The e ectiveness of the management of the learning process depends on the adequacy and objectivity of evaluations of the learning outcomes, for which it is necessary to determine latent learning parameters, such as the training level of the trainer and the di culty of the training and testing tasks. However, the existing meth-ods of processing test results based on the Rasch model allow us to determine not the latent parameters of training themselves, but only the relationships between them. In connection with this, it is proposed to use the Markov model of training, which in-cludes three states: the state of knowledge transfer, the state of the training, and the state of knowledge control using testing. The author shows that application of the time spent by trainees to solve tasks in the framework of this model provides addi-tional information on the knowledge, abilities, and skills of trainees, assess the values of latent parameters and, based on this, better manage the learning process. The paper presents a mathematical apparatus used within the framework of the proposed model, including formalized methods of information theory.
The goal of any education system is the management of the learning process. To solve this problem, it is necessary to create a mathematical model that can ade-quately describe and analyze, and then control the learning process [1]. The result of the operation of the education system should be an increase in the level of knowledge, abilities and skills of the trainee. One of the important management problems in modern education system is the control and evaluation of acquired knowledge, abilities and skills. The categories of knowledge, abilities and skills in an automated teaching system in pedagogical work are usually determined in the following way.
Knowledge is the elements of information, in which learning is understood as facts, concepts, rules, algorithms, heuristics, the basic laws of the subject area, as well as decision-making strategies in this area, allowing people to solve speci c production, scienti c and other tasks. However, knowledge does not mean that they will be used.
Abilities are usually understood as the abilities necessary to perform a function in the work. A necessary, but not su cient condition for the manifestation of abilities is the possession of appropriate knowledge.
Skills are the ability to quickly and easily perform the actions necessary during the work. Speaking about the skill, it is necessary to indicate a certain normative level, the achievement of which is necessary for the e ective performance of the work.
A rather detailed review of these methods is given in [
The main goal of this work is the development of a method for determining latent training parameters. A study has been made of the connection between the time-based learning model and the Rasch model, and describes the algorithm for managing the learning process based on the results of training and testing, taking into account the time spent.
Consider a mathematical model of learning that allows you to manage the work of the learning system in di erent modes. This model describes the behavior of the components of the learning system as a function of time.
The object of management in this model is the learning process. The purpose of management is to improve the quality of education.
To solve the problem of managing the learning process, we present the model of the learning system as a system with several discrete states and continuous time. We describe this model as a random Markov process consisting of the following states: the state of knowledge transfer, the state of the solution of training tasks, or training, and the state of control of knowledge, or testing.
Figure 1 shows the Markov model of a training system of three states. The Markov model of a learning system consists of three states: E0 -{ state of knowledge transfer E1 -{ state of training E2 -{ state of testing 0 -{ transition intensity from E0 to E1 0 -{ transition intensity from E1 to E0 1 -{ transition intensity from E1 to E2 1 | transition intensity from E2 to E1
Let the probability that at time t the system is in the state Ei is equal to Pi(t); i = 0; 1; 2. Then the system of Kolmogorov equations describing the behavior of the system in time will have the form 8 dP0(t) = > < dPd1t(t) = > dPd2t(t) = : dt
0 and probabilities must satisfy the normalizing condition P0(t) + P1(t) + P2(t) = 1.
This system is a Markov process of birth and death with three states with continu-ous time and constant intensities.
The paper considers three models of training: 1. The model of learning from one state, which allows you to evaluate the solution to the learner's task or series of tasks in the state of training or testing; 2. The model of learning from two states: active learning (transfer of knowledge and training) and knowledge control; 3. Model of learning from three states: knowledge transfer, training and knowledge control. 2
and the estimate for solving the problem for a time not less than t is [
t p (t) = 1
F (t) = e t
When t = 0 this estimate is equal 1, the rate of change of the estimate decreases with time, the evaluation curve itself is exponential and monotonically decreasing, and t ! 1 tends asymptotically to 0, which indicates a slow-asymptotic nature of the process, and the random variable T obeys the exponential law.
We represent the process of solving a problem as an information processing pro-cess or as a sequence of
elementary Data Transformation Operations (DTO) [
The distribution density of a random variable T is
and the di erential entropy, or measure of the average information processed in the process of solving the problem,
is [
t H(T ) = ln(e= ) = ln e = = e
If minutes are selected as the unit of time, then is measured in min, in - dto=min where dto is one DTO,
and { in logits. Values and corresponding to it are continuous latent variables [
Let the problem be decided by n trainees. In [
The disadvantage of this model is that it takes into account only the measure of the di culty of the task and does not take into account the level of training of the trainee. 3
In this model it is assumed that 1 = 1 = 0
This model describes the learning system as a random Markov process of the "birth-death" type, consisting
of two states: the state of active learning (knowledge transfer and training) and the state of knowledge control
[
Solving the system (1) for P0(0) = 0 and P1(0) = 1, we obtain
P0(1) = lim P0(t) = 0=( 0 + 0); P1(1) = lim P1(t) = 0=( 0 + 0) (3) t!1 t!1 limit values of process parameters.
Imagine the learning process as a process of information processing or as a sequence of elementary DTO [
( dP0(t) = dPd1t(t) = dt is the probability of completing the active learning process in a time not less than t, and is the probability of completing the process of monitoring knowledge in a time not less than t.
From (4) follows the independence of the processes of active learning and knowledge control. According to the
local independence axiom of Lazarsfeld [
F 0 (t) = 1 p 0 (t) = 1 e
0t ; f 0 (t) = 0e 0t: Then the di erential entropy, or the average amount of information, is Accordingly, the distribution function of knowledge control time is If 0 and 0 are measured in min, then 0 and 0 in dto=min, and and in logits.
As a result, the limit value of the probability of successful solution of tasks can be represented by the formula P0(1) = 0=( 0 + 0) = 1=(1 + 0= 0) = 1=(1 + 0= 0) = 1=(1 + e ) of i(k) and (k) are found using iterative formulas j
Values 0 , 0 , 0 , 0 , and are continuous latent variables [
(k) i(k) +i (jk)
(k) i(k) +i (jk)
Pm
j=1 tij ; i = 1; 2; :::; n Pn i=1 tij ; j = 1; 2; :::; m (5) (6)
In the one-parameter model of Rasch's training [
Up to notation fomula (7) coincides with the formula for P0(1) , and [
Let the training be conducted in three stages, and the education system can be in one of three states: the transfer of knowledge, the development of abilities through training and testing skills through testing. Abilities suggest that the learner simply knows how to solve the problem, and skills { that the problem should be solved as quickly as possible. In this case, the maximum permissible or desirable time of solving the problems can be speci ed, in their totality and/or for each task.
To test knowledge, a set of control questions on the topic of my study is used. The development and testing of skills is carried out in the course of training under the guidance of the teacher with the obligatory solution of all training tasks. Testing skills in the testing process presupposes an independent solution for the learner of a set of test tasks. They should be arranged in order of increasing di culty and, in addition, be adequate to the levels of training of trainees. In the case of training, the complexity of tasks should not be lower than the level of training of trainees, because as a result, this level should be increased. The complexity of test tasks cannot be higher than the level of training of trainees, since these tasks include tasks that the learner must already be able to solve, and when checking skills, the speed of problem solving is important. The choice of the relationship between the level of complexity of tasks and the level of training of trainees at the stages of training and testing is an independent task. These relationships can be selected, for example, by one of the iterative methods.
Let n students participate in the learning process. After each stage of the training, you should check the result with the help of test tasks. Let these tasks consist of m questions or tasks. Based on the results of the tests, we will compile two tables consisting of n rows and m columns: table A and table T . Table A contains the results by the dichotomous principle: aij = 1 if the i-th trainee correctly performed the j-th job and, 0 otherwise. Table T contains a set tij { i.e. times spent by the i-th trainee for the execution of the j-th task.
Let us consider the peculiarities of processing the obtained results in the Rasch model and in the model of the training system from two states.
The Rasch model for determining latent parameters is one of the sections of the theory of latent analysis
of Lazarsfeld [
Thus, the Rasch model is not suitable for evaluating the results of the training, in the course of which the learner solves all problems, and the results table A consists of one units.
In the case of testing, in which skills are assessed, the time of solving the problem becomes a decisive factor in obtaining an assessment, because skills require not only and not so much the ability to correctly solve problems (as re ected in table A), but also solve them in an acceptable time (as re ected in table T ). Therefore, you can expect table A of the test results to contain many rows and columns of only one unit. This makes the Rasch model less suitable for evaluating the test results than our model of the two-state training system. 6
Based on the results obtained, the following procedure for operating the training system is proposed.
Before starting the training with the help of an expert group, tasks are selected, problems solved and their
complexity determined [
In carrying out the training, we will adopt the Yerkes-Dodson law as the axiom, which states that "as the
intensity of motivation increases, the quality of activity changes along a bell-shaped curve: rst it rises and then
gradually decreases" [
After the training, testing is performed with the help of tasks, the di culty of which is equal to the current level of training of trainees and the nal level of training of trainees is determined.
The results of the research presented in this article will be used by the author in developing the automated training system at the rate of mathematical logic and the theory of algorithms. 7
Since the work of any training system is divided, as a rule, into two stages, we present the work of the model of training from three States in the form of two interrelated processes: rst, the process of knowledge transfer and training, and then the process of training and testing.
In the process of transferring knowledge and training, the system operates in the states E0 and E1. (see Figure 4), it is assumed that 1 = 1 = 0 .
In the process of training and testing, the system operates in the states E1 and E2. (see Figure 5), it is assumed that 0 = 0 = 0 .
Conclusions 1. Using the Rasch model, in which the evaluation of learning outcomes is carried out by processing responses to test tasks presented in dichotomous and ordinal scales, it is impossible to nd the values of all latent learning parameters, but only the ratio between these parameters. 2. The learning model described in the work, taking into account the time spent on the tasks, allows you to nd more objective values of the assessments for the trainees, determine their level of preparation and the complexity of the tasks.
3. Ultimately, this allows you to more e ectively manage the learning process. Danilova C. D. Adaptivnaya nechetkaya model otsenivaniya rezultatov avtomatizirovannogo testirovaniya s razdeleniem zadaniy po urovnyam usvoeniya dis. kand. tekhn. nauk. { Ulan-Ude: VSHTU, 2005. { 122 S. [17] Serbin V. I. Markovskaya model upravleniya protsessom obucheniya. XII Vserossiyskoe soveshchanie po problemam upravleniya VSPU-2014. Moskva, 16-19 iyanya 2014g.: Trudy [Elektronnyi resurs] Moskva: Institut Problem Upravleniya im. V.A. Trapeznikova RAN, 2014, s. 9489-9497.