1. Affecting and Documenting Shifts in Secondary Precalculus Teachers’ Instructional Effectiveness and Students’ Learning Marilyn P. Carlson Arizona State University Kevin C. Moore University of Georgia Katheryn Underwood Katheryn Underwood Arizona State University RIMSE: Research Innovations in Mathematics and Science Education

2. Pathways Phase I Goals: Greater student learning  Understandings of Key Ideas AND achievement on standardized exams Increased student continuation in STEM course taking  Improved mathematical practices

3. Major Findings from Phase I: 5 key variables that impacted teaching and student learning are teachers who have strong and connected mathematical content knowledge of key ideas of the content they’re teaching (e.g., precalculus level mathematics) (Carlson & Rasmussen, 2009) believe that students can solve novel applied problems and understand the mental processes involved in doing so. (Moore & Carlson, 2012) implement pedagogy that supports students’ construction of mathematical practices (sense making, explaining thinking, persistence) (Clark, Moore, Carlson; 2008) participation in Pathways Professional Learning Communities (PLCs) (Carlson, Moore, et al.; 2007). Use of inquiry-based and conceptually oriented instruction (includes supports for teachers) (Carlson, Oehrtman; 2011)

4. Pathways Phase II is scaling (and studying the scaling) of the Pathways Professional Development Model for Teaching Secondary Mathematics -Workshops/graduate courses focused on developing mathematical content knowledge for teaching key ideas and reasoning abilities of algebra through precalculus -PLC’s (“Effective PLC’s (FOP) (Carlson, Moore, et al., 2007) - Professional support tools for teachers (student level tasks, embedded research knowledge, conceptually oriented assessments.) (Carlson, Oehrtman, Moore, Strom)

5. Developing the Pathways to Calculus Professional Development Model Review research to provide the theoretical grounding --Cognitive Models of Learning Key Ideas -Models of How Mathematical Practices are Acquired --Models for Effective Learning Communities -Models of how MKTP Develops -Implement -Study Implementation Revise Frameworks Design Implementation (Revise)

6. Foundational Understandings and Reasoning Abilities Needed to Understand Calculus Covariation and Quantification Proportionality Function (Composition and Inverse) Linear Functions Exponential Functions Polynomial Functions Rational Functions Trigonometric Functions (Precalculus)

7. Acquiring Productive Mathematical Practices: (Rules of Engagement) The teacher expects (and acts on the expectation that) students: Engage in Meaning Making Conceptualize quantities and their relations Persist in making sense Attempt to make logical connections Base conjectures on a logical foundation Students are expected to express their thinking and speak with meaning about their problem solutions pose meaningful questions when they don’t understand

8. The Precalculus Concept Assessment Instrument (PCA) Assesses understanding of key ideas of precalculus that are foundational for learning calculus 25 Item Multiple Choice  Item choices are based on common responses that have been identified in students (clinical interviews) Validated to correlate with success in calculus  Tracking 277 students in beginning calculus, 80% of students who scored 13 or above on PCA at the beginning of the semester received an A, B, or C in calculus.  85% of the 277 students who scores 11 or below received a D, F or withdrew.

9. Student PCA Performance PCA administered to 550 college algebra and 379 pre- calculus students at a large southwestern university Also administered to 267 pre-calculus students at a nearby community college Mean score for college algebra: 6.8/25 Mean score for pre-calculus: 9.1/25

10. Summary of student PCA pre- and post-test gains in Pathways Precalculus Courses Pre-test MeanPost-test Mean 8.215.2 (Previous Best Post Mean ScoreL 10.4)

11. How can teachers be supported to realize shifts in their teaching that affect the above described shifts in students? Workshops and PLCs that support teachers’ development of  Mathematical content knowledge for teaching precalculus (Silverman & Thompson) Understanding of key connecting ideas and reasoning abilities Image of how students acquire these understandings and reasoning abilities Instructional tools—mini learning theory  Pedagogical Practices and Pedagogical Choices What they do in the classroom to promote student thinking How they react to their students’ expressed thinking

12. What do we Mean by Mathematical Content Knowledge for Teaching Proportionality (MKTP)? How Does Mathematical Content Knowledge for Teaching Proportionality Interact with a Teacher’s Pedagogical Actions?

13. Copyright ©2010 Carlson and Oehrtman 13 PHOTO ENLARGEMENT TASK 1. A photographer has an original photo that is 6 inches high and 10 inches wide and wants to make different-sized copies of the photos so that new photos are not distorted. a. If the photographer wants to enlarge the original photo so that the new photo has a width of 25 inches, what will the height of the new photo need to be so the image is not distorted? Explain the reasoning you used to determine your answer. W#3

14. TWO VARYING QUANTITIES ARE RELATED BY A CONSTANT RATIO Let h = the height of the new photo (in inches) Let w = the width of the new photo (in inches) Then, as h and w vary together, their ratio stays fixed Copyright ©2010 Carlson and Oehrtman 14 W#3

15. CONSTANT MULTIPLE & SCALING h is always the same multiple of w. h 1 =6″ is 6/10 as long as w 1 =10″ So h 2 should be 6/10 as long as w 2 =25″ h 2 = (6/10)  25″ Copyright ©2010 Carlson and Oehrtman 15 If we scale w by some factor, we should also scale h by the same factor. w 2 =25″ is 25/10 as long as w 1 =10″ So h 2 should be 25/10 as long as h 1 =6″ w 2 =(25/10)  6″ Why? W#3

16. PHOTO ENLARGEMENT – TABLE Width in inchesHeight in inches 0 1 2 3 4 10 11 25 16 ×3/5 ×5/2 ×1/10 0 3/5 6/5 9/5 12/5 6 33/5 15 W#3 Copyright ©2010 Carlson and Oehrtman

17. What do we mean by MKT Proportionality?

18. Analogous Problem: Given that a line contains the point (3.35, 8.3) and has a rate of change (slope) of -2.2, determine the formula for the line.

19. A Solution

20. A Second Solution The candle’s length decreases at a constant rate of 2.2 inches per hour. 3.35 hours have passed, which is 3.35 times as large as 1 hour. The length that has burned in this time is 3.35 times as large as 2.2 inches (7.37 inches). The original length was this amount plus the 8.3 inches left.

21. Two Solutions? The candle’s length decreases at a constant rate of 2.2 inches per hour. 3.35 hours have passed, which is 3.35 times as large as 1 hour. The length that has burned in this time is 3.35 times as large as 2.2 inches (7.37 inches). The original length was this amount plus the 8.3 inches left.

22. Solution as a Consequence of PR. The candle’s length decreases at a constant rate of 2.2 inches per hour. 3.35 hours have passed, which is 3.35 times as large as 1 hour. The length that has burned in this time is 3.35 times as large as 2.2 inches (7.37 inches). The original length was this amount plus the 8.3 inches left.

23. Why PR as a Focus? Trigonometry?

24. Why PR as a Focus? s r r

28. Differential Equations?

29. Why PR as a Focus?

30. Rate of change is proportional to amount…

31. Why PR as a Focus? Rate of change is proportional to amount… …is related to precalculus?

32. Why PR as a Focus? Consider a mass of bacteria that grows continuously at a rate of 25%.

33. Why PR as a Focus? Consider a mass of bacteria that grows continuously at a rate of 25%. In precalculus, this means use “Pert”:

34. Why PR as a Focus? Consider a mass of bacteria that grows continuously at a rate of 25%. In precalculus, this means use “Pert”: But we are really saying:

35. How do we support MKT Proportionality?

36. Student PCA Performance PCA administered to 550 college algebra and 379 pre- calculus students at a large southwestern university Also administered to 267 pre-calculus students at a nearby community college Mean score for college algebra: 6.8/25 Mean score for pre-calculus: 9.1/25

37. Summary of student PCA pre- and post-test gains in Pathways Precalculus Courses Pre-test MeanPost-test Mean 8.215.2 (Previous Best Post Mean ScoreL 10.4)