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ama_calculus_day_2009 - The Fundamental Theorem of the...

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The Fundamental Theorem of The Fundamental Theorem of the Calculus the Calculus Mike Thomas The University of Auckland
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Overview Overview Why is the fundamental theorem of calculus (FTC) important?? Brief theory Using Geogebra to build understanding of the FTC Classroom materials?
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Integration Integration This is often introduced by using an antiderivative to calculate areas. The question is how do we persuade students that these two processes, antidifferentiation and finding area under graphs by Riemann sums are related in this way?
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The The F F undamental undamental T T heorem of heorem of Calculus Calculus I I If f is continuous on [ a, b ], then the function F defined by is continuous on [ a, b ] and differentiable on ( a, b ), and F ( x ) = φ ( τ 29 δτ α ξ F ( x ) = f ( x )
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The The F F undamental undamental T T heorem of heorem of Calculus Calculus II II If f is a function continuous on [ a, b ], and F is an antiderivative of f then This is what is usually used. f ( x ) dx = Φ ( β 29 - Φ ( α 29 α β
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The Three Worlds of The Three Worlds of Mathematics Mathematics Embodied Symbolic Formal The embodied is where we make use of physical attributes of concepts, combined with our sensual experiences to build mental conceptions. The symbolic world is where the symbolic representations of concepts are acted upon, or manipulated, where it is possible to “switch effortlessly from processes to do mathematics, to concepts to think about.” (Tall, 2004a, p. 30). The formal world is where properties of objects are formalized as axioms, and learning comprises the building and proving of theorems by logical deduction from the axioms.
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Tall’s 3 worlds of thinking Tall’s 3 worlds of thinking
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Tall’s 3 worlds of thinking Tall’s 3 worlds of thinking
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A conclusion A conclusion A curriculum that focuses on symbolism and not on related embodiments may limit the vision of the learner who may learn to
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