The majority of educational software is specifically designed to make the maximum use of a dialogue between the computer and the student. By carefully graded questions or by multi-choice techniques they are led through the steps of a particular learning process. In the event of an error or difficulty the user is brought back to an earlier stage and then encouraged to try again. By repetition of this technique it is anticipated that the student will gain an understanding of the principles and thought processes involved and will commit the essential facts to memory. Those teachers who have introduced computer assisted learning into their classroom will be familiar with this technique.

The use of a menu at the start of a program is also a well established and helpful feature of most educational software; as is the program which is a half-way house between a computer game and a purely academic exploration. How far learning made easy has inherent dangers is an interesting pedagogic question which, although outside the scope of this article, does raise important philosophical issues which we ignore at our peril.

Effective though this approach can be, I suggest that there is also an important place in the educational field for programs which are written specifically for class demonstration purposes. Such programs have a valuable contribution to make to the learning process and can be designed to be attractive to the class. The demonstration program is especially useful where the associated graphics are somewhat complicated and so virtually impossible to reproduce on a conventional blackboard. In most cases the listing of such a program would be reasonably brief and the running time need not take more than a few minutes of the lesson. One important asset of such an approach is that, when presenting an academic subject or topic, pictures are generally more readily recalled by the student than, say, written statements, laws or mathematical formulae. Further, following a demonstration, the program can be run through again by the individual student just as easily as can the more familiar teaching software.

The argument is sometimes advanced that a demonstration program is merely an alternative to an educational film. To an extent this is true, but a computer program does have certain advantages over the film. For example, the listing can be amended to illustrate important ideas and facets of the theme being presented. Often very simple changes may alter the whole nature of a graphic and the impact of this upon a class can be most rewarding. Again, a menu may be used, just as in the more traditional program, to enable the user to select various paths to understanding the theme of the lesson. Few of us possess the expertise or the money to make our own films, but increasing numbers of teachers are able to write (or, at least, to amend) a computer program.

The listing, Curves, at the end of this article, which is intended for use with a class of students studying mathematics, is an example of a demonstration program. The fundamentals of trigonometry are often difficult to understand. In particular, the reason why the algebraic signs associated with the sine, cosine and tangent functions differ in each of the four quadrants can be hard to grasp. Also, students tend to forget which functions are positive and which are negative in the various quadrants.

The object of the program, therefore, is to clarify the logic and to act as an aid to memory.

Before reading further, type in and run the program, selecting speed 10 and the sine curve.

Now let's examine the graphics in detail. The sine function is plotted as the projection of the rotating unit vector on the 'j' or 'y' direction. Figures 1 and 2 illustrate this for two possible positions of the rotating vector. Since OB is unity, the value of sine(AOB) is numerically the length of BP. In the case of the acute angle the algebraic sign is positive; and for the fourth quadrant angle, PB is in the negative direction of the conventional y axis. In the program, the vector passes through its entire sweep of 360 degrees and the sine curve is plotted by projecting BP parallel to itself to measure off the value of each particular angle-sine as the vector rotates. Two circles are used merely for convenience. The graphic would be too confused if all the angle-sine measures were taken from the one circle.

The cosine could have been treated in a similar manner but, instead, the opportunity is taken to derive the curve by using the idea of the rotating vector leading the sine function vector by 90 degrees. This is achieved by starting the rotating vector along the 'j' or 'y' axis.

The tangent function is plotted from the intercept on the tangent at (1,0) of the unit circle by the rotating (and extended) vector. Figure 2 illustrates this.

This demonstration program could not be produced on a blackboard; and the graphics are visually attractive. Figure 3 shows the final frame for the sine curve option, but the gradual construction of the graphic needs to be viewed to appreciate the theory which the program illustrates.

For class use there should be some preliminary explanation, but this is inherent in most demonstration programs. Discussion after the running of the program should clarify the following.

1. The algebraic signs of the three functions in each of the quadrants. 2. The sine and cosine, in modulus, are less than or equal to unity. 3. The sine of (say) 120 degrees is equal to the sine of 60 degrees. 4. Names: cosine= COmplimentarySINE, tangent is derived from the intercept on the TANGENT to the unit circle.

The listing is written for the model B or Master 128, and is structured with five main procedures. PROCtime enables the selection of a variable rate for the rotating vector. PROCmienu enables the selection of the sine, cosine or tangent function. PROCcircle draws the two circles which are common to each of the options. PROCaxes draws the relevant axes and calls up the particular procedure to plot the chosen trig function. PROCend presents a choice of a re-run, or a different rate or exit.

PROCline, PROCfirst and PROCsecond produce the rotating vector and the lines for projection or for the tangent intercept. The latter is in red to make it easier to follow what is happening.

There are a huge range of concepts that can be supported with this electronic blackboard approach; it would be interesting to see what ideas other readers can come up with.