The Mathematical Practices in the CCSSM

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� Don�t 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 Won�t 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 Webster�s 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 Author�s 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: Isn�t math about the right answer? What�s 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