Mtf example

middlestart
The National Forum for
Middle Grades Reform
Mathematics
Improvement Toolkit
The Mathematical Task
Framework in Action
Recognizing Why We Rarely Get the
Level of Student Understanding We Want
Friday, November 19, 2010

middlestart
Mathematics
Improvement Toolkit
Why do students struggle with
challenging concepts and problems?
• Studies suggest low engagement in mathematics instruction
and practice
• What if we use quality instructional materials?
• Still poor higher order thinking skills suggests other issues in
students’ instruction
• Let’s check out an example

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Mathematics
Improvement Toolkit
An Example in Practice
Find a pattern for the sum of the interior
angles of a polygon.
xº
xº
xº

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Mathematics
Improvement Toolkit
Method 1:
Find a pattern for the sum of the
interior angles of a polygon.
• Teachers pose the problem and ask students to
measure angles in various triangles, quadrilaterals, and
polygons.
• Students are asked to identify possible patterns in their
data for different polygons.
• Students predict the sum of interior angles for polygons
that they did not measure or diagram.
• Students try to generate a rule for the sum of all angles
based on the number of sides.

middlestart
Mathematics
Improvement Toolkit
xº
xº
yº
zº
xº zº
wº yº yº zº
wº
vº
x + y + z ≈ 180˚
w + x + y + z ≈ 360˚
v + w + x + y + z ≈ 540˚

middlestart
Mathematics
Improvement Toolkit
Method 2:
Find a pattern for the sum of the
interior angles of a polygon.
• Teachers pose the problem and ask students to
measure angles in various triangles, quadrilaterals, and
polygons.
• Students are shown that any polygon can be “broken
down” into triangles by drawing diagonals from one
“corner” of the object, making two less triangles than
sides.
• Students are told that there are 180º in a triangle, and
so the sum is 180 x (the number of sides - 2)

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Mathematics
Improvement Toolkit
xº

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Mathematics
Improvement Toolkit
Higher Order Thinking Skills

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Mathematics
Improvement Toolkit
• Bloom’s Taxonomy

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Mathematics
Improvement Toolkit
• Tasks vary in their
cognitive demand

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Mathematics
Improvement Toolkit
cognitive demand
• More complicated
tasks need to be
practiced and
internalized.

middlestart
Mathematics
Improvement Toolkit
cognitive demand
tasks need to be
practiced and
internalized. Knowledge

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Mathematics
Improvement Toolkit
cognitive demand
tasks need to be
practiced and
Comprehension

middlestart
Mathematics
Improvement Toolkit
cognitive demand
tasks need to be
practiced and
Comprehension
Application

middlestart
Mathematics
Improvement Toolkit
cognitive demand
tasks need to be
practiced and
Comprehension
Application
Analysis

middlestart
Mathematics
Improvement Toolkit
cognitive demand
tasks need to be
practiced and
Comprehension
Application
Analysis
Synthesis

middlestart
Mathematics
Improvement Toolkit
cognitive demand
tasks need to be
practiced and
Comprehension
Application
Analysis
Synthesis
Evaluation

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Mathematics
Improvement Toolkit
The Mathematics Task Framework
from Stein and Smith, 1998

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Mathematics
Improvement Toolkit
Tasks as
they
appear in
curriculum
materials

middlestart
Mathematics
Improvement Toolkit
Tasks as
they
appear in
curriculum
materials
Tasks as
set up by
teachers

middlestart
Mathematics
Improvement Toolkit
Tasks as
they
appear in
curriculum
materials
Tasks as
set up by
teachers
Tasks as
enacted by
teachers
and
students

middlestart
Mathematics
Improvement Toolkit
Tasks as
they
appear in
curriculum
materials
Tasks as
set up by
teachers
Tasks as
enacted by
teachers
and
students
Student
learning

middlestart
Mathematics
Improvement Toolkit
MTF in Action
Tasks as
they appear
in curriculum
materials
Tasks as set
up by
teachers
Tasks as
enacted by
teachers and
students Student
learning
Method 2
Method 1

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Mathematics
Improvement Toolkit
MTF in Action
Tasks as
they appear
in curriculum
materials
Tasks as set
up by
teachers
Tasks as
enacted by
teachers and
students Student
learning
Curriculum Teachers Enactment Understanding
Method 2
Method 1

middlestart
Mathematics
Improvement Toolkit
Insights from cognitive biology...
• Brain activity and the ability to effectively apply prior knowledge
is related to the complexity of a task.
• Tasks that are too difﬁcult and beyond the scope of
conceptualizing cause the cognitive centers to “shut down”
• Similarly, simple tasks, such as memorization, simple
application, and the use of algorithms to “solve” a problem,
generally cause the brain to disengage (i.e. boredom)
• Endorphins are released when the brain is engaged over time
in relatively complex tasks (“the pleasure of ﬁnding things out”)

middlestart
Mathematics
Improvement Toolkit
Insights from educational research
• Signiﬁcant research addresses the need to support “high-level”
thinking and problem solving in schools, showing clear beneﬁts
to learning mathematics
• TIMSS Video Study explored international variation
- In high achieving countries, when problems addressing both
skill and conceptual knowledge were posed, students
studied the relationships and connections often (about 50%)
- In the U.S., students never got the chance - over 99% of
teachers stepped in to do the conceptual work, leaving only
skill development.
- U.S. students spend all of their time focusing on skills.

Mtf example

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