2. Reflecting on Practice: Making Connections that Support Learning
Unit 3, Session 3
2016
Reflecting on Practice
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3. Park City Mathematics Institute
“Metacognitive functioning is also facilitated by shifting from a focus of right or wrong to a more detailed focus on ‘debugging’ a wrong answer, that is, finding where the error is, why it is an error, and correcting it” (How Students Learn: Mathematics in the Classroom, p. 239).  
Reflecting on Practice
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4. Park City Mathematics Institute
Using Comparison
1. Comparing how different problems can be solved with the same method 
Example: Choosing examples for the Cover Up Method  
2. Comparing problems involving different contexts but the same mathematical structure 
Example: Triangular Numbers 
3. Different problems in same context solved with different methods   
Example: Equation of a line, choosing most appropriate strategy  (Star approach) 
4. Comparing different approaches to the same problem.  
Example: Mast problem & Japanese angle task 
5. Comparing correct and incorrect solutions to the same problem. 
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5. Research on learning algebra: Making links to the classroom
1988 NCTM Yearbook on Algebra: Common Mistakes in Algebra (Marquis, 1988)
a2 b5 = (ab)7 (x+4)2 = x2+ 16
x+y y x r x+r
x+z = z y s = y+s
x+y - 3(z+w) = x+y - 3z+w
If (2-z) < 4, then z < -2
Gail slide 3 of 19 5

6. Instructional Goal: Students recognize that a line of symmetry with certain properties in a triangle produces an isosceles triangle.

7. Park City Mathematics Institute
What does dynamic geometry bring to helping students think about errors?
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8. Park City Mathematics Institute
Highlighting Errors
As you watch the video, reflect on what evidence of metacognition you see.
Reflecting on Practice
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9. Park City Mathematics Institute
Highlighting Errors
What evidence of metacognition did you see in the video?
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10. Park City Mathematics Institute
Highlighting Errors
What evidence of metacognition did you see in the video?
What norms did Leah create to establish a class culture of learning from mistakes?
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11. Students correcting “error exemplary” work
As you watch the video, reflect on what evidence of metacognition you see
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12. Park City Mathematics Institute
What evidence of metacognition did you see in the video?
What norms did Susie create to establish a class culture of learning from mistakes?
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13. Park City Mathematics Institute
"Finding the part that is wrong and understanding why it is wrong can be a powerful aid to understanding and promotes metacognitive competencies"
(How Students Learn Mathematics, p. 224)
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14. Park City Mathematics Institute
In your table groups.
What are some additional strategies that you have found successful in your own classroom?
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15. Park City Mathematics Institute
In your table groups.
What are some additional strategies that you have found successful in your own classroom?
Grindelwald -
St. Moritz -
Davos -
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16. Park City Mathematics Institute
Journal: It is important that we are metacognitive as learners & practitioners ourselves. How have your thoughts about your own practice changed since starting PCMI? Exit Ticket: If we had one more week at RoP, what are 3 things that you would want us to focus on?
Reflecting on Practice
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