Cognitive Modeling of Learning Process

Cognitive model of the educational process

The educational process in an educational institution is a complex, semi-structured system that depends on a large number of vaguely defined factors. As is known (Mikryukov & Mazurov, 2022), one of the ways to describe such systems is the use of cognitive modeling, in which the description of the relationship between system parameters is given in the form of cognitive maps.

Cognitive Map (Cognitive Map from cognitio - knowledge, cognition) is a tool of cognitive modeling methodology designed for analysis and decision making in ill-defined situations. It is based on modeling the subjective perceptions of experts about the situation and includes:

• methodology for structuring the situation;
• a model for representing expert knowledge in the form of a signed digraph (cognitive map)CM={F,W}; where F is the set of basic factors (concepts) of the situation, W is the set of cause-and-effect relationships between the factors of the situation.

A cognitive map can be understood as a schematic description of the worldview of an individual related to a given problem situation. Therefore, with the construction of cognitive maps, one can begin the study and modeling of complex semi-structured systems to find alternatives to a decision and make it.

A cognitive map can be visualized as a graph containing many vertices, each of which corresponds to one factor or element of the individual's worldview. The labeled arc connecting vertices and corresponds to a causal relationship $A\Rightarrow B$, where is the cause, is the effect. A relationship $A\Rightarrow B$, is called positive (marked with a “+”) if an increase in leads to an increase (strengthening) of, and a decrease in leads to a decrease in, all other things being equal. The “–” sign above the arc means that the relationship is negative, i.e. ceteris paribus, an increase in leads to a decrease (inhibition) of and a decrease in leads to an increase in. On a cognitive map, in addition to the sign, one can indicate the relative degree of influence of one factor on another in the form of a numerical assessment (for example, in the range from +1 to - 1). In this case, we obtain fuzzy cognitive maps.

Cause-and-effect relationships can also be described as a weight matrix, which displays the influence of each factor on all the others.

As an example, let's consider simplified versions of cognitive maps compiled by the authors for the aforementioned area of bachelor's training on 09.03.01 (we will take universal and professional competencies for analysis) based on a survey of specialists working with students in this area (Tables 1 and 2). These maps provide a list of factors of the educational process and consider their interaction, the influence on each other, including the level of competence.

The map for assessing universal competencies includes 3 competencies from the list given in table 1. It is compiled using a continuous scale of assessments of the influence of factors lying in a continuous range from -1 to +1.

The map for professional competencies contains three groups of competencies from the specified list, and the assessment of the influence of factors is carried out using three values: -1, 0 and +1.

As experience shows, the compilation of cognitive maps is a rather complicated process that requires careful selection of experts and gradual coordination of their opinions. However, this issue is not discussed in this work.

Based on the above tables, weight matrices were compiled for universal and professional competencies - respectively and.

As shown in Moskaleva and Dorrer’s (2012) research, the analysis of these matrices makes it possible to judge the stability of the process described by this cognitive map. To do this, it is necessary to analyze the eigenvalues of the matrix, and if they all lie inside the unit circle on the complex plane, then the system is stable, but if at least one eigenvalue is greater than one in absolute value, then the system is unstable.

In particular, for the considered cognitive maps, we obtain the following results.

For the cognitive map of universal competencies, the maximum eigenvalue is 4.34, which indicates the instability of the system. In the case under consideration, the instability of the process described by the cognitive map indicates the positive dynamics of the system, its ability to self-develop.

For one of the blocks of the cognitive map, the maximum eigenvalue is 1. This indicates that the system is on the border of stability and is not capable of self-development.

Model of educational system dynamics.

For further analysis of the system and building a model of the control system, all factors included in the cognitive map can be divided into control and manageable. Those factors that are available for variation and that can influence the rest should be referred to as control factors, and the rest of the factors, including the target ones, become controllable. The latter include levels of competence. Considering that the acquisition of competencies is a dynamic process, it should be represented as a discrete time equation based on a cognitive map

Turning to the construction of a model of the educational process, we will consider it as a sequence of certain stages (for example, semesters) during which learning takes place, i.e. increasing the competence of trainees.

Let us introduce the following notation:

$t=\mathrm{0,1},2,\dots ,T$ - discrete time (months, semesters, years - depending on the specifics of the task), - depth of process planning;

$x=\mathrm{0,1},\dots ,N+1$ – stages of the educational process (for example, semester numbers); corresponds to the applicant, $x=N+1$ corresponds to the graduate;

$P\left(t,x\right)$ –vector of factor values that determines the current state of the system (including the level of competence of trainees), where n is the number of factors taken into account in some conventional units.

Let's single out the input and output factors in the list of factors. Renumbering the factors, if necessary, so that the input factors are at the beginning of the list, we represent the column vector in the following form

$P(t,x)={\left[\begin{array}{c}U(t,x)\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}S(t,x)\end{array}\right]}^T$, (1)

Where $U\left(t,x\right)$ –vector of input factors,–-vector of controlled factors. The sign means matrix transposition.

The weight matrix can be represented in the following block form

$W=\left[\begin{array}{cc}{D}^{\mathrm{\text{'}}}& C\\ B& A^{\mathrm{\text{'}}}\end{array}\right]$. (2)

Here:

is a matrix of dimension, which determines the impact of input factors on the controlled ones,

’ – matrix of dimensions which determines the mutual influence of controlled factors,

- matrix of dimensions, which determines the influence of controlled factors on the input,

' is a matrix of dimensions, which determines the mutual influence of input factors.

We assume that the matrix is unchanged for the entire period of study and that the discrete time coincides with the stage of the educational process, i.e. with the semester number, so.

Then the dynamics of the educational system, which develops both according to its own internal laws and under the influence of input influences, can be represented by the equation

$P(t+1)=P\left(t\right)+\mathrm{\Delta }P\left(t\right)+\xi \left(t\right)=P\left(t\right)+W\cdot P\left(t\right)+\xi \left(t\right)$,(3)

where $\mathrm{\Delta }P\left(t\right)=W\cdot P\left(t\right)$ is the increment of the parameter vector in one time step (per one stage of the educational process), which is also called an impulse in the literature (National Qualifications System (NSC) National Qualifications Development Agency, 2022), $\xi \left(t\right)$ n is the vector of random noise affecting the educational process.

Equation (3), taking into account the structure of the vector (1), the matrix (2), and the rules for working with block matrices, takes the form

$\begin{array}{c}P(t+1)={\left[\begin{array}{cc}U(t+1)& S(t+1)\end{array}\right]}^T=\\ =\left[\begin{array}{c}U\left(t\right)\\ S\left(t\right)\end{array}\right]+\left[\begin{array}{cc}{D}^{\mathrm{\text{'}}}& C\\ B& A^{\mathrm{\text{'}}}\end{array}\right]\cdot \left[\begin{array}{c}U\left(t\right)\\ S\left(t\right)\end{array}\right]=\left[\begin{array}{cc}D+I_m& C\\ B& A^{\mathrm{\text{'}}}+I_{n-m}\end{array}\right]\cdot \left[\begin{array}{c}U\left(t\right)\\ S\left(t\right)\end{array}\right]=\\ ={\left[\left(D+I_m\right)\cdot U\left(t\right)+C\cdot S\left(t\right)+{\xi }_{1}B\cdot U\left(t\right)+\left(A^{\mathrm{\text{'}}}+I_{n-m}\right)\cdot S\left(t\right)+{\xi }_2\right]}^T.\end{array}$ (4)

Here and are identity matrices of the corresponding dimension.

Denote

$\left(A^{\mathrm{\text{'}}}+I_{n-m}\right)=A,\hspace{0.5em}{D}^{\mathrm{\text{'}}}+I_m=D$ (5)

Since we are interested in controlled factors (including target ones - indicators of competence), and the influence of matrices and can be ignored, then the vector equation (4) should consider only the second component. As a result, we obtain an equation for the dynamics of controlled factors in the form of a standard linear equation for a controlled system in discrete time:

$S(t+1)=A\cdot S\left(t\right)+B\cdot U\left(t\right)+{\xi }_2\left(t\right)$, (6)

Here

t=0,1,…,T, ${\xi }_2\left(t\right)$,– n-m - random noise vector.

Vector is the control and is set during the study according to the scenario of the system development. Matrices and allow us to evaluate the controllability of the system, i.e. its ability, due to the choice of the control action, to obtain the required values of the controlled parameters (Zhilov, 2016).

System (6) should be considered under the initial conditions:

At $t=0,\hspace{0.5em}S\left(0\right)=S_0$ (7)

where $S_0$ is the given initial distribution of controlled factors.

Models of the educational system management process

Consider the procedure for assessing control actions. Since both control and controlled factors are vectors -and, respectively, the task of managing the educational process is a multicriteria and multivariate task. Therefore, to study the dynamics of the system, a scenario approach will be applied, in which the values of the control factors are set expertly and, using a mathematical model, the values of the output factors are calculated. In this case, the control law of the system depends on its dynamic properties determined by the eigenvalues of the matrix. If the system is stable, i.e. Since all eigenvalues of the matrix A lie inside the unit circle, then additional external influences are required to transfer it to a new level. If the system is unstable, i.e., capable of self-development (the matrix has eigenvalues, modulo greater than 1), then to move to a new level, it is necessary to turn off the stabilization mechanism and set the initial conditions correctly.

So, two modes of operation of the system will be considered:

Let us consider the control models for the indicated modes.

With a steady process, this function should ensure the maintenance of a given level of controlled factors providing negative feedback on the deviation from the required level:

$U\left(t\right)=K\left(t\right)\left(S\left(t\right)-S_u\left(t\right)\right)$, (8)

Where $K\left(t\right)-m^{*}\left(n-m\right)$ is the matrix of influence coefficients that determine the speed at which the level of controlled factors at all stages of the educational process approaches the required level Su when the system is unbalanced.

It is easy to see that if there is a property of controllability of the system, a small amount of interference, and after a sufficient time has elapsed, the educational process will “attract” to the required level and will fluctuate around it under the influence of interference. The meaning of the coefficients of the matrix is that they determine the efficiency of the system self-regulation process, i.e. level of management and management.

Let us now consider the case when the educational process needs to be transferred from some achieved level to a new, higher level. Here an important role is played by the ability of the system to self-development, which is determined by the properties of the matrices W and, in particular, by their own numbers, and, ultimately, by cognitive maps that reflect the opinion of experts.

If the educational system is capable of self-development, then control actions should only be of a corrective nature to bring priority competencies to the required level.

In the case when the system is not capable of self-development, certain external efforts are required to move to a new level. The system needs to overcome the attraction of the level and then move to the area of attraction of the level Then, after a certain period of time, the process will be established in the region of the new center of attraction. The features of such control are considered in (Dorrer et al., 2008).

Model for the growth of universal competencies

Let us consider the dynamics of managing the growth of universal competencies in the educational system based on the cognitive map (Table 4) and the model outlined above.

Based on the list of factors given in the cognitive map, we choose the following as control factors (components of the vector).

- Qualification of teachers,

- Educational and methodological support,

- Hardware and software,

- Living conditions,

- Pre-university education.

We will choose the following competencies as controlled factors (components of the vector).

- Ability to work in a team,

- Ability to build oral and written speech,

- Proficiency in a foreign language,

- Ability to work with documents,

- Motivation for learning,

- Social adaptation.

Having compiled a matrix of mutual influences of factors in accordance with Table 4, presenting it in block form in accordance with formulas (2) and (3), we obtain the matrices and:

$A=\left(\begin{array}{cccccc}1& 0.6& 0.3& 0.3& 0.5& 0.5\\ 0.9& 1& 0.7& 0.8& 0.2& 0.5\\ 0.2& 0.8& 1& 0.4& 0.6& 0.3\\ 0.1& 0.7& 0.6& 1& 0.7& 0.2\\ 0.9& 0.8& 0.5& 0.3& 1& 0.7\\ 0.8& 0.7& 0.4& 0.7& 0.7& 1\end{array}\right),\hspace{0.5em}B=\left(\begin{array}{ccccc}0.5& 0& 0.4& -0.2& 0.1\\ 0.7& 0& 0.1& -0.3& 0.6\\ 0.3& 0& 0.5& -0.3& 0.8\\ 0.4& 0.3& 0.4& -0.1& 0.2\\ 0.3& 0.6& 0& -0.8& 0.7\\ 0.3& 0.5& 0& -0.7& 0.7\end{array}\right)$

Maximum eigenvalue of matrix ${\lambda }_{\text{max}}=3.732$, which indicates a high ability of the system under consideration for self-development.

The level of competence will be assessed in some arbitrary units, which allow us to evaluate the dynamics of these indicators over time. In this case, the zero value $S_i=0,\hspace{0.5em}\left(i=\mathrm{1,2},\dots ,6\right)$ indicates a lack of relevant competence, and negative values indicate a negative attitude of students to this factor.

Let us set the initial state of the competence vector in the following form

$S_0=\left[\begin{array}{cccccc}0.1& 0& 0& 0& -0.1& -0.1\end{array}\right]$.

The level of control actions will be evaluated in relative units, lying in the range from -1 to +1. In this case, the positive value of the factor $U_k\left(k=\mathrm{1,2},\dots 5\right)$ indicates its positive impact on competence levels, a negative value indicates - a negative impact of this factor, and zero - indicates a slight or no effect.

The system control scenario is given as a set of vectors of control actions in each semester of the educational process (Table 3).

The results of modeling the dynamics of competence indicators according to formulas (6), (7) are shown in Table 4.

We see that due to the high ability of the modeled system for self-development, which was determined by experts when compiling a cognitive map, competencies grow at a high rate with minimal control actions, especially in the last semesters. This corresponds to the real state of affairs, since the specialist is especially intensively formed precisely in the last years of training.

Model for the growth of professional competencies

Let us turn to the cognitive map of professional competencies according to the methodology discussed above.

As control factors, we will leave the same ones that were chosen in the previous example.

Controlled factors in accordance with Table 3 are selected as follows:

- Design and technological activities

- Research activities

- Scientific and pedagogical activity

- Motivation for learning

Representing the matrix of mutual influences of factors (Table 3) in block form in accordance with formulas (2) and (3), we obtain matrices and:

$\text{A}=\left(\begin{array}{cccc}1& 0& -1& 1\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 1& 1\end{array}\right),\hspace{0.5em}\text{B}=\left(\begin{array}{cccc}1& 1& 1& 0\\ 1& 1& 1& 1\\ 1& 1& 1& 1\\ 1& 0& 0& 1\end{array}\right)$

Maximum eigenvalue of matrix A ${\lambda }_{\text{max}}=1$, which indicates the inability of the system under consideration to self-development.

Let us set the initial state of the competence vector in the following form

$S_0={\left[\begin{array}{cccc}0& 0.1& 0.5& 0\end{array}\right]}^T$.

The system control scenario is given as a set of control action vectors in each semester of the educational process (see Table 5).

The results of modeling the dynamics of competence indicators according to formulas (6), (7) are shown in Table 6.

In this example, to obtain the values of controlled factors comparable with the previous example in order of magnitude, the maximum possible control actions were required (Table 3), which confirms the conclusion that the system is incapable of self-development.

It should be noted that in both examples the picture of the growth of competencies is qualitatively similar to the data in Tables 1 and 2.