Solving methods in training programs

In this article, we will discuss a proportion between the general and special solving methods in training programs. These notes lean upon reviewing of the existing training programs and upon my own experience. Programming is a difficult job and therefore I do not dare to criticize or praise anyone, and thus I do not include any reference to any concrete program.
To make an issue clearer, I shall offer you a few examples.

• In trigonometry, there exists the so-called "universal substitution" which allows reducing any trigonometric equation to algebraic equation. After this substitution is made, the resulting algebraic equation can appear so complicated that it cannot be solved. On the other hand, the initial trigonometric equation may be solvable with the help of special methods.
• There are general solution methods for equations of the third and fourth degree. In some countries, this topic enters into school program. On the other hand, all school programs include special kinds of equations of the third and fourth degree, solvable by special methods. It may seem strange - to learn a set of special methods and types of equations solvable by these methods when there is a universal way guaranteeing the result. Are special methods easier? No, they are not. The significance of special methods lies in the fact that a few narrow classes of high-degree equations admit solutions by these methods whereas general methods of solution of high-degree equations do not exist.
• Did you ever ask yourself why you were so intensively "tortured" at school with trigonometry? In fact, this subject is necessary for astronomers and geodesists only; just 5-6 formulas are used in physics. The point is that trigonometry is full of special methods.

Now, let us draw obvious conclusions:

• General algorithms are not better than special ones and can give no result at all in some cases
• Special methods of solution are necessary for training
• In training programs, a proportion between general and special solving methods should be observed. This proportion was verified by almost a century of pedagogical practice, and there is no serious reason to break it.

Let us consider this question from the programming point of view.
In the article "Organization of material in training programs in math", we discussed the existing approaches in development of training programs. Two main approaches were indicated there, denoted as "hypertext" and "symbolical".
In "hypertext" programs, the proportion between the general and special methods of solution is observed. All decisions on a choice of examples and solving methods are taken by that part of the team, which is responsible for the training functions of the program. I shall remind you that in such programs the solutions are not produced during the work of the program but introduced beforehand. The development scheme can be presented approximately as follows:

• The list of contents is made, i.e. the list of topics to be included
• The list of examples is made
• The examples are solved by those methods, which are expedient from the point of view of training process.
• The next step is the work of a programmer, which is hardly interesting.

The scheme above indicates that the teachers work first. This means that the discussed proportion will not be skewed.
In "symbolical" approach, the situation is somewhat different. The scheme of development can be presented as follows:

• The algorithm of solution is developed
• Examples corresponding to the developed algorithm are created with the help of either the generator of examples or the system of examples entry
• The structure of contents is created
• Next, the teachers are embodied, if they are embodied at all. Often, I have an impression that there are only mathematicians and programmers in such teams of developers.

The examples are solved in a course of program work. The program determines the solution method. It's lapsus calami - in fact, the programmer determines and codes the program. Again, a number of questions arise:

• General algorithm covers a very large body of tasks. Should we choose rather general algorithm?
• If the chosen type of tasks admits both general and special algorithm of solution, which algorithm should we prefer?
• If the chosen type of tasks admits both general and special algorithms and both algorithms have been developed, then which criteria should be used to prefer this or that algorithm in the course of solution?
• The system of search over algorithms should be simple; otherwise, the process of solution can draw out considerably.
• The smaller is the total number of algorithms, the faster the program works. After some threshold, the program begins "to sleep".

These questions make it clear how the program will "skew": the preference will be given to the general methods of solution. After all these developments, a teacher appears on the scene. However, he cannot help with "skewing" anymore. The program cannot think, it just enumerates very quickly. If the program is designed to solve an example of a particular type by a certain method, then time after time it will solve all examples of this type by this method. Suppose, the teacher gives a recommendation: (1) to discriminate a subclass admitting solution by a special method, and (2) to teach the program to solve examples from this subclass sometimes by general method and sometimes by a special method. This recommendation can be taken into account but it is not so easy to implement.
Predominance of general methods is not a big defect of a "symbolical" program. The developers should just write somewhere in the corner of the program's title: "general methods of solution". The program teaches general methods - that is good enough.
I am an advocate of the "symbolical" approach. It is possible to avoid "skewing" described above but you have to sacrifice universality. The "solving" program does not care what method to apply, and the universality is its main feature. In the training program, the needed method of solution should be applied at the needed spot.