Date post: | 19-Jul-2015 |
Category: |
Documents |
Upload: | middlestart |
View: | 223 times |
Download: | 0 times |
middlestartThe 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
middlestartThe National Forum for Middle Grades Reform
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
middlestartThe National Forum for Middle Grades Reform
Mathematics Improvement Toolkit
An Example in PracticeFind a pattern for the sum of the interior angles of a polygon.
xº
xº
xº
middlestartThe National Forum for Middle Grades Reform
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.
middlestartThe National Forum for Middle Grades Reform
Mathematics Improvement Toolkit
An Example in PracticeFind a pattern for the sum of the interior angles of a polygon.
xº
xºyº
zº
xº zº
wº yº yº zº
wºvº
x + y + z ≈ 180˚w + x + y + z ≈ 360˚
v + w + x + y + z ≈ 540˚
middlestartThe National Forum for Middle Grades Reform
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)
middlestartThe National Forum for Middle Grades Reform
Mathematics Improvement Toolkit
An Example in PracticeFind a pattern for the sum of the interior angles of a polygon.
xº
middlestartThe National Forum for Middle Grades Reform
Mathematics Improvement Toolkit
An Example in PracticeFind a pattern for the sum of the interior angles of a polygon.
xº
middlestartThe National Forum for Middle Grades Reform
Mathematics Improvement Toolkit
Higher Order Thinking Skills
middlestartThe National Forum for Middle Grades Reform
Mathematics Improvement Toolkit
Higher Order Thinking Skills• Bloom’s Taxonomy
middlestartThe National Forum for Middle Grades Reform
Mathematics Improvement Toolkit
Higher Order Thinking Skills• Bloom’s Taxonomy
• Tasks vary in their cognitive demand
middlestartThe National Forum for Middle Grades Reform
Mathematics Improvement Toolkit
Higher Order Thinking Skills• Bloom’s Taxonomy
• Tasks vary in their cognitive demand
• More complicated tasks need to be practiced and internalized.
middlestartThe National Forum for Middle Grades Reform
Mathematics Improvement Toolkit
Higher Order Thinking Skills• Bloom’s Taxonomy
• Tasks vary in their cognitive demand
• More complicated tasks need to be practiced and internalized.
middlestartThe National Forum for Middle Grades Reform
Mathematics Improvement Toolkit
Higher Order Thinking Skills• Bloom’s Taxonomy
• Tasks vary in their cognitive demand
• More complicated tasks need to be practiced and internalized. Knowledge
middlestartThe National Forum for Middle Grades Reform
Mathematics Improvement Toolkit
Higher Order Thinking Skills• Bloom’s Taxonomy
• Tasks vary in their cognitive demand
• More complicated tasks need to be practiced and internalized. Knowledge
middlestartThe National Forum for Middle Grades Reform
Mathematics Improvement Toolkit
Higher Order Thinking Skills• Bloom’s Taxonomy
• Tasks vary in their cognitive demand
• More complicated tasks need to be practiced and internalized. Knowledge
Comprehension
middlestartThe National Forum for Middle Grades Reform
Mathematics Improvement Toolkit
Higher Order Thinking Skills• Bloom’s Taxonomy
• Tasks vary in their cognitive demand
• More complicated tasks need to be practiced and internalized. Knowledge
Comprehension
middlestartThe National Forum for Middle Grades Reform
Mathematics Improvement Toolkit
Higher Order Thinking Skills• Bloom’s Taxonomy
• Tasks vary in their cognitive demand
• More complicated tasks need to be practiced and internalized. Knowledge
Comprehension
Application
middlestartThe National Forum for Middle Grades Reform
Mathematics Improvement Toolkit
Higher Order Thinking Skills• Bloom’s Taxonomy
• Tasks vary in their cognitive demand
• More complicated tasks need to be practiced and internalized. Knowledge
Comprehension
Application
middlestartThe National Forum for Middle Grades Reform
Mathematics Improvement Toolkit
Higher Order Thinking Skills• Bloom’s Taxonomy
• Tasks vary in their cognitive demand
• More complicated tasks need to be practiced and internalized. Knowledge
Comprehension
Application
Analysis
middlestartThe National Forum for Middle Grades Reform
Mathematics Improvement Toolkit
Higher Order Thinking Skills• Bloom’s Taxonomy
• Tasks vary in their cognitive demand
• More complicated tasks need to be practiced and internalized. Knowledge
Comprehension
Application
Analysis
middlestartThe National Forum for Middle Grades Reform
Mathematics Improvement Toolkit
Higher Order Thinking Skills• Bloom’s Taxonomy
• Tasks vary in their cognitive demand
• More complicated tasks need to be practiced and internalized. Knowledge
Comprehension
Application
Analysis
Synthesis
middlestartThe National Forum for Middle Grades Reform
Mathematics Improvement Toolkit
Higher Order Thinking Skills• Bloom’s Taxonomy
• Tasks vary in their cognitive demand
• More complicated tasks need to be practiced and internalized. Knowledge
Comprehension
Application
Analysis
Synthesis
middlestartThe National Forum for Middle Grades Reform
Mathematics Improvement Toolkit
Higher Order Thinking Skills• Bloom’s Taxonomy
• Tasks vary in their cognitive demand
• More complicated tasks need to be practiced and internalized. Knowledge
Comprehension
Application
Analysis
SynthesisEvaluation
middlestartThe National Forum for Middle Grades Reform
Mathematics Improvement Toolkit
The Mathematics Task Frameworkfrom Stein and Smith, 1998
middlestartThe National Forum for Middle Grades Reform
Mathematics Improvement Toolkit
The Mathematics Task Framework
Tasks as they
appear in curriculum materials
from Stein and Smith, 1998
middlestartThe National Forum for Middle Grades Reform
Mathematics Improvement Toolkit
The Mathematics Task Framework
Tasks as they
appear in curriculum materials
from Stein and Smith, 1998
middlestartThe National Forum for Middle Grades Reform
Mathematics Improvement Toolkit
The Mathematics Task Framework
Tasks as they
appear in curriculum materials
Tasks as set up by teachers
from Stein and Smith, 1998
middlestartThe National Forum for Middle Grades Reform
Mathematics Improvement Toolkit
The Mathematics Task Framework
Tasks as they
appear in curriculum materials
Tasks as set up by teachers
from Stein and Smith, 1998
middlestartThe National Forum for Middle Grades Reform
Mathematics Improvement Toolkit
The Mathematics Task Framework
Tasks as they
appear in curriculum materials
Tasks as set up by teachers
Tasks as enacted by teachers
and students
from Stein and Smith, 1998
middlestartThe National Forum for Middle Grades Reform
Mathematics Improvement Toolkit
The Mathematics Task Framework
Tasks as they
appear in curriculum materials
Tasks as set up by teachers
Tasks as enacted by teachers
and students
from Stein and Smith, 1998
middlestartThe National Forum for Middle Grades Reform
Mathematics Improvement Toolkit
The Mathematics Task Framework
Tasks as they
appear in curriculum materials
Tasks as set up by teachers
Tasks as enacted by teachers
and students
Student learning
from Stein and Smith, 1998
middlestartThe National Forum for Middle Grades Reform
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 2Method 1
middlestartThe National Forum for Middle Grades Reform
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 2Method 1
middlestartThe National Forum for Middle Grades Reform
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 2Method 1
middlestartThe National Forum for Middle Grades Reform
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 2Method 1
middlestartThe National Forum for Middle Grades Reform
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”)
middlestartThe National Forum for Middle Grades Reform
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.