Implementing High Cognitive
Demand Tasks
in Mathematics Classrooms
Based on a NCTM’s seminar series presentation by
Margaret Smith, University of Pittsburg
Relationship among various task-related variables and students’
learning outcomes
Mathematical task
as represented in
curricular/instructional
materials
Mathematical task
as set up
by the teacher
in the classroom
• Task features
• Cognitive demands
Mathematical task
Students’
Learning
Outcomes
Henningsen & Stein, 1997
Four levels of cognitive demand
1. Memorization
2. Procedures without connections to concepts or meaning
3. Procedures with connections to concepts or meaning
4. Doing mathematics
Cognitive Demands at Set Up
in the QUASAR* Project
N
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W
O
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PR
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W
IT
H
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AT
H
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O
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45
40
35
30
25
20
15
10
5
0
Stein, Grover, & Henningsen, 1996
QUASAR = Quantitative Understanding: Amplifying Student Achievement and Reasoning
The Fate of Tasks Set Up as Doing
Mathematics
10%
Doing Mathematics
14%
37%
Unsystematic
Exploration
No Mathematics
Procedures
WITHOUT
Other
17%
22%
Stein, Grover, & Henningsen, 1996
The Fate of Tasks Set Up as Procedures WITH
Connections to Meaning
2%
2%
Procedures
WITHOUT
Procedures WITH
43%
53%
Memorization
No Mathematics
Stein, Grover, & Henningsen, 1996
Relationship among various task-related variables and students’
learning outcomes
Mathematical task
as represented in
curricular/instructional
materials
Mathematical task
as set up
by the teacher
in the classroom
Mathematical task
as implemented
by students
in the classroom
• Task features
• Cognitive demands
• Enactment of task
features
• Cognitive processing
Students’
Learning
Outcomes
Henningsen & Stein, 1997
The level cognitive demand declines in many
classrooms
… because …
1. Challenges become non-problems
(64% of 61 tasks)
2.
3.
4.
5.
6.
7.
Tasks are inappropriate for students (61%)
Focus shifted to correct answer (44%)
Too much or too little time (38%)
Classroom-management problems (18%)
Lack of student accountability (21%)
Others (8%)
Stein & Henningsen, 1996
Factors that are likely to assist students to engage
at high-level thinking
1. Task builds on students’ prior knowledge
(82% of 45 tasks that have been set up as high-level)
2. Appropriate amount of time (71%)
3. High-level performance modeled (71%)
4. Sustained pressure for explanation and meaning
(64%)
5.
6.
7.
8.
Scaffolding (58%)
Student self-monitoring (27%)
Teacher draws conceptual connections (13%)
Others (9%)
Stein & Henningsen, 1996
Relationship among various task-related variables and students’
learning outcomes
Mathematical task
as represented in
curricular/instructional
materials
Mathematical task
as set up
by the teacher
in the classroom
Mathematical task
as implemented
by students
in the classroom
• Task features
• Cognitive demands
• Enactment of task
features
• Cognitive processing
Factors
influencing setup
Teachers’ goals
Teachers’ knowledge of
subject matter
Teachers’ knowledge of
students
Factors
influencing student
implementation
Classroom norms
Task conditions
Teachers’ instructional
dispositions
Students’ learning
disposition
Students’
Learning
Outcomes
Does Maintaining
Cognitive Demand Matter?
YES
Research shows . . .
• The QUASAR Project
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).
• TIMMS Video Study
Higher-achieving countries implemented a greater percentage
of high level tasks in ways that maintained the demands of
the task (Stigler & Hiebert, 2004).
Patterns of Set up, Implementation, and
Student Learning
Task Set Up
Task Implementation
Student Learning
A.
High
High
High
B.
Low
Low
Low
C.
High
Low
Moderate
Stein & Lane, 1996
TIMSS Video Study
• Approximately 17% of the problem statements in the U.S.
suggested a focus on mathematical connections or relationships.
This percentage is within the range of many higher-achieving
countries (i.e., Hong Kong, Czech Republic, Australia).
• Virtually none of the making-connections problems in the U.S.
were discussed in a way that made the mathematical
connections or relationships visible for students.
Mostly, they turned into opportunities to apply procedures.
Or, they became problems in which even less mathematical
content was visible (i.e., only the answer was given).
TIMSS Video Mathematics Research Group, 2003
Conclusion
• Different tasks provide different opportunities
for students to learn
• High level tasks are difficult to carry out in a
consistent manner
• Engagement in cognitively challenging tasks
leads to the greater learning gains for students
Open Question
• How can we maintain the high-cognitive
demand of challenging tasks during
implementation?
• More specifically, how can we orchestrate
productive classroom discussions, build on
student thinking, and guide them towards the
mathematics we want students to develop?
Open Question
• How can we maintain the high-cognitive
demand of challenging tasks during
implementation?
• More specifically, how can we orchestrate
productive classroom discussions, build on
student thinking, and guide them towards the
mathematics we want students to develop?
Orchestrating Discussions in
Math Classrooms
(Part 1)
Based on a MTMS article, Smith’s presentation, and
the 5 Practices for Orchestrating Productive Math Disposition book.
Why dedicate class time for discussion?
Students have the opportunity to …
• Share ideas and clarify understandings
• Develop convincing arguments
(How things work? Why?)
• Develop a language for expressing mathematical ideas
• Learn to see things for other people’s perspective
Possible Challenges in Facilitating Discussion
• Lacks experience
• Requires flexible, deep, and interconnected knowledge
of content, pedagogy, and students
• Reduces teachers’ perceived level of control
• Requires split-second decision-making
The Five Practices
0. Setting Goals and Selecting Tasks
1. Anticipating
2. Monitoring
3. Selecting
4. Sequencing
5. Connecting
0. Setting Goals & Selecting Tasks
• Identify the math you want students to understand
• Being specific (make the math idea explicit)
Compare and contrast these 3 goals
Goal A
Students will learn the Pythagorean theorem (c2 = a2 + b2)
Goal B
Students will be able to use the Pythagorean theorem
(c2 = a2 + b2) to solve a series of missing value problems
Goal C
Students will recognize that the area of the square build
on the hypotenuse of a right triangle is equal to the sum
of the areas of the squares build on the legs and will
conjecture that c2 = a2 + b2
0. Setting Goals & Selecting Tasks
• Identify the math you want students to understand
• Being specific (make the math idea explicit)
“Without explicit learning goals, it is difficult to know what
counts as evidence of students’ learning, how students’
learning can be linked to particular instructional activities,
and how to revise instruction to facilitate students’ learning
more effectively. Formulating clear, explicit learning goals set
the stage for everything else.”
(Hiebert, Morris, Berk, & Jansen, 2007, p. 51)
0. Setting Goals & Selecting Tasks
• Identify the math you want students to understand
• Being specific (make the math idea explicit)
How to set goals?
• Think about how students come to know and
understand math as they engaged in the lesson,
not just what students will do
• Consult resources to unpack big ideas
• Collaborate with other teachers
0. Setting Goals & Selecting Tasks
In Elementary and Middle School Mathematics Teaching
Developmentally , the authors (Van De Walle, Karp, and
Bay-Williams)
• Identify a set of big ideas for each content chapter
• Make clear how understanding of the big ideas develops
• Provide tasks that can be used to help students develop
this understanding)
(Smith & Stein, 2011, p. 15)
0. Setting Goals & Selecting Tasks
• Are aligned with the lesson goals
• Demand students’ engagement with concepts and
stimulate students to make connections
• Are rich enough to support a discussion
0. Setting Goals & Selecting Tasks
These two tasks differ in their instructions.
What can you say about them in terms of the level of
thinking required of the students?
Students are not pressed to
figure out why the pattern
works.
Students are asked to
explain why the conjecture
is true; students need to dig
into the math underlying
the observed pattern.
0. Setting Goals & Selecting Tasks
• Are aligned with the lesson goals
• Demand students’ engagement with concepts and
stimulate students to make connections
• Are rich enough to support a discussion
How to select tasks?
• Use the Task Analysis Guide (Smith & Stein, 1998)
to analyze the potential of tasks to support
students’ thinking and reasoning
0. Setting Goals & Selecting Tasks
• Are aligned with the lesson goals
• Demand students’ engagement with concepts and
stimulate students to make connections
• Are rich enough to support a discussion
How to select tasks?
• Use the Task Analysis Guide (Smith & Stein, 1998)
to analyze the potential of tasks to support
students’ thinking and reasoning
• Match tasks with their goals for student learning
Goal B
Students will be able to use the Pythagorean theorem
(c2 = a2 + b2) to solve a series of missing value problems
Goal C
Students will recognize that the area of the square build
on the hypotenuse of a right triangle is equal to the sum
of the areas of the squares build on the legs and will
conjecture that c2 = a2 + b2