1. Reflecting on Practice: Using Inquiry to Build Thinking Classrooms
Unit 1, Session 3
2017
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3. Park City Mathematics Institute
Key chains
These are 7/8 Graders in an algebra class
Teacher’s goals were to look at patterns to create equations, connect equations to real world context and see multiple representations of functions
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4. Park City Mathematics Institute
Video Norms
Video clips are to spur discussion, not criticism
Video clips are to spur inquiry, not judgement
Video clips are to provide a snapshot of a particular moment
Be sure to cite specific examples (evidence) from the clip
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5. Park City Mathematics Institute
Key Chains
As you watch the video make note of what student thinking you see in the lesson.
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6. Park City Mathematics Institute
Key Chains
As you watch the video make note of what student thinking you see in the lesson.
At your table discuss the student thinking you saw and how it was promoted/hindered by the task itself.
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7. Park City Mathematics Institute
Keychains
How can we improve this task to increase the cognitive demand and promote more student thinking?
Consider the question for a minute or so and generate a few modifications as a table.
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8. Park City Mathematics Institute
Cognitive Demand
Higher-level demands (Doing Mathematics):
Require complex and nonalgorithmic thinking—a predictable, well- rehearsed approach or pathway is not explicitly suggested by the task, task instructions, or a worked-out example.
Require students to explore and understand the nature of mathematical concepts, processes, or relationships.
Demand self-monitoring or self-regulation of one’s own cognitive processes.
Require students to access relevant knowledge and experiences and make appropriate use of them in working through the task.
Require students to analyze the task and actively examine task constraints that may limit possible solution strategies and solutions.
Require considerable cognitive effort and may involve some level of anxiety for the student because of the unpredictable nature of the solution process required.
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Smith & Stein, 1998

9. Park City Mathematics Institute
Systems Version I
Draw a line through the two points (-4, 3) and (2, 6). Write the equation of the line.
Draw a line through the points (-3, -6) and
(2, 4). Write the equation of the line.
Using the equations you found above, solve for the point of intersection of the two lines.
 (Herbel-Eisenmann & Crillo, 2013)
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Systems Version II
If the scale for each axis is one unit, find the intersection of the two lines, if it exists.
Find more than one way to solve the problem.
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Two versions
Think for a minute about the different ways the two tasks shaped the discussion and your own thinking.
In particular how did the way the tasks were posed relate to the characteristics of worthwhile tasks that contribute to a thinking classroom?
Reflecting on Practice
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12. Park City Mathematics Institute
Cognitive Demand
Higher-level demands (Doing Mathematics):
Require complex and nonalgorithmic thinking—a predictable, well- rehearsed approach or pathway is not explicitly suggested by the task, task instructions, or a worked-out example.
Require students to explore and understand the nature of mathematical concepts, processes, or relationships.
Demand self-monitoring or self-regulation of one’s own cognitive processes.
Require students to access relevant knowledge and experiences and make appropriate use of them in working through the task.
Require students to analyze the task and actively examine task constraints that may limit possible solution strategies and solutions.
Require considerable cognitive effort and may involve some level of anxiety for the student because of the unpredictable nature of the solution process required.
Reflecting on Practice
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Smith & Stein, 1998

13. Park City Mathematics Institute
A story about change
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Modeling Change, TLP

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What mathematical thinking emerged in the discussion?
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Where are we now?
What makes tasks worthwhile?
Wordle.
Refined to a few essential elements.
Cognitive demand is one essential element.
Today was about adapting tasks to make them worthwhile.
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16. Park City Mathematics Institute
Homework
For tomorrow, we would like for you to read the Sanchez article, Open Ended Questions and the Process Standards [handout].
As you read, identify two strategies for modifying tasks so they promote thinking and reasoning.  
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17. Park City Mathematics Institute
References
Harper S., & Edwards, M. (2011). A New Recipe: No More Cookbook Lessons. The Mathematics Teacher, 105(3), pp Reston VA: National Council of Teachers of Mathematics .
Herbel-Eisenmann, B. & Crillio, M. (2013). Two versions of same task adapted from Association of Mathematics Teacher Educators presentation, Mathematics discourse in secondary classrooms: A case- based professional development curriculum. Orlando FL
Liljedahl, P. (under review). On the edges of flow: Student problem solving behavior. In S. Carreira, N. Amado, & K. Jones (eds.), Broadening the scope of research on mathematical problem solving: A focus on technology, creativity and affect. New York, NY: Springer.
Modeling Change. (2004) PCMI Teacher Program Lesson Study.
Sanchez, W. (2013). Open ended questions and the process standards. The Mathematics Teacher, 107(3),
Solving Equations, (2005). Break Through Mathematics. Lesson Lab. Pearson Education Company.
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