An educational material, both practical and theoretical, can be arranged in different ways, depending on methodical aims. Practical materials - math examples, answers, solutions - can be grouped by:
- theoretical attribute: math problems are selected to illustrate the studied theoretical material. In bookshops, this manner of material presentation corresponds to tutorials. A typical name for such book would be "Algebra Course", "Trigonometry Course" etc.
- solution methods: math problems are grouped according to methods applied to solve them. In bookshops, this is usually the optional literature. A typical name for such book would be "Method of intervals in tasks", "Method of math induction" etc.
- types of tasks: problems of certain type collected in one book. These collections are typically published by organizations holding competitions. Total amount of such books approaches infinity.
On a certain phase of math training, a certain way of material presentation can be optimal. Therefore, for a quality study, one needs a number of different books.
Using a few books at once is very inconvenient and besides, math books are very expensive. On the other hand, one training program can combine different approaches to organization of an educational material. Technically, different ways of material organization corresponds to different criteria of sorting and sampling. Computer is the best tool for such job. If, in addition, the program offers training options and feedbacks then an advantage of software over books becomes obvious. Presently, however, this advantage is not very much pronounced because of an adolescent age of training software.
A short note about technical approaches to the development of training mathematics software. Let us indicate two main approaches:
- A "hypertext" approach: a developer creates databases of math problems, solutions, formulas etc, and various search systems. An essential defect of this approach is the large size of basic files and the small total amount of incorporated material. An essential advantage is the simplicity of introduction of various internal relations.
- A "symbolic" approach: a developer creates "solving" algorithms, and input-output systems of examples entry and solution writing. An essential advantage of this approach is the small size of files and an opportunity to create large bases of examples with wide subject coverage. An essential disadvantage is the complexity of development, difficulty in creating various relations and an apparent universality.
The "hypertext" programs are more suitable for presentation of educational material grouped by a theoretical principle, and they also allow organizing material by two other attributes. However, combining all three attributes in one "hypertext" program is difficult because this combination requires very large databases of examples. Local computers cannot support such databases.
The "symbolic" programs are naturally suited for organization of material by solution methods as solving algorithms are in the core of these programs and everything else is built around them. Symbolic programs are not large. The gained resource could be used for the development of databases and interfaces. Up to now, however, little is done in this direction.