Vol.2, No.8, 915922 (2010) Natural Science http://dx.doi.org/10.4236/ns.2010.28113 Copyright © 2010 SciRes. OPEN ACCESS Semantic model and optimization of creative processes at mathematical knowledge formation Victor Egorovitch Firstov Department of Mechanics and Mathematics, Saratov State University, Saratov, Russia; firstov1951@gmail.com Received 26 February 2010; revised 24 April 2010; accepted 29 April 2010. ABSTRACT The aim of this work is mathematical education through the knowledge system and mathemat ical modeling. A net model of formation of ma thematical knowledge as a deductive theory is suggested here. Within this model the formation of deductive theory is represented as the de velopment of a certain informational space, the elements of which are structured in the form of the orientated semantic net. This net is properly metrized and characterized by a certain system of coverings. It allows injecting net optimiza tion parameters, regulating qualitative aspects of knowledge system under consideration. To regulate the creative processes of the formation and realization of mathematical knowedge, stochastic model of formation deductive theory is suggested here in the form of branching Markovian process, which is realized in the corresponding informational space as a seman tic net. According to this stochastic model we can get correct foundation of criterion of opti mization creative processes that leads to “great main points” strategy (GMPstrategy) in the pro cess of realization of the effective control in the research work in the sphere of mathematics and its applications. Keywords: The Cybernetic Conception; Optimization of Control; Quantitative and Qualitative Information Measures; Modelling; Intellectual Systems; Neural Network; Mathematical Education; the Control of Pedagogical Processes; Creative Pedagogics; Cognitive and Creative Processes; Informal Axiomatic Theory; Semantic Net; Net Optimization Parame ters; The Topology of Semantic Net; Metrization; the System of Coverings; Stochastic Model of Cre ative Processes at the Formation of Mathematical Knowledge; Branching Markovian Process; Great Main Points Strategy (GMPStrategy) of the Crea tive Processes Control; Interdisciplinary Learning: Colorimetric Barycenter 1. INTRODUCTION “The book of nature is written by the mathematical lan guage” [1]. This Galilei`s manifesto determined the me thodology of natural science development on the basis of observations and experiments, the result of these obser vations and experiments are interpreted within the frames of corresponding mathematical model. The latter implicitly assumes the mechanism of science integration thus, this methodology of development have been suc cessfully realized during the last century beginning with famous Newton’s “Mathematical Priciples of Natural Philosophy” (1687) [2]. The reason of this success is easily explained by L. assertion in his book [3]:” The theory is more practical, it is quan tumessence of experiment.” One can understand in the sence that quantumessence is the system of postulates, which is in the foundation of theoretical model of the object under consideration. In 1948 N. Wiener [4] had stated main cybernetics’ positions. He had done it on C. information theory (1948) [5]. According to this theory, any process of control system represents some transformation of systemic information. Information is basis notion of cy bernetics and has an abstract quantitative measure. It allows to measure both material and nonmaterial as pects of the object under consideration. Thus, possibili ties of mathematical modelling have extended including processes in social and humanitarian fields [6]. With the appearance of computers in the middle of the XX century, a separate direction connected with the realization of intellectual systems (IS) has been formed in cybernetics [7,8]. Accents in this case focused on learning of cognitive processes. In 1980s this led to the building of learning ACTtheory [9] and neural network associative model of J. Hopfild [10]. Modern level of IS development represents adaptive learning systems [11]. A half century experience of IS development shows that their possibilities are determined by mathematical model parameters, which are in the ground of IS. Thus, the questions of IS efficiency lead to the parameters op
V. E. Firstov / Natural Science 2 (2010) 915922 Copyright © 2010 SciRes. OPEN ACCESS timization of corresponding mathematical models which describe given cognitive processes. Formally, it assumes the investigation of topology of semantic nets which realize given cognitive processes. This investigation allows determining the system of net parameters with the help of which one can influence the knowledge system. Thus, optimal control by cognitive processes is carried out, including creative search and interdisciplinary learning. 2. SEMANTIC MODEL OF INFORMAL AXIOMATIC THEORY Let S = (M; ) is a mathematical structure, where M = {M1;…;Mk} is a system of ground sets, representing main objects of structure S, and = { } is a system of axiom, describing ground relations between them. The informal theory Th(S) of the structure S represented denumerable set, the elements of which are ordered by definite rules of conclusion. One can speak of the theory Th(S) as of the structural space information [12], which is formed within given concluded rules in the form of constructively infinite recursive procedure, that is to say, the space Th(S) is some kind of universe analogue. Theory space Th(S) is given by the digraph , representing the mathematical structure S in the form of semantic net, which realizes the passing of definite ob ject information. At this, the set Th(S) determines the nodes of the object net domain , and its arcs (orientated edges) are given with the help of the com mutator I – set of functions f of following type: f: , (1) where , and symbol means in formal logical consequence of the statement T from . Digraph is appeared by the pair (V; E), where the set of nodes V and the set of arcs E is defined by the expressions: V= Th(S)I; E(Th(S) I) (I ), (2) where is the complement of the system axioms till Th(S), and, without community restrict, the axiom system may be considered independent. For a given digraph , the system of nodes Th(S) represents the sources and, thus, represents of the semantic net, which determines the information space structure of axiomatic theory Th(S). 3. TOPOLOGICAL PROPERTIES OF THE SEMANTIC NET In order to investigate semantic net properties , topological presentations, realized according to two di rections, are used [12]. The first direction determines routes, distances and connection between subject nodes of the semantic net . Let on the digraph (2) the nodes are dis tinguished , which form the sequence of arcs 001 121 ( ;...;):( ; )(;)...(;), n nn Lvvvvv vvvE − ∈ (3) where 11 ( ;)( ;)(;); iii iii vvv ffv ++ = . Thus, the oriented route is given, connecting the node v0 with vn (it means, that the node vn is reached from the node v0), and the nodes vi , , are called inter mediate ones on the route (3), and this fact is expressed in the form of: . The route length (3) and the distance from the node v0 to the node vn are, corres pondingly, determined by correlations:   = n ;   = inf  , (4) where   is the set of all route lengths, connect ing the node v0 with vn. The second direction of semantic net investi gations is carried out with the help of special system of coverings, which is formed by the following way. For arbitrary node T Th(S) the set is determined: U(T)={TiTiT Ti = T, Ti;T Th(S), iN} Th(S), (5) the elements of which represent the nodes, for which the node T is reached on the digraph . The set U(T) is called the sphere of dominating of the node T in the space Th(S), and its power U(T) determines the capacity of the dominating sphere U(T). Capacity U(T) in given case represents topological of information quantity ac cording to N. Rashevsky [13]. The following topological properties are set up for dominating spheres: 1) (6 ) moreover, the equality at is equivalent to the fact, that the nodes are linked by the cycle. 2) If and sets are not linked by implication, then among nodes , at least one of them, is a point of net branching . Let is the set of routes, leading from axioms to the node T. Then with the help of (4), the distance from to T and the diameter of the sphere U(T), are correspondently, determined:   = inf  , d(U(T))= sup  . (7) Within the conception capacity (5) and distance (7), a recursive procedure is determined, for forming the sys
V. E. Firstov / Natural Science 2 (2010) 915922 Copyright © 2010 SciRes. OPEN ACCESS tem of inclusion coverings in the theory structure Th(S). Theorem 1. In the space Th(S) one can always under line a denumerable assemblage of subsets, which form the chain 12 ( )( )( )...,Th SThSThS⊃⊃⊃ (8) thus, the chain of inclusion coverings of the space Th(S) is formed: h(S) … , (9) where h(S) = , { } ( ):(),(;),1;2;... i ThSTTTh SrTii=∈Σ≥ = Thus, algorithm of knowledge generalization is given in the semantic net : if a certain section is chosen in chain (9), then an object sphere of knowledge Th(S) is covered by the system of spheres and in each of such sphere the dominated node presents the generalization of the rest ele ments of sphere . As a result, the spheres are classified by some system of characteristics in order to form an appropriate conception. Accordingly, the property (9) in the process of generalization realizes the principle of “matryoshki”, and as a result the level of conception is gradually increasing. It means that the process of getting knowledge is con nected with the development of intellect. 4. OPTIMIZATION OF THE DEDUCTIVE CONCLUSION ON SEMANTIC NETS Let among the statements T Th(S) there is finite set of positions for which in the net there is the only function with the range of definition Dom { }. This function realizes the in formal logical conclusion: T. (10) If Dom , then each of the nodes similarly (10) has a corresponding Inode and also occurs the result, onevalued following from corresponding positions of predicate universe Th(S) and so on, till we come to the proofs: 1111 12221r :;:;...; :, rr fTfT fTΣ→ Σ→Σ→ ( 11) where 5. Thus, in general case, procedure’s proof of the statement T Th(S) is a partially ordered set B(T), which is made up of predicate nodes, structured through functions (10),(11). It is evident, that B(T) U(T) and, hence, the sphere of dominating U(T) present a union of the all possible proofs` of the statement T. Let is a set of routes from the axioms to the node T in the proof B(T) (10),(11). Then, the length   of proof B(T) presents critical way on B(T) and is determined through recursion:   = max ( ;…; )+1= d(B(T)), (12) where d(B(T))―the diameter of the proof B(T), which is determined similarly (7). Apart from the length (12), the proof B(T) is characte rized by the capacity B(T). Through regulation by these parameters of the conclusion the tasks of the optimiza tion while forming the knowledge, in the form of the theory Th(S) are considered in the net . Let B1(T);…;Bj(T) are different proofs of the statement T Th(S), having the lengths  ;…;  and capaci ties  ;…; . Then, the following tasks of the optimization are determined on the net : B0(T) = opt(B1(T); …; Bj(T))   = min( ; …;  ), (13) B0(T) = opt(B1(T); …; Bl(T))   = min( ; …;  ) (14) Each of the tasks (13), (14) present the optimization of the statement proof T Th(S), accordingly, by the mini mization of its length or capacity. On the whole, these tasks may be considered together. This statement of op timal tasks means the simplification of the proof through reducing the volume of the analysed information. On the whole, it is coordinated with the principles of the infor mation theory. For example, the optimization of Pythagorean theo rem proofs was carried out according to criteria (13); (14), through the analysis of existing variants in school geometry: Euclidean classical proof, the proofs of in dian mathematician Bhascara and vectorial method of proof with the help of scalar product [12]. The parame ters of the proofs of Pythagorean theorem Т in the Euc lid’s, Hillbert’s and Weyl’s axiomatics are given in the Table 1. Table 1. Metric characteristics of main versions of proofs Pythagorean theorem in different axiom system. Proof Axiomatics i     Euclid (IV B.C.) Bhascara (proofI, ~1150) Bhascara (proofII, ~1150)
V. E. Firstov / Natural Science 2 (2010) 915922 Copyright © 2010 SciRes. OPEN ACCESS As it is seen from the Table 1, the optimization of proofs according to the criteria (13), (14) the preference is on the side of the vectorial proof in Weyl’s axiomatics. However, the rise of abstraction level happens in the teaching process, which is in the inverse dependence with didactic principles of accessibility and visual teaching aids. This fact has its reflection in Russian teaching literature on elementary geometry. The analysis of this literature for the period of 17682000 shows, that in the second half of the XIX century, Euclid’s classic al proof is practically nowhere met in school textbooks and proofs are mostly used, because they are more vivid and accessable [12]. 5. FORMALIZATION OF CREATIVE PROCESSES DURING THE ASSIMILATION OF THE SPACE Th(S) AND THE RANGE SIGNIFICANCE OF THE ELEMENTS IN THE NET One can regard creative processes in the training as the generalization of knowledge at the metalevel with the help of heuristics. The link between metalevel know ledge and the known object sphere of knowledge is for mally expressed in the fact, that the space Th(S) is formed with the help of endless recursive procedure and, hence, one can say of current statement , for which , then knowledge metalevel is introduced with the help of partition Th(S) , (15) where is the complement till Th(S), which determines the knowledge metalevel in the form of nodes. To these nodes relation of incidence in the net is heurictically established. The partition (15) in the net induces set partition of functional nodes , where is the set of functions, realiz ing the conclusion to metalevel: . (1 6 ) Correlations (15) and (16) present a formalisational description of the knowledge generalization procedure to metalevel in the process of axsiomatical theory Th(S) formation. The optimization procedure of this process is forming in the range of presentation about node signi ficance in the semantic net. Let U(T) is the sphere of dominating of the statement T and B1(T); …;Bn(T) are possible proofs of this state ment, having lengths  ;…; . Let us call the quantity min ( ;…; ) (17) some logical distance from axiom`s Th(S) to the statement Т. In fact, logical distance coincides with algorithmical determination of information quantity according to A.N. Kolmogorov [14]. Formally, any significance presents a partial order in the space Th(S) and it is given in the form of domination relation according to Pareto:     D( ;T ) D( ; ), (18) where one of the inequalities is strictly done. In the case of defining (18), the statement Т is considered to be sig nificant than and it means that more important ele ments of the space Th(S) are more influential (the first inequality in (18)), and they are closer to the information system sources (the second inequality in (18)). One can also interpret the significant elements as huge main points of the net , which are closer to its sources. 6. GMPOPTIMIZATION STRATEGY OF CREATIVE PROCESSES IN THE FORMATION OF MATHEMATICAL KNOWLEDGE The reasons of relationships significance (18) is carried out in the language of random processes [15]. As in the process of creative searching the moments of revealing time t of new theory form statements Th(S) are not determined, then its creation presents a random process Th(S;t) with continuous time t 0 and with de numerable set of the states, the moments of transititions between them are distributed in the interval t > 0 by chance. Theorem 2. The random process Th(S;t) is a hetero geneous branching Markovian process. Heterogeneous in time, the branching Markovian process Th(S;t) is determined as a process, the transition probability of which satisfy the Kolmogorov Chapman equation with branching condition: Pik( ;t) = 12 1 r1 (;)(;)... (;) i i lr lrlr +...+r =k+i(l) PtPtPt ττ τ ∑ , (19) where Pik ( ;t) is the probability of the condition having i elements at the moment , to the moment t it will con tain k i l =   elements and the evolution Pik( ;t) is described by Kolmogorov system of differen tial equations for the heterogeneous branching Marko vian process [16]. The reason of Paretooptimization procedure by signi ficance criterion (18) means that when the information space theory Th(S) is mastered by chance and is realized by branching Markovian process Th(S;t) then more sig nificant theory statements Th(S) have higher probabili ties of transitions between the process statements Th(S;t) and the result of this is as follows: let is
V. E. Firstov / Natural Science 2 (2010) 915922 Copyright © 2010 SciRes. OPEN ACCESS the solution of Kolmogorov equations under condition (19). That will be enough to examine an example k = i + 1 and then from the branching condition (19) you will get: 11 1 ,1 (1 ) ii i,il lllllll PiPPiPP −− ++ ==− . (20) With the increasing the length of the interval , the meaning Pll 0 and the function Pi,i+1 is increasing in the wide range l i< . Since the meaning i in this case is linked with the quantity Th(S), which cor responds to istate of the process Th(S;t), then it goes without saying, that, under other similar conditions, the dominating sphere U(T) with more capacity  has more chances to widen, because the probability of proof of new statements at forming theory Th(S) is increasing. Thus, the reason of the first inequality in the Pare tooptimization procedure (18) is given. However, the growth of transition`s probability in this model occurs not only with the growth i, but also with the de crease , because in this case the interval length is increasing. The decrease is equivalent to the decrease of logical distance (17) that leads to the reasons of second inequality in Paretooptimization (18). Thus, the conception of the range significance of the statements` theory Th(S) by means of Paretodomination (18) has a sufficient reason. According to this reason, the optimum control by creative processes in training comes to effec tive control of definite random process according to the criterion of significance (18). In this case, the effective strategy of theory formation Th(S) in the creative process leads to the conception of “great main points” in the net or GMPstrategy. This conception intends the investigation, coming out from significant statements T01;…;T0k Th(S), have been chosen according to the net criterion of the significance (18). The inductive hy pothesis Н, given on the base of these statements, has more chances “to be materialized” as a logical generali zation of starting positions. The idea of existing “great main points” is in tune with the modern psychological conceptions in the field of intellectual theory [17]. Ac cording to these conceptions, “great main points”, which have an increased sensitivity to certain semantic influ ence, can qualitatively change the character of under standing` problem situation. 7. GMPSTRATEGY AND HILBERT’S MATHEMATICAL PROBLEMS A vivid illustration of optimization creative search with in the framework of GMPstrategy presents the solution of wellknown problems in the theory of numbers, and also problems (1900) [18]. The chronology of their position and solution is exactly known (Table 2). The analysis shows, if the period of decision , and problems in the theory of numbers makes up hundreds of years, then problems has a unique result, which occurs by 12 less. It is important to say, that the choice of 23 problems from a wide manifold of mathematical ones, appearing in the XIXXX centuries according to report at the II International Congress of Ma thematicians (1900), supposed quite certain “rules of selection”. The sense of them is the realization GMPstrategy. Those problems are interesting, the deci sion of which is possible at a given level of mathemati cal development. These problems can give a further progress to mathematics. 8. THE EXPERIMENT OF GMPSTRATEGY IN INTERDISCIPLINARY LEARNING The possibilities of GMPstrategy are not limited only by the optimization of the mathematical researches but are spread over the interdisciplinary training level within the framework of category. In this case, the author ’s experience shows, that the realization of GMPstrategy usually takes place on the basis of some Table 2. The problematic optimization of the research work in mathemati cs. Period of problem decision, years Last Theorem of 1630 A.Wiles 1995 365 problem: n=p1+p2+p3 n=p1s +…+ pks 20 problems are solved during the period of 19012007
V. E. Firstov / Natural Science 2 (2010) 915922 Copyright © 2010 SciRes. OPEN ACCESS general scientific methodology for example, in the course of a canon or the conception of centrism. The conception of canon development is often ob served in social sciences for example, in the states theory. As a canon here, the ideals of a democracy state justice can be taken into consideration dating back to the repub lic of Ancient Rome (510/509 B.C.) and di alogues (427347 B.C.). In this case, GMPstrategy is started through the axiomatization of the justice canon, on the basis of which the deductive theory is formed. Nowadays, this theory represents a separate sector in mathematics, known as cooperative games, within the framework of which, it became possible to explain the peculiarities of modern democracy [19]. The centrism conception represents a general principle of methodology, the grounds of which dates back to an cient times, having a reflection in antiquity doctrines, traditions and religions. According to Archimedean in terpretation, this principle leads to the conception about center of gravity (barycenter) of a material corpse. On the ground of this principle, modern classical mechanics was formed. GMPstrategy on the basis of barycenter conception is formed by the following way. In 1827, А.Möbius gave mathematical grounds to the barycenter of the material points, that is, he came to the grounds of coordinates, which turned out to be projection ones [20]. Thus, mechanic concep tion of barycenter acquired an abstract interpretation within projection geometry. Furthermore, GMPstrategy can have a lot of variations, for example: 1). With the help of coordinates is given the interpretation of Hardy law in the genetic population [21,22] and on this ground, a power interdisciplinary direction in the form of mathematical genetics has appeared [23]; 2). In the end of the ХХ century, a famous american specialist in the field of psychology of the arts R.Arnheim, propounded a thesis [24], according to which, the psy chology of professional artists differs by rather intuitively sensitive color perception and as a result, they, anyhow, see the distribution of color shades on the pictorial field as a balanced one. This thesis is proved by the experience within the conception about the colorimetric barycenter of paintings, developed in the works [2528], where it is in troduced through mapping: Im F W, (21) which to every point of the pictorial image Im, depend ing on its color F, is brought to conformity with a nonnegative number from W set, which is designated the colorimetric mass of this point. The mapping (21) determines structural colorimetric of a pictorial work. The colorimetric barycenter of the picture, with the posi tion of which the compositional peculiarities of a given picture are connected, can then be calculated on the basis of the wellknown mechanics formulas. As investiga tions have shown [2528], the colorimetrical barycenter is a balancing color point of a pictorial work and this fact has been proved by the formation of barycenter en semble for the large collection of paintings. For this, barycenter ensemble coordinates of pictures are mapped on the unit square and dotting image of ensemble re ceived like this gives an idea about barycenter dispersion relatively to central position. Figure 1 shows such colo rimetric barycenter ensemble from 1174 pictures of painters of the ХХth (white dot indicates the average position of the barycenter for the ensemble). It is evident, that artists representing a variety of compositional ge nres in many cases try to avoid significant deflections from equilibrium of colorimetric mass in the picture. This GMPstrategy within the conception of colorimetric barycenter represents an important component in teach ing mathematics in the field of humanity education [29]. In pedagogics GMPstrategy is carried out on the ba sis of cybernetic conception [30]. Objectively, it is caused by the fact that in the field of didaktikos, peda gogics is based on the theory of cognitive processes, which realize transformation and transfer of information from generation to generation. Cybernetics promotes the development of pedagogical science helping solve aris ing contradictions between its content and form not only through experience, but also within the category of morphism with the help of modeling and optimization of pedagogical processes. The realization of cybernetic conception while making up the fundamental theory of mathematical models in order to regulate effectively cognitive processes in teaching comes from information nature of pedagogical processes. The control in this case can go on through aim influence on quantitative or qua litative aspects of information, realized in the teaching
V. E. Firstov / Natural Science 2 (2010) 915922 Copyright © 2010 SciRes. OPEN ACCESS Figure 1. Colorimetric barycenter ensemble from 1174 pic tures of painters of the XXth century. process. Models, describing the control by cognitive processes through aim influence on quantitative information aspect of corresponding educational content, are formed on the basis of metric functions, which, in this case, have an explicit mathematical form. The procedure of optimiza tion in these models has a universe character, as abstract quantitative information measures are in its ground and the optimal control in this case leads to the improvement of the systems organization of the teaching process through the search of optimal configuration of informa tion nets and flows in this process according to mini mum criterion of information entropy. The class of basic models, the control of which is carried out according to the quantitative measures of information, includes: dialogue, testing, classlesson system of teach ing, organization of group cooperation during the teach ing process and the procedure of subject planning of the teaching process [3134]. Given models make up the basis of a number of information technologies, which have been approved in the teaching process. In particular, at optimization of group cooperation in the teaching process the observations have shown the rise of progress standards of education contingent up to 2025%. Models, describing the control of cognitive processes through aim influence on qualitative (semantic) informa tion aspect of educational content, are formed within ac cepted model of knowledge interpretation. In accepted cognitological model (point 2) the system of knowledge is introduced within informal axiomatic theory in the form of semantic net. This net is properly metrized and is characterized by a definite system of coverings in point 3. It makes possible to introduce net parameters of optimi zation, controlling qualitative aspects of the knowledge system under consideration (points 36). To the class of basic models, the control of which is carried out by in fluence on information semantic aspect, models of for mation of education content, creative pedagogics and realization of teaching on interdicipline level are included. The criteria of such control optimization are formulated in points 3;4 and at optimization of deductive conclusion leads to reducing of volume of analyzed information within criteria (13),(14), and at optimization of creative processes they are controlled by criterion of significance (18). These theoretical statements are confirmed not only by experienced data of Tables 1, 2, but also are effec tively realized by the author in his teaching work while training teachers of mathematics at Saratov State Univer sity after N.G. Chernyshevski (Russia) since 1997. 9. CONCLUSIONS As a matter of fact, GMPstrategy based on Pare tooptimization (18) is treated as a control by definite random process Th(S;t), modeling the creative search while developing the space Th(S). This search turns to be more effective, if it comes from more significant posi tions. This thesis is acknowledged not only by theoreti cal premises (points 36) and by the success in solving Hilbert’s problems (Table 2), but by the whole process of historical mathematical development. It is difficult to overrate Pithagor’s theorem in Euclid’s geometry, De sargues’s and Pascal’s theorems in projective geometry, Euclid’s algorithm in the theory numbers, Viete’s theo rem in the algebra of polynomial, Cayley’s theorem in the theory of groups and so on. Nowadays the significance of specified states is actual as well. Recently, in the work [35] within the framework of GMPstrategy for determining of Pythagorean triples, a special matrix transformation semigroup’s transforma tion of primitive pairs has been made up. Thus, a close link between Pithagor’s theorem and Euclid’s algorithm is stated and original approaches to the solution of the oldest mathematical problem about the power of a set of twin primes numbers. It is important to say that, except the original mathematical results, an effective training to the methods of the mathematical creation takes place within the framework of GMPstrategy. The possibilities of GMPstrategy are not limited only by the control of creative processes in the sphere of ma thematical education. In the morphism’s category, GMP strategy is spread on the level of interdisciplinary teach ing. Thus, the integration of mathematical knowledge in the field of natural and humanitarian sciences takes place. In this case, GMPstrategy realizes a certain variant of common scientific methodology (for example, canon or centrism). This variant is axiomatized and then theoreti cal model of the object or phenomenon is formed. It is important to say, that in the process of interdisciplinary GMPstrategy, effective training to the methods of ma thematical creation takes place. So, the original reflexive conception in the fields of creative pedagogics is rea lized on the basis of GMPstrategy. 10. ACKNOWLEDGMENTS The author thanks Alaitseva V. V., an English teacher of Gymnasia №5 (Saratov, Russia), for the help in translating this work paper. 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