1. MODELING AND USING TOOLS: MATHEMATICAL PRACTICES OF THE COMMON CORE Statewide Instructional Technology Project

2. Introduction This is the fourth in a series of five webinars on the Standards of Math Practice. Links to resources, including recordings of previous webinars can be found at ( http://tinyurl.com/StandardsMathPractice )

3. Official ADE documents  2010 Arizona Mathematics Standards 2010 Arizona Mathematics Standards  Overview of the 2010 Mathematical Standards PDFPDF  Standards for Mathematical Practices PDFPDF  Mathematics Introduction (Coming Soon)  Mathematics Glossary PDFPDF  Summary of Updates to Explanations and Examples PDFPDF

4. Mathematical Practices

5. Standards of Math Practice Teacher Centered Student Centered TeachersFountains of Knowledge Create an environment that supports learning StudentsReceptive LearnersActive Learners Teacher/Student Relationship AdversarialCollaborative Reflect Changes in Teaching and Learning

6. MP. 4 Model with mathematics MP. 5 Use appropriate tools strategically Model and Use Tools

7. MP. 4 Model with Mathematics “__________________ is seen as the process towards a solution where sometimes an exact answer does not exist”

8. MP. 4 Model with Mathematics “___________________ activities are inherently social experiences, where students work in small teams to develop a product that is explicitly sharable.”

9. MP. 4 Model with Mathematics “_____________________________ is a means to develop competencies … that are important in highly technological societies, and for making decisions that are imperative for the maintaining and further development of democracy.”

10. MP. 4 Model with Mathematics Mathematically proficient students can…  Choose an appropriate and efficient model  Model, represent and solve real-world problems.  Simplify by making assumptions and approximations.  Interpret results and revise the model if necessary. Mathematics is the language with which God has writ the universe.” Galileo

11. MP. 4 Model with Mathematics Choose an appropriate and efficient model Models are systems of elements, operations, relationships, and rules that can be used to describe, explain, or predict the behavior of some other experienced system.

12. MP. 4 Model with Mathematics Why model?  Reframe math from a set of abstract formal structures into a useful tool for making sense of the world.  Apply math to problems and situations of real interest to children  Give students experiences that lead them to see and interpret our world using the language and lens of mathematics.  Demonstrate the necessity and usefulness of simplifying assumptions and approximations. About how many children are in our school? $1.23 x 0.78 = ?  Help children develop spatial understanding. (Think maps, diagrams, and physical models with cross-curricular applications)

13. MP. 4 Model with Mathematics Modeling can look like:  Counters to solve early number problems involving grouping and separation are direct modeling  An array  A number line  Model-drawing method to solve word problems  Mapping the structure of a problem situation and the corresponding symbolic expression  A physical structure

14. MP. 4 Model with Mathematics Modeling can also look like: A Problem or Situation in which students – are confronted with the need to develop a model clearly recognize the need to revise or refine their current ways of thinking about the given problem are challenged to express their understandings in ways they can test themselves and revise as often as necessary develop models that can be shared with others and applied in other situations ( Lesh & Yoon, 2004 ) Lesh & Yoon, 2004

15. Real World Problem Problem Representation Mathematical Representation Real World Solution MP. 4 Model with Mathematics Modeling can also look like a Mathematical Modeling Process Interpreting Problem Situation Reevaluating Application Deducing Reevaluating AbstractingReevaluating

16. Great Model-Eliciting Problems include:  Real world contexts and real world solutions  Model exploration and application where students can build, consolidate and refine their conceptual systems  Multiple approaches, multiple solutions - which provides opportunities to study, reflect, interpret, and re-interpret the problem information  Shared responsibility and 21st Century Skills  A variety of end products so students can explore creative approaches and representational fluency  High level cognitive engagement MP. 4 Model with Mathematics

17. to unlearn relearn enhance skills that make students’ thinking visible relinquish some control fully understand content Teachers Need….

18. MP. 4 Model with Mathematics Coach students Persevere in solving problems Communicate with precision Reason and makearguments Teachers should….

19. MP. 4 Model with Mathematics Talk about thinking Draw connections Start with simple situations Facilitate Teachers should….

20. MP. 4 Model with Mathematics Research Shows…  "subjects who worked cooperatively spent more time working on practice exercises and reported greater satisfaction than those who worked individually.“ Effects of cooperative learning Effects of cooperative learning James D. Klein and Doris R. Pridemore  "Studies have shown that groups out perform individuals on learning tasks, and further that individuals who work in groups do better on later individuals assignments as well” (Barron, 2000b, 2003; O'Donnell & Danserau, 1992)." Powerful Learning by Linda Darling-Hammond, page 19.Powerful Learning

21. What we DON’T want!!

24. MP. 4 Model with Mathematics Mathematically proficient students can…  Identify mathematical tools and recognize their strengths and weaknesses.  Select and use appropriate tools.  Use estimation to predict reasonable solutions and/or detect errors.  Identify and successfully use external mathematical resources.  Use a variety of technologies, including digital content, to explore, confirm, and deepen conceptual understanding.

25. MP. 4 Model with Mathematics Strategic tool use  Make tools available  Introduce tools  Allow exploration  Demonstrate how the tool can be used  Keep tools continuously available  Know your students, different developmental levels need different tools

26. MP. 4 Model with mathematics MP. 5 Use appropriate tools strategically Model and Use Tools

27. MP. 5 Use appropriate tools strategically. Resources to support

35. (Password = SITP)

36. Mathematical Practices Expertise that we each seek to develop in our students  What does it mean to do mathematics?  What does it mean to understand mathematics? As teachers, our goal is to provide regular and consistent opportunities to develop and build these habits of mathematical thinking.

37. Mathematical Practices How to transition  Focus on the Mathematical Practices How do your students model the Mathematical Practices? In what ways do your classroom strategies foster development of the Mathematical Practices?  Implement the Critical Ideas  Look at the ADE website for standards, crosswalk, and summary of changes.

38. Resources for Further Exploration  The Illustrative Mathematics Project http://illustrativemathematics.org/ http://illustrativemathematics.org/  Math Common Core Coalition http://www.nctm.org/standards/mathcommoncore/ http://www.nctm.org/standards/mathcommoncore/  Achieve the Core http://www.achievethecore.org/http://www.achievethecore.org/  National Council Teacher of Mathematics http://www.nctm.org/standards/content.aspx?id=23273 Guiding Principles for Mathematics Curriculum and Assessment http://www.nctm.org/standards/content.aspx?id=23273  Mathematics Problem Solving http://jwilson.coe.uga.edu/emt725/PSsyn/Pssyn.html http://jwilson.coe.uga.edu/emt725/PSsyn/Pssyn.html  Mathematics Through Problem Solving http://www.mathgoodies.com/articles/problem_solving.html http://www.mathgoodies.com/articles/problem_solving.html

39. Resources for Further Exploration  Inside Mathematics http://www.insidemathematics.org/index.php/standard-1 http://www.insidemathematics.org/index.php/standard-1  Curriculum Exemplars from EngageNY http://engageny.org/resource/curriculum-exemplars/ http://engageny.org/resource/curriculum-exemplars/  Indiana Dept of Ed Implementing the Standards for Mathematical Practice  Indiana Dept of Ed - Implementing the Standards for Mathematical Practice http://media.doe.in.gov/commoncore/2011-05-10- StandforMath.htmlhttp://media.doe.in.gov/commoncore/2011-05-10- StandforMath.html  Tools for the Common Core Standards http://commoncoretools.me/http://commoncoretools.me/

40. Resources for Further Exploration  New Jersey Center for Teaching & Learning Progressive: Mathematics Initiative http://njctl.org/programs/ Free digital course content for over twenty courses, these initiatives span K-12 mathematics and high school science. http://njctl.org/programs/  Wiki on Standards of Practice http://enhancingmypractice.wikispaces.com/Standards+of+Math+Practice http://enhancingmypractice.wikispaces.com/Standards+of+Math+Practice

41. To view the official ADE documents  2010 Arizona Mathematics Standards 2010 Arizona Mathematics Standards  Overview of the 2010 Mathematical Standards PDFPDF  Standards for Mathematical Practices PDFPDF  Mathematics Introduction (Coming Soon)  Mathematics Glossary PDFPDF  Summary of Updates to Explanations and Examples PDFPDF

42. More Questions  Contact ADE Mary Knuck Mary.Knuck@azed.govMary.Knuck@azed.gov Suzi Mast suzi.mast@azed.govsuzi.mast@azed.gov