Ivy Kidron ivy kidronweizmann ac il Important role

Ivy Kidron ivy. kidron@weizmann. ac. il

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.

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

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

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

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)

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

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

in red in green Click to view the animation

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

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

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.

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

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

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

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

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

Former animations were present in the students’ minds when they were generating new animations, and sometimes it was a source of conflict. 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. 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.

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

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

"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.

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

(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. 68% expressed the formal definition: to every positive number , there is a positive number such that…

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