Animation and Epistemology
Ivy Kidron
ivy.kidron@weizmann.ac.il
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 created…the 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 families of functions with parameters. For
example, we can see the periods of Cos (n x) for different values of n.
Figure 1 Cos(nx) for n=1, 2, 3, 4.
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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
f(x  h)  f(x)
h0
h
f ' (x)  lim
By means of animation, the students visualize the process
f(x  h)  f(x)
 f ' (x) for
h
decreasing values of h (a finite number of discrete values).
As
an
example,
f(x  h)  f(x)
 3 x2,
h
f(x)  x 3
and
the
students
visualize
the
process
- 2  x  2 for decreasing values of h from 0.3 to 0.05 with step –
0.05. In the last plot (h = 0.05) the derivative 3 x 2 and the quotient difference
2
f(x  0.05)  f(x)
0.05
seemed to coincide. This could reinforce the misconception: lim
be replaced by
y
for Δx very small (how much small?)
x
y
can
Δx  0 x
Figure 2: Visualizing the process
f(x  h)  f(x)
 f ' (x) for decreasing values of h from 0.3 to 0.05
h
with step – 0.05.
Mathematica might be used in order to overcome some of the misleading images:
Graphically we can plot the difference 3 x 2 -
f(x  0.05)  f(x)
0.05
numerically to calculate values of the difference 3 x 2 -
for
f(x  0.05)  f(x)
0.05
- 2  x  2 or / and
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 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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H
LH
L
Sin x
H
LH
L
and P 1 x
Sin x
H
LH
L
Sin x
H
LH
L
and P 3 x
Sin x
H
LH
L
and P 7 x
Sin x
and P 5 x
H
LH
L
and P 9 x
Sin x
and P 11 x
Figure 3 Sin(x) and the approximating Taylor polynomials of degree 1, 3, 5, 7, 9, 11
The students computed the expansion of sin (x) around x = 0 up to exponent 5. The error
( f(x) - Pn (x) ) - the remainder of Lagrange - is
f (6) (c) x 6
for some c value between 0 and
6!
the current x value. The absolute value of the error as a function of x and c with
π x  π ,
- π  c  π was plotted. Because the c value in
error
- π  c  π that
analysis : n = 5
corresponds to the exact error is an unknown number, the
students were requested to look at all pairs (x, c) such that
 π  x  π , - π  c  π.
1
Z
0.5
The following 3-dimension graphics represents the
error (in fact, an upper estimate on the error) as a function
of the two variables x and c.
2
0
0
-2
0
C
-2
X
2
Figure 4 the error as a function of x and c
Later, they have seen an illustration of R n ( x)  0 where R n ( x) is the error ( f(x) - Pn (x) )
n
which
is
done
when
we
take
n= 3
the
approximating polynomial Pn (x) instead of
n= 5
4
Z 3
2
1
0
f(x) (the Lagrange Remainder).
2
0 C
-2
0
We observed the upper estimate of the
4
Z 3
2
1
0
X
2
0 C
-2
0
-2
X
2
-2
2
n= 7
n= 9
absolute value of the error.
In the example illustrated in the lab the
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Z 3
2
1
0
function was f(x) = sin(x).
2
0 C
-2
0
X
-2
2
4
Z 3
2
1
0
2
0 C
-2
0
X
-2
2
Figure 5 "animation" illustrating R n (x)  0 in the case f(x) = Sin(x)
n 
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In sin(x)’ example, when n was increasing the error was steadily decreasing for every n.
n=3
this was not the case for other examples
such as cos(2 x).
Z
R n ( x)  0 as
Z
C
C
X
R n (x) is monotonically
n=7
Z
In order to overcome the misleading image:
if
n= 5
C
X
n=9
X
n= 11
n=13
decreasing for every n the students had to
Z
Z
revise the different dynamic images that
C
C
X
produced the conflict.
Figure 6
Z
X
C
X
"animation" illustrating R n (x)  0 in the case f(x) = Cos(2 x)
n 
(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 ε1 , ε 2 , ε 3 ,.... find the
appropriate δ1 , δ 2 , δ 3 ,.... ”. They had no difficulty in proceeding step by step through a
discrete sequence of ε1 , ε 2 , ε 3 ,.... finding the appropriate δ1 , δ 2 , δ 3 ,....
“To every  n there is δ n “(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 ε … It was difficult for them to reverse the order!
This source of difficulties is represented in students’ expressions:
“  is not dependent on ε , ε is dependent on  ”.
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.
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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
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