides
not six. When asked to find the perimeter of the 10th train without constructing the train, the students
do not know what to do. One student asks if he can follow the steps. Ms. Gorman is pleased that
the student finds the steps helpful. Most students do not find the perimeter of the 10th train.
© 2013 UNIVERSITY OF PITTSBURGH
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Factors Associated with Maintenance and Decline of High-level
Demands
Factors Associated with the Decline
Factors Associated with the Maintenance
of High-Level Cognitive Demands
of High-level Cognitive Demands
Routinizing problematic aspects of the task.
Scaffolding of student thinking and reasoning.
Shifting the emphasis from meaning, concepts, or
understanding to the correctness or completeness
of the answer.
Providing a means by which students can monitor
their own progress.
Providing insufficient time to wrestle with the
demanding aspects of the task or so much time that
students drift into off-task behavior.
Modeling of high-level performance by teacher or
capable students.
Engaging in high-level cognitive activities is
prevented due to classroom management problems.
Pressing for justifications, explanations, and/or
meaning through questioning, comments, and/or
feedback.
Selecting a task that is inappropriate for a given
group of students.
Selecting tasks that build on students’ prior
knowledge.
Failing to hold students accountable for high-level
products or processes.
Drawing frequent conceptual connections.
Providing sufficient time to explore.
Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standards-based
mathematics instruction: A casebook for professional development. New York: Teachers College Press.
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Selected References Related to Mathematical Tasks
Tools for Professional Development
Smith, M. S. & Stein, M. K. (1998). Selecting and creating mathematical tasks: From research
to practice. Mathematics Teaching in the Middle School, 3 (5), 344-350.
Stein, M. K. & Smith, M. S. (1998). Mathematical tasks as a framework for reflection: From
research to practice. Mathematics Teaching in the Middle School, 3 (4), 268-275.
Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standardsbased mathematics instruction: A casebook for professional development. New York: Teachers
College Press.
Smith, M. S., Stein, M. K., Arbaugh, F., Brown, C. A., & Mossgrove, J. (2004). Characterizing
the cognitive demands of mathematical tasks: A sorting activity. Professional Development
Guidebook for Perspectives on the Teaching of Mathematics (Companion to 2004 yearbook of the
National Council of Teachers of Mathematics), (pp. 45-72). Reston, VA: National Council of
Teachers of Mathematics.
Smith, M. S., Silver, E. A., Stein, M. K., Boston, M., & Henningsen, M. A. (2005). Improving
instruction in rational numbers and proportionality: Using cases to transform mathematics teaching
and learning, Volume 1. New York: Teachers College Press.
Smith, M. S., Silver, E. A., Stein, M. K., Henningsen, M. A., Boston, M., & Hughes, E.K. (2005).
Improving instruction in algebra: Using cases to transform mathematics teaching and learning,
Volume 2. New York: Teachers College Press.
Smith, M. S., Silver, E. A., Stein, M. K., Boston, M., Henningsen, M. A., & Hillen, A. F. (2005).
Improving instruction in geometry and measurement: Using cases to transform mathematics
teaching and learning, Volume 3. New York: Teachers College Press.
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Research on Tasks and Curriculum
Boaler, J. & Staples, M. (in press). Creating mathematical futures through an equitable teaching
approach: The case of Railside School. Teachers College Record.
Henningsen, M. & Stein, M. K. (1997). Mathematical tasks and student cognition: Classroombased factors that support and inhibit high-level mathematical thinking and reasoning. Journal for
Research in Mathematics Education, 29, 524-549.
Stein, M. K., Grover, B. W., & Henningsen, M. (1996). Building student capacity for
mathematical thinking and reasoning: An analysis of mathematical tasks used in reform
classrooms. American Educational Research Journal, 33 (2), 455-488.
Stein, M. K. & Lane, S. (1996). Instructional tasks and the development of student capacity to
think and reason: An analysis of the relationship between teaching and learning in a reform
mathematics project. Educational Research and Evaluation, 2, 50-80.
Stein, M. K., Remillard, J., & Smith. M. S. (2007). How curriculum influences student learning.
In F. K. Lester, Jr. (Ed.), Second handbook of research on mathematics teaching and learning, pp.
319-369. Greenwich, CT: information Age Publishing.
Stigler, J. W. & Hiebert, J. (2004). Improving mathematics teaching. Educational Leadership, 61
(5), 12-16.
Tarr, J. E., Reys, R. E., Reys, B. J., Chavez, O., Shih, J., & Osterlind, S. J. (in press). The
impact of middle grades mathematics curricula on student achievement and the classroom learning
environment. Journal for Research in Mathematics Education.
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Supporting Rigorous Mathematics
Teaching and Learning
Enacting Instructional Tasks:
Maintaining the Demands of the Tasks
Tennessee Department of Education
High School Mathematics
© 2013 UNIVERSITY OF PITTSBURGH
LEARNING RESEARCH AND DEVELOPMENT CENTER
Using the Assessment to Think About
Instruction
In order for students to perform well on the Constructed
Response Assessments (CRAs), what are the
implications for instruction?
• What kinds of instructional tasks will need to be
used in the classroom?
• What will teaching and learning look like and sound
like in the classroom?
© 2013 UNIVERSITY OF PITTSBURGH
LEARNING RESEARCH AND DEVELOPMENT CENTER
-23-
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). By analyzing the classroom actions and
interactions of five teachers enacting the same highlevel task, teachers will begin to identify classroombased factors that are associated with supporting or
inhibiting students’ high-level engagement during
instruction.
Session Goals
Participants will:
• learn about characteristics of the written tasks that
impact students’ opportunities to think and reason
about mathematics; and
• learn about the factors of implementation that
contribute to the maintenance and decline of
thinking and reasoning.
© 2013 UNIVERSITY OF PITTSBURGH
LEARNING RESEARCH AND DEVELOPMENT CENTER
-24-
Overview of Activities
Participants will:
• discuss the differences between two written tasks
and their relationship to the CCSSM;
• discuss how tasks are implemented in classrooms
and the impact on students’ opportunities to learn;
and
• make connections to what research says about task
implementation.
© 2013 UNIVERSITY OF PITTSBURGH
LEARNING RESEARCH AND DEVELOPMENT CENTER
Comparing Two Mathematical Tasks
© 2013 UNIVERSITY OF PITTSBURGH
LEARNING RESEARCH AND DEVELOPMENT CENTER
-25-
Comparing Two Mathematical Tasks
Compare the two tasks. How are they similar and how
are they different?
• The Hexagon Pattern task
• The Square Tiles Task
© 2013 UNIVERSITY OF PITTSBURGH
LEARNING RESEARCH AND DEVELOPMENT CENTER
The Hexagon Pattern Task
Trains 1, 2, 3, and 4 (shown below) are the first 4 trains in the hexagon
pattern. The first train in this pattern consists of one regular hexagon. For
each subsequent train, one additional hexagon is added.
1. Compute the perimeter for each of the first four trains.
.
2. Make some observations that help you describe the perimeter of larger
trains.
3. Determine the perimeter of the 25th train without constructing it.
4. Write a function that can be used to compute the perimeter of any train
in the pattern. Explain how you know it will always work.
Extension
How can you find the perimeter of a train that consisted of triangles?
Squares? Pentagons? Can you write a general description that can be
used to find the perimeter of a train of any regular polygons?
(Adapted from Visual Mathematics Course I, Lessons 16-30 published by The Math Learning Center.
© 1995 by The Math Learning Center, Salem, Oregon.)
-26-
The Hexagon Pattern Task
Below are some of the possible expressions that can be
used to compute the perimeter of any train in the pattern if n
represents the train number. Explain how each expression
relates to the diagrams.
a. 10 + 4(n – 2)
b. 4n + 2
c. 6 + 4(n – 1)
d. 6n – 2(n – 1)
© 2013 UNIVERSITY OF PITTSBURGH
LEARNING RESEARCH AND DEVELOPMENT CENTER
The Square Tiles Task
Using the side of a square pattern tile as a measure, find
the perimeter (i.e., distance around) of each train in the
pattern block figure shown below.
train 1
train 2
train 3
(Adapted from Visual Mathematics Course I, Lessons 16-30 published by The Math Learning Center.
© 1995 by The Math Learning Center, Salem, Oregon.)
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Comparing Two Mathematical Tasks
(Whole Group Discussion)
What are the similarities and differences between the
two tasks?
• The Hexagon Pattern task
• The Square Tiles Task
© 2013 UNIVERSITY OF PITTSBURGH
LEARNING RESEARCH AND DEVELOPMENT CENTER
Similarities and Differences
Similarities
• Both require prior
knowledge of perimeter
Differences
• Way in which the
perimeter is used
• Both use geometric
shapes
• The need to generalize
• The amount of thinking
and reasoning required
• The number of ways the
problem can be solved
© 2013 UNIVERSITY OF PITTSBURGH
LEARNING RESEARCH AND DEVELOPMENT CENTER
-28-
The CCSS for Mathematical Content
CCSS Functions Conceptual Category
Building Functions
F-BF
Build a function that models a relationship between two quantities.
F-BF.A.1
Write a function that describes a relationship between two quantities.
F-BF.A.1a
Determine an explicit expression, a recursive process, or steps for
calculation from a context.
Interpreting Functions
F-IF
Understand the concept of a function and use function notation.
F-IF.A.2
Use function notation, evaluate functions for inputs in their domains,
and interpret statements that use function notation in terms of a
context.
F-IF.A.3
Recognize that sequences are functions, sometimes defined
recursively, whose domain is a subset of the integers. For example,
the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1)
= f(n) + f(n-1) for n ≥ 1.
Common Core State Standards for Mathematics, 2010
The CCSS for Mathematical Content
CCSS Functions Conceptual Category
Interpreting Functions
F- IF
Interpret functions that arise in applications in terms of the context.
F-IF.B.5
Relate the domain of a function to its graph and, where
applicable, to the quantitative relationship it describes. For
example, if the function h(n) gives the number of person-hours
it takes to assemble n engines in a factory, then the positive
integers would be an appropriate domain for the function.
Analyze functions using different representations.
F-IF.C.8
Write a function defined by an expression in different but
equivalent forms to reveal and explain different properties of
the function.
F-IF.C.8a
Use the process of factoring and completing the square in a
quadratic function to show zeros, extreme values, and
symmetry of the graph, and interpret these in terms of a
context.
Common Core State Standards for Mathematics, 2010
-29-
The CCSS for Mathematical Content
CCSS Algebra Conceptual Category
Seeing Structure in Expressions
A-SSE
Interpret the structure of expressions.
A-SSE.A.1
Interpret expressions that represent a quantity in terms of its
context.
A-SSE.A.1a Interpret parts of an expression, such as terms, factors, and
coefficients.
A-SSE.A.1b Interpret complicated expressions by viewing one or more of
their parts as a single entity. For example, interpret P(1+r)n as
the product of P and a factor not depending on P.
A-SSE.A.2
Use the structure of an expression to identify ways to rewrite
it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing
it as a difference of squares that can be factored as (x2 –
y2)(x2 + y2).
Common Core State Standards for Mathematics, 2010
The CCSS for Mathematical Content
CCSS Algebra Conceptual Category
Seeing Structure in Expressions
A-SSE
Write expressions in equivalent forms to solve problems.
A-SSE.B.3
Choose and produce an equivalent form of an expression to
reveal and explain properties of the quantity represented by
an expression.
Creating Equations
A-CED
Create equations that describe numbers or relationships.
A-CED.A.2
Create equations in two or more variables to represent
relationships between quantities; graph equations on
coordinate axes with labels and scales.
Common Core State Standards for Mathematics, 2010
-30-
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
Mathematical Tasks:
A Critical Starting Point for Instruction
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
-31-
A Glimpse into Students’
Opportunities to Think and Reason
about Mathematics
© 2013 UNIVERSITY OF PITTSBURGH
LEARNING RESEARCH AND DEVELOPMENT CENTER
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 M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standards-based mathematics
instruction: A casebook for professional development, p. 4. New York: Teachers College Press.
-32-
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 M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standards-based mathematics
instruction: A casebook for professional development, p. 4. New York: Teachers College Press.
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 M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standards-based mathematics
instruction: A casebook for professional development, p. 4. New York: Teachers College Press.
-33-
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 M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standards-based mathematics
instruction: A casebook for professional development, p. 4. New York: Teachers College Press.
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 M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standards-based mathematics
instruction: A casebook for professional development, p. 4. New York: Teachers College Press.
-34-
The Enactment of the Task
(Private Think Time)
• Read the vignettes.
• Consider the following question:
What are students learning in each classroom?
© 2013 UNIVERSITY OF PITTSBURGH
LEARNING RESEARCH AND DEVELOPMENT CENTER
The Enactment of the Task
(Small Group Discussion)
Discuss the following question and cite evidence from
the cases:
What are students learning in each classroom?
© 2013 UNIVERSITY OF PITTSBURGH
LEARNING RESEARCH AND DEVELOPMENT CENTER
-35-
The Enactment of the Task
(Whole Group Discussion)
What opportunities did students have to think and
reason in each of the classes?
© 2013 UNIVERSITY OF PITTSBURGH
LEARNING RESEARCH AND DEVELOPMENT CENTER
Research Findings:
The Fate of Tasks
© 2013 UNIVERSITY OF PITTSBURGH
LEARNING RESEARCH AND DEVELOPMENT CENTER
-36-
Linking to Research/Literature:
The QUASAR Project
How High-Level Tasks Can Evolve During a Lesson:
• Maintenance of high-level demands.
• Decline into procedures without connection to
meaning.
• Decline into unsystematic and nonproductive
exploration.
• Decline into no mathematical activity.
© 2013 UNIVERSITY OF PITTSBURGH
LEARNING RESEARCH AND DEVELOPMENT CENTER
Factors Associated with the Maintenance
and Decline of High-Level Cognitive
Demands
Decline
• Problematic aspects of the
task become routinized.
• Understanding shifts to
correctness, completeness.
• Insufficient time to wrestle
with the demanding aspects
of the task.
• Classroom management
problems.
• Inappropriate task for a given
group of students.
• Accountability for high-level
products or processes not
expected.
© 2013 UNIVERSITY OF PITTSBURGH
LEARNING RESEARCH AND DEVELOPMENT CENTER
-37-
Factors Associated with the Maintenance
and Decline of High-Level Cognitive
Demands
Maintenance
• Scaffolds of student thinking
and reasoning provided.
• A means by which students
can monitor their own
progress is provided.
• High-level performance is
modeled.
• A press for justifications,
explanations through
questioning and feedback.
• Tasks build on students’ prior
knowledge.
• Frequent conceptual
connections are made.
• Sufficient time to explore.
© 2013 UNIVERSITY OF PITTSBURGH
LEARNING RESEARCH AND DEVELOPMENT CENTER
Factors Associated with the Maintenance
and Decline of High-Level Cognitive
Demands
Decline
Maintenance
• Problematic aspects of the
task become routinized.
• Scaffolds of student thinking
and reasoning provided.
• Understanding shifts to
correctness, completeness.
• A means by which students
can monitor their own
progress is provided.
• Insufficient time to wrestle
with the demanding aspects
of the task.
• Classroom management
problems.
• Inappropriate task for a given
group of students.
• Accountability for high-level
products or processes not
expected.
• High-level performance is
modeled.
• A press for justifications,
explanations through
questioning and feedback.
• Tasks build on students’ prior
knowledge.
• Frequent conceptual
connections are made.
• Sufficient time to explore.
© 2013 UNIVERSITY OF PITTSBURGH
LEARNING RESEARCH AND DEVELOPMENT CENTER
-38-
Linking to Research: The QUASAR
Project
• Low-Level Tasks:
– Memorization.
– Procedures Without Connections.
• (The Square Tiles Task)
• High-Level Tasks:
– Doing Mathematics.
– Procedures With Connections.
• (The Hexagon Pattern Task)
© 2013 UNIVERSITY OF PITTSBURGH
LEARNING RESEARCH AND DEVELOPMENT CENTER
Cognitive Demands at Set-Up
45
40
35
30
25
20
15
10
5
0
© 2013 UNIVERSITY OF PITTSBURGH
LEARNING RESEARCH AND DEVELOPMENT CENTER
-39-
The Fate of Tasks Set Up as
Doing Mathematics
Doing Mathematics
10%
37%
14%
Unsystematic Exploration
No Mathematics
17%
Procedures WITHOUT
22%
© 2013 UNIVERSITY OF PITTSBURGH
Other
LEARNING RESEARCH AND DEVELOPMENT CENTER
The Fate of Tasks Set Up as Procedures
WITH Connections to Meaning
2% 2%
Procedures WITHOUT
Procedures WITH
43%
53%
Memorization
No Mathematics
© 2013 UNIVERSITY OF PITTSBURGH
LEARNING RESEARCH AND DEVELOPMENT CENTER
-40-
Linking to Research/Literature:
The QUASAR Project
Task Set-Up
A.
B.
C.
Task Implementation Student Learning
High
High
Low
Low
High
Low
High
Low
Moderate
Stein & Lane, 1996
Mathematical Tasks and Student Learning
• Students who performed the best on project-based
measures of reasoning and problem-solving were in
classrooms in which tasks were more likely to be set up
and enacted at high levels of cognitive demand (Stein &
Lane, 1996; Stein, Lane, & Silver, 1996).
• Higher-achieving countries implemented a greater
percentage of high-level tasks in ways that maintained
the demands of the task (Stiegler & Hiebert, 2004).
• The success of students was due in part to the high
cognitive demand of the curriculum and the teachers’
ability to maintain the level of demand during enactment
through questioning (Boaler & Staples, 2008).
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