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Overview of SessionOverview of Session
• Challenge and Goals
• Resources: Tasks and Cases
• Task Analysis
• Research That Informs Our Work
• Mathematical Task Framework (MTF)
Goals of the SessionGoals of the Session
• To understand what characterizes a
cognitively demanding task
• To become familiar with findings from
research on the critical role tasks
play in instruction
Core ChallengeCore Challenge
• Teachers must decide “what aspects of
a task to highlight, how to organize
and orchestrate the work of the
students, what questions to ask to
challenge those with varied levels of
expertise, and how to support students
without taking over the process of
thinking for them and thus eliminating
the challenge.”
• NCTM, 2000, p.19
Goals for Professional Goals for Professional DevelopmentDevelopment
• To provide opportunities for teachers to deepen
the mathematical knowledge used in teaching
mathematics in the middle grades
• To build a collective vision of meaningful
interactions among teachers, students, and
mathematical tasks that positively influence
student learning
• To develop teachers who will have the skills and
knowledge to create more effective mathematics
learning environments for their students
Central Resources Central Resources
• Tasks - complex mathematical tasks
that provoke examination of
underlying mathematical concepts
• Cases - accounts of mathematics
instructional episodes that depict
interactions that occur when a
teacher uses a complex mathematical
task in the classroom
Why Focus on Tasks?Why Focus on Tasks?
• Classroom instruction is generally organized and
orchestrated around mathematical tasks.
• The tasks with which students engage determine
the mathematics they learn.
• Teachers’ facilitation of tasks determine how
students learn the mathematics within the tasks.
• The inability to enact challenging tasks well is
what distinguished teaching in the U. S. from
teaching in other countries that had better
student performance on TIMSS.
Tasks & CasesTasks & Cases
• Emerge from the activity of classrooms
• Provide opportunities for teachers to
become involved in critical discussions
of actual teaching situations (Loucks-
Horsley, 1998)
• Promote reexamining our assumptions
about what “understanding mathematics”
really means
(Schifter, Russell, & Bastable, in press)
Mathematical TasksMathematical Tasks
“Not all tasks are created equal,
and different tasks will provoke
different levels and kinds of
student thinking.”•(Stein, Smith, Henningsen, & Silver, 2000)
Comparing TasksComparing Tasks
• Task 1 - Shade 6 small squares in a 4 x 10
grid. Using the grid, explain how to
determine each of the following: a) the
percent area that is shaded, b) the decimal
part that is shaded, and c) the fractional
part that is shaded.
• Task 2 - Convert the fraction 3/8 to a
decimal and percent.
• What similarities and differences do you
notice about the two tasks?
Levels of Cognitive Levels of Cognitive Demand:Demand:Low levelLow level
• Procedures without connections
•Algorithmic and use of procedureis specifically called for
•Have no connection to the concepts or meaning that underlie the procedure being used
• Memorization
•Reproducing previously learned facts
•Committing facts, rules, formulae to memory
Levels of Cognitive Levels of Cognitive Demand: High levelDemand: High level
• ‘Doing’ mathematics
• Requires complex and nonalgorithmic thinking
• Requires students to explore and understand the nature of mathematical concepts, processes, or relationships
• May involve some level of anxiety
• Procedures with connections
• Focuses student’s attention to the use of procedures for the purposes of developing deeper levels of understanding
• Suggest pathways that have close connections to underlying mathematical ideas
• Usually represented in multiple ways (e.g. visual diagrams)
Quantitative Quantitative Understanding:Understanding:
Quantitative Understanding: Amplifying
Student Achievement and Reasoning (QUASAR)
&
Trends in International Mathematics and
Science Study (TIMSS)
TASKS as
presented in
curriculum/ materials
TASKS as
presented in
curriculum/ materials
TASKS as set up by the teachers
TASKS as set up by the teachers
TASKS as
enacted by
students and
teacher in the
classroom
TASKS as
enacted by
students and
teacher in the
classroom
Student
Learning
Student
Learning
(Adapted from Stein, Smith, Henningsen, & Silver, 2000)
The Mathematics Task The Mathematics Task FrameworkFramework
MTF - The Bottom MTF - The Bottom LineLine
• Tasks are important, but teachers also
matter!
• Teacher actions and reactions …
•influence the nature and extent of
student engagement with challenging
tasks,
• and
•affect students’ opportunities to
learn from and through task
engagement.
Challenges in Using Challenges in Using HighHighLevel TasksLevel Tasks
• Resisting the persistent urge to tell and
to direct
• Allowing time for student thinking
• Knowing when/how to ask questions and to
provide information to support rather than
replace student thinking
• Helping students accept the challenge of
solving worthwhile problems and sustaining
their engagement at a high level
(Stein, Smith, Henningsen, & Silver, 2000)
Factors that Undermine Factors that Undermine the the Cognitive Demand of a Cognitive Demand of a TaskTask• Routinizing problematic aspects of the
task
• Shifting the emphasis from meaning,
concepts, or understanding, to the
correctness or the completeness of the
answer
• Not providing enough time
• Not holding students accountable for
high-level products or processes
(Stein, Smith, Henningsen, & Silver,
2000)
Factors that Support Factors that Support Maintaining Maintaining High Cognitive DemandHigh Cognitive Demand
• Building on students’ prior knowledge of
concepts and meanings
• Scaffolding students’ thinking
• Providing appropriate amount of time
• Modeling high-level performance
• Sustaining pressure for explanation and
meaning
• Drawing frequent conceptual conceptions
(Stein, Smith, Henningsen, & Silver,
2000)
ReflectionReflection
• What were the important mathematical
ideas you encountered today?
• Did this experience generate any
insights/connections relate to
teaching? (What about the day
prompted these?)
The MMSTLC ProgramThe MMSTLC Program
The MMSTLC program and these materials
were developed under a Mathematics and
Science Partnership grant awarded by
the Michigan Department of Education.