Dr. April Strom Scottsdale Community College Dr. Ted Coe Grand Canyon University Dr. Trey Cox

1. Dr. April Strom Scottsdale Community College Dr. Ted Coe Grand Canyon University Dr. Trey Cox Chandler-Gilbert Community College NCTM Las Vegas 10-24-13 This work was supported in part by MSP grant #1103080 through the National Science Foundation. Opinions expressed are those of the authors and not necessarily those of the NSF.

2. Arizona Mathematics Partnership • Targeted MSP • 5-year, $8.7M • Partners • 3 community colleges • Math Ed Faculty/Researchers • Mathematicians • Master’s and Doctoral Students • Undergrad Researchers • Pre-service Teachers • 7 school districts • 300 middle school teachers (expanded to 5th-9th grade) • Currently 63 teachers in Cohort 1 • 20+ school administrators • RMC Research • Arizona State Univ

3. Teacher Quality, Quantity, & Diversity Focus on… • Meaningful professional development (Summer Institutes, Saturday Workshops, CCOLs) • Enhancing pre-service teacher experiences in CCs, developing service learning field experiences • Improving developmental mathematics

4. Challenging Courses & Curricula Institutes/Workshops

5. What is a Collaborative Communityof Learners (CCOL)? • A “type” of Professional Learning Community (PLC) • A community of teachers that self-directs its investigative focus and collaborative efforts toward matters that the participants find problematic in their own teaching environment. • The goal of a CCOL is to enhance classroom instruction so that student learning is improved. The CCOL is ultimately about the STUDENTS!

6. What is a Collaborative Communityof Learners (CCOL)? • A "better" form of professional development (workshops, conferences, meetings, coaching, etc.)  • Same: curriculum, students, standards, district/school assessments, parents, socio-economic climate, etc.  • Different: focuses on different ideas and has control (with facilitator input/veto) of direction of investigation  • Ongoing learning 

7. Teacher Expectations • 18 face-to-face hours  • 12 "homework" hours • (Plus the Summer Institute and Saturday Workshops) • Collaborative venture/equal participation • Encouragement and constructive criticism in a safe environment • Implement new ideas and techniques  • Ongoing assessment and revision  • Make practice public - open classroom door to visitors  • Instructional Rounds • Show-n-Tell presentation 

8. Facilitator Expectations • Insure effective use of time - relevant to classroom teachers  • Establish collaborative venture/equal participation • Become expert in the CCSS Content Standards and Math Practices • Observe teacher classrooms and provide input/suggestions • Set monthly agendas and action items • Attend quarterly training sessions • Be flexible, provide encouragement, support, and appropriate challenges to CCOL teachers • FACILITATORS ARE A MAJOR KEY TO SUCCESS 

9. A Multiplicative Thread • Ways of Doing • Ways of Thinking • Habits of Thinking

10. Broomsticks Activity You have three broomsticks: The RED broomstick is 3feet long The YELLOW broomstick is 4feet long The GREEN broomstick is 6feet long

11. LEARNING TRAJECTORIES IN MATHEMATICS: A Foundation for Standards, Curriculum, Assessment, and Instruction, pp48-49.

12. Sequencing: LEARNING TRAJECTORIES IN MATHEMATICS: A Foundation for Standards, Curriculum, Assessment, and Instruction, pp48-49.

13. Measurement What do we mean when we talk about “measurement”?

14. “Technically, a measurement is a number that indicates a comparison between the attribute of an object being measured and the same attribute of a given unit of measure.” • But what does he mean by “comparison”? • Van de Walle (2001) Elementary and Middle School Mathematics: Teaching Developmentally, 4th ed.

15. How about this? • Determine the attribute you want to measure • Find something else with the same attribute. Use it as the measuring unit. • Compare the two: multiplicatively.

16. From Fractions and Multiplicative Reasoning, Thompson and Saldanha, 2003. (pdf p. 22)

18. The circumference is about how many times as large as the diameter? • The diameter is about how many times as large as the circumference?

19. http://tedcoe.com/math/circumference

20. Angles • What attribute are we measuring when we measure angles? CCSS Grade 4 p. 31

21. Radian Measure http://tedcoe.com/math/trigonometry/radius-unwrapper

23. A measure of the area of whole square is 4r^2 A measure of the area of red square is 2r^2 A measure of the area of circle is…

24. Similar Triangles

25. The Pythagorean theorem http://tedcoe.com/math/geometry/pythagorean-and-similar-triangles

26. Constant Rate http://tedcoe.com/math/algebra/constant-rate

27. Similar TRiangles & Trigonometry • Right Triangle Trigonometry http://tedcoe.com/math/geometry/similar-triangles

28. Bank Accounts • Consider a bank account that grew from $95 to $120 over THREE years. • Use an additive comparison for these two numbers. What does your answer mean? • Use a multiplicative comparison for these two numbers. What does your answer mean? what? what?

29. Bank Accounts • If the terms of growth remained constant, ($95 to $120, 3 years) • What is the annual additive growth for this account? What does your number mean? • What is the annual multiplicative growth (factor) for this account? What does your number mean? what? what?

30. Exponential Functions…

31. What’s next? • Where could you go next? • What other mathematical ideas can be built upon multiplicative thinking?