1. Pathways to Achieving High Gains in Secondary Mathematics Students’ Learning: The Role of Curriculum, Teachers, and Instruction Marilyn P. Carlson marilyn.carlson@asu.edu Professor, School of Mathematical and Statistical Sciences Arizona State University RIMSE: Research Innovations in Mathematics and Science Education

2. Framing the Problem: The Percent of 2012 College Graduates Earning STEM Degrees 5.6% USA 28.1% Germany 46.7% China 32.8% Korea

3. Passing rates in first-semester calculus by students who had passed the prerequisite precalculus ranged from 17%-37% at the six colleges and universities (Schattschneider’s, 2006). In another study, less than 20% of students who received an A, B, or C ) in precalc (and had a declared major that required calculus) went on to complete beginning calc. and less than 3% of these students completed calc II (Thompson, 2008) Where are we losing our STEM students?

4. Select research results reveal sources of the problem Precalculus level students are not developing the foundational understandings and reasoning abilities that are necessary for learning and using ideas of calculus (e.g., Making the Connection: Research and Teaching in Undergraduate Mathematics Education, Carlson & Rasmussen, 2009; Kaput, 1992; Dubinsky 2002; Monk, 1992; Thompson 1994; Carlson, 1998; Jacobs, 2000; Carlson, Oehrtman, Engelke, 2010) Teachers are not receiving the support they need to teach for meaning (MKT) (e.g., Carlson & Moore; in press; Thompson, 2011; Thompson, Carlson & Silverman: 2007; Thompson 2014) Curriculum materials have a strong focus on methods for getting answers (Oehrtman, 2009; Teuscher, 2010; Thompson, 2008)

5. Common Conceptions Held by Precalculus Level Students at the End of the Course Variable: Something to “solve for” (Schoenfeld, 1989; Jacobs, 1999) Function/Formula:  Two expressions separated by an equal sign (Thompson, 1994; Carlson, 1995) Function Graph:  Picture of an event (Monk, 1992; Kaput: 1992; Carlson, 1998) Constant Rate of Change: m or the speed stays the same Average Rate of Change on an interval of a function’s domain: Add them up and divide (Carlson, 1998) The speed the car is going most of the time (Carlson & Oehrtman)

6. Give an example of an exponential function? What does the phase, “write a function that expresses s in terms of t” mean? there must be an s and t and an equal sign What does f(g(x)) mean? f times g times x What does f -1 (x) represent? (1/x), 1/(f(x)), 1/f (Carlson, 1998; Carlson, Frank, Fowler, in press)

7. Problem Solving Behaviors : Classroom Rules of Engagement: Individuals are expected to: Persist in sense making/meaning making  Attempt to make logical connections Exhibit Mathematical Integrity  Don’t pretend to understand when you don’t  Base conjectures on a logical foundation Students are expected to express their thinking and speak with meaning about their problem solutions Students are expected to listen to the meaning being conveyed by their peers and ask questions when the meaning is not clear

8. Research on calculus learning revealed that students’ weak covariational reasoning abilities and weak conceptions of ideas of quantity, variable, function, and rate of change are obstacles for learning and understanding key ideas of calculus Rate of Change, Changing Rate of Change: (Thompson, 1992; Carlson, 1998, 2002; Zandieh, 2004) Limit (Cottrill, Brown, Dubinsky, 1998; Jacobs, 1999; Oehrtman, 2009; Williams, 2002) Derivative (Tall, 1992; Zandieh, 2004) Accumulation (Carlson, Oehrtman, & Thompson 2009; Smith, 2009; Carlson, Larsen, Jacobs; 2007) Related Rate Problems (Engelke, 2008; Carlson, 2007) Fundamental Theorem (Thompson, 1994; Carlson, Oehrtman, Thompson, 2009; Smith, 2009)

9. Covariational Reasoning: Conceptualizing two quantities and how they change togeter Quantitative Reasoning: Identifying attributes of an object that can be imagined as being measured (Thompson, 1989) Variable: A letter that represents the varying values that a quantity may assume

10. Essential Ways of Thinking: The Bottle Problem Imagine this bottle filling with water. Sketch a graph of the water’s height as a function of the amount of water that’s in the bottle. Explain the thinking you used to construct your graph.

11. Covariational Reasoning (Carlson et al., 2002) MA1: Imaging 2 quantities that are changing together MA2: Imaging how the direction of the 2 quantities are changing together MA3: Imaging how one quantity is changing while imagining changes in the other quantity MA4: Imaging how the average rate of change of the output quantity with respect to the input quantity is changing while imaging uniform increments of change in the input quantity.

12. A Research Based Response for Precalculus Level Mathematics: Over-arching Ideas and Reasoning Abilities Quantity, Variable, Formula, Function, Function Composition, and Function Inverse Constant Rate of Change & Average Rate of Change Exponential Functions Polynomial Functions Rational Functions Trigonometric Functions (Algebra II and Precalculus) Pathways: Other Cross-Cutting Ideas/Methods Evaluating Representational equivalence Solving

13. Copyright © 2010 Carlson and Oehrtman 13 The number of teachers in a school varies with the number of students in the school so that there are seven times as many students as teachers. a.Write a formula that defines the number of students s in terms of the number of teachers t. Investigation #1 QUANTITIES & CO-VARIATION OF QUANTITIES

14. Copyright © 2010 Carlson and Oehrtman 14 The number of teachers in a school varies with the number of students in the school so that there are seven times as many students as teachers. a.Write a formula that defines the number of students s in terms of the number of teachers t. Investigation #1 QUANTITIES & CO-VARIATION OF QUANTITIES Number of teachers in the school Number of students in the school 1 2 3 7 14 21

15. Copyright © 2010 Carlson and Oehrtman Investigation #4 CONSTANT RATE & LINEARITY a. Write a formula to represent the distance between Alan and Alissa in terms of the number of seconds since they spotted each other. 6.Alan and Alissa are attempting to find each other in a crowded mall. They finally spot each other near the food court, at which point they are 140 feet apart. They begin walking towards each other, Alan at a constant rate of 5 feet per second, Alissa at a constant rate of 4 feet per second. 15 Alan Alissa ? ft. 5 ft/s4 ft/s Constant Quantities 140 ft. Varying Quantities ? seconds

16. Copyright © 2010 Carlson and Oehrtman Investigation #4 CONSTANT RATE & LINEARITY Write a formula to represent the distance between Alan and Alissa in terms of the number of seconds since they spotted each other. Alan and Alissa are attempting to find each other in a crowded mall. They finally spot each other near the food court, at which point they are 140 feet apart. They begin walking towards each other, Alan at a constant rate of 5 feet per second, Alissa at a rate of 4 feet per second. 16 Alan Alissa d = 140 – 9t ft. 5 ft/s4 ft/s 140 ft. t seconds d 1 = 5t ft. d 2 = 4t ft.

17. Copyright © 2010 Carlson and Oehrtman Investigation #4 CONSTANT RATE & LINEARITY How many seconds it will take for Alan and Alissa to reach one another? 6.Alan and Alissa are attempting to find each other in a crowded mall. They finally spot each other near the food court, at which point they are 140 feet apart. They begin walking towards each other, Alan at a constant rate of 5 feet per second, Alissa at a rate of 4 feet per second. 17 Alan and Alissa reach one another when the distance (in feet) between them is zero. Thus, we let d = 0 and solve for t: 0 = 140 – 9t –140 = –9t t = 15.556. Therefore it will take Alan and Alissa 15.556 seconds to reach each other.

18. Constant Rate of Change Item Buddy runs straight down the track at a constant rate of change, traveling a distance of 15 meters every 2 seconds. How long will it take Buddy to run 5 meters? 10 meters? 60 meters?120 meters? How far did Buddy travel over any 3 second interval? 1 second interval? 10 second interval?

19. Understanding Rate of Change What does it mean to understand constant rate of change?

20. What does it mean for two quantities to change together at a constant rate of change? The changes in the two quantities are proportional  As the change in distance and change in time CHANGE TOGETER, the ratio of these changes remains constant The change in distance is always some constant times as large as the change in time

21. Given that the x and y axes have the same scale, what is the approximate rate of change of y with respect to x for the linear functions whose graphs are given?

22. MODELING THE CO-VARIATION OF QUANTITIES IN FOUR FUNCTION CONTEXTS: Problem context (a box), a formula, a table (numeric) and a graph 1. a. Create an open box (without a top) by: i) cutting four equal-sized square corners from an 8.5 by 11-inch sheet of paper and ii) folding up the sides and taping them together at the edges. Define a function that expresses the volume of the box in terms of the length of the side of the square cutoout Copyright © 2009 Carlson and Oehrtman 22 W#1

23. Occurred after he had created a diagram and determined V=13*11*x  Travis appeared to focus on the object of the box, but not the attributes of length and width of the box  The length and the width of the box had fixed measurements. The various references to length were not distinct in Travis’ mind. What are the dimensions of a box with the maximum volume that can be formed from cutting equal squares from the corner of an 11” X 13” piece of paper?

24.  Travis appeared to conceive of the various quantities (attributes of the box that could be measured) and how they change together  Tavis defined and conceptualized variables and then constructed a formula (for the box’s volume in terms of the cutout length) that was meaningful to Travis. Shifting the Instructional Approach After Instruction that supported students in imagining relevant quantities and how they change together …

25. Process of Developing and Refining Pathways Interventions Review research to provide the theoretical grounding --Cognitive Models of Learning Key Ideas --Develop Curriculum and Aligned Assessments -Models of how MKTP Develops -Implement -Study Implementation Revise Frameworks Design Implementation (Revise)

26. Teachers’ MKT for teaching concept X Teacher Goals for Student Learning Teacher Beliefs Variables that Impact Teachers and Teaching

27. Learning Goals for Students: (Influenced by teachers’ MKT, beliefs, etc.) Teaching & Reflection on Teaching Materials Teachers Use/Design A Framework For Supporting Shifts in Teaching and Student Learning

28. Learning Goals for Students Teaching & Reflection on Teaching Materials Teachers Use/Design The Didactic Triad: A Framework for Supporting Shifts in Teaching and Learning Looking at each in relation to the others

29. Learning Goals for Students Teaching & Reflection on Teaching Materials Teachers Use/Design The Didactic Triad Looking at each in relation to the others

30. Learning Goals for Students: (Influenced by teachers’ MKT, beliefs, etc.) Teaching & Reflection on Teaching Instructional Materials The Didactic Triad Looking at each in relation to the others

31. Learning Goals for Students: (Influenced by teachers’ MKT, beliefs, etc.) Teaching & Reflection on Teaching Materials You Use/ Design The Didactic Triad Your understanding of the content Your understanding of students’ mathematics Your understanding of the content Your understanding of students’ mathematics Your understanding of the content Your understanding of students’ mathematics

32. Curriculum that is facilitating shifts in teachers’: i) MKT, ii) teaching practices, iii) students’ learning  One Example—Project Pathways: Conceptually Oriented Materials that facilitate teacher development of mathematical and pedagogical knowledge, and ongoing reflection on student learning Pathways materials include instructional tools such as: i) Carefully scaffold in-class activities ii) Detailed teacher notes and videos; v) Animations to support student learning of key concepts iv) PPts to support in-class discussions v) Formative assessments

33. Laying the Foundation for Success in Calculus (and Beyond)  Tasks support students in developing foundational reasoning abilities and understandings  Classroom tasks engage students in meaning making and promote the development of students’ problem solving abilities.  Teacher Support Tools Help Teachers Acquire “Content Knowledge for Teaching”  PowerPoint Slides with Dynamic Animations and Links to Applets Help Novice Teachers Engage Students in Conceptual Conversations  Worksheets (with detailed solutions) help teachers promote critical reasoning abilities and understandings in students

34. Classroom Rules of Engagement: Individuals are expected to: Persist in sense making  Attempt to make logical connections Exhibit Mathematical Integrity  Don’t pretend to understand when you don’t  Base conjectures on a logical foundation Students are expected to express their thinking and speak with meaning about their problem solutions Students are expected to listen to the meaning being conveyed by their peers and ask questions when the meaning is not clear

35. Comparison Data: Precalculus Concept Assessment Instrument  A 25 item multiple choice exam  Based on the body of literature related to knowing and learning the concept of function (Carlson, 1998, 1999, 2003; Dubinsky et al., 1992; Monk,1992; Thompson and Thompson, 1992; Thompson, 1994; Sierpinska, 1992)  Validated and refined over a 12 year period

36. Predictive Potential of PCA How does PCA score relate to course grade in calculus?  83% of 277 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.

37. Student Responses A ball is thrown into a lake, creating a circular ripple that travels outward at a speed of 5 cm per second. Express the area A of the circle in terms of the time t in seconds that have passed since the ball hits the water. a)A = 25πt (21%) b)A = πr 2 (7%) c)A = 25πt 2 (17%) d)A = 5πt 2 (34%) e)None of the above (20%) WHY ARE WORD PROBLEMS SO DIFFICULT FOR STUDENTS?

38. 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 Post Mean PCA Score at 5 universities Pre: 8.4; Post: 13.6

39. Precalculus: Pathways to Calculus Summary of PCA pre- and post-test Academic Yr. Pre-test Post-test 2009-108.215.2 2010-119.115.8 2011-129.715.9 2012 –1310.216.3