ADVANCED ADVANCED MATHEMATICAL THINKING MATHEMATICAL THINKING

(AMT) IN THE COLLEGE (AMT) IN THE COLLEGE CLASSROOMCLASSROOM

Keith NabbKeith NabbMoraine Valley Community CollegeMoraine Valley Community College

Illinois Institute of TechnologyIllinois Institute of TechnologyMarch 2009March 2009

AGENDAAGENDA

Background on AMTBackground on AMT FoundationsFoundations Diverse PerspectivesDiverse Perspectives

Classroom ExamplesClassroom Examples AlgebraAlgebra CalculusCalculus Differential EquationsDifferential Equations

Challenges facing students and Challenges facing students and teacherteacher

FOUNDATIONSFOUNDATIONS

WhatWhat is advanced? is advanced? Concept image and concept Concept image and concept

definitiondefinition Learning ObstaclesLearning Obstacles Process/Concept DualityProcess/Concept Duality

IMAGE & DEFINITIONIMAGE & DEFINITIONConcept imageConcept image is defined as “the total cognitive structure that is is defined as “the total cognitive structure that is

associated with the concept, which includes all the mental associated with the concept, which includes all the mental pictures pictures

and associated properties and processes” and associated properties and processes” (Tall & Vinner, 1981)(Tall & Vinner, 1981)

LEARNING OBSTACLESLEARNING OBSTACLES

Didactic obstaclesDidactic obstacles Epistemological obstacles Epistemological obstacles (Brousseau, (Brousseau,

1997; Harel & Sowder, 2005; Sierpińska, 1987)1997; Harel & Sowder, 2005; Sierpińska, 1987)

PROCESS/CONCEPT PROCESS/CONCEPT DUALITYDUALITY

““Instead of having to cope consciously with the duality of concept Instead of having to cope consciously with the duality of concept and process, the good mathematician thinks ambiguously about the and process, the good mathematician thinks ambiguously about the

symbolism for product and process. We contend that the symbolism for product and process. We contend that the mathematician simplifies matters by replacing the cognitive mathematician simplifies matters by replacing the cognitive

complexity of process-concept duality by the notational complexity of process-concept duality by the notational convenience of process-product ambiguity.” convenience of process-product ambiguity.” (Gray & Tall, 1994)(Gray & Tall, 1994)

Dubinsky & Harel, 1992; Harel & Kaput, 1991; Dubinsky & Harel, 1992; Harel & Kaput, 1991; Schwarzenberger & Tall, 1978; Sfard, 1991Schwarzenberger & Tall, 1978; Sfard, 1991

DIVERSE PERSPECTIVESDIVERSE PERSPECTIVES

Criteria for AMTCriteria for AMT Linking informal with formalLinking informal with formal Advancing Mathematical Practice: Advancing Mathematical Practice:

A Human ActivityA Human Activity Professional MathematicianProfessional Mathematician

CRITERIA FOR AMTCRITERIA FOR AMT

Thinking that requires deductive and rigorous Thinking that requires deductive and rigorous reasoning about concepts that are inaccessible reasoning about concepts that are inaccessible through our five senses through our five senses (Edwards, Dubinsky, & (Edwards, Dubinsky, & McDonald, 2005)McDonald, 2005)

Overcoming epistemological obstacles Overcoming epistemological obstacles (Harel & (Harel & Sowder, 2005)Sowder, 2005)

Reconstructive generalization Reconstructive generalization (Harel & Tall, 1991)(Harel & Tall, 1991) ““The concept image has to be radically The concept image has to be radically

changed so as to be applicable in a broader changed so as to be applicable in a broader context.” context.” (Biza & Zachariades, 2006)(Biza & Zachariades, 2006)

LINKING INFORMAL AND LINKING INFORMAL AND FORMAL IDEASFORMAL IDEAS

““The move to more advanced mathematical thinking The move to more advanced mathematical thinking involves a difficult transition, from a position where involves a difficult transition, from a position where concepts have an intuitive basis founded on concepts have an intuitive basis founded on experience, to one where they are specified by experience, to one where they are specified by formal definitions and their properties re-constructed formal definitions and their properties re-constructed through logical deductions.” through logical deductions.” (Tall, 1992)(Tall, 1992)

Mathematical Idea Analysis Mathematical Idea Analysis (Lakoff and Núñez, 2000)(Lakoff and Núñez, 2000) Concept image and concept definition Concept image and concept definition (Tall & Vinner, (Tall & Vinner,

1981)1981) Horizontal and vertical mathematizing Horizontal and vertical mathematizing (Rasmussen (Rasmussen

et al., 2005)et al., 2005)

ADVANCING ADVANCING MATHEMATICAL PRACTICEMATHEMATICAL PRACTICE

Horizontal and vertical mathematizing Horizontal and vertical mathematizing (Rasmussen et al., 2005)(Rasmussen et al., 2005)

Teaching proof through debateTeaching proof through debate (Hanna, 1991) (Hanna, 1991) Pedagogical toolsPedagogical tools

Didactic engineeringDidactic engineering (Artigue, 1991) (Artigue, 1991) Computer algebra systemsComputer algebra systems (Dubinsky & Tall, 1991; (Dubinsky & Tall, 1991;

Heid, 1988)Heid, 1988) Pedagogical content tools Pedagogical content tools (Rasmussen & (Rasmussen &

Marrongelle, 2006)Marrongelle, 2006) ““Play first, operationalize later”Play first, operationalize later”

THE PROFESSIONAL THE PROFESSIONAL MATHEMATICIANMATHEMATICIAN

““The working mathematician is using many processes in The working mathematician is using many processes in short succession, if not simultaneously, and lets them short succession, if not simultaneously, and lets them interact in efficient ways. Our goal should be to bring our interact in efficient ways. Our goal should be to bring our students’ mathematical thinking as close as possible to that students’ mathematical thinking as close as possible to that of a working mathematician’s.” of a working mathematician’s.” (Dreyfus, 1991)(Dreyfus, 1991)

““To observe and reflect upon the activities of advanced To observe and reflect upon the activities of advanced mathematical thinkers is in principle the only possible way mathematical thinkers is in principle the only possible way to define advanced mathematical thinking.” to define advanced mathematical thinking.” (Robert & (Robert & Schwarzenberger, 1991)Schwarzenberger, 1991)

““Mathematical point of view” or “mathematical way of Mathematical point of view” or “mathematical way of viewing the world” viewing the world” (Schoenfeld, 1992)(Schoenfeld, 1992)

““What comes first to mind is being alone in a room and What comes first to mind is being alone in a room and thinking . . . I almost always wake up in the middle of the 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 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 half hour thinking, not because I decided to think; it just comes.” comes.” (Paul Halmos, 1990 interview)(Paul Halmos, 1990 interview)

CLASSROOM EXAMPLESCLASSROOM EXAMPLES

AlgebraAlgebra Calculus IICalculus II Differential EquationsDifferential Equations

ALGEBRAALGEBRA

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

CALCULUSCALCULUS

Product rule for differentiationProduct rule for differentiation(Brannen & Ford, 2004; Dunkels & Persson, (Brannen & Ford, 2004; Dunkels & Persson,

1980; Maharan & Shaughnessy, 1976; 1980; Maharan & Shaughnessy, 1976; Perrin, 2007)Perrin, 2007)

Alternating Series TestAlternating Series Test

STUDENT BENEFITSSTUDENT BENEFITS

Nature of mathematicsNature of mathematics ““Where do I start?”Where do I start?” Casting mathematics in a positive lightCasting mathematics in a positive light

OwnershipOwnership The Stevenian SeriesThe Stevenian Series ““This is so cool because it’s mine!”This is so cool because it’s mine!”

Multiplicity of SolutionsMultiplicity of Solutions AuthenticityAuthenticity Research orientedResearch oriented

MotivationMotivation

TEACHER CHALLENGESTEACHER CHALLENGES

Risk-taking: “Can I do this?”Risk-taking: “Can I do this?” Uncertain outcomeUncertain outcome (Initial) Student (Initial) Student

resistance/unwillingnessresistance/unwillingness

STUDENT FEEDBACKSTUDENT FEEDBACK

““This drove me nuts. I had trouble This drove me nuts. I had trouble stopping thinking about it.”stopping thinking about it.”

““I have never worked so hard on one I have never worked so hard on one problem.”problem.”

““Hmmm, I’ll never see AST the same Hmmm, I’ll never see AST the same way.”way.”

““Is this like what Newton did?”Is this like what Newton did?”

DIFFERENTIAL DIFFERENTIAL EQUATIONSEQUATIONS

Chris Rasmussen’s Inquiry-oriented Chris Rasmussen’s Inquiry-oriented Differential Equations (IO-DE)Differential Equations (IO-DE)

Rasmussen, C., Zandieh, M., King, K., & Teppo, A. (2005). Advancing Rasmussen, C., Zandieh, M., King, K., & Teppo, A. (2005). Advancing mathematical activity: A practice-oriented view of advanced mathematical mathematical activity: A practice-oriented view of advanced mathematical thinking. thinking. Mathematical Thinking and Learning, 7 Mathematical Thinking and Learning, 7 (1), 51-73.(1), 51-73.

Rasmussen, C. & Marrongelle, K. (2006). Pedagogical content tools: Rasmussen, C. & Marrongelle, K. (2006). Pedagogical content tools: Integrating student reasoning and mathematics in instruction. Integrating student reasoning and mathematics in instruction. Journal for Journal for Research in Mathematics EducationResearch in Mathematics Education, , 37 37 (5), 388-420.(5), 388-420.

Rasmussen, C. & King, K. (2000). Locating starting points in differential Rasmussen, C. & King, K. (2000). Locating starting points in differential equations: A realistic mathematics education approach. equations: A realistic mathematics education approach. International Journal International Journal of Mathematical Education in Science and Technologyof Mathematical Education in Science and Technology, , 31 31 (2), 161-172.(2), 161-172.

Rasmussen, C., & Kwon, O.N. (2007). An inquiry-oriented approach to Rasmussen, C., & Kwon, O.N. (2007). An inquiry-oriented approach to undergraduate mathematics. undergraduate mathematics. Journal of Mathematical BehaviorJournal of Mathematical Behavior, , 2626, 189-, 189-194.194.

Wagner, J.F., Speer, N.M., & Rossa, B. (2007). Beyond mathematical content Wagner, J.F., Speer, N.M., & Rossa, B. (2007). Beyond mathematical content knowledge: A mathematician’s knowledge needed for teaching an inquiry-knowledge: A mathematician’s knowledge needed for teaching an inquiry-oriented differential equations course. oriented differential equations course. Journal of Mathematical BehaviorJournal of Mathematical Behavior, , 2626, , 247-266.247-266.

STUDENT FEEDBACKSTUDENT FEEDBACK

STUDENT FEEDBACKSTUDENT FEEDBACK

HOW CAN THESE HOW CAN THESE TASKS BE DEVELOPED?TASKS BE DEVELOPED?

Open-ended and/or unusual exercisesOpen-ended and/or unusual exercises Study the very Study the very contentcontent of mathematics of mathematics Why do mathematicians use the tools Why do mathematicians use the tools

that they use?that they use? Tasks share an element of invention Tasks share an element of invention

(something new—thinking like a (something new—thinking like a mathematician) mathematician)

Thanks for listening!Thanks for listening!

nabb@morainevalley.edunabb@morainevalley.edu