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Depending on methodical aims, educational material in elementary mathematics can be organized in different ways. In this article, we discuss a relation between theoretical and practical material. How this relation is incorporated into training programs? What are the benefits of various ways of organization of educational material?
We can separate out the following ways of organization of practical material:
- Material grouped by theoretical attribute: math problems are selected specially to illustrate the investigated theoretical material, such as a theorem, formula, and definition. In bookshops, this manner of material presentation corresponds to tutorials. A typical name for such book would be "Course of Algebra", "Course of Trigonometry" etc.
- Material grouped by solution methods: math problems are grouped depending on 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.
- The target task collections: problems of certain type collected in one book. These collections are produced and printed prior to a competitive examination when it is apriori known that this examination includes problems of certain types. Such books are typically published by organizations holding competitions. Total amount of such books approaches infinity.
On a certain phase of training, a certain way of material presentation can be most efficient. In fact, for the quality study of elementary math one needs a number of different books. The use of a number of books at once is very inconvenient.
Math books are not an easy reading. One has to work with them, meaning to spend lots of time and efforts to get "used" to them. Thus, if one training program (or several programs with similar control system) combines different approaches to organization of educational material, then the gaining in time "of getting used" is obvious. Another advantage of programs over books is that pressing buttons is much faster and easier than turning pages over. In addition, books are very expensive. If, in addition, the program offers training feedbacks then an advantage of programs over books becomes obvious because no paper book can offer any feedback.
It is clear that technically, different ways of organization of educational material corresponds to different criteria of sorting and sampling. Computer is the best tool for tasks of that sort.
Presently, however, an advantage of training programs over books is not very much pronounced. The reason is not limitations of computer technologies but an adolescent age of training programs.
A short note about technical approaches in the development of training programs in mathematics.
Let us indicate two main approaches, although there are many hybrids as well.
- The "hypertext" approach: a developer writes "pieces" of mathematical text in any editor, and then creates relations between the "pieces". In fact, he creates a specific database. Similar technology is used in help files. An essential defect of this approach is the large size of basic files and, as a consequence, the small total amount of incorporated material. An essential advantage is the simplicity of introduction of various internal relations and a relative simplicity of development.
- The "symbolical" approach: a developer creates "solving" algorithms, and input-output systems of examples entry and solution writing. Sometimes, steps of solution are provided with theoretical substantiation. An essential advantage of this approach is the small size of files and, as a consequence, an opportunity to create large bases of examples covering the large scope of topics. An essential disadvantage is the complexity of development, difficulty in creating various relations and an apparent universality.
Time will show which approach is better. Here, we are interested in how the chosen approach affects the organization of educational material in the training program. From above, we draw the following conclusions:
- The "hypertext" programs are more suitable for presentation of educational material grouped by a theoretical principle, and they allow organizing material by two other attributes. However, it is difficult and currently hardly possible to combine all three attributes in one "hypertext" program. The reason is simple: to make this combination effective, one needs large database of examples. Local computers cannot support such database, whereas the low data transfer rate limits an allocation of the database in the Internet. Besides, the development of an online program working with databases requires considerable investments.
- The "symbolical" programs are naturally suited for organization of material by solution methods. The backbone of these programs is the solving algorithm. Everything else is built around it. These programs are not large. The gained resource can be used for the development of databases and interfaces. Up to now, however, little is done in this direction.
Now let us go back to the main question of this article: is it possible to combine different ways of organization of educational material in one program, at the given level of computer technologies? Sure. On what kind of basis? A "symbolical" environment + large database of educational tasks + various systems of search. Within the "hypertext" approach, this problem may be also solved if a break in the data transfer rate occurs.
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