BULLETIN OF THE POLISH ACADEMY OF SCIENCES TECHNICAL SCIENCES, Vol. 64, No. 4, 2016 DOI: 10.1515/bpasts-2016-0103
Teachers’ and students’ motivation model as a strategy for open distance learning processes
O. ZAIKIN1*, R. TADEUSIEWICZ2, P. RÓŻEWSKI3, L. BUSK KOFOED4, M. MALINOWSKA5, and A. ŻYŁAWSKI1 1Warsaw School of Computer Science, 17 Lewartowskiego St., 00-169 Warsaw, Poland 2AGH University of Science and Technology, 30 Mickiewicza St., Krakow, Poland 3Faculty of Computer Science and Information Systems, West Pomeranian University of Technology in Szczecin, 49 Żołnierska St., 71-210, Szczecin, Poland 4Department of Architecture, Design &Media Technology, Aalborg University Copenhagen, A.C. Meyers Vænge 15, Denmark 5Faculty of Management and Economics of Services, University of Szczecin, 8 Cukrowa St., 71-004, Szczecin, Poland
Abstract. Motivation and social aspects as interaction among teachers and students are essential for a successful learning process in an Open Distance Learning environment, and the goal of this paper is to develop an interactive model to manage motivation between the participants of that process. The article contains the concept of developing a motivation model aimed at supporting activity and cooperation of both the students and the teacher. The structure of the motivation model and formal assumptions are presented. The proposed model constitutes a theo- retical formalization of a new situation in which a teacher and the students are obligated to elaborate on a didactic material repository content in accordance with the competence requirements. A mathematical method based on game theory and simulation is suggested. The conditions for applying simulation to the model are analyzed in two simulated cases.
Key words: motivation modelling, open distance learning (ODL), task repository, simulation.
1. Introduction The educational experience is constituted through sustained interaction and communication between and among learners, During the last 20 years, distance learning has changed consid- teachers, and learning objects. In ODL this process can be facil- erably, and online teaching and learning have become routine itated through asynchronous and synchronous communication practices at many universities [1]. The technological develop- and participation through students’ teamwork [6, 7]. ment, at the same time, has given distance education a new Motivation is a key factor in learning in case of the face-to- appeal to students as well as teachers. Due to the increasing face educational contexts [5] as well as in online learning en- popularity of distance education, there is a number of defini- vironments [8]. Traditionally online learners have been under- tions connected to the aspects of teaching and learning in online stood as independent, self-directed and intrinsically motivated educational systems which need to be clarified [2]. [6], but this is not the case. Motivation of the learner is com- Open Distance Learning (ODL) is often used as a synonym plex, multifaceted and sensitive to situational conditions [8]. of e-learning and is nowadays considered the most viable means The implementation of ODL assumes modification of the for broadening educational access. The ODL approach supports traditional organization of learning processes. The special atten- improving quality of online educations as well as advocating the tion is subject to a change in designing the structure and content peer-to-peer collaboration. ODL is giving the learners a greater of the specific educational elements. However, the opportunities sense of autonomy and responsibility for their own learning [3]. of relations between students and the teacher seem to be crucial. Moreover, ODL is one of the most popular method to provide The ODL, combined with the new possibilities and strategies opportunities to meet the needs of a growing and increasingly of distance learning, provides a basis for active collaboration diverse student population [4]. ODL has a number of potential between teachers and students, between students and students, benefits, e.g., the possibility of overcoming the temporal and as well as between students and the university [9]. spatial restrictions of traditional educational settings like space, These new possibilities urge universities to consider the new time, access etc. [1]. Despite all the benefits of ODL, different status of the potential students. This means that, on the one factors have been identified as crucial to the success of online hand, students must be able to achieve knowledge, skills and education. Two of the main factors are motivation and the social competences connected to the goals in their curriculum. On aspect (interaction among students and teacher) of learning [5]. the other hand, the universities have to provide the students with an active role in the learning process, to motivate them to work on an independent study program under the supervision *e-mail: [email protected] of a teacher [10].
Bull. Pol. Ac.: Tech. 64(4) 2016 943 O. Zaikin, R. Tadeusiewicz, P. Różewski, L. Busk Kofoed, M. Malinowska, and A. Żyławski
In this paper we argue that the motivation and social as- The teacher might be aware of the problem and can take pects of learning, such as the interaction among teachers and it into account during direct contact with the students. students are essential for a successful learning process in an 3. The main conventional tool for establishing motivation ODL environment, and our aim is to develop an interactive used by the university systems is knowledge assessment model to manage motivation related to this process. The paper process (tests, exams, grades etc.). proposes the concept of developing a motivation model aimed 4. The design and relevance of didactic materials have at supporting activity and cooperation of both the students and a secondary role in a traditional teacher’s professional the teachers in the process of implementing and using ODL role (research is priority one), but it is still the main en- systems. The structure of the motivation model and formal as- gine in a student’s cognitive learning processes. sumptions are presented. The proposed model constitutes the Cognitive processes are characterized by high entropy. theoretical formalization of a new situation in which a teacher However, during direct contact with a student the traditional and students are obligated to elaborate on a didactic material teacher using his/her professional competence can reduce en- repository in accordance with the competence requirements. It tropy as a result of certain didactic or pedagogical methods, or covers two motivation areas: the teacher’s and the student’s, in other words, be able to manage a student’s cognitive pro- which describe their interests in the knowledge repository de- cesses within certain limits. In this case efficiency of teaching/ velopment. The interpretation of the cooperation between the learning management depends on the intensity of the contacts, teacher and the students is described based on game theory. abilities of students, as well as of the teacher’s pedagogical and A game situation, in which the teacher plays the role of the communicative skills [12]. game leader and the students can play either in an organized Różewski and Zaikin [9] find that the importance of the sub-group or can play individually, is presented. Moreover, this role of didactic materials in ODL is greatly increased, as in cooperation is analyzed (time, workload) based on the proposed ODL teachers have to substitute direct contact and exchange simulation approach. The possibility of assessing several vari- of information between teacher and student. This imposes new ants of teacher’s work in different educational situations by requirements on didactic materials as sources of information not using the simulation model and the simulation software is de- only on learning topics, but also on teaching and learning strat- scribed. The level of students’ motivation can be determined. egies as well as the whole learning process as a combination Finally we can conclude that there is a possibility to compare of communication, interaction and knowledge production [15]. the expected costs that the teacher will bear considering the as- The big challenge for ODL teachers is to design and prepare sumed repository development (working time) with the results the teaching materials for ODL. The needed resources can be achieved in the didactic process (number of students with a high listed: professional knowledge related to the subject, pedagog- level of competence – participating in the repository, number of ical skills, knowledge and experience about teaching within the students with an average level of competence etc.). frame of ODL, time for developing and designing new material The presented paper is an extension of [9] and [19]. Both which will fit the ODL demands. A solution could be to actively refer to the model of motivation. The current version of moti- involve the students in establishing knowledge as a common vation model allows for more accuracy in student-teacher co- process of teaching/learning activities. operation. Ontology is widely used for knowledge representation and there are ontology description languages, and programs to operate them [11]. More detailed design of didactic materials 2. The need for active cooperation of students based on ontology models of subject knowledge is described and teachers in ODL in [13]. Design, preparing and access to the didactic materials in an ODL computer environment (creating e.g. a repository) In terms of significant reduction or absence of direct contact require specialized knowledge and time from the involved between the teacher and students, the work scope and the way teachers. Therefore, the following problems appear when de- they perform in the teaching and learning processes as well as in veloping ODL: how can teachers be motivated to carry out research and educational activities varies considerably. Changes additional work to establish and maintain didactic materials in occur due to various reasons [9]: repository up to date. A repository is defined here as a virtual 1. In a traditional learning situation didactic (teaching) or digital library of didactic materials, assigned for learning of material is a support for accumulating knowledge which a subject or domain area [16]. is seen as necessary for a professional career, and which One way to minimize the problem is to encourage students will remain relevant for a long time. This cannot be to supplement the repository with new materials on the proposed said for newer education dealing with new technologies teacher topic within a pre-defined ontological domain model. such as computer science, management and business, In the ODL situation the students independently could develop productions areas, banking and new media technolo- the ontology of the proposed theme and then make it com- gies etc. [14]. patible with the conceptual structure developed by a teacher. 2. In traditional teaching approach learning might look Performing this task requires extensive use of highly demanding like “the process of intelligent production”, except when operations such as generalization, classification, induction, de- a student finds out that the learned knowledge lacks being duction, and promotes deep mastery of the conceptual apparatus reflected in the context, and he/she has no skills to use it. of the subject area. Coombs et al. [17] discuss the advantages
944 Bull. Pol. Ac.: Tech. 64(4) 2016 Teachers’ and students’ motivation model as a strategy for open distance learning processes of using an ontological approach to learning and teaching. The might result in a significant increase of students’ activity. main advantage of this method is recognized as the development The time costs of the teacher are comparable to the first of analytical skills of the students and a systematic vision of case. But in the second case the students not only acquire wide fields of knowledge objects and their applications. When knowledge, but also competence, and the teachers in re- performing this task, the students involved should be granted turn obtain new improved didactic material to be used in priority when requesting the teacher’s consulting. For students the ontological approach and in game tools. who work together with highly skilled teachers, this activity For evaluation and assessment the teacher can use the lin- promotes self-esteem and motivates self-education activity. guistic scales to evaluate the proposed alternatives. In each Another way to involve students in active learning is to situation the teacher can give preference to any of these alter- create a game situation [18], which requires a connection be- natives, as each of them is connected with its various temporal tween different kinds of knowledge. A distinctive feature of and intellectual costs as well as other preferences. The teacher’s the ontology developed for educational purposes is that the intellectual cost is related to checking and evaluation of the ontology graph contains fragments corresponding to different complex tasks. The more students choose the complex tasks, types of subject knowledge: theoretical (what is this?) and the the more intellectual cost the teachers have to spend for their procedural (how to do this?). Theoretical vertices describe the checking and evaluation. semantics of concepts and their relationships, and the proce- One of the impact factors of the teachers’ choice is the eval- dural ones – test tasks associated with the corresponding path uation to get the final result at the end of the learning cycle in in the graph. The project task, to develop the domain ontology the form of a specific indicators such as: with both types of vertices, can be represented as a game with ● average index of student performance; total win and distribution points depending on student partici- ● index of students’ activity and independence of choosing pation and performing. Overall gain considered is the number types of task, e.g. the traditional way of learning and of vertices of both types, added to the domain ontology graph, testing, the work together with the teacher on the use of which are stored in the repository. The teacher plays the role of the ontological approach for repository development, an the head of the game, students are combined into sub-groups engaging in joint or individual learning projects. or they can play individually. Teacher’s motivation is the pos- ● the index of repository filling by didactic materials is sibility of an extension and updating of the repository by in- created on the principle of “actual domain ontology” plus dependent work of students. Students’ motivation is the joint the base of learning projects presented as “best practices”. study of the subject under supervision of a teacher – live chat, Availability of such data allows for further research, not cognition through competition, stress reduction compared to only in teaching methods, but also in the field of artificial in- traditional testing, choice possibility, etc. The game can be telligence and the development trends of corresponding knowl- carried out remotely [19]. Final assessment depends on the task edge domains. This in turn is a problem raised for discussion complexity, participation in the project, the number of ECTS in numerous papers [20]. points etc. The students will progress from a simple task to The modern system of assessment of the teacher’s profes- more complicated task due to the mechanism of motivation sional activities requires wide participation in international sci- triggered by a teacher. entific research. In the proposed game this can play an important role for evidence based evaluation of the learning process. Scoring system used for evaluation of students’ work is the 3. Interpretation of the motivation model teacher’s prerogative and it can be addressed to the student in learning processes groups in the beginning of the learning process, but a student’s choice will also depend on their personal preferences: In traditional teaching, based on traditional didactic materials, ● leadership skills, the average activity of students’ involvement in the learning ● the possibility of increasing the total average Diploma process is weak, and the accumulation of material in a repos- score, itory for subsequent use is small. However, the teacher’s time ● possibility of participation in individual grants due to costs are lower. In ODL the student can be involved in learning personal contribution to “best practices”. process in two ways: The proposed approaches to learning allow for realizing 1. Independent development and update of the repository motives and preferences of both students and teachers, so we by students. Then the repository is using as a base di- can form two motivation models and examine their mutual in- dactic material for developing knowledge in ODL. The fluence. costs of the teachers time will increase significantly, but Choice making is a cognitive process and it cannot be di- it is justified by the possibility of obtaining results for rectly observed. This means that the relationship between a per- further improvement and modernization of the learning son’s preferences in motives when choosing the result of his/ processes. her actions is very difficult to define. Nevertheless, the choice 2. Student is taught how to use the ontological approach of motive can be evaluated by registering the outcomes. for representation and development of knowledge in A motivation model in the described learning situation can any domain area, and how to organize games as a way be represented as a game scenario, in which the activities of of self-learning in a given subject area. This approach the teacher and students will respond to their preferences in
Bull. Pol. Ac.: Tech. 64(4) 2016 945 O. Zaikin, R. Tadeusiewicz, P. Różewski, L. Busk Kofoed, M. Malinowska, and A. Żyławski choosing the above alternatives. The proposed model is based to their own time and intellectual resources. Every student on the following assumptions: intends to use the maximum of teacher’s time of supervision ● the elements of the described chain and its alternatives competing for it with other students in this group. are predefined, In case a) the teacher and students with minimal effort will ● choice of a job in the game is made once, complete the learning process, but indicators, such as average ● for each task selected by a student, the degree of its com- achievement and filling of repository will be low, although plexity and the corresponding number of ECTS is known, satisfying the low boundary conditions (formal compliance ● the content of selected task can be displayed on an on- schedule and personal quota for supervision). In case b) the tology graph of the studied subject as a set of vertices teacher and students will spend far more effort, but a great part with the theoretical and procedural knowledge, which of successfully completed tasks will be included into the repos- constitute a certain portion of competence, itory and the average achievement will be higher. At the same ● students and teachers will have the possibility to know time in case b) learning will be aimed not only at understanding the results of the partners’ choices in the game. the theoretical material, but also the ability to operate them (i.e. mastering of didactic material at the level of competence). Thus, the aim of the game is the maximum extension of 4. Statement of the motivation problem repository with original results of teacher’s and students’ joint in a particular learning situation in ODL work under restrictions on their time resources and compliance with preferences of all participants of the game. Motivation In the proposed game scenario each participant has the possi- problem is considered in particular educational situations, when bility of free choice among the available alternatives. The main we are dealing with special didactic material for particular par- result of the game is to develop an ontology graph of studied ticipants of the learning process. For this we will use the term subject by means of adding new vertices connected with the “learning situation”, which is represented in Fig.1. existing ones. These new vertices submit new theoretical knowl- The formal model which describes the competence-oriented edge and/or skills acquired in task performing. The goal of the education process [15] has the following structure. teacher is to attract as many students to this kind of tasks under the following constraints: the total number of hours assigned for Basic components of the educational situation subject in scheduling and time interval allotted for the game. a) participants of the learning process {N,S}, where Each student defines the result of the game in the range of N – teacher, the following features: from lowest allowable assessment on the S = {sj} – set of students/project team, where subject (in the traditional way of studying) to the highest result j = 1, 2, j* – index of student. for active and successful participation in the project and “best The process of students’ arrival to ODL system can be practices”. A teacher distributes the supervision time among described by students at his/her discretion, but at the condition of minimum π(S) = {χ, λ}, where mandatory quota for each student. In this situation it may be x – distribution law, assumed that a teacher is interested to spend the most of his/her λ – intensity of arrival, time and intellectual resources on joint work with best students. They will contribute to the updating and expansion of the reposi- tory, which provides the basis for compiling the results in studied subject, with an option to use grants or publications. This aspect motivates the teacher to advance students to more complex work and to spend more resources on their performance. In this case there is a possibility to use the results in teacher’s and students’ common research. Thus, the proposed scenario allows the teacher and each student to formulate the conditions to choose their strat- egies in the learning process based on individual preferences. The teacher’s objective will be to maximize possible ex- tension of the repository through the involvement of a larger number of students for competitive performance of complex project tasks under the restrictions on total time of supervision. Considering students’ preferences it may be defined within two extreme cases: a) the students prefer to perform the simplest tasks, minimiz- ing their time and intellectual costs and getting the lowest possible scores. In fact, these students do not participate in the repository extension. b) the students prefer to participate in the repository extension Fig. 1. Process of choice and execution of project tasks by students of knowledge acquired in the project task under restrictions (educational situations)
946 Bull. Pol. Ac.: Tech. 64(4) 2016 For this we will use the term ‘learning situation’, which is k A(ri ) – topicality of the task – characteristic of a represented on fig.1. task, assigned by the teacher, which can be
expressed with a binary scale,
Teacher N ͳǡ The( ) teacherൌ places an up-to-date, properly solved Ͳǡ task in the task repository R; Motivation function N (r ) (X (r ), Z(r )) i i i f) repository of the projects assigned by teacher for Students S Students’ workflow (S) {,} saving of subject knowledge X (s1, t1) (si, ti) k Repository of tasks k For this we will use the term ‘learning situation’, which is X {rˆi }, where Task 1 r R, i 1,2.....i * Task i A(ri ) – topicality of the task – characteristic of a represented on fig.1. i task, assignedk by the teacher,k which can be rˆi - executed task ri , included in repository Repository of projects expressed with a binary scale, j Project 1 rˆ Rˆ, j 1,2,..... j * Project j g) state of the repository X in time t, GX (t) i Teacher N ͳǡ GX (t) {WX (t),LX (t)}, where The( ) teacherൌ places an up-to-date, properly solved Ͳǡ C X D j - set of vertices of graph G (t) , Repository of subject G (t) G G(rˆi ) WtaskX (t) in theW task repository R; X Motivation function N (r ) (X (r ), Z(r )) i i i f) Lrepository(t) L of-set the of projects arcs/edges assigned of graph by teacher G ( tfor) , Students S Students’ workflow (S) {,} savingX of subjectC knowledge X X (s1, t1) (si, ti) Fig. 1. Process of choice and Repository execution of tasks of project tasks by students k t [t0,Tc ] -time Xinterval{rˆ } ,assigned where for acquiring of the i (educational situations) Task 1 ri R, i 1,2.....i * Task i competencek k rˆi - executed task ri , included in repository The formal model whichRepository of projectsis describing the competence- j Project 1 rˆ Rˆ, j 1,2,..... j * Project j g) state of the repository X in time t, GX (t) oriented education processi [15] has the following structure: Decision parameters G (t) {W (t), L (t)} , where Basic components of the educational situation a) student’sX decisionX parameter:X binary function of C k Repository of subject G X (t) GD G(rˆ j ) W (t) W - set of vertices of graph GX (t) , a) participants of the learning processi {N,S}, where X the task choice y(ri ,s j ) -set of arcs/edges of graph G (t) , N - teacher, LX (t) LC k X k y(r ,s ) = 1,if student s j chooses task ri Fig. 1. Process of choice and execution of project tasks by students i j { S {s j} - set of students/project team, where t [t0,Tc ] -time interval assigned for acquiring of the (educational situations) 0,otherwise j=1,2,.j*- index of student. competenceb) teacher’s decision parameter: binary function of TheThe formal process model of which students’ is describing arrival the to competence ODL system- assignment of the task/project to repository orientedcan education be described process by [15] pattern has the following structure: Decision parametersN k Basic components of the educational situation a) student’s ( rˆdecisioni ) parameter: binary function of (S) {,} , where k a) participants of the learning process {N,S}, where the task choice y(ri ,s j ) N- -distribution teacher, law, k k y(r ,s ) = 1,if student s j chooses task ri S {s } - set of students/project team, where 𝑁𝑁 i 𝑘𝑘 j { 𝑝𝑝 - intensityj of arrival, 𝑖𝑖 ͳǡ 0, otherwiseȀ D δ (𝑟𝑟̂ ) ൌ b) ontologyj=1,2,.j* graph- index of of thestudent. domain area G b) Criterionteacher’s decision and Ͳobjectiveǡ parameter: functions binary function of The process GofD students’{W D , LarrivalD}, where to ODL system Let’sassignment define theof followingthe task/project variables: to repository can be described by pattern a) N ontk ology graph of the task/project WD - nodes/concepts/learning objects of the graph, (rˆi ) (S) {,} , where k k k LD – graph edges (relations between concepts), G(rˆi ) {W(rˆi ), L(rˆi )} ,where (1) - distribution law, C c) ontology graph of the course G k - intensity of arrival, 𝑁𝑁 𝑘𝑘W(rˆi ͳ)ǡ -set ofȀ the𝑝𝑝 vertices of the graph, C C CTeachers’ and students’C DmotivationC D model as a strategy for𝑖𝑖 open distance learning processes , where D δ (𝑟𝑟̂ ) ൌ b) ontologyG {W graph, L of} the domainW areaW G ,L L k C CriterionW and(rˆi Ͳobjective)ǡ W functions d) set/repository ofD the tasksD D R G {W ,L }, where Let’s define thek following variables: k C k L(rˆ ) - set of the arcs of the graph, L(rˆ ) L R {r } , where D a) ontology igraph of the task/project i k b) ontologyWD - graphinodes/concepts of the domain/learning area objects G of the graph, b) knowledge gain under replacement of task/project r ̂ i in re- D D D b) knowledgek gaink k under kreplacement of task/project kLD – graph edges= (relations between concepts), ( ) r – task ‘i’ consistingG {W , forL }competence, where portion ‘k’, positoryG (X,rˆi Δ)GX{Wr ̂ i (rˆi ), L(rˆi )} ,where (1) i C k k W – nodes/concepts/learning objects of the graph, rˆk in repository X, G (rˆ ) c)D ontology graph of the course G ˆi k-set of the verticesk X ofi the graph,k i=1,…, i* – index of task, ΔWG(ri(r ̂) ) = G (t) G(r ̂ ) = W (t) W(r ̂ ), LD – graphC edgesC (relationsC betweenC concepts),D C D X i { X i X N ki (2) G {W , L }, where W W ,L L d) teacherk resourcesC \ expenditures \ , e.g. k=1,…,k* – index of Cacquired competence/ k Z (rˆi ) (s ) = c) ontology graph of the course G LW((tr)ˆi ) LW(r ̂ ) j d) set/repository of the tasks R X i } 𝑘𝑘 𝑘𝑘 competenceC C portionC C D C D k \ k k C (2) i* k* G = {Wk , L }, where W W , L L consultationL(rˆ ) - set oftime 𝑥𝑥the for𝑖𝑖arcs task/project of the𝑥𝑥 graph, rˆL𝑖𝑖(erˆxecuting) L R {r } , where kµ ¶ i Δ𝐺𝐺 (𝑟𝑟̂ ) = 𝐺𝐺 (𝑡𝑡) ∩ 𝐺𝐺(i𝑟𝑟̂ )i = k N k C k d)e) set/repositoryparametersi of of the tasks tasks R c) c)numerical numerical characteristic characteristic of knowledge𝑘𝑘 of knowledge gain 𝑘𝑘in repository gain in (ri ) 𝑥𝑥N k 𝑖𝑖 k)𝑥𝑥 𝑖𝑖 y(ri ,s j ) (rˆi ) W W (rˆi ) kk b) knowledgek = gain{𝑊𝑊 (under𝑡𝑡) ∩ 𝑊𝑊replacementk(𝑟𝑟̂ ), 𝐿𝐿 ( 𝑡𝑡of) task/project∩ 𝐿𝐿(𝑟𝑟̂ )} R = rr –, task where ‘i’ consisting for competence portion ‘k’, ΔG (r ̂ ) Z (rˆi ) N G(rˆi i1 k 1 {i i } k k k k Xrepositoryi GX (rˆi ) k , (4) k (ri ) {Q(ri ),A(ri )}, where j rˆi in repositoryj X, GX (rˆi ) N k ri – task ‘i’ consisting for competence portion ‘k’, d) teacher resources expenditures Z (rˆ ) , e.g. S k i=1,…, i* – index of task, where N k C k) i - (s j ) = k k W(rˆi ) , (9) i = 1, …, i*Qk=1,…,k*(r ) – –the index –level index of oftask, complexityof acquired of competence/a task, which GX (rˆi ) (rˆi ) W W(rˆi ,, where (3) i d) teacher - weight resources coefficient expenditures of teacher expendituresk N k , e.g.(3) i* k* k = 1, …, k* – index of acquired competence/competence consultationN time𝑘𝑘 for task/project𝑘𝑘 rˆi executingZ (rˆi ) (s ) = competence portion C 𝑥𝑥 k) 𝑥𝑥 (2) where k N k j C k can be expressed in a binary scale C W kW (𝑖𝑖rˆ - number of 𝑖𝑖common vertices in the k Δ(𝐺𝐺 )(𝑟𝑟̂ i )N = k𝐺𝐺 𝑘𝑘(𝑡𝑡) ∩ 𝐺𝐺(𝑟𝑟̂ k)) =𝑘𝑘 y(ri ,s j ) (rˆi ) W W (rˆi ) portione) parameters of tasks k c) Wnumerical W r ̂ i characteristic – number NofN ofkcommonk knowledge vertices kgain in the in ontology N ki* k* (ri ) d)d) teacherteacher resourcesresources expendituresexpenditures𝑥𝑥 Z (rˆ )𝑖𝑖ZZ (rˆ(i Nrˆ𝑥𝑥i) ,)G k, (e.g.rˆe.g. 𝑖𝑖 k 1,if taskk r is a complex one d) teacher consultationresourcesj \ C expendituresj timek i for task/projectNC , i e.g.k rˆ executing (s(sj )j )==i (1rˆi k)1- number of ECTS points assigned by e) parameters of tasks Π(ri ) i graphs= G{𝑊𝑊 i( G𝑡𝑡)(r ̂∩i )𝑊𝑊k(𝑟𝑟̂ N),Z𝐿𝐿k ((rˆi𝑡𝑡) ∩ 𝐿𝐿(, 𝑟𝑟 ̂ )}i N k (4) (s ) = k N k C k Q(r ) k k k d) teacher resourcesObjectived) repositoryteacher ontologyexpenditures function resourcesG graphs( rofˆ )k theZk (expenditures Grˆteacher) , i Ge.g.(rˆ i ) Z (rˆ ) , e.g. j i k k k , where X i i N k i i*i* k*k* (s j ) = yS(r,(sk ) = (krˆ ) W W (rˆ ) Π(r(r)i =) Q{Q(r(r)i, A),(Ar(r)i ,)} where consultationconsultationd) timeteacher time for for resourcestask/project task/projectN expenditures r ˆrkˆexecutingekxecuting Z (r ̂k)), Ne.g.k consultation time - i jj i i i 0,otherwise{ i i } consultationd)a) teachertotalwhere time forgainresources task/project of repositoryki expendituresi ˆexecuting X oni the , intervale.g. i*of k* k k NNthek kteacherCC for thek k task/W( projectrˆ ) ˆ , (9) k k Z Nk (rˆik )ri C N G(rˆki ) Z (rˆki ) i* k* k ˆNˆ k C(sˆ ˆ) = k i , ri k consultation timeN Nfork k task/projectk)k) rˆ executing yy(r(iri,s,sj )j ) (r(iri)i)*WWi k1* WWk(r(1ijri)) Q(ri )Q –( rthe) – level the level of complexity of complexity of aof task, a task, which which can be forconsultation task/projectGX(rˆi ) time r ̂ i (Nexecutingrˆii fork) W task/projectW(rˆi , ,where ˆ executing (4) y(rk ,s ) N (rˆk ) W C W (rˆ k) or in inumerical scale (e.g. number ofd) conceptsteacher resourcesZ Zacquir(expendituresrˆ(NiNrˆi)- )weightkingN NG ofG(rˆ( ircoefficienttheˆi k )competence, e.g. of teacher kG ( r0expendituresi ,T ) (3)i i11 k k11 i j i i Z (rˆ ) G(Zrˆ, , ( rˆi ) (4(4) ) X 0 y(r(s,i*)s=) k* (rˆC) W Wk (krˆ ) N k C k expressed in a Dbinary scale Nconsultationk i timeN k) fori task/project rˆ executing i1 k 1 i j j wherei i - S ˆ k ˆ can be expressed in a binary scale Z where(rˆ C) Gk)(r-ˆ numberN ,k of common(ki4) ) vertices in the S S k k W kW (rˆyiN()ri k,-s jnumerical)C (ri )ˆ kWcharacteristicW (ri )of (9) from set W included in the task), wherewhere Wi WN(rˆi i k i* k*i1 k 1 - - N k W(ri ) , consultation time for task/project NrˆZ ,e kxecuting (rˆi ) N kG) ((4r)ˆi , (4) -WWS (r(ˆirˆi)),k ,y (ri , s j ) ((9)rˆi(9) ) W W (rˆi ) k where - weightZ icoefficient(rˆ)G (0G,T(rˆ of) = teacher , expenditures(4) k N kW(rˆ Ci) (1, rˆ k) 1k- number(9) of ECTS points assigned by k 1,if task ri is a complex one - -weight weight coefficient coefficientN of of teacher teacheri expenditures CexpendituresX N 0ki - iS1ˆ k 1 ik i ˆ k Q(r ) whereN N -Objective Nweightkwhereontology coefficient function k )graphs of teacher ofG thei expenditures G teacher(rˆ ) , (4) y (ri ,s j ) (riW) (Wrˆ ) W (ri ) S(9) k i NZ (rˆ ) i*G(rkˆ* j* i wherewhere gaini where,of repository- for task/project rˆ , i whereN i i1 k where1 - S k W(rˆ ) k i (9) 0,otherwise N - weighta) coefficient wheretotal= gain of teacher, of N repositoryexpendituresk (4)k X on the intervalk NN kofk the teacherN fork ˆ the task/ projecti , (9)ˆ , αN – weight coefficient (rˆ ) ofy( rteacher, s ) G expenditures.(t) G(rˆ ) (r(ˆNrˆ ))k- -number number of of ECTS ECTS points points assignedW assigned(ri ) ,by by ri D - weight coefficienti i of j teacherX expendituresi wherei i - S k k ˆ - number of ECTS points assigned by or in numerical scale (e.g. number of conceptswhereObjectiveObjective function function of Nof- theweight the teacher teacher coefficient of teacher expenditures (rˆi ) -W number(rˆ ) of ECTS points(ri(9) assigned) by or in numerical scale (e.g. number of concepts from set W acquirN i 1 ingk 1 ofj 1 the competence GX (0,T0 ) N(5) k i , Wwhere(rCˆi ) - numericalk characteristic of complexity D ObjectiveObjective function of function the teacher of the teacher where k k a)Na) - weighttotaltotal gain coefficientgain ofof repositoryrepository of teacher X expendituresX onon thethe intervalinterval ofof thethe teacher teacher(rˆi ) for -for number the the task/ task/ of project project ECTS r ˆ rˆpoints, , k assignedˆ - bynumerical characteristic of includedfrom in set the W task), included in the task), a) total b)Objectivegain numericof repository function characteristic ofX theon theteacher ofinterval total knowledgeof where gainthe teacherin for theN task/k projectWN i i k W, (ri ) k Objectiveacquiracquir function inga)ing of of thetotal ofthe competencethe competence gainteacher of repository X on the interval of the rˆteacheri kfor the task/ project ˆ , k GGX X(0(,0T,0T)0 ) CC k k (rˆi ) - number(rˆi )k of- numberECTS points of ECTS assigned points by assignedri by A(ri ) – topicality of the task – characteristic ofa) a tasktotal as gainacquir- a)ofing total ofrepository the gain competence of repositoryX on theGG X ( 0X(interval,0 Ton ,)T 0 the) = intervalof of acquiringN kthe teacher of the for the task/of project task/project , rˆi , ObjectiveObjectiverepository function functionG of( the0of,T X theteacher) 0teacher (rˆ W)W- numberCWW(r(ˆirˆ iof)) kECTS - - numericalnumerical points assigned characteristiccharacteristicrˆi by ofof k 5 acquir ing of theΣX competence0 G (0,T ) i ˆ - numerical characteristic ofk Objective function of the teacheri* k* j* ( ) X 0 W W (ri ) gain of repositoryC fork task/project krˆi , signed by the teacher, which can be expressed with a acquirbinarying ofa)a) thecompetencetotal totalcompetenceGG gain (0gain(0,T, T of)Δ )= Gof= X Grepository X0, (repository0T,T00 ) X on X theon intervalthe interval of of the teacher forS the task/ project rˆ , X X 0 0 N k k k C k thek teacherWk k forW the(rˆ task/)i project- numerical rˆi , characteristic of a) total gain of repository= G X ( 0on,T 0the) = interval of k the teacher W for theW task/(rˆ ) project - numerical rˆ , - weight characteristic coefficienti of of student’s expenditures scale, i*i* k*k* j*j*acquiring of the competence(rˆi ) y(ri ,s j ) GX (t) G(rˆi )gaingain of of repository repositoryi for for task/project task/projecti kr ˆirˆi , , k acquir ing of the competence ˆ GX (=0 ,T0 ) gain of repository forC task/projectk , acquiring== of ithe* kcompetence* j*N N k k k k GkXk ( ri ) GX ( 0,T0 ) C rˆi k k G (rˆi((rˆ1)0)y,(Tykr(1r),Gs,=sjX) (1)G0G,T(0t)()tG) GXG(rˆ(rˆ0) ),T0 ) = C (5) k W WW(r(ˆrˆi) ) - - numerical numerical characteristic characteristic of of complexity k 1, if task ri is chosen by the teaher to develop the repository= Xi Ni k i0 i jk j X X i i k k k Wi Wk (rˆ ) - numerical characteristick of A(r ) = i1i1 k k11 j1j1 (rˆi ) y(ri , s j ) GX (t) G(rˆi ) W Wˆ(ˆrˆ k)- - numerical numerical- numerical characteristic characteristic characteristic of of complexity complexity ofi i j* j* (5)(5) WgainW(r(i rofi)i) repository for task/projectgain ofrˆ ,repository for task/project rˆ , ( 0, otherwise i* ) k* i1 b)k 1 numericj1 ii** k* characteristicj*GX (0,T0 ) of= total knowledge gainW (inrˆ ) - numerical characteristic iof complexity i N k =k k (5) i (5) Constraints k k b)b) = numeric5numeric characteristic characteristicGX =(0ˆ,T0 ) of of total total knowledge knowledgeNNGXk ˆ(k0, Tgain k0gain)k= in in C k k gain of repositoryk for task/project , ==(rii*) y(kr*i , sjj*) GX (t) Gˆ (ri ) ˆ k k k of task/project rˆi , rˆi k b) numeric characteristicrepository of total knowledge(r(i rˆi) y )(yri(r, isgainj,)sG jin)X W(t) gainGW( ofrofi of (repository )rtask/projectˆ itask/project) for task/project, , k rˆ , k The teacher places an up-to-date, properly solved taski* i k1*ink j1* j1 =i* k* j*GN X (k0,T0 )k (5) k W (rˆ ) - numericalrˆirˆi a) characteristicsummarygaini of repository ofresources complexity for task/project (time-related, rˆ , technical, repositoryrepositoryNGkG(0(,0Tk,T) ) (rˆ )k y(r , s ) G (t) G(rˆ ) of task/projecti rˆ , i = repository X Xii110 0 k 11 jj11 iN k i j k X i k (5) i k S W (rˆi ) - numerical characteristic of complexity the task repository R; (rˆi )yG=(rXi (,0s,jT)0G) X (t) G(rˆi ) k S S didactic, staff) offered to students for solving b) numeric characteristici1 k of1 totalj1 knowledge (rˆi ) y(r igain, s j ) inG X (t) G(rˆ(5)i ) Wk (rˆi ) - numerical- weightk characteristic coefficient of of complexity student’s expenditures i1 k 1 j1 kk (5) k W (rˆi ) -- -weightnumericalS weight coefficient coefficient characteristic of of student’s student’s of complexity expenditures expenditures f) repository of the projects assigned by teacher for saving of b)b) numericnumerici1 characteristick 1characteristic j1 k ofG total of(rˆ knowledge total) = knowledge gain inof repository task/project(6)gain- weight in coefficientrˆi , Wof student’s(rˆ ) - expendituresnumerical kcharacteristic of complexity repository b)G numeric(,0 where,T ) G G XcharacteristicX(r(ˆirˆi))== of totalX i knowledge gain in (5) tasks: i k b) numeric characteristicX Σof0 totalG knowledge(rˆ ) = gain in k of task/project ˆ , X i S of task/project rˆ , k ri k N subject knowledge X i*i* k*kb)* j*repositoryΔrepositoryj*numericGXC(0, T0) characteristickG (0,T ) of total knowledge of task/project gain in rˆ , i repository G (0,T ) i* k* kGXj*(0X,T-0the) 0number of commonConstraintsConstraints vertices - weight i coefficient ofZ student’s zexpenditures(r (ks ))y (r (s )) Z , k i* X k*j 0 Wj* NN k Wk j( pk ik ) CC k k Constraints SConstraintsof otask/projectS i rˆ , j i J X = {r ̂ i }, where == G(r(ˆNrXˆ)()yrkˆiy(r()r,=s, ks))WNW kCWW(r(ˆrˆ)k) k C S k i repository= i i ii GjX j(0,T(r0ˆ) ) ky(iri ,s ) W a)a) W - summaryweightsummary(rˆ ) coefficient resourcesresources of student’s- weight(time(time-related, -coefficientexpendituresrelated,s - S weighttechnical,technical, of student’s coefficient expenditures of student’s expenditures k k = k(rˆi ) y(ri ,s j ) Wi C Wi(rkˆi k)j a) summaryi resourcesa) summary(time-related,j resources technical, (time-related, technical, r ̂ i – executed task ri , included in repository ii11 k k11 jj11G (rˆ ) = GX (rˆi ) = S (10) i* ki*1 jk*1 inj 1ontologyiX1 i k 1 jgraphs1 GGiX G(r(ˆipi))= didactic,didactic, staff)staff) offeredoffered toto studentsstudents forfor solvingsolving N k k C k k Constraintsdidactic, staff) offereddidactic, to students- weight staff) for coefficient solvingoffered ofto student’sstudents expendituresfor solving g) state of the repository X in time t, GX(t) , ,where where j* (6)(6) tasks:tasks: where i=* k* j* i*(rˆi )k*y(ri ,s j ) W GWX((rˆriˆi))(6)= a) summary (6) Constraintsresources (time-related, technical, , where c) N totalk i* expenditureskk* j* CN k ofk thek teacherC Constraints on intervalk tasks:(6) of k k tasks:k k NN GX(t) = {WX(t), LX(t)}, where i1 CkC1 j1 k k, where Constraintsk = WW CWW( (p =p()=rˆ)i k-)the-ythe( rnumberi number,s j ) W of of com (comrˆWiNmon)(monryˆi(k)r ivertices vertices,s j ) kW a) Wsummary(Crˆi ) , didactic,ZZ resourcesk staff)za)z( r( rsummary( (times(offeredks))))y-yrelated,(r( rto(s(resources ksstudents)) ))technical,ZZ , for,(timeN solving-related, technical, C i i i*-thek number* j* of common vertices o o i i j j izi(ri J(Js j )) - resources, k appointedk to studentN s j for ( ) ( ) i1 k 1 Wj1 W (=p i ) C k ˆ Zˆo z(ri (Constraintss j ))y(ri (sJ )) Z WX t W – set of vertices of graph GX t , acquiringi 1 k 1 jof1 the -competencethe( rnumberi ) y((6)ri Z of,s comj ) Wmon verticesW(rsi jsj)S S a) Z summary z( r resources(s ))y(r (s(time)) -related,Z , technical, µ , where W CWC ( pi k)k N k k N Cdidactic,tasks: staff)k sofferedS didactic, to students staff)o for offered solving to (10)i(10) students j fori solvingJ inin ontology ontology graphs graphs= G G i iGC G( p( p) ) k ˆ ˆ j k LX(t) LC – set of arcs/edges of graph GX(t), i1 k1 ji1i (ri (6)) y (ri ,s j ) W (6)W (ri ) k a) summaryk resourcesN (10) (time-related, technical, , whereC in ontologyk , wheregraphs G i G( pi ) tasks:where where tasks: solvingdidactic,s taskj S staff), offered to students for solving µ W W ( p ) -theC i 1numberk i1k* jofk1* comj*monC verticesk Zwhere z(r (s ))y(r (s )) Zri , (10) [ ] c)c) C totaltotal expenditures kexpendituresi in ontology of of( the the) teacher graphsteacher on onG interval intervali of of o k i kj i NJ t t0, Tc – time interval assigned for acquiring of the com- W C W pi k – the number ofG common(kpi ) verticesk in kontologyk didactic,k staff) koffered toN students for solving W c) Wtotal( pi ) expenditures-the numberZ = ofof thecom -teachermonthe number vertices on interval of com ofmon verticesZo z((6)sr i- -(Sresources sresourcesj ))y(ri appointed (appointedsJ )) Ztasks: to to ,student student forfor , 2 j ,W where\ NCWC(pij )kk N y(ri ,s j ) W(rˆi ) z(zr(iri(s(jsk))j ))j Zo wherez(kri (s j ))y(rsisj j(s(10)J )) Z petence. in ontologyacquiringacquiring graphsof of graphsthe the competence competenceG Gi i G(pZiZ) s zS((6)ri (s j )) - resources appointedtasks: to student s j for c) total, Cwhere expenditures Gki(1 pik )N1N j1 of the teacher on intervalj of s yS(ri (sJ )) {1,0}- kbinary functionk of choiceN acquiring of theC competencek ZN C k where(7) k k j k (10) (10) in ontology graphs G i G( p i ) k k N , c)i* i* totalinkW*k *ontology expendituresj*Cj*W (graphspik) of -G thethei teacherGnumber( pi ) on ofinterval com monof solvingacquiringsolving vertices task task of r r , ,k z(r (sZ))o - resourcesz(ri appointed(s j ))y(r toi (studentsJ )) sZjfor c) total expenditures of the teacherΣ - theon intervalnumber of of comwheremon vertices i i Zi j k z(r (s ))y(r (s )) Z , c) total expendituresi*, acquiring ofwhereWk *the j*teacher W of( pk k itheon) intervalcompetencek k of Z solvingk task rwherei , o i j i J Decision parameters ZZ == the competence y(yr(r, s,ZskX) )WW(rˆ(rˆ) ) k N z(r (s )) - resources appointedthe task tor student sby Sstudent s for s , NNZ c)= total expendituresN N i i j j of ithei ˆC teacher onk intervalk of i-k kresourcesj appointed to students i S forj k j j (10) acquiring ofNi i1the1k k1competence1jj11 N yZ(rNi ,s j ) W(rCi ) k z(ri (syj ))y(r(iri(s(ksJ J)))){1{,10,0}}-k - binarybinary functionfunctionj s ofj of choicechoice a) student’s decision parameter: binary function of the task in ontology-weight graphscoefficientj* G i G(7(7)(of )p i ) the teacher’s z(r (s- ))binarysolving - resources function task appointed rof , choice to student s for (10) acquiring of the competencei1inkN ontology1 j1 ZiN* graphsk* G i G( p(7)) y(ri (sJ )) k {1,i0} j i j k acquiring of the competence ZNk i k solvingk taskk k , where choice y(ri , sj) , ,where where i * k* j* Z = , (7) k ri Zwhere- summaryk resources for the subject lead by the ,i *wherec)k* j*totalexpenditures, N expendituresk k ofN they(ri teacher,s j ) W( solvingrˆion) intervalthethe task task task r ir i r,i of by by student student s sj ,j , k k = c) total expenditures of the teacher on intervalthe taskof bysolving student task s , , ZN Nk yi*(ri k,*s j) jkW* (rˆi ) k ri teacher.yj(rri (s k )) {1,0}- binary function of choice k 1, if student sj chooses tast ri ZN =NN-weight-weight coefficientcoefficienty(kr ,si )1 Wofk(ofr ˆ1 )thej the1 teacher’steacher’s k (7) ki J - resources appointed to student s for y(ri , sj) = -weight N coefficienti j- the iofnumber the k ofteacher’s vertices k in ontologyy(r (s )) {1,0}- binaryz(r z (s(functionri))( s- j ))resources of choice appointed to student s for j (0, otherwise Ni 1 k 1 Wj Z1(Np=i ) y(r ,s(7))W (rˆ ) y(r (ZsZ- ))-summary isummary{1J,0} resources- resources binaryk forfunction for the thei subject subject ofj choice lead lead by by the the j expenditures,expenditures,i1 k 1 jwhereacquiring1 acquiring ofof thethe competenceN (7)i j ZZNi i J - summary resources for the subjectk lead by the , wherei 1 k 1 j1 N Z k y(rb)i (sJcalendar)) {1,0 }interval- binary function[0 ,ofT 0choice] , appointed to , whereexpenditures, k (7teacher.) teacher.k the task r by kstudentk s , b) teacher’s decision parameter: binary function of assignment, where k k αN – weight coefficient of the teacher’s expenditures,theteacher. task by student s , i j WW( p( pi i) ) k - - thethegraph numbernumber of task/project ofof verticesvertices in in G ontologyontology(rˆ ) the task r by studentri s , j solving task , N k -, wherethek -numberweight i*i* ofkk **verticescoefficientjj* in ontologyi of the teacher’si j studentssolvingk taskfor choosing ri , ri and solving tasks -weightW ( pi ) coefficientN of the teacher’s b)b) calendarcalendar intervalintervalthe task[ 0[0,T, byT00] student], , appointedappointed , toto of the task/project to repository δ (r ̂ i ) -weightN coefficientW(pi ) – theof k numberk the teacher’s of vertices k ink ontologyk graphkb) calendar of task/ interval ri [0,T ] , appointeds j to N graphgraph of of task/project task/projectZZ = = y(r ,s ) W(Nrˆ ) - summaryk 0 k resources for the subjectk lead by the k j -weightNjN GG(krˆ( irˆi))coefficientk NN ofy( iri the,sj j ) Wteacher’sZ(i rˆ-i summary)studentsstudentsZ - summary forresources for choosing choosing resources for andthe and subjectsolving Zforsolving the lead taskssubject tasksk by the lead by the N k 1, if project/task pi is selected for repository graphd) of projecttask/projectobjectiveexpenditures,N G(r ̂ i function)G(rˆ ) of the teacher : total gain min (r (s )) 0 , max (r (s )) T , ( ) = expenditures,expenditures, i1 i k 1 j1 students for choosing andy (solvingriy(sr Jtasks))i(s ))j{1,0}{-1 ,binary0}- binary functioni functionj of choice0 of choice δ r ̂ i i1 k 1 jN1N N teacher.teacher. (7)( 7 )k k Z - summaryteacher.j resourcesk k i Jfor the subject leadj by the (0, otherwise d)d) objectivekobjectivek function functionexpenditures, of of kthe the teacher teacher : N:total total gain gain , , , , W (Wp ()p -) the-d) the numberobjective ofnumberW (repository pof ) functionverticesof - verticesthe inXofnumber ontologythe onin teacherontologythe of interval verticesΦ : total gainin ontology ofmin minreposito of( r( iri-(s(ksj ))j )) 00 maxmax (r(iri(s(ksj ))j )) TT0 0 d)i objectivei function, wherei of the teacher : total gain [t0,Tc ]j minj (ri (steacher.j )) 0 ,where j maxj (ri (ks j )) k T0 , , wherek [ ] b) calendarb) calendar intervalj interval [ 0b),T 0calendar][, 0jappointed,T0 ] , intervalappointed to to [0,T ] , appointed(11) to ofof repositoryrepositoryry W XX( X onp oni on )kthe the-the intervalthe intervalinterval number t 0, [Tt [ctof, T,of Tvertices ]acquiring] ofof in ofontology the competence, the task r by student s j , 0 graphgraph of oftask/projectof task/projectrepository G (X rˆ )on k the interval 0 0[t c,cTk ] of wherewhere b) calendar intervalthe taski r[i(11)0(11), byT ]student, appointed s j , to Criterion and objective functions takingacquiringgraph -intoiweightG of(r ˆitask/projectaccount of) thecoefficient competence,expenditures G0(rˆ c) of oftaking the theteacher studentsinto teacher’saccountstudents forwhere choosing for choosing and solving and taskssolving tasks (11)0 acquiringacquiring of of the theN competence,N competence,-weight taking takingcoefficient into into kaccount accounti of the teacher’s studentsk for choosingk and solving tasks acquiring graphofexpenditures the ofcompetence, task/project ofN thetaking N Gteacher( rˆintoi ) account k k k k Let us define the following variables: d) objective function of the teacher : total gain N min (rk k (s )) 0studentsk, kmax (forrZ( r(choosing-s summary())s))T kand,, (resources solvingr (s )) tasks- for appropriate the subjectk moments lead by theto d) objectiveexpendituresexpenditures function of expenditures,of the ofthe teacherthe teacher teacher : total gain (r(minri (s(ks))j ())r,i,(rs(rj ))(s(ks))))0- -,appropriate appropriatemaxi i Zj (-jr i summarymoments (moments0s j )) i Tto to0 resources, j for the subject lead by the expendituresd) objective of the teacher function of the teacherN : totalj i gaini j j j i i jjj kminj (r (s )) 0k, max (r (s )) T , expenditures, N (ri (s j )) , (ri (s j ))teacher.- appropriate i momentsj to i j 0 a) ontology graph of the task/project of repository d)X objectiveonNN the kinterval function [= tof, Tthe] teacher of - : total= gain min (ri (sj j )) 0 , max (ri (s j ))jk T0 , of repository X =W=Non(G pGkXtheX)(0( 0,-Tinterval, T0the)0 )- -Z 0numberZ G[=ct=X,T(0 ,]ofT 0ofvertices) ZN wherein ontologywhere j k k teacher.(11) j of repository= i G -( 0the,T X)number -N NonZ 0 =thec ofinterval vertices instart startontology and and of end end solving solving task task startr r by kbyand student student end s solvings . . (11) task r by student s . k k k acquiring of the competence,ofW repository( pi ) takingX X 0into¡on accounttheN interval [t ,T[t]0 ,Tofc ] b) calendarwherei i intervalj j [0i ,T ] , appointedj (11) to G(r ̂ i ) = {W(r ̂ i ), L(r ̂ i )}, acquiring(1) i*i* ofk*k *thej* j*competence,i* k* takingj* into account 0 c start and end solvingwhere task ri by student s j . 0(11) j* k k k b) calendar interval [0,T ] , appointed to expenditures ofi* thek* teachergraphk kof task/projectN N k k C Ck kˆk N k C TwoTwok extreme kextreme casescasesk of of the the teacherteacher and and student student 0 acquiringacquiringy(yr(iri,s, sjk)ofj{) {of the (therˆ( iNrcompetence,ˆi) )Wcompetence,kW CWGW(rˆ(ir ˆi)takingi )k)k taking into accountinto(r (accounts )) , ( r (s )) - appropriateTwo momentsextreme to cases of the teacher and student expenditures of the= teachery(r ,s ){ (rˆy()rWi ,sj )W{(rˆ ()rˆi ) W Wi (rˆijTwo()r ( sextreme(8)i)) ,j ( rcases(s ))ofstudents - theappropriate teacher for andchoosingmoments student to and solving tasks where ii11Nk k11jgraphj11 ofi task/projectj i G(rˆii ) motivationmotivationi j functions functionsi arek arej shown shown kin in figk fig 5. 5. k i1= Nkexpenditures1expendituresj1 i1 k 1- ofj ofthe1= the teacher teacher N k motivationstudents functionsk for choosing are shown and in solving figk 5. tasks k k C GX (0,T0 ) ZN motivation functions (ri ( sarej)) shown(,ri ((ris inj( ))sfigj )), 5.- (appropriateri (s j )) - momentsappropriate to moments to W(r ̂ i ) – set of the vertices of the graph, W(r ̂ i ) W d) =objectiveGX (0,k Tfunctionk0 ) - ZN =of the teacher start: totaland end gain solving task r by studentmink s(r. i (s j )) 0 , max (ri (s j )) T0 , - - Nk N N i j j k j k k k C µ i* k* j* -NNWW(r(ˆirˆi))}} k (8(8) ) start and end solving task ri by student s j . ( ) ( ) d) objective W function(rˆ =-) }=GX Gof(0 ,theT(0 ), teacherT- Z()8N)- = Z = : total gain min (ri k (s j )) 0k , max (ri (s j )) T0 , L r ̂ i – set of the arcs of the graph, L r ̂ i L i* k* j* ofk repositoryN N k i C XN onWk X ( therˆi ) }0interval N Two extreme(8 ) of cases ofstart the and teacher end solvingand studentj task by student . j µ y(ri ,¡s j )k{ (rˆi )N Wk WC(rˆi ) k [t0,Tc ] startwhere and end solvingri task r s byj student s . j* Two extreme cases of the teacher and student i j (11) i1 k 1 j1 y(rii*,s j k)*{ (rˆij*) W W (rˆi ) motivation functions are shown in fig 5. of repositoryi* k* Xk on Nthek intervalC k [t0,Tc ] of where ObjectiveObjectivei 1 kfunction function1 j1 acquiring of of the the student student of the competence,k N takingk C into motivationaccountk functionsTwo extreme are shown cases in fig of 5. the teacher and student (11) Objective function ofk the student y(r i ,s j ){ (rˆi ) W W (rˆi ) Two extreme cases of the teacher and student - y(ri ,s j ){ (rˆi ) W W (rˆi ) k k objectiveobjective function functionObjective Nacquiring Wexpendituresof of( irˆ i1student )studentfunctionk}1 k ofj 1 theof( ofs (the scompetence, )the): teacher:number ( numberstudent8) of of taking into account motivation functions are shown in fig 5. Bull. Pol. Ac.: Tech. 64(4) 2016 - i1 k 1 j1 j j 947 motivation (ri (s j )) functions, (ri (s arej )) -shown appropriate in fig 5.moments to objective function Wof ( rˆstudent) } (s j ) : number(8) of N i - N k k k ECTSECTS points, points,objective taking takingexpenditures into into function account account ofexpenditures expendituresof theW student( teacherrˆ ) } of kof the the (s ) : number(8) of (r (s )) , (r (s )) - appropriate moments to ECTS points, taking into account - N =expendituresGi X (0 , Tof0 )the- Zj N = i j i j k Objective function of the student k k W (rˆ ) } (8) studentstudent for for execution execution of of task/project task/projectN r ˆrNˆ k i start and end solving task ri by student s j . ECTS points, takingj* = intoi i account expenditures- = of the k objectiveObjectivestudent function function for of execution ofstudent thei *ofstudent task/projectk*(s ) : number rˆiG X of( 0,T0 ) ZN j k N k C k start and end solving task r by student s . objective functionObjective of student function of they(: rstudent number,s ){ of(r ˆ ) Wk W (rˆ ) Two extreme cases of the teacheri and student j ECTS points, takingstudent into account fori* execution kexpenditures* j*(s j )of itask/projectof thej i rˆi i objectiveObjective functioni 1functionk 1 ofj1 ofstudent thek student (sN ) :k numberC of k motivation functions are shown in fig 5. ECTS points, taking into account kexpendituresy(r ,s of){ thej (rˆ ) W W (rˆ ) Two extreme cases of the teacher and student student for execution of task/project rˆi i j k i i ECTSobjective points, function taking - intoofk account student expenditures (s j ) : ofnumber the of i 1 k 1 j 1 N W (rˆi ) } (8) motivation functions are shown in fig 5. student for execution of task/project rˆi k studentECTS points,for execution taking of into task/project accountk rˆ expenditures of the - i 66 N W (rˆi ) } 6k (8) studentObjective for function execution of of the task/project student rˆi
objective function of student (s ) : number of Objective function of the student6 j 6 ECTS points, taking into account expenditures of the objective function of student 6 (s j ) : number of student for execution of task/project rˆk 6 ECTS points, taking into account expendituresi of the k 6 student for execution of task/project rˆi
6
6 d) teacher resources expenditures Z N (rˆk ) , e.g. i (s j ) = k i* k* consultation time for task/project rˆi executing y(r k ,s ) N (rˆk ) W C W (rˆk ) Z N (rˆk ) G(rˆk) i j i i i N i , (4) i1 k 1 where - S k W(rˆi ) , (9) - weight coefficient of teacher expenditures O. Zaikin, R. Tadeusiewicz, P. Różewski, L. Busk Kofoed, M. Malinowska, and A. Żyławski N where N k (rˆi ) - number of ECTS points assigned by Objective functiond) teacher of theresources teacher expenditures N k , e.g. Z (rˆi ) Objective function of the student(s ) = k a) total gain of repository X on the interval of the teacher for the task/ projectj ˆ , k objective function of student Φ(s ): numberri of ECTS points, consultation time for task/project ˆ executing i* k* j acquiring of the competence G (0,T ) ri k N k C k X 0 taking intoC accountk expenditures of the student for execution N k k) ˆ y(r i -, snumericalj ) (rˆi ) WcharacteristicW (rˆi ) of N k W W (kri ) Z (rˆi ) N G(rˆi d) teacher resources expenditures Z (rˆi ) , e.g. of task/projecti1 k r ̂ i1 = G (0,T ) = , (4) (s j ) X 0 S k k where k gain of repositoryi* k* - for task/projectˆ ˆ , (9) i* k* j* consultation time for task/project rˆi executing W(ri ) , ri = N- weightk coefficientk of teacherk expenditures k k N Nk k C C k k N (rˆ ) y(r , s ) G (t) G(rˆ ) Φ(sj) = y(yr(ir, i s,js)jγ)(r ̂ i ()rˆWi ) W W(rW ̂ i )( rˆi ) i i jN ˆkX iˆk) wherek i1 k 1 j1 Z (ri ) N G(ri ˆ i- 1numericalk 1 characteristicj of\ complexityj ¡ (9) , (5) (4) W (ri N) k (rˆ ) - number of ECTS points assigned by b) numericObjective characteristicwhere function of totalof the knowledge teacher gain in i S ( k) - S k β W r ̂ i k , W(rˆi ) , (9) of task/project¡ rˆi , k repositorya) Gtotal(0 N,gainT- weight) of coefficientrepository ofX teacheron the expendituresinterval of the teacherj for thej task/ project rˆi , X 0 where whereS acquiring of the competence GX (0,T0 ) C k N k- weightN coefficientk of student’s expenditures k γ (r ̂ )W – number Wˆ (r ˆofi -) ECTSnumber - numerical points of ECTS assigned characteristic points by assigned the ofteacher by for i (ri ) Objective GfunctionX (rˆi ) of= the teacher k GX (0,T0 ) = the task/ project r ̂ , k a) total gain of repository X on the interval of the teacheri for the task/ project k , i* k* j* i* k* j* C gain of krepository for task/project rˆi rˆi, ConstraintsW W ( r ̂ ) – numerical characteristic of gain of repository = acquirN ingk of N the(krˆk )competencey(r k , s C) G ( t) GG(k(0rˆ,kT) ) i = (rˆ ) y(r i,s ) i Wj X W(rXˆ )i 0 j \ k Cj k k i1 k 1 i j1 i j i (5) a) summaryfor task/projectW (rˆWi ) resources - numerical rW ̂ i ,(rˆi ) characteristic(time- numerical-related, ofcharacteristic complexitytechnical, of i1 k 1 j1 k b) numeric characteristicGX of(0 total,T0 ) =knowledge gain in didactic,W(r ̂ i ) – numericalstaff) offeredk characteristic to students of complexity fork solving of task/ of task/projectgaink of repository rˆ , for task/project rˆ , repositoryi* k* j* (6) tasks:projectj j r ̂ i , i i , where = GX (0,TN0 ) k k k (rˆ ) y(r , s ) G (t) G(rˆ ) S S C k i i j X i β – weight coefficientk k of student’sk expendituresN i1 k 1 j1 k -W weight(rˆ ) coefficient - numerical of characteristic student’s expenditures of complexity group of non-ambitious students W W ( pi ) -the number of common vertices (5) Zo zi (ri (s j ))y(ri (sJ )) Z , GX (rˆi ) = b) numeric characteristic of total knowledge gain in s S C k j k (10) i* k* j* of task/project rˆ , in ontology graphsrepository G i GG( p(i0,)T ) ConstraintsConstraints i NX k 0 k C k where = (rˆi ) y(ri ,s j ) W W(rˆi ) a) a)summary summary resourcesS resources (time-related, (time-related, technical, technical, didactic, staff) c) total expenditures of the teacher on kinterval of - weight coefficient of student’s expenditures i1 k 1 j1 G (rˆ ) = offeredkdidactic, to students staff) foroffered solving to studentstasks: for solving X i z(ri (s j )) - resources appointed to student s j for acquiring of the, where competence Z (6) tasks: i* k* j* N C k N k k C k Constraintsk k k N W = W ( p ) -the number of common vertices solvingZ task z(, r (s ))y(r (s )) Z ,, (10) i* k* j* i (rˆi ) y(ri ,s j ) W W(rˆi ) a) osummaryri iresourcesj i (timeJ -related, technical, k k s S = i1 k 1 j1 C k j (10) ZN in ontology Ngraphsy(ri , s j )iW (rˆi ) k didactic, staff) offered to students for solving G G( pi ) - binary function of choice i1 k 1 , wherej1 (7) (6) wherey(riwhere(stasks:J )) {1,0} c) total expenditures of the teacher on interval of ̄ k C k z(ri (sj)) k– resourcesk appointedk to kstudent sj forN solving task , where W W ( p ) -the number of common vertices z(r (sZ)) - resourcesz(r (appointeds ))y(r (tos student)) Z s,j for i thek taski rj o by studenti sjj , i J acquiring of the competence ZN ri , i s S C k j k (10) N -weight coefficientin ontology graphsof Gthei G ( pteacher’s) k k i* k* j* i y(ri (solvingsJ)) = { task1, 0 }r i– , binary function of choice the task ri by k k Z - summarywhere resources for the subject lead by the expenditures,c) ZtotalN = expenditures ofy( rthe,s teacher) W(rˆ )on interval of student sk, N i j i j k k i1 k 1 j1 teacher. ̄ y(ri (sJ )) -{ 1resources,0}- binary appointed function to ofstudent choice s for - the number of vertices in ontology (7) Z – summaryz(ri (s resourcesj )) for the subject lead by the jteacher. W ( pi ) acquiring of the competence ZN , where b) calendar intervalk k [0,T ] , appointed to b) calendarthe task interval r by τ student [0, T 0]s, appointed, 0 to students for choos- k j* solvingi task r , j graph of task/project -weight G i*(rˆcoefficientk)* of the teacher’s 2 i N Z = i y(r k ,s ) W(rˆk ) studentsing and solving for choosing tasks and solving tasks N N i j i Z - summaryk resources for- binary the subject function lead byof thechoice expenditures,i 1 k 1 j1 N (7) y(rik (sJ )) {1,0} k d) objective function of the teacher : total gain minteacher. (ri (s j )) 0 , max (ri (s j )) T0,, (11) W,( wherepk ) - the number of vertices in ontology j k j i the task r by student s , of repository X on the interval [t ,T ] of b) calendar intervali [0,Tj 0 ] , appointed to k 0 c where (11) graph N of-weight task/project coefficient G(rˆ ) of the teacher’s i ( k(students)) ̄ (- ksummaryfor( ))choosing resources and solving for the tasks subject lead by the acquiring of theexpenditures, competence, taking into account τ ri sj , Zτ ri sj – appropriate moments to start and end N k kk k k expendituresd) objective of the teacher kfunction of the teacher : total gain solvingmin teacher.task (rrii by (s jstudent)) 0 , smaxj. (ri (s j )) T0 , W ( p ) - the number of vertices in ontology (ri (sjj )) , (ri (s j )) - appropriatej moments to group of-ambitious students i ofN repository X on the interval [t0,Tc ] of Twob) whereextreme calendar cases interval of the teacher[0, Tand0 ] ,student appointed motivation to = - =k k (11) graphG ofX task/project(0,T0 ) Z GN(rˆ ) functions are shown in Fig. 2. acquiring of the competence,i taking into account start andstudents end solving for choosing task randi by solving student tasks s j . i* k* expendituresj* of the teacher N Motivation (r k (s of)) students, (kr k (s in)) both- appropriate, cases, is describedmomentsk to as ,a linear Fig. 2. The shape of the motivation function during the teaching process d) objectivek functionN k of theC teacher k : total gain Two extremei minj cases(ri i(s j of))j the0 teachermax ( riand(s j ))studentT0 y(ri ,s j )N{ (rˆi ) W W (rˆi ) function. Dependingj on whether it isj the motivation ambitious = - = k i1 k 1 j1 of repository XG Xon(0 ,theT0 ) intervalZN [t0,Tc ] of motivationwhere functions are shown in fig 5. or non-ambitiousstart and end students, solving task this r i function by student is srespectivelyj . (11) in- acquiringi* k* jof* k the competence, taking into account - k N k C k creasing or decreasing. The nature of the motivation function indicates that it is W (rˆ )y}(r ,s ){ (rˆ ) W (8W) (rˆ ) Two extremek cases kof the teacher and student expendituresN i of thei jteacheri i (r (s )) , (r (s )) - appropriate moments to i1 k 1 j1 Identifyingmotivation ithe functions teacherj imotivationare shownj in assumes fig 5. the function of non-linear. The shape of this function results to the rules for N - = GXk (0,T0 ) - ZN = this motivation will be built for each groupk of students. Its shape summarizing the functions of teacher and student motivation. N W (rˆi ) } (8) start and end solving task ri by student s j . Objective function of thei* studentk* j* will depend on which group of students the teacher will work Both components of the objective functions depends on the y(r k ,s ){ N (rˆk ) W C W (rˆk ) Two extreme cases of the teacher and student objective function of student (is j )j : numberi of i at a given time. The curve describing the motivation function same argument as opposed. The summarizing function of stu- Objective functioni1 k 1 ofj1 the student of the teachermotivation in the general functions case are takesshown the in figform 5. of non-linear dent motivation reaches its maximum value when making the ECTS points,objective taking intofunction account of- studentexpenditures k(s )of: numberthe of N W (rˆi ) } j (8) and convex. In the case of non-ambitious students, the teacher optimal choice task – optimal level of the task complexity. student for execution of task/project rˆk ECTS points, taking into accounti expenditures of the motivation function is increasing due to the need for contin- k uous motivating students solving more difficult tasks and re- studentObjective for execution function of of task/project the student rˆ i quiring more time. Otherwise, the students will be ambitious 5. Motivation model interpretation in terms objective function of student (s j ) : number of and strongly determined to achieve a high score, which will of games theory ECTS points, taking into account expenditures of the result in a decreasing function. Due to existing restrictions the k student for execution of task/project rˆi teacher must inhibit the motivation of students. Distribution of Some initial remarks about the motivation model interpreta- 6 this function is related to the time constraints which are con- tion in terms of games theory can be found in [19]. Following 6 nected the teaching process. that publications we can assume that proposed model refers
9486 Bull. Pol. Ac.: Tech. 64(4) 2016 Teachers’ and students’ motivation model as a strategy for open distance learning processes
to the class of games with a defined number of steps and full his/ her reward obtained from the centre γN and the losses con- information about participants’ activities. Moreover, based on nected to solving the task ws game theory terminology the motivation model can be seen as incentive task, where motivation management signifies direct ΦS = γN wS (16) rewarding an agent (student) for his actions. ¡ The game model which is described in [19] and can be If income of centre X = 0, and reward of each agent only N S solve by algorithmsThe game model described which in is [20] described and [24] in has[19] the and following can be compensates his/her estimated costs γ = w , then the objective S structure:solveThe by game algorithms model whichdescribed is described in [20] and in [19] [24] and has can the be function of each agent Φ = 0. M – afollowingsolve set of by acceptable algorithmsstructure: motivation described inmethods, [20] and [24] has the Y(σ) –followingM a- seta set of of structure:game acceptable solutions motivation (strategy methods, of agents having bal- Step 3. On the base of result of the agents’ selection the centre M - a set of acceptable motivation methods, ance inY (their ) - amotivation set of game method solutions σ). (strategy of agents having can adjust the function of motivation to achieve its maximum Y ( ) - a set of game solutions (strategy of agents having Managementbalance in their (motivation) motivation methodeffectiveness ). means obtaining value of the objective function. In addition the centre can verify balance in their motivation method ). maximumManagement value of the objective(motivation) function effectiveness U(σ) on anmeans appro - restrictions on the total resources provided by agents Management (motivation) effectiveness means priate obtainingset of game maximum solutions. value of the objective function U ( ) obtaining maximum value of the objective function U ( ) N Σ on an appropriate set of game solutions. Φ = X ZN = max on an appropriate set of game solutions. ¡ (17) U( ) max f ( , y),, (12)(12) Σ ̂ U( ) maxyY ( ) f ( , y) , (12) ZN ZN. yY ( ) ∙ Fig. 2. The shape of the motivation function during the teaching process where is a motivation function of the centre, y is a Fig. 2. The shape of the motivation function during the teaching processwhere where σ is a motivation is a motivation function function of the of centre,the centre, y is ya binaryis a Motivation of students in both cases, is described as a linearargument binary of argumentagent’s choice. of agent’s choice. Step 4. Agent makes the final selection of a task on the base Motivation of students in both cases, is described as a linear binaryThe argument task of optimizationof agent’s choice. is about searching for an function.. Depending on whether it is the motivation The taskThe of task optimization of optimization is about issearching about searching for an acceptable for an of his/her preferences while maximizing its objective function ambitiousfunction.. Depeor nonnding-ambitious on whether students, it isthis the functionmotivation is acceptable motivation function with maximum ambitious or non-ambitious students, this function ismotivation effectiveness:acceptable function motivation with maximum function effectiveness: with maximum respectively increasing or decreasing. effectiveness: ΦS = γN wS = max. (18) Identifyingrespectively the increasing teacher motivationor decreasing. assumes the function of * Arg maxU( ). (13) ¡ Identifying the teacher motivation assumes the function of * Arg maxM U( ).. (13)(13) this motivation will be built for each group of students. Its M At the moment of the decision making (motivation shapthis motivatione will depend will on be which built forgroup each of group students of students. the teacher Its function At the for moment the centre of andthe student’sdecision decisionmaking function(motivation for willshap ework will dependat a given on which time. group The ofcurve students describing the teacher the At the moment of the decision making (motivation function thefunction agent) for the the objective centre andfunc student’stions and decisionacceptable function actions for of motivationwill work functionat a given of the time. teacher The in thecurve general describing case takes thefor the centre and student’s decision function for the agent) the allthe participantsagent) the objective are known. func Let’stions define and acceptable the game actionsscenario: of 6. Simulation model of teacher themotivation form of function non-linear of the and teacher convex. in the In generalthe case case of takesnon- objectiveStepall participants functions1 areand known. acceptable Let’s defineactions the of game all participants scenario: and student collaboration ambitiousthe form ofstudents, non-linear the and teacher convex. mot Inivation the case function of non is- are known.StepThe 1 Let centre us define has the the right game of scenario: the first move, when it increasingambitious students,due to thethe needteacher for motcontinuousivation functionmotivating is choosesThe centrea motivation has the function, right of beforethe first the move,agents whenchoose it The assumed approach to modelling the process of acquiring studentsincreasing solving due tomore the difficultneed for tasks continuous and requiring motivating more Step activitieschooses1. The centrea thatmotivation optimizehas the function, righttheir of objectivebefore the firstthe function agentsmove, s.choose when The it competences can be depicted as a set of three components: the time.students Otherwise, solving more the difficultstudents tasks will andbe requiringambitious moreand choosesactivities a motivation that optimize function, their before objective the agents function chooseN s. Theactiv - model of defining competences, the motivation model and the stronglytime. Otherwise, determined theto achieve students a highwill score; be ambitious which will and be centre’s (teacher’s) objective function N is the centre’s (teacher’s) objective function is the astrongly decreasing determined function. to achieve Due to a highexisting score; restrictions which will the beities thatdifference optimize between their objectiveexpected income functions. X (gain The of centre repository) (teach - simulation model [25]. Interrelationship between these models difference between expectedN income X (gain of repository) teachera decreasing must function.inhibit Duethe tomotivation existing restrictionsof students. theer’s) objectiveand the summary function reward Φ is paidthe differenceto the agents between (sharing expected one’s is presented in Fig. 3. teacher must inhibit the motivation of students. and the summary reward paid to the agents (sharing one’s Distribution of this function is related to the timeincome own X (gainresources of repository) Z ) and the summary reward paid to Distribution of this function is related to the time own resources ZN ) Σ constraints which are connected the teaching process. the agents (sharing one’sN own resources ZN ) constraints which are connected the teaching process. N =X - Z (14) The nature of the motivation function indicates that it is N =X - ZN (14) nonThe- linear.nature Theof the shape motivation of this functionfunction resultsindicates to thethat rulesit is The centre’s choiceN of a motivationNΣ function takes place Φ = X ZN. (14) fornon -summarizinglinear. The shape the offunctions this function of teacher results andto the student rules in Thecondition centre’s of choice uncertainty of a¡ motivation and foreseeingfunction takes random place motivation.for summarizing Both componentsthe functions of ofthe teacher objective and functions student in condition of uncertainty and foreseeing random Thecharacteristics centre’s choice of ofthe a motivationbasic students’ function knowledge takes place and in motivation. Both components of thek objective functions parameterscharacteristics of theof process the basic of their students’ arrival. Thereforeknowledge centre and depends on the same argument rik as opposed. Thecondition parameters of uncertainty of the process and foreseeing of their arrival. random Therefore characteristics centre depends on the same argument ri as opposed. The can use the principle of minimal guaranteed result and summarizing function of student motivation reaches itsof thecompensatory canbasic use students’ the principle function knowledge ofof minimalrestit andution parameters guaranteed , that means of result the lack process and of summarizing function of student motivation reaches *its compensatory function of restitution , that means lack of maximum value when making the optimal choice task r *of , theircentre’s arrival. income. Therefore In that centre case centre’s can use objective the principle function of has min - maximum value when making the optimal choice task i , centre’s income. In that case centre’s objective function has * ri imal guaranteed result and compensatory function of restitution Q(r ) - optimal level of the task complexity). negative value and represents the minimal costs of centre Q(ri* ) - optimal level of the task complexity). , that whennegativemeans working lack value of withand centre’s representsagents income. the minimalIn that case costs centre’s of centre ob - i when working with agents jective function X=0has negative , N = -value Z N andmin represents the minimal(15) 5. Motivation model interpretation in the X=0 , N = - Z N min (15) 5. Motivation model interpretation in the costs ofTherefore centre whenthe centerworking creates with agentsmotivation function in terms of games theory Therefore the center creates motivation function in terms of games theory condition of minimal guaranteed result, Z N min , condition Xof = minimal 0, ΦN =guaranteed Z N = min result, Z N min(15), Some initial remarks about the motivation model which cannot exceed admissible¡ centre’s costs Z N . interpretationSome initial in the remarks terms of about games the theory motivation can be found model in which cannot exceed admissible centre’s costs ZN . interpretation in the terms of games theory can be found inTherefore Step the2 center creates motivation function in condition of [19]. Following that publications we can assume that Step 2 proposed[19]. Following model refersthat topublications the class of wegames can with assume a defi thatnedminimal Agentsguaranteed choose result, their strategiesZ N = min independently, which cannot and do exceed not proposed model refers to the class of games with a defined exchangeAgents information choose their or strategies¡Σ wins, this independently signifies that and we do notare number of steps and full information about participants’admissible exchange centre’s information costs orZN .wins, this signifies that we are activities.number of Moreover, steps and based full informationon game theory about terminology participants’ the dealing with a relational¡ dominant strategy (RDS). By activities. Moreover, based on game theory terminology the dealing with a relational dominant Sstrategy (RDS). By motivation model can be seen as incentive task, whereStep 2.objective Agents function choose oftheir each strategies agent independentlyS we understand and the do motivation model can be seen as incentive task, where objective function of each agent we understand the motivation management signifies direct rewarding an agent difference between his/her reward obtained from the centre motivation management signifies direct rewarding an agentnot exchange information or wins, this signifies that we are (student) for his actions. differenceN between his/her reward obtained from theS centre (student) for his actions. dealing withN and a relationalthe losses connecteddominant tostrategy solving (RDS). the task By w objectiveS Fig. 3. Process of choice and execution of project tasks by students and the losses connectedS to solving the task w function of each agent ΦS we understandN S the difference between (educational situations) [28] S = N - wS (16) = - w (16)
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7 7 O. Zaikin, R. Tadeusiewicz, P. Różewski, L. Busk Kofoed, M. Malinowska, and A. Żyławski
Integration of the three components mentioned above allows tion of a training session can be calculated using the following for defining the scope and goal of learning, the conditions of indicators: the percentage of correctly interpreted jobs, the cooperation between the participants of the learning process, percentage of correctly identified tasks, the percentage of cor- and the possibilities of analysing the process and influencing the rectly associated pairs of typical tasks and typical algorithms, teacher’s existing strategy of working with the students. Thus the number of modules placed in the repository of knowledge, the proposed approach to modelling the process of acquiring the number of sessions needed to perform a specific task, the competences appears to be a solution in which: number and complexity of the problems included in one job. ● the structure of competence and the method of assessing The described indicators allow measuring teaching, based on the development of competence (with the use of an on- skills of mastering theoretical knowledge and skills of its use tology graph and a repository) is defined, in solving design tasks. ● conditions of competence development are defined (with The procedure includes several steps, and the algorithm of the use of the motivation model), its implementation depends on the software environment in ● there is a possibility to compare the expected costs that which the repository will be created (Fig. 4). the teacher will bear considering the assumed repository Input data of the procedure: development (working time) with the results achieved in 1. Scope: subject / topic of teaching the didactic process 2. The model of theoretical knowledge areas: a hierarchi- The proposed formalisation of cooperation between partic- cally ordered the basic concepts of the semantic network ipants of the learning process during repository complementa- (graph). tion indicates that quantitative analysis of this problem from the point of view of expected results and incurred costs is a difficult task. This difficulty is mainly caused by the fact that this process is related to behaviours and preferences of humans, which can change and are difficult to predict. For this reason simulation seems an adequate mechanism, which allows for obtaining information about the analysed object of study and is the basis for further activities related to the functioning of this object. On the basis of the developed simulation model it is possible to study the adapted strategy of cooperation between the teacher and a group of students. The aim of the simulation is to assess if the expected repository development plan is possible to realise with the teacher working in a distance mode, with a specified distribution of student arrival, and a specified distribution of grading.
6.1. The procedure for the acquisition of personal compe- tence. Within psychology there is a distinction between the mastery of operation and mastery of theoretical knowledge and acquisition of competences. Based on Woolfolk [26], we will try to present the process of acquiring competences based on a certain level of domain knowledge. The aims of the execution procedure is an acquisition of competence training for students and collection of statistics, allowing for clarification of the individual process management model of teaching in the open distance learning system. The training needed to get the student through the mechanisms of the repository’s virtual lab with the set of triplets: part of the description fields – a typical task – a typical solution, and the corresponding test task which is used to describe knowledge as an extended ontological model. The task of testing should be interpreted in the terms con- tained in the proposed triplet. For proper identification of test task a disclosure of knowledge necessary to prepare a solution will be needed. The student’s solution can be compared with the solution typically stored in triplet. Measurement of teaching consists not only of collecting statistics of personal data, but Fig. 4. The algorithm acquisition of competencies during a structured also analysing the interpretation of test tasks. The implementa- training
950 Bull. Pol. Ac.: Tech. 64(4) 2016 Teachers’ and students’ motivation model as a strategy for open distance learning processes
3. The reference model to enable the use / assumption of proposed structure for the fulfilment of the repository has sev- the taxonomy studied during training problems; the pre- eral advantages: sentation depends on the specific areas (Sp). ● corresponds with the cognitive meaning of the compe- 4. Repository of solved tasks: typical task – a typical al- tency structure, gorithm – the number of tasks. At the start of the session ● represents a system approach to a knowledge ontological the repository can be empty or filled, depending on the model, topic and the student’s personal learning goals. ● enhance the ability to structure theoretical knowledge and The procedure for the acquisition of competencies during connect it with results of student’s own experience a structured training is done according to the following steps: A similar idea is used in simulators, the aim of which is 1) Analysis of the research problem. Determining whether to teach the use of complex equipment repair and machinery. the issue belongs to specific areas will allow to interpret The proposed method of domain area describing the structure the problem and to present it in terms of a particular and the content of repository, and allows students to master the model, and take into account existing knowledge of tax- complex theoretical knowledge while the teacher is absent. In onomy – (Zi, Ai), 1, … i the formulation of the input this case, a chain “typical task-typical solution” is represented problem. XDepending on the educational situation the at the level of a mathematical reference model, which depends first step can be done in two ways: on the specific subject and purpose of education. a) independent formulation of the task by the student This procedure is a model skeleton in the course of training based on the ontological model Gp and reference related to a specific topic. The ontological model, repository model of typical tasks; and the task should be completed with relevant teaching ma- b) analysis and identification of the tasks set by the terials. An additional variant of the described approach to the supervisor of the project. structure and contents of the repository is associated with the 2) Analysis and systematizing of experience. A compar- formation of a virtual laboratory. It is designed to conduct simu- ison between the input tasks with the tasks placed in the lation experiments, when the assumption of the task simulation, repository. The result is to determine the path that must execution and interpretation of the results of the experiment be used to solve the problem either the way to develop needs the cooperation of experts located at a distance. an algorithm. 3) Typing of input tasks. Preparation of the task pass- 6.2. Ontological graph of the subject/course consistent with port in the repository language (meta information in the structure of competence. The first ontologies were built an XML). for the needs of engineering knowledge and they began to 4) The collection of the input task passport in the working emerge in the 80s of XX century. Intensive works on ontology memory of training session. resulted in the development of its various categories. Proposed 5) Developing of own algorithm for solving the task. The by Guarino [27] division includes: general ontology (a top-level algorithm can be described using standard pseudocode ontology), domain ontology, task ontology, and application on- or presented as a simulation task. tology. Each of them has a different level of generalization. Do- 6) Execution of algorithm. The input data should be se- main ontology describes domain-area knowledge characteristic, lected directly from the text of the analysed problem e. g. medicine, pharmacy, law, music. The terms used in domain or pulled during its interpretation. ontology are the result of specialization of concepts defined 7) Evaluation of the effectiveness of the algorithm. At in general ontology. Task ontology describes the dictionary of this stage the interpretation of results is made in the a particular task or activity, e. g. diagnosing, scheduling, spe- context of the output of algorithm (in terms of solved cializing terms stemming from the general ontology. In contrast problem). to the domain ontology, in this case for solving the problem 8) Typing of the developed algorithm. Preparation of the concepts from various fields can be used. Application ontology solution algorithm passport in the repository (meta in- includes the concepts that are required for the description of formation in an XML document). knowledge for individual applications. It specializes the con- 9) The collection of algorithm passport in working memory cepts of domain and tsk ontologies for the given application. of training session. For example of ontological graph the subject/course “Dis- 10) Preparing the knowledge module in the form of record crete mathematics”, provided for specialty “Computer Science” in repository. At this stage teacher has to fill in a repos- is represented in Fig. 5. itory form, including: track keywords of the domain Practical part of the course/ subject includes a number of knowledge, matching the given problem, the task pass- tasks. There are two kinds of the tasks: simple and complex port and passport algorithm. ones. They differ in labour complexity of implementation and 11) Supplementing of existing repository. Required level of reward-number of ECTS, which student will get for correct supplement depends on the subject, purpose and stage execution of the task. In general the teacher can define any of teaching and must be given by the teacher to each number of classes of tasks, which differ in degree of complexity. student. Described training is one of the traditional tools to collect 6.3. The supporting tools for determining the level of stu- statistics in the course of self-education students. However, the dent’s motivation is the linguistic data base. Recognition of
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Fig. 6. Place of a tool to support the process of identifying students’ motivation
Fig. 5. Ontological graph of a subject/course “Discrete Mathematics” teacher and student motivation is the basis to determine whether the existing restrictions on the competence acquisition process they can achieve required competence level and at what cost. Checking these constraints is done using simulation. Fig. 7. Structure of linguistic data base for the purpose of identifying Basis to estimate input value of simulation model is ex- the student’s motivation [25] pected activity of participants resulting from their motivation. Therefore, recognition of this motivation is necessary to per- form the simulation/experiments [28]. Teacher motivation identification is the task so easy that he is one and could recognize his motives. More complicated issue is to assess students’ motivation, because there are many students and each of them has a different level of motivation. Therefore, it is proposed to use the mechanism to determine the motivation in the form of linguistic knowledge base. In the presented example it includes a set of student’s motivation pa- rameters, which can be based on the experience of the teacher. A linguistic data base aimed at supporting the process of iden- tifying students’ motivation is presented in Fig. 6. The structure of the linguistic data base created in order to support the process of identifying students’ motivation is presented in Fig. 7. The supporting tools which determine the level of student’s motivation is the linguistic knowledge base [29]. For selected factors influencing the motivation we developed distribution functions of linguistic quantifiers. Two distribution functions of the features are presented in Fig. 8. Fig. 8. Two distribution functions of the features [25]
952 Bull. Pol. Ac.: Tech. 64(4) 2016 Teachers’ and students’ motivation model as a strategy for open distance learning processes
Table 1. Examples of conditions for conducting simulation experiment
Simulation Experiment 1 Simulation Experiment 2 Determination of the number of students waiting for service by Determination of the total time of the teacher assigned for a teacher at a certain interval of time, at a certain probability checking of tasks with a certain probability distribution of outputs, distribution outputs, distribution of student services and the distribution of students service time and the forecasted time of tasks estimated time of verification of tasks checking
Distribution of the arrival of Poisson distribution Distribution of the arrival of Poisson distribution tasks (students): tasks (students):
Distribution of time for student Triangular: minimum – 10 min, Distribution of time for student Triangular: minimum – 10 min, services: most desirable – 15 min, services: most desirable – 15 min, maximum – 30 min maximum – 30 min
Number of service channels: 1 Number of service channels: 1
The time interval: 6 days The time interval: Indefinite
Time assigned for work with 3h/1days Time assigned for work with 3h/1days students (checking of tasks): students (checking of tasks):
Number of students: 55 students (55 complex tasks) Number of students: 55 students (55 complex tasks)
Probability of students’ going completion – 70%, a repository Probability of students’ going to completion – 70%, a repository to one of the outputs: – 15% correction – 15%; one of the outputs: – 15% correction – 15%;
Time to correct the task (delay): 1 day Time to correct the task (delay): 1 day
Results of simulation experiment 1 Results of simulation experiment 2
Name of the counter Achieved value Name of the counter Achieved value
Teacher time spent for checking 60 Teacher time spent for checking 75 all the tasks all the tasks
Total number of tasks addressed 18 Total number of tasks addressed 20 to correction to correction
The number of tasks at the 55 the number of tasks at the 55 beginning beginning
The number of tasks at the 8 The number of tasks at the 9 repository repository
Output of completed tasks 34 Output of completed tasks 46
The number of students who at a certain time do not complete the Time interval allowed completing the cycle of students’ competence cycle of competence acquisition 13 acquisition 8 days
Individual aggregation allows categorizing students within bour input, information about the level of student motivation presented space of the motivation function. If each student de- can be used, which can be helpful in determining e. g. time termines proper motivation and the student are placed in the de- of task execution by a student. Simulation experiments can be picted space, this information may be used to estimate the initial an indication for the teacher whether the expected strategy of setup of simulation e.g. forecasting of students service time (stu- working with the group of students allows for obtaining the dents use of the teacher) or forecasting the probability of outputs assumed level of competence development. On the basis of from the model for different degrees of competence development. the obtained results the teacher can introduce changes to the adopted strategy and check the new conditions in the simulation 6.4. Example of a simulation model for a teacher working model. An example of such activity is presented in Table 1. For with group of students in the process of acquiring compe- conducting experiments a queuing system model in Kendall’s tence. Carrying out simulation experiments can be performed notation M/G/1 was used. It was assumed that students (tasks) in a simulation package Arena. An example of an simulation arrive according to Poisson’s distribution (inter-arrival time dis- model allows considering different values of input parameters tribution), and are served by one server (teacher) with triangular and their impact on earnings. To determine the value of la- service time distribution [30].
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Distribution The forecast distribution of incoming taska of service time
Return of the task to be improved Determine the probability with a specified time of delay of model′s outputs
Fig. 9. An example diagram of a simulation model of the teacher’s work with a group of students in the process of acquiring competence (adapted from [25])
7. Conclusions sults achieved in the didactic process (number of students with a high level of competence – participating in the repository, Building a repository of students’ knowledge can be an im- number of students with an average level of competence, etc.). portant motivational factor for students and for teachers. Stu- Simulated experiments can be a useful indicators for teachers’ dents have the opportunity to actively work with their acquired to know whether their pedagogical strategy works for obtaining knowledge and competences in a way where they can follow the the assumed level of competence development for the students results visually and discuss them with their peer-students and within certain amount of work and time. teachers. The motivation for the teachers can be an element of the learning management and a pedagogical strategy of ODL. The presented approach developed a new way of teaching and References teacher’s involvement, during the process the teacher observes whether the students get the relevant knowledge and compe- [1] A.W. Bates, Technology, E-learning and Distance Education, 2nd ed., Routledge Falmer, New York, 2005. tence goals. The students and the teacher can follow the process [2] O. Zawacki-Richter and T. Anderson, Online Distance Education and relate to the outcome and make the necessary adjustments Towards a Research Agenda, AU Press, Athabasca University, to the desired goals. Finally it is possible for the teacher to 2014. balance the workload with the students’ results. [3] J. Larreamendy-Joerns and G. Lainhardt, “Going the distance The motivation model includes both teacher’s and students’ with online education”, Review of Educational Research 76 (4), interest in extension of the knowledge repository as they both 567–605 (2006). can share and develop the repository into new functions. The [4] G. Rumble and C. Latchem, “Organizational models for open and measure of success in cooperation between the teacher and the distance learning”, in Policy for open and distance learning, eds. students, according to the presented scenario, is the level of H. Perraton and H. Lentell, Routledge Falmer, London, 2004. the repository content development in a given time. The pro- [5] J. Brophy, Motivating Students to Learn, 3rd ed., Routledge, New posed motivation model can be solved on the basis of one of the York, 2010. [6] S. Hrastinski, “Participating in syncronous online education”, known algorithms realizing a cooperative game with dominant Lund Studies in Informatics 6, Lund, Sweden, 2007. strategy (RDS). [7] L. Busk Kofoed and M. Stachowicz, “Problem-based learning To analyse the constrains of this cooperation, the simula- principles in two pedagogical models”, Proceedings from ICIT tion approach is proposed. The simulation allows to compare conference on Information and Communication Technologies in the expected costs that the teacher will bear considering the Education, Manufacturing and Research, Saratov University of assumed repository development (working time) with the re- Technology, Russia, 2012.
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