Description | Aim | Process | Evaluation | Contact

Description

Fractals are useful for modelling (for example in engineering) and are particularly suited to modelling nature. The difficulty in teaching the mathematical concepts of chaos and fractals is that the concrete realisation of the maths needs to be seen and manipulated. This has previously been too complex for students to attempt. This course utilises computer technology to provide an experiential learning environment. The students can get immediate feedback on their understanding by generating their own fractals.

Aims

The original aim was to teach fractals. The question that arose immediately for me was - `how do you make an extremely complex subject both understandable and enjoyable?'

Process

This is a third year course. Lectures are given in Aldus Persuasion, as there is a good deal of imagery in the lecture material. This is obviously important here as fractals are visually intricate sets. Regular sessions (one every three weeks) are held in the computer lab where the students work on their own programs. Simple Pascal programs (it doesn't have to be Pascal, Turtle Logo would do, for instance) have been written and are run using CodeWarrior software, along with written instructions. Students are motivated to get both the maths and programming right. If either their mathematical understanding or the programming are incorrect, the fractal construction will fail. Students are encouraged to experiment, and are assured that mistakes in this kind of endeavour are an integral part of learning. While it is true that programming is difficult for some students, the level required here is very simple, and as an exercise it is excellent for logical thinking. Following is a segment of the program for generating a Twin Christmas Tree fractal, along with a simulation of the fractal as it would be if the input was correct.

program TwinXmasTree; {Constructs Twin Christmas Tree}

uses
TurtleGraphics, Types, QuickDraw, QuickdrawText, Fonts, OSUtils, Events;

var
Order: Integer;
Color: RGBColor;

procedure Twin (Size: Real; Level: Integer); {change of name here}

begin {We now make changes in this part of the program; compare this with
the last Sierpinski program. We place the three 'blueprint' squares
in different orientation}
pu;
fd(Size/2);
lt(90);
pd;
Twin(Size / 2, Level - 1);
{The iteration takes place in the position of Square 1}
pu;
fd(Size / 2);
rt(180);
pd;
Twin(Size / 2, Level - 1);
{The iteration takes place in the position of Square 2}
pu;
lt(90);
bk(Size / 4);
pd;
Twin(Size / 2, Level - 1);
{The iteration takes place in the position of Square 3}
pu;
bk(Size/4);
rt(90); {Now bring the turtle back to its initial position and heading}
fd(Size / 2);
lt(90);
end;

Evaluation

A Student Evaluation of Teaching (SET) found that the subject was generally enjoyed, although some students would have preferred it to be more mathematical. Responses to the question 'what was most useful in this course?' included:

`Lectures, because of the use of computer and screen to ... provide actual running examples of fractals etc.'

`The use of computers to aid teaching; it would have been difficult to see what was going on without them.'

According to SET the students also found the course challenging, the lectures and practicals valuable for understanding the subject, and that the use of teaching materials enhanced learning.

Contact