2. Reflecting on Practice: Making Connections that Support Learning
Unit 3, Session 1
2016
Reflecting on Practice
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3. Park City Mathematics Institute
How People Learn
1.Teachers must draw out and work with the pre-existing understandings that their students bring with them.
Students come to the classroom with preconceptions about how the world works. If their initial understanding is not engaged, they may fail to grasp the new concepts and information that are taught, or they may learn them for purposes of a test but revert to their preconceptions outside the classroom.
Reflecting on Practice
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NRC, 2001

4. Park City Mathematics Institute
How People Learn
2.Teachers must teach some subject matter in depth, providing many examples in which the same concept is at work and providing a firm foundation of factual knowledge.
To develop competence in an area of inquiry, students must: (a) have a deep foundation of factual knowledge, (b) understand facts and ideas in the context of a conceptual framework, and (c) organize knowledge in ways that facilitate retrieval and application.
Reflecting on Practice
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NRC, 2001

5. Park City Mathematics Institute
How People Learn
3.The teaching of metacognitive skills should be integrated into the curriculum in a variety of subject areas. A "metacognitive" approach to instruction can help students learn to take control of their own learning by defining learning goals and monitoring their progress in achieving them.
Reflecting on Practice
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NRC, 2001

6. Park City Mathematics Institute
How would you expect students to find the equation of the line through the points (-2, 5) and (1, 3)?
Reflecting on Practice
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7. Park City Mathematics Institute
At your tables discuss the question: Should students be given one general all purpose strategy for solving a problem or should they be given a variety of strategies? Explain your reasoning.
Reflecting on Practice
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8. Possible forms of a line
Slope-intercept form: y = mx + b
Point-slope form v.1: y – y1 = m(x-x1)
Point-slope form v.2: y = y1 + m(x-x1)
General form: ax + by = c
Slope definition: (y – y1)/(x – x1) = m
Intercept form: x/a + y/b = 1
Factored form: y = a(x-b)
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9. Park City Mathematics Institute
In pairs, decide how you would expect students to write the equation of the line in each of the following. Explain your thinking.
the line through (0,5) and (3,-1)
a line perpendicular to 5x+3y=6
a line parallel to 5x+3y = 6
a line parallel to y=-1- ½ (x -3)
the line where [f(3)-f(7)]/3-7= -12
the tangent line to a curve at (3, -1) passing through (7, -3)
Reflecting on Practice
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10. Park City Mathematics Institute
Researchers have been focusing on the value of “flexible procedural knowledge” as an important part of learning with understanding.
What do you think they mean by flexible procedural knowledge?
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11. Park City Mathematics Institute
At your table, take a few minutes and discuss how you would explain the meaning of “mean” of a set of data.
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12. Park City Mathematics Institute
Researchers have some evidence that having students deliberately compare strategies using reflective questions is effective in helping students develop flexible procedural knowledge (Maciejewski & Star, 2016).
We are going to have you experience this as a student first and then later from a teacher's perspective.
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13. Park City Mathematics Institute
Gabe & Jenn…
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14. Park City Mathematics Institute
Let’s look at a short clip of interpretations of the mean
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15. Mean as Fair Share

16. Mean as fair share “ Leveling” Pooling and dividing the pool
Building Concepts: Statistics and Probability, 2016

17. Park City Mathematics Institute
Which is better?
Find the mean: 70, 77, 90   
Jen’s solution:
I added the three numbers to get 237, then I divided by 3 to get 79.
Gabe’s solution
I took 10 from 90 and gave it to the 70, so the numbers were 80, 77, and 80. Then I took 1 from each 80 and gave it to the 77 to get all three numbers equal to 79.
Reflecting on Practice
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18. Park City Mathematics Institute
“Frequently there are viable alternative methods for solving a problem, and discussing the advantages and disadvantages of each can facilitate flexibility and deep understanding of the mathematics involved.” (How Students Learn, p. 233).
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19. Park City Mathematics Institute
The Star Approach
Phase 1
Present two different strategies
Phase 2
Students respond to 3 types of questions (Understand, Compare, Make Connections)
Phase 3
Students apply their thinking
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20. Reflection questions: Pushing and probing student thinking
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21. Park City Mathematics Institute
Using your example (arithmetic, algebra, calculus), work through the questions on your own and then compare with your partners. Agree on a common set of responses, then post your agreed on responses on the wall.
Reflecting on Practice
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22. Park City Mathematics Institute
Gallery Walk
Visit the posters, and using post-it notes leave:
Compliments
Comments
Considerations
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23. Park City Mathematics Institute
Metacognition
Metacognition might be thought of as “thinking about your own thinking”.
How can deliberate comparisons among forms for representing mathematical solutions, strategies for finding solutions, language used to describe approaches, etc. help students develop metacognitive skills?
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24. Park City Mathematics Institute
"Comparison can bring dimensions of variation of the concept or procedure to the learner's attention"
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25. Park City Mathematics Institute
Today we started our focus on Key Finding 3: Metacognition. Please read the small excerpt about this key finding for tomorrow's class. When you read...
Keep track of 2 key ideas
One thing that surprised you
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