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Dieudonne (1992) wrote in his inspiring book "Mathematics-The music of reason":
Nothing which is taught at a secondary school was discovered later than the year 1800.
He added that new mathematical objects have been createdthe far more severely abstract nature of these objects (since they are not supported in any way by visual pictures) has deterred those who did not see any use in them.
In order to teach central ideas in analysis, ideas that are rather abstract, maybe we have to support our teaching by visual pictures (using dynamic graphics).
This presentation is focused on the question: how animations might affect the teaching and learning of some subjects in Mathematics?
Some examples are taken from a research that checked students transition from visual interpretation of the limit concept to formal reasoning. In the laboratory, the students used the Mathematica software (Wolfram Research) and a Classnet system that permits transmitting the content of the screen of each computer to all the computers in the classroom.
First, what is an animation?
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 (Gray and Glynn, Exploring Mathematics with Mathematica (1991).
We will describe some usages of animations:
Animation and Generalization
Animations can be used in order to explore HYPERLINK "parabola.html" families of functions with parameters. For example, we can see the periods of HYPERLINK "period.html" Cos (n x) for different values of n.
Animation and Visualization of Dynamic Processes
Animations can be used in order to visualize the process of differentiation or convergence processes.
By applying the symbolic manipulation capability of Mathematica in order to generate animations, the students are doing algorithmic reasoning.
In the case of dynamic processes, it is not always easy to produce the animation command. This is especially true in examples in which the animation is obtained with the domain as a variable. The effort is rewarding.
As Knuth (1974) mentioned: the attempt to formalize things as algorithms leads to a much deeper understanding
We mention here some important advantages of applying animations while learning Mathematics.
We were interested in the important role played by the usage of animations in students process and object understandings, and found that under some conditions
using animations could reinforce some existing misconceptions or generate new misleading images.
The teacher has to be aware of the complexity of the instrumentation process with regard to animations.
Following are some examples.
Example 1 The process of Differentiation
The derivative is defined as EMBED Equation.3 EMBED Equation.3
By means of HYPERLINK "derivative.html" animation, the students visualize the process EMBED Equation.3 for decreasing values of h (a finite number of discrete values).
As an example, EMBED Equation.3 and the students visualize the process EMBED Equation.3 for decreasing values of h from 0.3 to 0.05 with step 0.05. In the last plot (h = 0.05) the derivative EMBED Equation.3 and the quotient difference EMBED Equation.3 seemed to coincide. This could reinforce the misconception: EMBED Equation.3 can be replaced by EMBED Equation.3 for EMBED Equation.3 very small (how much small?)
Mathematica might be used in order to overcome some of the misleading images:
Graphically we can plot the difference EMBED Equation.3 - EMBED Equation.3 for EMBED Equation.3 or / and numerically to calculate values of the difference EMBED Equation.3 - EMBED Equation.3 for different x.
Example 2 The transition from process to object understandings of the limit concept
Some animations could help to reinforce the intuitive dynamic conception of limit as the result of a process of motion. It might be a source of difficulty when the students will be required to grasp the precise definition of limit.
We observed the fact that former animations were present in the students minds when they were generating new animations, and sometimes it was a source of conflict.
Our main mathematics subject was approximation of functions by polynomials. The students have seen by means of HYPERLINK "sin-1.html" 2- dimensional animations that the different approximating polynomials shared more ink with the function when the degree of the Taylor polynomial increased. Later, they have seen an HYPERLINK "remainder.html" illustration of EMBED Equation.3 where EMBED Equation.3 is the error ( EMBED Equation.3 ) which is done when we take the approximating polynomial EMBED Equation.3 instead of f(x) (the Lagrange Remainder).
We observed the upper estimate of the absolute value of the error.
In the example illustrated in the lab the function was f(x) = sin(x).
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 HYPERLINK "matan.html" cos(2 x).
In order to overcome the misleading image: if EMBED Equation.3 as EMBED Equation.3 is monotonically decreasing for every n the students had to revise the different dynamic images that produced the conflict. (It might be a good idea for the teacher to initiate such a conflict by giving the students an example of a convergent decreasing sequence that is not monotonically decreasing.
Example 3 The discrete/ continuous strand
We noticed that 81% of the students (N = 84) were helped by animations in order to visualize the process described by the formal definition of limit, and that they were able to translate visual pictures to analytical language: We are given EMBED Equation.3 find the appropriate EMBED Equation.3 . They had no difficulty in proceeding step by step through a discrete sequence of EMBED Equation.3 finding the appropriate EMBED Equation.3
To every EMBED Equation.3 there is EMBED Equation.3 (sequential thinking).
Example 4 Difficulty to reverse the order worked in the laboratory
The dynamic graphics produced by the animations were present in the students' minds even when the computer was turned off. The students remembered also the order in which they worked in the laboratory - beginning with domain and finding the error. The limit definition begins with EMBED Equation.3 It was difficult for them to reverse the order!
This source of difficulties is represented in students expressions:
EMBED Equation.3 is not dependent on EMBED Equation.3 , EMBED Equation.3 is dependent on EMBED Equation.3 .
The above few examples demonstrate that the students have to be confronted with different well-chosen examples (sometimes with conflicting images) in order to bring them to interact with the dynamic graphics and have control on the dynamic representations. Actions on the dynamic representations could aid the students in developing their own reasoning. Only by carefully instruction we can help the students progress from process to object understanding.
References
Dieudonne, J., 1992, Mathematics- The Music of Reason, Springer-Verlag, Berlin.
Knuth,D.,E., 1974,Computer Science and its relation to Mathematics, AMS- Monthly 81
Ivy Kidron HYPERLINK "mailto:ivy.kidron@weizmann.ac.il" mailto:ivy.kidron@weizmann.ac.il
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