Mtf example

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



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



An Example in PracticeFind a pattern for the sum of the interior angles of a polygon.

xº

xº

xº



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.




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˚



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)




xº




xº



Higher Order Thinking Skills



Higher Order Thinking Skills• Bloom’s Taxonomy




• Tasks vary in their cognitive demand





• More complicated tasks need to be practiced and internalized.





• More complicated tasks need to be practiced and internalized.





• More complicated tasks need to be practiced and internalized. Knowledge











Comprehension






Comprehension






Comprehension

Application






Comprehension

Application






Comprehension

Application

Analysis






Comprehension

Application

Analysis






Comprehension

Application

Analysis

Synthesis






Comprehension

Application

Analysis

Synthesis






Comprehension

Application

Analysis

SynthesisEvaluation



The Mathematics Task Frameworkfrom Stein and Smith, 1998



The Mathematics Task Framework

Tasks as they

appear in curriculum materials

from Stein and Smith, 1998




Tasks as they






Tasks as they


Tasks as set up by teachers





Tasks as they







Tasks as they



Tasks as enacted by teachers

and students





Tasks as they




and students





Tasks as they




and students

Student learning




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



MTF in Action



Tasks as set up by

teachers

Tasks as enacted by


learning

Curriculum Teachers Enactment Understanding

Method 2Method 1



MTF in Action



Tasks as set up by

teachers

Tasks as enacted by


learning


Method 2Method 1



MTF in Action



Tasks as set up by

teachers

Tasks as enacted by


learning


Method 2Method 1



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”)



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.

Documents

Mtf example