1. ADVANCED MATHEMATICAL THINKING (AMT) IN THE COLLEGE CLASSROOM Keith Nabb Moraine Valley Community College Illinois Institute of Technology March 2009

3. AGENDA Background on AMT Background on AMT Foundations Foundations Diverse Perspectives Diverse Perspectives Classroom Examples Classroom Examples Algebra Algebra Calculus Calculus Differential Equations Differential Equations Challenges facing students and teacher Challenges facing students and teacher

4. FOUNDATIONS What is advanced? What is advanced? Concept image and concept definition Concept image and concept definition Learning Obstacles Learning Obstacles Process/Concept Duality Process/Concept Duality

5. IMAGE & DEFINITION Concept image is defined as “the total cognitive structure that is associated with the concept, which includes all the mental pictures and associated properties and processes” (Tall & Vinner, 1981)

6. LEARNING OBSTACLES Didactic obstacles Didactic obstacles Epistemological obstacles (Brousseau, 1997; Harel & Sowder, 2005; Sierpińska, 1987) Epistemological obstacles (Brousseau, 1997; Harel & Sowder, 2005; Sierpińska, 1987)

7. PROCESS/CONCEPT DUALITY “Instead of having to cope consciously with the duality of concept and process, the good mathematician thinks ambiguously about the symbolism for product and process. We contend that the mathematician simplifies matters by replacing the cognitive complexity of process-concept duality by the notational convenience of process-product ambiguity.” (Gray & Tall, 1994) Dubinsky & Harel, 1992; Harel & Kaput, 1991; Schwarzenberger & Tall, 1978; Sfard, 1991

8. DIVERSE PERSPECTIVES Criteria for AMT Criteria for AMT Linking informal with formal Linking informal with formal Advancing Mathematical Practice: Advancing Mathematical Practice: A Human Activity Professional Mathematician Professional Mathematician

9. CRITERIA FOR AMT Thinking that requires deductive and rigorous reasoning about concepts that are inaccessible through our five senses (Edwards, Dubinsky, & McDonald, 2005) Thinking that requires deductive and rigorous reasoning about concepts that are inaccessible through our five senses (Edwards, Dubinsky, & McDonald, 2005) Overcoming epistemological obstacles (Harel & Sowder, 2005) Overcoming epistemological obstacles (Harel & Sowder, 2005) Reconstructive generalization (Harel & Tall, 1991) Reconstructive generalization (Harel & Tall, 1991) “The concept image has to be radically changed so as to be applicable in a broader context.” (Biza & Zachariades, 2006) “The concept image has to be radically changed so as to be applicable in a broader context.” (Biza & Zachariades, 2006)

10. LINKING INFORMAL AND FORMAL IDEAS “The move to more advanced mathematical thinking involves a difficult transition, from a position where concepts have an intuitive basis founded on experience, to one where they are specified by formal definitions and their properties re-constructed through logical deductions.” (Tall, 1992) “The move to more advanced mathematical thinking involves a difficult transition, from a position where concepts have an intuitive basis founded on experience, to one where they are specified by formal definitions and their properties re-constructed through logical deductions.” (Tall, 1992) Mathematical Idea Analysis (Lakoff and Núñez, 2000) Mathematical Idea Analysis (Lakoff and Núñez, 2000) Concept image and concept definition (Tall & Vinner, 1981) Concept image and concept definition (Tall & Vinner, 1981) Horizontal and vertical mathematizing (Rasmussen et al., 2005) Horizontal and vertical mathematizing (Rasmussen et al., 2005)

11. ADVANCING MATHEMATICAL PRACTICE Horizontal and vertical mathematizing (Rasmussen et al., 2005) Horizontal and vertical mathematizing (Rasmussen et al., 2005) Teaching proof through debate (Hanna, 1991) Teaching proof through debate (Hanna, 1991) Pedagogical tools Pedagogical tools Didactic engineering (Artigue, 1991) Didactic engineering (Artigue, 1991) Computer algebra systems (Dubinsky & Tall, 1991; Heid, 1988) Computer algebra systems (Dubinsky & Tall, 1991; Heid, 1988) Pedagogical content tools (Rasmussen & Marrongelle, 2006) Pedagogical content tools (Rasmussen & Marrongelle, 2006) “Play first, operationalize later” “Play first, operationalize later”

12. THE PROFESSIONAL MATHEMATICIAN “The working mathematician is using many processes in short succession, if not simultaneously, and lets them interact in efficient ways. Our goal should be to bring our students’ mathematical thinking as close as possible to that of a working mathematician’s.” (Dreyfus, 1991) “The working mathematician is using many processes in short succession, if not simultaneously, and lets them interact in efficient ways. Our goal should be to bring our students’ mathematical thinking as close as possible to that of a working mathematician’s.” (Dreyfus, 1991) “To observe and reflect upon the activities of advanced mathematical thinkers is in principle the only possible way to define advanced mathematical thinking.” (Robert & Schwarzenberger, 1991) “To observe and reflect upon the activities of advanced mathematical thinkers is in principle the only possible way to define advanced mathematical thinking.” (Robert & Schwarzenberger, 1991) “Mathematical point of view” or “mathematical way of viewing the world” (Schoenfeld, 1992) “Mathematical point of view” or “mathematical way of viewing the world” (Schoenfeld, 1992) “What comes first to mind is being alone in a room and thinking... I almost always wake up in the middle of the night, go to the john, and then go back to bed and spend a half hour thinking, not because I decided to think; it just comes.” (Paul Halmos, 1990 interview) “What comes first to mind is being alone in a room and thinking... I almost always wake up in the middle of the night, go to the john, and then go back to bed and spend a half hour thinking, not because I decided to think; it just comes.” (Paul Halmos, 1990 interview)

13. CLASSROOM EXAMPLES Algebra Algebra Calculus II Calculus II Differential Equations Differential Equations

14. ALGEBRA Invent your own coordinate system. Explain any advantages and/or disadvantages of this system. Define clearly any letter(s) you are using. Also provide a picture so the context is clear. Invent your own coordinate system. Explain any advantages and/or disadvantages of this system. Define clearly any letter(s) you are using. Also provide a picture so the context is clear.

17. CALCULUS Product rule for differentiation (Brannen & Ford, 2004; Dunkels & Persson, 1980; Maharan & Shaughnessy, 1976; Perrin, 2007) Alternating Series Test

18. STUDENT BENEFITS Nature of mathematics Nature of mathematics “Where do I start?” “Where do I start?” Casting mathematics in a positive light Casting mathematics in a positive light Ownership Ownership The Stevenian Series The Stevenian Series “This is so cool because it’s mine!” “This is so cool because it’s mine!” Multiplicity of Solutions Multiplicity of Solutions Authenticity Authenticity Research oriented Research oriented Motivation Motivation

19. TEACHER CHALLENGES Risk-taking: “Can I do this?” Risk-taking: “Can I do this?” Uncertain outcome Uncertain outcome (Initial) Student resistance/unwillingness (Initial) Student resistance/unwillingness

20. STUDENT FEEDBACK “This drove me nuts. I had trouble stopping thinking about it.” “This drove me nuts. I had trouble stopping thinking about it.” “I have never worked so hard on one problem.” “I have never worked so hard on one problem.” “Hmmm, I’ll never see AST the same way.” “Hmmm, I’ll never see AST the same way.” “Is this like what Newton did?” “Is this like what Newton did?”

21. DIFFERENTIAL EQUATIONS Chris Rasmussen’s Inquiry-oriented Differential Equations (IO-DE) Rasmussen, C., Zandieh, M., King, K., & Teppo, A. (2005). Advancing mathematical activity: A practice-oriented view of advanced mathematical thinking. Mathematical Thinking and Learning, 7 (1), 51-73. Rasmussen, C., Zandieh, M., King, K., & Teppo, A. (2005). Advancing mathematical activity: A practice-oriented view of advanced mathematical thinking. Mathematical Thinking and Learning, 7 (1), 51-73. Rasmussen, C. & Marrongelle, K. (2006). Pedagogical content tools: Integrating student reasoning and mathematics in instruction. Journal for Research in Mathematics Education, 37 (5), 388-420. Rasmussen, C. & Marrongelle, K. (2006). Pedagogical content tools: Integrating student reasoning and mathematics in instruction. Journal for Research in Mathematics Education, 37 (5), 388-420. Rasmussen, C. & King, K. (2000). Locating starting points in differential equations: A realistic mathematics education approach. International Journal of Mathematical Education in Science and Technology, 31 (2), 161-172. Rasmussen, C. & King, K. (2000). Locating starting points in differential equations: A realistic mathematics education approach. International Journal of Mathematical Education in Science and Technology, 31 (2), 161-172. Rasmussen, C., & Kwon, O.N. (2007). An inquiry-oriented approach to undergraduate mathematics. Journal of Mathematical Behavior, 26, 189-194. Rasmussen, C., & Kwon, O.N. (2007). An inquiry-oriented approach to undergraduate mathematics. Journal of Mathematical Behavior, 26, 189-194. Wagner, J.F., Speer, N.M., & Rossa, B. (2007). Beyond mathematical content knowledge: A mathematician’s knowledge needed for teaching an inquiry- oriented differential equations course. Journal of Mathematical Behavior, 26, 247-266. Wagner, J.F., Speer, N.M., & Rossa, B. (2007). Beyond mathematical content knowledge: A mathematician’s knowledge needed for teaching an inquiry- oriented differential equations course. Journal of Mathematical Behavior, 26, 247-266.

22. STUDENT FEEDBACK

24. HOW CAN THESE TASKS BE DEVELOPED? Open-ended and/or unusual exercises Open-ended and/or unusual exercises Study the very content of mathematics Study the very content of mathematics Why do mathematicians use the tools that they use? Why do mathematicians use the tools that they use? Tasks share an element of invention (something new—thinking like a mathematician) Tasks share an element of invention (something new—thinking like a mathematician)

25. Thanks for listening! nabb@morainevalley.edu