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• 1. The Mathematical Task Framework in Action Recognizing Why We Rarely Get the Level of Student Understanding We Want The National Forum for Middle Grades Reform Mathematics Improvement Toolkit middlestart Friday, November 19, 2010
• 2. 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 The National Forum for Middle Grades Reform Mathematics Improvement Toolkit middlestart Friday, November 19, 2010
• 3. An Example in Practice Find a pattern for the sum of the interior angles of a polygon. xº xº xº The National Forum for Middle Grades Reform Mathematics Improvement Toolkit middlestart Friday, November 19, 2010
• 4. Find a pattern for the sum of the Method 1: 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. The National Forum for Middle Grades Reform Mathematics Improvement Toolkit middlestart Friday, November 19, 2010
• 5. An Example in Practice Find a pattern for the sum of the interior angles of a polygon. vº yº xº wº xº zº wº yº yº zº v + w + x + y + z ≈ 540˚ x + y + z ≈ 180˚ xº zº w + x + y + z ≈ 360˚ The National Forum for Middle Grades Reform Mathematics Improvement Toolkit middlestart Friday, November 19, 2010
• 6. Find a pattern for the sum of the Method 2: 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) The National Forum for Middle Grades Reform Mathematics Improvement Toolkit middlestart Friday, November 19, 2010
• 7. An Example in Practice Find a pattern for the sum of the interior angles of a polygon. xº The National Forum for Middle Grades Reform Mathematics Improvement Toolkit middlestart Friday, November 19, 2010
• 8. An Example in Practice Find a pattern for the sum of the interior angles of a polygon. xº The National Forum for Middle Grades Reform Mathematics Improvement Toolkit middlestart Friday, November 19, 2010
• 9. Higher Order Thinking Skills The National Forum for Middle Grades Reform Mathematics Improvement Toolkit middlestart Friday, November 19, 2010
• 10. Higher Order Thinking Skills • Bloom’s Taxonomy The National Forum for Middle Grades Reform Mathematics Improvement Toolkit middlestart Friday, November 19, 2010
• 11. Higher Order Thinking Skills • Bloom’s Taxonomy • Tasks vary in their cognitive demand The National Forum for Middle Grades Reform Mathematics Improvement Toolkit middlestart Friday, November 19, 2010
• 12. Higher Order Thinking Skills • Bloom’s Taxonomy • Tasks vary in their cognitive demand • More complicated tasks need to be practiced and internalized. The National Forum for Middle Grades Reform Mathematics Improvement Toolkit middlestart Friday, November 19, 2010
• 13. Higher Order Thinking Skills • Bloom’s Taxonomy • Tasks vary in their cognitive demand • More complicated tasks need to be practiced and internalized. The National Forum for Middle Grades Reform Mathematics Improvement Toolkit middlestart Friday, November 19, 2010
• 14. Higher Order Thinking Skills • Bloom’s Taxonomy • Tasks vary in their cognitive demand • More complicated tasks need to be practiced and internalized. Knowledge The National Forum for Middle Grades Reform Mathematics Improvement Toolkit middlestart Friday, November 19, 2010
• 15. Higher Order Thinking Skills • Bloom’s Taxonomy • Tasks vary in their cognitive demand • More complicated tasks need to be practiced and internalized. Knowledge The National Forum for Middle Grades Reform Mathematics Improvement Toolkit middlestart Friday, November 19, 2010
• 16. Higher Order Thinking Skills • Bloom’s Taxonomy • Tasks vary in their cognitive demand • More complicated tasks need to be practiced and Comprehension internalized. Knowledge The National Forum for Middle Grades Reform Mathematics Improvement Toolkit middlestart Friday, November 19, 2010
• 17. Higher Order Thinking Skills • Bloom’s Taxonomy • Tasks vary in their cognitive demand • More complicated tasks need to be practiced and Comprehension internalized. Knowledge The National Forum for Middle Grades Reform Mathematics Improvement Toolkit middlestart Friday, November 19, 2010
• 18. Higher Order Thinking Skills • Bloom’s Taxonomy • Tasks vary in their cognitive demand • More complicated Application tasks need to be practiced and Comprehension internalized. Knowledge The National Forum for Middle Grades Reform Mathematics Improvement Toolkit middlestart Friday, November 19, 2010
• 19. Higher Order Thinking Skills • Bloom’s Taxonomy • Tasks vary in their cognitive demand • More complicated Application tasks need to be practiced and Comprehension internalized. Knowledge The National Forum for Middle Grades Reform Mathematics Improvement Toolkit middlestart Friday, November 19, 2010
• 20. Higher Order Thinking Skills • Bloom’s Taxonomy • Tasks vary in their cognitive demand Analysis • More complicated Application tasks need to be practiced and Comprehension internalized. Knowledge The National Forum for Middle Grades Reform Mathematics Improvement Toolkit middlestart Friday, November 19, 2010
• 21. Higher Order Thinking Skills • Bloom’s Taxonomy • Tasks vary in their cognitive demand Analysis • More complicated Application tasks need to be practiced and Comprehension internalized. Knowledge The National Forum for Middle Grades Reform Mathematics Improvement Toolkit middlestart Friday, November 19, 2010
• 22. Higher Order Thinking Skills • Bloom’s Taxonomy • Tasks vary in their Synthesis cognitive demand Analysis • More complicated Application tasks need to be practiced and Comprehension internalized. Knowledge The National Forum for Middle Grades Reform Mathematics Improvement Toolkit middlestart Friday, November 19, 2010
• 23. Higher Order Thinking Skills • Bloom’s Taxonomy • Tasks vary in their Synthesis cognitive demand Analysis • More complicated Application tasks need to be practiced and Comprehension internalized. Knowledge The National Forum for Middle Grades Reform Mathematics Improvement Toolkit middlestart Friday, November 19, 2010
• 24. Higher Order Thinking Skills • Bloom’s Taxonomy Evaluation • Tasks vary in their Synthesis cognitive demand Analysis • More complicated Application tasks need to be practiced and Comprehension internalized. Knowledge The National Forum for Middle Grades Reform Mathematics Improvement Toolkit middlestart Friday, November 19, 2010
• 25. The Mathematics Task Framework from Stein and Smith, 1998 The National Forum for Middle Grades Reform Mathematics Improvement Toolkit middlestart Friday, November 19, 2010
• 26. The Mathematics Task Framework from Stein and Smith, 1998 Tasks as they appear in curriculum materials The National Forum for Middle Grades Reform Mathematics Improvement Toolkit middlestart Friday, November 19, 2010
• 27. The Mathematics Task Framework from Stein and Smith, 1998 Tasks as they appear in curriculum materials The National Forum for Middle Grades Reform Mathematics Improvement Toolkit middlestart Friday, November 19, 2010
• 28. The Mathematics Task Framework from Stein and Smith, 1998 Tasks as they Tasks as appear in set up by curriculum teachers materials The National Forum for Middle Grades Reform Mathematics Improvement Toolkit middlestart Friday, November 19, 2010
• 29. The Mathematics Task Framework from Stein and Smith, 1998 Tasks as they Tasks as appear in set up by curriculum teachers materials The National Forum for Middle Grades Reform Mathematics Improvement Toolkit middlestart Friday, November 19, 2010
• 30. The Mathematics Task Framework from Stein and Smith, 1998 Tasks as Tasks as they Tasks as enacted by appear in set up by teachers curriculum teachers and materials students The National Forum for Middle Grades Reform Mathematics Improvement Toolkit middlestart Friday, November 19, 2010
• 31. The Mathematics Task Framework from Stein and Smith, 1998 Tasks as Tasks as they Tasks as enacted by appear in set up by teachers curriculum teachers and materials students The National Forum for Middle Grades Reform Mathematics Improvement Toolkit middlestart Friday, November 19, 2010
• 32. The Mathematics Task Framework from Stein and Smith, 1998 Tasks as Tasks as they Tasks as enacted by appear in set up by teachers curriculum teachers and Student materials students learning The National Forum for Middle Grades Reform Mathematics Improvement Toolkit middlestart Friday, November 19, 2010
• 33. MTF in Action Tasks as Tasks as Tasks as set they appear enacted by up by in curriculum teachers and teachers materials students Student learning Method 2 Method 1 The National Forum for Middle Grades Reform Mathematics Improvement Toolkit middlestart Friday, November 19, 2010
• 34. MTF in Action Tasks as Tasks as Tasks as set they appear enacted by up by in curriculum teachers and teachers materials students Student learning Method 2 Method 1 Curriculum Teachers Enactment Understanding The National Forum for Middle Grades Reform Mathematics Improvement Toolkit middlestart Friday, November 19, 2010
• 35. MTF in Action Tasks as Tasks as Tasks as set they appear enacted by up by in curriculum teachers and teachers materials students Student learning Method 2 Method 1 Curriculum Teachers Enactment Understanding The National Forum for Middle Grades Reform Mathematics Improvement Toolkit middlestart Friday, November 19, 2010
• 36. MTF in Action Tasks as Tasks as Tasks as set they appear enacted by up by in curriculum teachers and teachers materials students Student learning Method 2 Method 1 Curriculum Teachers Enactment Understanding The National Forum for Middle Grades Reform Mathematics Improvement Toolkit middlestart Friday, November 19, 2010
• 37. 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”) The National Forum for Middle Grades Reform Mathematics Improvement Toolkit middlestart Friday, November 19, 2010
• 38. 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. The National Forum for Middle Grades Reform Mathematics Improvement Toolkit middlestart Friday, November 19, 2010