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- 1. 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
- 2. middlestart The 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
- 3. middlestart The National Forum for Middle Grades Reform Mathematics Improvement Toolkit An Example in Practice Find a pattern for the sum of the interior angles of a polygon. xº xº xº
- 4. middlestart The 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.
- 5. middlestart The National Forum for Middle Grades Reform Mathematics Improvement Toolkit An Example in Practice Find a pattern for the sum of the interior angles of a polygon. 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˚
- 6. middlestart The 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)
- 7. middlestart The National Forum for Middle Grades Reform Mathematics Improvement Toolkit An Example in Practice Find a pattern for the sum of the interior angles of a polygon. xº
- 8. middlestart The National Forum for Middle Grades Reform Mathematics Improvement Toolkit An Example in Practice Find a pattern for the sum of the interior angles of a polygon. xº
- 9. middlestart The National Forum for Middle Grades Reform Mathematics Improvement Toolkit Higher Order Thinking Skills
- 10. middlestart The National Forum for Middle Grades Reform Mathematics Improvement Toolkit Higher Order Thinking Skills • Bloom’s Taxonomy
- 11. middlestart The National Forum for Middle Grades Reform Mathematics Improvement Toolkit Higher Order Thinking Skills • Bloom’s Taxonomy • Tasks vary in their cognitive demand
- 12. middlestart The 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.
- 13. middlestart The 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.
- 14. middlestart The 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
- 15. middlestart The 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
- 16. middlestart The 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
- 17. middlestart The 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
- 18. middlestart The 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
- 19. middlestart The 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
- 20. middlestart The 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
- 21. middlestart The 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
- 22. middlestart The 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
- 23. middlestart The 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
- 24. middlestart The 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 Evaluation
- 25. middlestart The National Forum for Middle Grades Reform Mathematics Improvement Toolkit The Mathematics Task Framework from Stein and Smith, 1998
- 26. middlestart The 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
- 27. middlestart The 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
- 28. middlestart The 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
- 29. middlestart The 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
- 30. middlestart The 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
- 31. middlestart The 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
- 32. middlestart The 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
- 33. middlestart The 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 2 Method 1
- 34. middlestart The 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 2 Method 1
- 35. middlestart The 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 2 Method 1
- 36. middlestart The 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 2 Method 1
- 37. middlestart The 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”)
- 38. middlestart The 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.