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The training programs in mathematics are rather young and rapidly developing phenomenon. A non-specialist can hardly estimate the perspectives of this development and see how the programs can actually help in a training process. The words "interactive" (I wonder how the training program can teach without any feedback), "interesting", etc., circulating in the Internet, are not informative. At best, you can guess that the reviewer liked the program, at worst, that he is busy with his own problems and did not see the program at all. The training program is a tool meant to be used for formation of concrete knowledge, abilities and skills, and not a watermelon, which you "like" or "dislike".
It is desirable to have some quantitative and qualitative parameters allowing evaluating capabilities of programs. From my own experience, both in math teaching and in programs development, I shall remark on this question.
Any training program reflects a model "student - teacher" which is too complex for programming. Any developer will inevitably simplify this model. A subsystem of relations, taken from this model as a basis for the program, determines the program's capabilities. This idea allows selecting criteria for evaluation of the program. For example, if the program does not include formulas as a separate structure or if the program's vocabulary consists of 50 phrases, then at any conditions this program cannot:
- Give a substantiation of solution
- Build up knowledge basic for studying internal mathematical cause-and-effect relationships
- Build up knowledge, abilities and skills necessary to solve examples independently
This does not mean that such program is "bad". This means only that this program does not offer the described user options. Surely, many users do not need these options at all. We do not intend to reveal the "best" training program. We want to specify certain quality criteria, in order to help the user to choose the program that suits him best.
We propose the following line of program evaluation, allowing specifying the desired criteria:
- Find out characteristics representative of technical capabilities of a program. These characteristics are easily deducible.
- Select characteristics reflecting pedagogical capabilities and determine the program's parameters necessary for their realization. This can be done with the help of the relation tables below.
- Relate training capabilities of the program to characteristics mentioned above.
This scheme helps to determine, based on technical parameters of the program, what abilities and skills this program can train and what abilities and skills it cannot train at all.
As an example, let us construct the discussed criteria for the programs of the complex EM Teachline. Below, shown are the lists of characteristics; the relation tables can be found under the references.
Technical parameters
Group of characteristics reflecting pedagogical potentials
Types of training capabilities
We did not trace all relations in the relation tables. There are internal relations between the accentuated knowledge, abilities and skills. For example, the absence of knowledge of formulas results unambiguously in the absence of problem-solving skills. These relations are accounted for if knowledge is acquired in the proper order. The "natural" order corresponds to that in the list, although not without deviations, as it is usual in pedagogics. There are numerous books and papers expanding on this topic. For the user, it is enough to understand: if the program is not capable to build up knowledge related with internal cause-and-effect relationships, then all talks that this program can teach problem-solving skills are nonsense.
All considered relations are necessary, meaning that if a relation is absent, then in no way the program can realize a corresponding training function.
The described characteristics and conditions allow evaluating potentials of any training program promptly, without any special knowledge in pedagogics or programming. Just find out general properties of a program, use the above-mentioned relations and make your own conclusions about training potentials. In short: if an input (technical parameters) lacks this and that parameter, then for sure an output (training capabilities) will lack this and that user option. Finally, the user has to decide: does he need the absent training options or not.
The concluding remark. The above criteria for evaluation of training capabilities of programs are certainly not unique or exhaustive. Any other team of developers can offer another system of criteria related to their understanding of the model "student - teacher" and how this model is represented in their training program. Given all subjectivity of developers, I am sure that their evaluation of their own "child" is always better than any evaluation by experts "in general questions".
Vladimir V. Kukoverov
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