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Training programs in mathematics are rather young and rapidly developing phenomenon. A non-specialist can hardly grasp the perspectives of this development and see how programs can actually help in the learning process. The words "interactive" (I wonder how the training program can teach without any feedback), "interesting", etc., circulating in the Internet, are uninformative. At best, you can guess that the reviewer liked the program, at worst, that he was busy with his own business 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 training programs. I shall remark on this issue based on my own experience, both in math teaching and in programs development.
Any training program is a reduced form of the model "student - teacher". Any developer will simplify this complex model in his own way. A subsystem of relations, taken from this model as a basis for a program, determines the program's capabilities. This idea allows selecting criteria for program evaluation. For example, if a 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:
- Provide 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 implies only that such program cannot provide the described user options. Surely, many users do not need these options at all. In this article, we do not intend to reveal the "best" training program. We want to specify certain quality criteria, which may 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 a program, which abilities and skills this program can train and which abilities and skills it cannot train at all.
As an example, let us deduce such quality criteria for the programs of the complex EMTeachline. Have a look at the lists of characteristics below; the relation tables can be found under the references.
Technical parameters
Group of characteristics representative of pedagogical potential
Types of training capabilities
Not all relations are traced. In addition, there are internal relations between the accentuated knowledge, abilities and skills. For example, an ignorance of formulas results unambiguously in the absence of problem-solving skills. Such internal 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 usual in pedagogics. There are numerous books and papers expanding on this topic. For the user, suffice it to understand: if a program is not capable to build up knowledge related with the internal cause-and-effect relationships, then all talks that this program can teach problem-solving skills are nonsense.
All considered relations are necessary. This means that if a relation is absent, then in no way the program can offer 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 conclusion 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 option or not.
A concluding remark. The above criteria for evaluation of training 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". 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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