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Ivy Kidron ivy.kidron@weizmann.ac.il

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Important role played by the usage of animations in students’ process and object understandings Using animations could reinforce some existing misconceptions or generate new misleading images.

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Exploring families of functions with parameters. Process of differentiation Convergence processes

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Taught at the secondary/tertiary interface. Age 16 – 17, N = 84

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The software (Wolfram Research) GraphicsGraphics Animation Symbolic tool Numeric tool Communications possibilities web classnet

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An animation is a sequence of pictures that you flip through quickly. If the pictures are related to each other in some sensible way, you get the illusion of motion. T. W. Gray and J. Glynn “Exploring Mathematics with Mathematica” (1991)

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a = -3, -2, -1, 0, 1, 2, 3 Click to view the animation

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n = 1, 2, 3, 4, 5 Click to view the animation

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in red in green Click to view the animation

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Discrete - Continuous By means of animation, visualization of the process for decreasing values of h (a finite number)

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the misconception: can be replaced byfor very small. The dynamic image produced by the animation could reinforce (how much small?)

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Mathematica might be used in order to overcome some of the misleading images: Graphically we can plot the difference - for Numerically we can calculate values of the difference for different x.

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Sin(x) and We fix n = 5 and we change the domain i is decreasing from 4 to 1 with step -1

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For anyin f(x)’s domain for some c between 0 and. (

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Expansion of f(x) = Sin(x) at x = 0 The error is up to exponent 5: for some c between 0 and the current x value.

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f(x) = Sin(x) n = 3, 5, 7, 9 Click to view the animation

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f(x) = Sin(x) We fix n = 5 and we change the domain i is decreasing from 4 to 1 with step -1

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Former animations were present in the students’ minds when they were generating new animations, and sometimes it was a source of conflict. In sin(x)’ example, when n was increasing the error was steadily decreasing for every n. This was not the case for other examples such as Tami’s example. The students have seen by 2- dimensional animations that the different approximating polynomials “shared more ink” with the function when the degree of the Taylor polynomial increased.

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as a function of x and c n = 1, 2, 3, 4, 5 All the graphs were surprising!

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The way Tami used Mathematica in order to check this surprising situation Tami expanded f(x) in power series and revised the visual pictures of the polynomials approximating better f(x) (in red) as n increases. f(x)=sin(x) cos(x) and the different polynomials which approximate it

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"dynamic" plot of f(x) and the approximating polynomials Tami "it seems that the polynomials better approach the function when n is bigger. It is strange! When we looked at the animation of the upper estimate of the remainder of Lagrange, the result was different: The maximal error was getting bigger when n increased." The surprising effect brought Tami to research the exact meaning of approximating better f(x) as n is increasing.

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N = 84, 81% proceeded step by step through a discrete sequence of finding the appropriateand were aware that the process is infinite. To everythere is (sequential thinking) Discrete - Continuous

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68% expressed the formal definition: to every positive number, there is a positive number such that… (beginning with domain and finding the error). is not dependent on. is dependent on is not dependent on the error, since is fixing the error: the nearer we approach the point x=0 about which the function was expanded, the smaller is the error.

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