ADVANCED MATHEMATICAL THINKING AMT IN THE COLLEGE CLASSROOM

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ADVANCED MATHEMATICAL THINKING (AMT) IN THE
COLLEGE CLASSROOM

• Keith Nabb
• Moraine Valley Community College
• Illinois Institute of Technology
• March 2009

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AGENDA

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

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FOUNDATIONS

• What is advanced?
• Concept image and concept definition
• Learning Obstacles
• Process/Concept Duality

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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)
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LEARNING OBSTACLES

• Didactic obstacles
• Epistemological obstacles (Brousseau, 1997 Harel
  Sowder, 2005 Sierpinska, 1987)

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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
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DIVERSE PERSPECTIVES

• Criteria for AMT
• Linking informal with formal
• Advancing Mathematical Practice
• A Human Activity
• Professional Mathematician

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CRITERIA FOR AMT

• 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)
• 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)

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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)
• Mathematical Idea Analysis (Lakoff and Núñez,
  2000)
• Concept image and concept definition (Tall
  Vinner, 1981)
• Horizontal and vertical mathematizing (Rasmussen
  et al., 2005)

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ADVANCING MATHEMATICAL PRACTICE

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

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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 mathematicians.
  (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)
• 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)

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CLASSROOM EXAMPLES

• Algebra
• Calculus II
• Differential Equations

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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.

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CALCULUS
Product rule for differentiation (Brannen Ford,
2004 Dunkels Persson, 1980 Maharan
Shaughnessy, 1976 Perrin, 2007) Alternating
Series Test
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STUDENT BENEFITS

• Nature of mathematics
• Where do I start?
• Casting mathematics in a positive light
• Ownership
• The Stevenian Series
• This is so cool because its mine!
• Multiplicity of Solutions
• Authenticity
• Research oriented
• Motivation

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TEACHER CHALLENGES

• Risk-taking Can I do this?
• Uncertain outcome
• (Initial) Student resistance/unwillingness

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STUDENT FEEDBACK

• This drove me nuts. I had trouble stopping
  thinking about it.
• I have never worked so hard on one problem.
• Hmmm, Ill never see AST the same way.
• Is this like what Newton did?

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DIFFERENTIAL EQUATIONS

• Chris Rasmussens 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. 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., 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
  mathematicians knowledge needed for teaching an
  inquiry-oriented differential equations course.
  Journal of Mathematical Behavior, 26, 247-266.

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STUDENT FEEDBACK
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STUDENT FEEDBACK
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HOW CAN THESE TASKS BE DEVELOPED?

• Open-ended and/or unusual exercises
• Study the very content of mathematics
• Why do mathematicians use the tools that they
  use?
• Tasks share an element of invention (something
  newthinking like a mathematician)

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Thanks for listening!

• nabb_at_morainevalley.edu