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Opening. As described: This session will unpack" the meaning of the eight mathematical practices and explore critically the key terms in each.Two meanings for unpack"Physical: Known contents of a box; unpacking means just getting at themMental: Vague content (e.g., a word); explore possible/lik
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1. The Mathematical Practices in the CCSSM 2012 Math In Action Conference
Grand Valley State University, Allendale campus
February 25, 2012
Jack Smith
STEM Project
College of Education
Michigan State University
2. Opening As described: This session will unpack the meaning of the eight mathematical practices and explore critically the key terms in each.
Two meanings for unpack
Physical: Known contents of a box; unpacking means just getting at them
Mental: Vague content (e.g., a word); explore possible/likely meaning(s)
This session will not be direct instruction
Dont believe telling supports deep learning
Practices are not well-defined
Objectives
Present my views: How to make sense of the practices
Engage you and encourage you as leaders 8/13/2012 2 Practices, Math in Action, 2012
3. Why Care about the Practices? I could be assessed in my implementation of them
They are brief enough to actually wrap my head around
I can use them to talk to my building colleagues (across grades)
They are a better investment than a list of grade-level content standards; closer to the goal of teaching thinking
They remind me of the process standards in the NCTM Standards
8/13/2012 Practices, Math in Action, 2012 3
4. Present & Future Lots of other people are working on unpacking the practices
From one paragraph (core meaning) to specific meanings in different content areas and grades/grade levels
Some organizations
National Council of Supervisors of Mathematics http://www.mathedleadership.org/ccss/materials.html
Illustrative Mathematics Project http://illustrativemathematics.org/
Common Core Tools site http://commoncoretools.me/
No national consensus on their meaning; detail is missing 8/13/2012 4 Practices, Math in Action, 2012
5. Where are you? (relative to the practices) Grade level: Elementary, Middle, Secondary, College?
Some positions (you and the practices):
1. Know almost nothing them, but am curious
2. Have looked at some of the descriptions, but not thought about them deeply
3. Have looked with interest at particular practices and thought about their meaning (or discussed them with colleagues)
4. Have been asked to do some professional development around them
5. Have heard knowledgeable others speak about them and what they do or could mean
6. Other 8/13/2012 Practices, Math in Action, 2012 5 Pass out the copies of the practicesPass out the copies of the practices
6. Plan Address four practices is some depth for 20 minutes
Focus on key terms that are essential to the practice
Wont be answers sorry!
Rationale for choices: Importance, lack of detail, Jack has thought about them
Remaining time to discussion, including attention to other practices
Short out the door activity (5 minutes) 8/13/2012 Practices, Math in Action, 2012 6
7. Pr-1: Make sense of problems & persevere in solving them Core issue: What is the meaning of problem?
If problem means task, this is not a new directive
Two meanings for problem (from Websters via Alan Schoenfeld)
In mathematics, anything required to be done, or requiring the doing of something. {task above]
a question that is perplexing or difficult
Problem as problematic for the student
Alternative: Exercises, the solution method has already been demonstrated 8/13/2012 Practices, Math in Action, 2012 7 Allow time to read the paragraph, based on the familiarity of the group
This is the mother practice, listed first for a reason
Allow time to read the paragraph, based on the familiarity of the group
This is the mother practice, listed first for a reason
8. Challenges to Pr-1 (for problematic problems) Challenge #1: How often should the class work on real problems?
Challenge #2: What is challenging for most/all the class, but still within their access?
Challenge #3: What do I need to do to prepare my class when they expect a steady diet of exercises?
Challenge #4: What happens when my class that is behind? Interference with grade-level content standards
8/13/2012 Practices, Math in Action, 2012 8
9. Pr-3: Construct viable arguments and critique the reasoning of others Authors assumption: Mathematics is social, public, and language-based
Written forms of argument
Oral forms of argument
Responses to the arguments of others
Arguments: A reason to believe a solution, not just proof at the secondary level
Both arguments and critique can take on a negative tone
We are having an argument => a bad social interaction
Critique must mean being critical of others
Authors intent: Focus on the mathematics, not the person
Model: The mathematical profession 8/13/2012 Practices, Math in Action, 2012 9 Allow time to read the paragraph, based on the familiarity of the group
Arguments are offered by students of all grades and abilitiesAllow time to read the paragraph, based on the familiarity of the group
Arguments are offered by students of all grades and abilities
10. Challenges to Pr-3 (prior experience: talk & argument) Challenge #1: Isnt math about the right answer? Whats there to talk about?
Challenge #2: Arguments and critiques can become personal
Challenge #3: Defensiveness; real listening is a hard skill to teach
Challenge #4: How to avoid right/wrong and make the repair of flawed solutions typical and expected?
Useful term: social norms for classroom mathematics work
No short-cut to a healthy learning community; you have to invest in norms you value (and do so early) 8/13/2012 Practices, Math in Action, 2012 10
11. Pr-5: Use appropriate tools strategically Focus on a wide range of physical tools and technology
Tools serve problem solving; more meaning when problems are problematic, not exercises
Key issue: What counts as a tool?
Tools include different kinds of reasoning, e.g., estimation
Tools include computational algorithms and formulas
Example: Formula for the area of a rectangle
Understanding formulas is a prerequisite to strategic use
8/13/2012 Practices, Math in Action, 2012 11 Allow time to read the paragraph, based on the familiarity of the group
Expect most readers of the title would not list pencil & paper
Technology is in quotes because it really means other technology
Note shift from reading the authors to my own interpretation (computational algorithms and formulae)
Allow time to read the paragraph, based on the familiarity of the group
Expect most readers of the title would not list pencil & paper
Technology is in quotes because it really means other technology
Note shift from reading the authors to my own interpretation (computational algorithms and formulae)
12. Challenges to Pr-5 (access & understanding) Challenge #1: What tools are available to me?
Challenge #2: How can I find useful internet resources?
National Library of Virtual Manipulatives http://nlvm.usu.edu/
STEM Project, measurement simulations (URL on last slide)
NCTM Illuminations http://illuminations.nctm.org/
Challenge #3: Teaching my students to think about algorithms and formulas as tools; as efficient means
Challenge #4: Strategic thinking cannot develop from steady diet of exercises 8/13/2012 Practices, Math in Action, 2012 12 Implicitly, we still have core challenge: How often do we address real problems?Implicitly, we still have core challenge: How often do we address real problems?
13. Pr-7: Look for and make use of structure The statement is rich in examples, yet.
What are patterns or structures? Which ones are worth paying attention to? On what basis, do I judge?
Some implicit answers:
Properties (e.g., commutative, distributive) of operations on whole numbers
Structure is often revealed in algebraic statements, e.g., (x + a)(x + b) = x2 + (a+b)x + ab
Structures support reading algebraic expressions
Importance: Mathematics IS the discovery and statement of structure
8/13/2012 Practices, Math in Action, 2012 13 Allow time to read the paragraph, based on the familiarity of the group
Allow time to read the paragraph, based on the familiarity of the group
14. More on Pr-7 (coverage & teacher judgment) Challenge #1: At the elementary level, are there structures and patterns outside of base 10 number & operations? What are they?
In Geometry or Measurement?
In Data?
Challenge #2: How do I focus students attention on structure (beyond particular answers and solutions)?
Challenge #3: If I can get my students to look for and nominate patterns or structures, how do I evaluate them, since there is no reason to think their structures will be commonly accepted ones? 8/13/2012 Practices, Math in Action, 2012 14
15. Out the Door For two minutes: Turn to a neighbor and decide:
One useful element of the presentation (make it shared, if possible)
One element missing from the talk (again shared, if possible)
For a discussion of the practices as they relate to measurement (K-5), The Educators Companion to Measurement in CCSSM, at: https://www.msu.edu/~stemproj/teaching.html
Slides are available at the conference site
Feedback on the presentation (beyond the above): jsmith@msu.edu
8/13/2012 Practices, Math in Action, 2012 15 Discussion of the practices is 4+ pages in a much longer documentDiscussion of the practices is 4+ pages in a much longer document