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Beware of Blue Boxes: Motivating Techniques in Calculus

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Beware of Blue Boxes: Motivating Techniques in Calculus

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    1. Beware of Blue Boxes: Motivating Techniques in Calculus AMATYC 2009 Robert Cappetta College of DuPage cappetta@cod.edu

    2. 2

    3. 3 Calculus Concerns Poor student work ethic Poor prerequisite knowledge from pre-calculus or AP calculus. Poor retention from one course to the next Inability to solve non-routine problems Inability to transfer knowledge from calculus to physics or engineering.

    4. Why do some students successfully learn mathematics and others do not?

    5. One theory is that good students Internalize mathematical processes Connect mathematical ideas Generalize ideas into other domains Develop a complete, personal understanding of concepts. Reverse mathematical processes.

    6. Question Can an instructor encourage all students to behave like good students?

    7. Dubinsky (1991) Piaget’s concept of reflective abstraction helps students in advanced mathematical thinking. Instructors should “induce students to make specific reflective abstractions” (p.123). Instructors must implement appropriate learning activities to increase the likelihood that students engage in reflective abstraction.

    8. Constructs of Reflective Abstraction Interiorization Coordination Generalization Encapsulation Reversal

    9. Initiates Individual Peer Instructor Curriculum

    10. 10 Working Definitions Interiorization A student performs the steps in a procedure, The student reflects on the procedure and begins to define a concept. Coordination A student examines two different processes and integrates them into a coordinated process that is used to analyze a mathematical concept. Encapsulation A student encapsulates a concept by constructing individual meaning. Encapsulation is the act of personifying a concept. An abstract notion or a collection of abstract notions becomes meaningful to an individual.

    11. 11 Working Definitions Generalization After an individual has encapsulated a notion, it is extended and applied to a wider collection of mathematical problems. Reversal A student constructs a new mathematical notion by reversing the steps of the original notion.

    12. 12 Definition of Derivative

    13. 13 Internalize the process of finding the derivative. Coordinate the notion of function, limit and tangent to develop the notion of derivative. Develop a personal understanding of the concept of derivative. Generalize the notion to differentiable function. Reverse the definition of differentiability to describe a non-differentiable function.

    14. 14 Derivatives of Inverse Functions

    15. 15 Blue Box

    16. 16 Internalize the process of finding the derivative of an inverse trig function. Coordinate the notions of implicit derivatives, triangle trigonometry, Pythagorean Theorem and inverse functions. Develop a personal understanding of the concept of the derivative of inverse trig functions. Generalize the notion to other inverse functions. Reverse the process to develop the trig substitution strategy for integrals.

    17. 17 Find the arc length of a function

    18. 18 Blue Box

    19. 19 Internalize the process of finding the arc length of a function. Coordinate the notions of the Pythagorean Theorem, Riemann sum and limit to develop the formula. Develop a personal understanding of the concept of the arc length of a function. Extend this notion to find arc lengths of parametric functions or vector-valued functions. Reverse the process to construct a function for a given arc length.

    20. 20 Taylor and Maclaurin Polynomials

    21. 21 Blue Box

    22. 22 Internalize the process of finding a Maclaurin polynomial for a function. Coordinate the notions of derivative at a point, function at a point, and equality of functions. Develop a personal understanding of the concept of Maclaurin polynomial. Generalize the notion of a Maclaurin polynomial to a Taylor polynomial. Construct the “blue-box” rule. Reverse the process to construct a transcendental function from a Taylor polynomial.

    23. 23 In mathematics you don't understand things. You just get used to them. John Von Neumann

    24. 24 “One cannot understand… the universality of laws of nature, the relationship of things, without an understanding of mathematics. There is no other way to do it.” Richard P. Feynman

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