The world around us has two sides-qualitative and quantitative. Quality is the totality of properties indicating that is a subject than he is. Quantity determines the size, the value of this quality. Quality is traditionally revealed through the description of features, and the quantity is identified with a measure, a number. Analyzing the quality, the researcher determines which class of already known phenomena belongs to this and what is its specificity, then establishes causal relationships between the phenomena. The task of quantitative analysis is to measure and account for the identified properties. Resolving it, looking for an answer to the question, what measure of quality is included in the established measure.
Until recently, in pedagogical science is clearly visible empirical part, reflecting the rich material of observations and experiments; there are theoretical generalizations, completing the systematization of the material, but there is no third logical part, characterizing the developed science, — mathematical (structural). It is known that science only reaches perfection when it manages to use mathematics. Complementing the high-quality presentation on the subject of formalized generalizations, pedagogical theory acquires the necessary rigor and sustainability.
On the way of quantitative research of pedagogical phenomena there are many obstacles, and among the main ones — the nature and nature of pedagogical phenomena. They're nonmetric. More precisely, they seem to us to be non-metric, because we do not yet have meters of these phenomena. The classical mathematical apparatus is not adapted for the analysis of phenomena of such complexity as pedagogical. This obstacle is overcome in two ways: by trying to present the phenomena in such a simplified form, which is available for analysis by traditional mathematical methods, or by developing and applying new ways of formalized description. Appearing, new methods immediately attract the attention of specialists.
It is necessary to distinguish two main directions in the use of quantitative methods in pedagogy: the first-for processing the results of observations and experiments, the second — for modeling, diagnosis, forecasting, computerization of the educational process. Methods of the first direction are widely used. Holding the palm of researchers developed a statistical method with known methods, which include:
- registration-identification of a certain quality in the phenomena of this class and counting the number of the presence or absence of this quality (for example, the number of successful and underachieving students);
- ranking-the location of the collected data in a certain sequence (decrease or increase of the recorded indicators), determining the place in this series of studied objects (for example, drawing up a list of students depending on the number of missed classes, etc.);
- scaling is the assignment of scores or other numeric indicators of the studied characteristics, thus achieving greater certainty. There are four main gradation of the measuring scales: the scale items (or nominal);
- scales of order (or rank); interval scales; scales of relations.
Scale items — the "weak", and other designations are used symbolically. In fact, they are the names of any class of objects. Their only mathematical characteristic is whether the object under study belongs to this class or not. Examples of nominal scales can be considered classifications on various grounds — a list of specialties, a list of characteristics of students, the name of the causes of failure, etc.
The ordinal (rank) scales set the order, the ratio of "more" and "less", the General hierarchy. Examples of their application is the ranking of the type "higher growth", "more than five", "less gaps", etc.
"Strong" scales-interval and scale of relations-have all the positive qualities of "weak" scales, but the interval scale provides for certain distances between individual (any two) numbers on the scale, and in the scale of relations, in addition, defined and zero point (reference point). Scale thermometers, voltmeters, of course,"strong".
Modeling is becoming an increasingly powerful transformative tool of pedagogical research. A scientific model is a mentally presented or materially realized system that adequately reflects the subject of research and is able to replace it so that the study of the model allows to obtain new information about the object. Modeling is a method of creating and researching models. Its main advantage is the integrity of information presentation. Hundreds of years pedagogy has been mostly due to the dismemberment of the whole into parts; synthesis as such is almost neglected. Modeling is based on a synthetic approach: they isolate integral systems and investigate their functioning.
The vast majority of the pedagogical models created today refer to didactic phenomena. Educational process where first of all it is necessary to direct a beam of epistemological modeling, are studied in models is clearly insufficient. The reason for this — the incredible complexity of education, hundreds of factors affecting its results, as well as understandable fear of formalization, threatening result in a" deserted " mathematized theory, which will be impossible to apply to real practice.
Modeling in didactics is successfully used to solve important problems of optimizing the structure of educational material, improving the planning of the educational process, management of cognitive activity and educational process, diagnosis, forecasting, design of training.
Modeling is a complex and controversial method. In essence, it serves three useful purposes: heuristic — for classification, designation, finding new laws, construction of new theories and interpretation of the obtained data; computational — to solve computational problems with the help of models; experimental — to solve the problem of empirical testing (verification) of the hypothesis by operating with these or other models. But, despite all its attractiveness, as well as the ability to cover the system as a whole, it is necessary to resort to conditional schemes, to introduce a lot of assumptions. As a result, there are models that have nothing to do with the simulated reality, distorting it.
The mathematization of pedagogy is of the great epistemological potential. It not only eliminates the science of one-sided qualitative description, but also arranges a strict revision of what has been achieved, providing for this objective methods of verification and a better language. For the complete success of formalization, important conditions must necessarily be met: a clear, consistent hypothesis based on scientifically proven positions; the following model, which includes the necessary number of variables; "playing" this model, and then filling it with experimental facts, polished objective measure. This sequence is the logical chain of dialectical transitions from the phenomenon to its mathematical description.