2.1 Competencies

The paradigm of acquisition of competencies recently adopted by the Mexican Higher Middle Educational System aims to the development of generic, disciplinary and professional competencies by the students (Figure 1). Pertaining explanations of terms appear in the document entitled Integral Reform of the Higher Middle Education (RIEMS in Spanish) [²].

The development of generic competencies requires to all graduates and is pertinent to all the subjects taught. The disciplinary competencies comprise two categories. They could be basic disciplinary competencies that are intended to be achieved in every subject composing the curricular design at a whole system level. There could be also extended disciplinary competencies whose accomplishment is required to be achieved by the enrolment in the educational scheme pertaining to constituting subsystems. Professional competencies also split into basic and extended categories and with similar associations to the whole system and subsystem levels. Finally, generic competencies comprise 11 different dimensions.

In the present study, we center on generic competencies because they ought to be acquired by every student. Certainly, their development could entail a successful performance in any professional field, thereby increasing hiring chances. Generic competencies compose 11 dimensions and each one integrates a set of attributes that every student must develop (Table 1).

The process of evaluating learning by competencies requires a comprehensive evaluation of the student’s attainment of affective, cognitive and psychomotor characteristics. In the affective aspect, the know how to be at coexistence is promoted, the individual is encouraged to be a good citizen so that he assumes his rights and duties with responsibility and builds her (his) personal identity; in the cognitive aspect, the known to know is stimulated so the student could seek to develop the ability to use a set of tools that allow processing and interpreting information, and provide solutions to situations that arise; in the psychomotor aspect, the know how to do is heartened by focusing on the individual's performance in real life situations, it induces the use of knowledge in a systemic and reflective way to achieve goals; know-how implies an individual's performance in a proficient way in the completion of an activity or in the solution of a problem based on planning and understanding the domain of the problem [²⁷].

An evaluation criterion should aim to assess the development of the student’s acquisition of characteristics associating with the cognitive, affective, and psychomotor domains [²⁹-³⁰]. Using linguistic appraisals of performance conditions, assessment procedures must allow establishing the level of development of competences. Tobón [³⁰] proposed as evaluation elements those corresponding to the affective, cognitive and psychomotor characteristics (Figure 2).

Concisely, an evaluation of acquisition of competencies amounts to collecting evidence through the analysis of learning activities and application of techniques aimed to their assessment. Gonczi and Athanasoub [³¹] establish that the development of competencies must be evaluated integrally by selecting the most suitable methods, for example, written tests, observation, problem-solving, or a combination of techniques depending on the skill or competency to be evaluated, more examples are presented in Table 2.

Regardless of the evaluation technique used by the teacher, for present aims we have considered that evidence of success in a learning activity should base on the display of following elements: knowledge (notions, concepts, prepositions or/and categories), the procedure or technique carried out, and the attitude shown by the student during achieving a certain process. In other words, according to our approach, the evaluation of the effectiveness of each learning activity should consider the formation of listed elements.

The traditional procedures for evaluating learning, for the most part, centers on the proficiency attained by the student at a specific moment and addresses only the cognitive aspect as the basis of the assessment.

The reason underneath adopting a fuzzy inference system is that it offers a scheme that allows evaluation according to a comprehensive perspective, wherein competence assessment considers affective, cognitive, and psychomotor aspects.

All affective aspects are essential; however, we consider that the student’s attitude turns out to be a key feature since it sets; the type of emotion displayed while carrying out a given activity, the way of interacting, or the acting manner in the face of a specific situation or stimulus [³²-³³]. Castillero-Mimenza [³⁰] classifies different types of attitudes according to various criteria (Table 3).

We consider attitude as evidence according to its practical value, how the subject values the environment or situation, positive or negative. The effects of a positive attitude favor optimistic interpretation regardless of the difficulties that may arise, bringing the subject closer to stimulation or action and to pursue achieving goals through a healthy, confident, and generally disciplined way [³⁰].

The effects of a negative attitude maximize the aversive experience, the subject could minimize the relevance of a circumstance, or directly fail to acknowledge the positive aspects of the situation.

A negative attitude can also induce them to withdraw from acting or acquire complaining behavior beyond what is rational, making it difficult to achieve goals [³⁰].

To acquire competencies, the student must achieve a certain number of learning activities. These activities are guided by the teacher, who should also aim to obtain sufficient evidence to determine how efficiently a student develops them.

The teacher can choose any technique (Table 2) to evaluate a learning activity, as long as the criteria include the ranks that the teacher assigns to knowledge, procedure, and attitude, such that the way in which these elements combine, coordinate, and integrate into determining the student's performance can be appraised [⁷].

2.2 Fuzzy Inference System

We use the notation convention described in the Appendix to construct the fuzzy inference system. We begin by defining the membership functions that characterize the fuzzy sets inherent to the fuzzy system’s antecedents and consequents.

Based in an expert system knowledge base, we consider three input variables, Knowledge = X1, Procedure = X2 and Attitude = X3, to be the antecedents of the fuzzy system and an output variable, Efficiency = Y, entailing the consequent. To the input variable X1 we associate the linguistic terms “Little”, “Enough” and “Much”, characterized by z-shaped (Equation (A6)), Gaussian (Equation (A4)) and sigmoidal (Equation (A5)) membership functions, respectively (see Figure 3).

Similarly, the input variable X2 describes by the linguistic terms “Not attempted”, “Incomplete” and “Achieved” and these characterize by means of the

z-shaped (Equation (A6)), Gaussian (Equation (A4)) and sigmoidal (Equation (A5)) membership functions, respectively (see Figure 3).

In turn, the input variable X_𝟑 characterizes by a trapezoidal membership function defined in the Equation (A7) (see Figure 3).In the same way, the linguistic terms “Terrible”, “Very Bad’, “Bad”, “Regular”, “Good”, “Very Good”, “Excellent” and “Outstanding” associate to the output variable Y and characterize by a triangular membership function defined by Equation (A4) (see Figure 3).

The process by which the IF-THEN rules composing the addressed fuzzy inference system are built must consider the relationship among the variables knowledge, procedure, attitude and efficiency (Table 4).

The fuzzy inference system implements through Matlab's fuzzy logic toolbox (version 2016b). This aims to evaluate the efficiency given the input values x1, x2 and x3 (see Appendix and Figure 4).

Building the fuzzy inference system relied on the Mamdani-Type method [³⁴] (see Appendix). The minimum operator modeled the AND expression (fuzzy intersection in Equation (A9)), and correspondingly the OR expression by the maximum operator (fuzzy union in Equation (A9)).

In turn, the implication defined by the THEN expression modeled by the minimum operator and the aggregation by the maximum operator. The defuzzification step relied on the center of gravity procedure [³⁵] defined in Equation (A14).

2.3 Alignment of Competencies Method

To line up the competencies with the objectives of the learning activities, an alignment matrix is initially constructed. This formalizes the mathematical relationship of the attributes of each competency with each one of the learning activities included in each unit of the study program. Relationship is made by indicating which attributes will be integrated in the evaluation of each learning activity, this relationship synthetically represents the alignment process.

From left to right, the attributes of the competencies are placed in the first column and from the second column the corresponding learning activities are placed in each unit. When an attribute is involved in a learning activity, the cell corresponding to the intersection of the attribute row with the column of the learning activity is marked, and this is done with the rest of the activities.

Once the alignment matrix is completed, it is easy to know which and how many attributes and competencies are integrated in a learning activity. This allows to calculate the value of the weight of each learning activity and each unit included in the study program. The grade of development of the competencies acquired by the student or by a group will be evaluated using named weight values. The grade is the percentage value of the development of the acquired competencies.

A study program can be structured by a number n of units and each one of them by another number m of learning activities (Figure 5). That is, let Ui, for i=1…n, be the ith unit and let Hij, for j=1…m, be the jth learning activity of the unit Ui.

To illustrate how the alignment method works, we will consider as an example a study program containing three units; unit U1 involving learning activities H1j, for j=1,2,3, unit U2 with learning activities H2j, for j=1,2, and unit U3 that integrates learning activities H3j, for j=1,2,3. Similarly, the h−th competence denotes by Ch, for h=1,2,…,11 (Table 1) and the k−th attributes of a competence Ch represents through Chk, that is, the index k designates the kth attribute of the competence Ch (Table 5), we create the alignment matrix and make the alignment indicating with the value of 1 that attribute Chk is integrated in the learning activity Hij, (Table 5).

So, let V(Hij) be the number of competence attributes integrated in the learning activity Hij, which can be expressed as:

where the symbol Chk(Hij) denotes the attribute k of the competence Ch integrated in the learning activity Hij (Table 5) and p represent the total number of attributes of the competence Ch. Likewise, V(Ui) represent the total number of attributes integrated to the unit Ui, which is obtained from the expression:

and the total sum of attributes V integrated in a study program will be calculated by:

Considering the information presented in Table 5, we take Equations (1) and (2) to obtain the total number of competence attributes integrated in Hij and Ui respectively (Table 6).

To finish the description of the alignment method, we calculate the values of the weights, vij and ui, corresponding to the alignment matrix. The weight value vij is obtained through:

And the weight value ui is obtained through:

For the addressed example, we calculate the weight values, vij and ui, that correspond to the alignment matrix (Table 5) through the Equations (4) and (5) and considering the total number of competence attributes integrated in the learning activities Hij and in the unit Ui from Table 6 (Figure 6).

The alignment method consists of establishing an implicit relationship between the attribute Chk to be evaluated and the learning activity Hij where it will be evaluated.

The teacher could select the most appropriate technique (Table 2) to evaluate the learning activity and could establish the criteria to collect the evidence (knowledge, procedure and attitude) required in our fuzzy system (Figure 4) to evaluate the efficiency with which the student developed the activity. Our method requires that the teacher indicates which competence attributes (into the alignment matrix, e.g., Table 5) are considered in each learning activity of the study program, then the educational evaluation system calculates the weights, vij and ui.

Keep in mind that in no way will it be possible to change the weights values, vij and ui, manually.

These values only can change when the alignment modifies. Also, it must be considered that the same competence attribute could be evaluated in one or more learning activities of one or more units.

Therefore, the student's progress, or in other words, the grade of development of the competencies acquired by the student is determined in a percentage way considering the tree forming by the resulting weight values (e.g., Figure 6) of the alignment matrix (e.g., Table 6).

2.4 Grade of Development of Acquired Competencies

The evaluation of the efficiency Eij bases on the quantitative and synthetic conception of the components explaining the performance displayed by the student in the courses of a learning activity Hij.

Then, through the resulting weight values (e.g., Figure 6) of the alignment matrix (e.g., Table 6) and the efficiency Eij, the educational evaluation system determines the grade of development of acquired competencies over a period within the study program. This is as follows: The grade of development of the competencies acquired by a student at the end of the learning activity Hij is denoted by Ds(Hij) and calculated through:

where s denotes the sth student, the efficiency value, Eij, is provided by the fuzzy system and vij is the weight value corresponding to the learning activity Hij. Likewise, Ds(Ui) denotes the grade of development of the competencies acquired by the sth student while completing unit Ui and obtained by:

where ui is the weight value corresponding to the unit Ui. In addition, the grade of development of the competencies acquired by the sth student while completing the study program, Ds, is obtained by:

Taking Equations (6-8) as a reference, it is possible to calculate a series of values that represent statistical parameters of the educational system that are relevant at obtaining information about students or groups.

For example, the number N+ of students who acquired their competencies at the end of the study program with a grade of development Ds greater than P or less than P, and correspondingly the number N− of those achieving a developmental grade smaller that P, can be calculated by:

where P is a perceptual value and ns is the number of students to be evaluated.

But if what we want is to calculate the percentage value associating to N%+ of N+, or that one N&− corresponding to N−, then we can use the following expressions:

In addition, we can obtain the average percentage of the grade of development of the competencies acquired by a group is obtained through:

Even more, it is possible to modify the Equation (9) if what we want is calculating the number NH+ of students who acquired their competencies at the end of a learning activity Hij with a grade of development greater than P, namely:

Similarly, if we want is calculating the number NH− of students who acquired their competencies at the end of a learning activity Hij with a grade of development less than P, then:

Respectively. Likewise, if what we want is calculating the number NU+ of students who

acquired their competencies at the end of the unit Ui with a grade of development greater than P, then:

or less than P:

3.1 Study Case Based on Simulated Data

To illustrate the performance of offered evaluation scheme, we first consider a try based on a simulated data set. We conceive a hypothetical group of 50 students, assumed to have enrolled in a study program structured in three units: unit U1 involving learning activities H1j for j=1,2,3, unit U2 composing learning activities H2j for j=1,2 and U3 with H3j for j=1,2,3 learning activities.

Following the procedure in Section 2.3, we create the associating alignment matrix, but first, we achieve this task randomly, on its course, indicating the Chk attributes integrated within the learning activity Hij (Table 7). As we mentioned, the teacher ought to achieve the alignment systematically, thereby only in extreme cases could they only align the competencies at random. However, the analysis of random alignment requires case attains relevance because it stands for the worst expected scenario for an inconsistent evaluation. In this sense, we can explore the suitability of the proposed method even in extreme cases of subjective alignment.

According with the alignment matrix presented in Table 7, we obtained the total number of attributes Chk integrated in Hij and Ui through Equations (1) and (2), respectively (Table 8).

In turn, weight values vij and ui calculated through Equations (4) and (5), respectively (Figure 7). Also, we considered simulated values for x1, x2 and x3 generated so that the highest concentrations of values distributed around a given mean (Tables 9, 10 and 11).

Figure 8 shows the distributions of the x1, x2 and x3 values fitted to a non-normal distribution curve. In Figure 8 domed lines differentiate the interval where the distributions of the input values x1, x2 and x3 concentrates for each learning activity Hij.

Shown concentration intervals help the teacher visualizing the possible effects of the combinations of x1, x2 and x3 values over the performance of the students during the development of their activities.

For example, we can observe in Figure 8b that in all cases knowledge rated insufficient, however, an attitude with very high values in x2 associates to a completed procedure. Contrary to Figure 8b, it is possible to observe in Figure 8a a non-attempted procedure, despite high values in attitude and knowledge. In this case, the teacher can analyze and determine, if the technique and the teaching-learning strategy are adequate to evaluate the efficiency.

An efficiency value Eij corresponding to each learning activity Hij calculates from running our fuzzy inference system (Figure 4).

In performing such a task, input values are those as simulated for the hypothetical group of 50 students (Tables 9, 10 and 11). We show resulting Eij values in Figure 9.

Figure 9 provides the teacher a clue on the performance of the students in each one of the learning activities Hij. For example, if most students obtained values of Eij greater that 50% then this would mean that knowledge was at least sufficient, that the procedure was attempted in half of the associated learning activities associated, and that the attitude was more positive than negative.

But, in our hypothetical study case, there is a lot of variability in the distribution of the Eij values presented in Figure 9. However, it is possible to identify the values x1, x2 and x3 belonging to those students who had the lowest Eij ratings, which allows to perform analytics and conceiving better teaching strategies aimed to improve learning.

Equation (6) allows calculation of Ds(Hij), interpreting as the grade of development of the competencies acquired by the sth student when performing the learning activity Hij, (Figure 10). Acquiring frequency distribution plots of resulting numbers permits visualizing the behavior of the grade of development of competencies.

Similarly, Equation (7) endures calculation of Ds(Ui) standing for the grade of development of the competencies acquired by the qth student in unit Ui, (Figure 11).

Additionally, Ds, identifying the grade of development of the competencies acquired by the 𝑠𝑡ℎ student at the end of the study program, calculates by means of Equation (8) (Figure 12). Then, adding the results at the individual level, we obtain the grade of development of the competencies acquired by the student’s group at the end of the study program.

Figure 10b shows the distribution of the grade of development of competencies concentrated in an interval with values very close to the maximum weight value corresponding to the learning activity H12. Only the grade of a few students places outside this range. Such a result implies that they all performed their activities with the best attitude but with little knowledge and with the procedure attempted most of the time (Figure 8b). All students completed the whole procedure due to the coincidence of a positive attitude that favored obstacle elimination, in this case the barely small knowledge, and which also promoted pursuing objectives.

Otherwise, in Figure 10h the distribution of the grade of development of competencies is concentrated in an interval with values above the mean of the maximum weight of corresponding to activity H33 (Figure 7), only the score of a few students is outside this range. This means that despite knowledge being predominantly sufficient, most of the students did not totally take on the procedure because the attitude was oscillating between the negative and the positive (Figure 8h).

There is a transition point between a negative and positive attitude, called a neutral attitude; it is not very common. Our fuzzy inference system characterizes the attitude with a small interval that defines the transition from negative to positive. The neutral attitude renders the subject's action ineffective, causing a loss of the notion of time or continuity from the left-off point.

The teacher can analyze the results of Figures 8-10 and being aware at detail what is happening with the elements that rate measure the performance of students while achieving the learning activities.

The teacher could identify problems in assessment techniques or at teaching-learning strategies with the advantage of modifying or changing them to achieve other objectives during the learning sequence. It is necessary to clarify that the results presented in Figure 10 and Figure 11 do not determine whether the students are competent or not. At the end of the study program, it is possible to assert know to what grade a student is competent or not competent (Figure 12).

In addition, it is possible to ascertain the competences that a student or a group developed through Table 7.

It is also possible to learn from Figure 12, that the maximum grade of development of the competences acquired by the students was less than 65%. Only 13 students who represent 26% of the total number in the group completed learning activities with a grade grater that 60% but less than 65%. A 75% of the total number of students developed the learning activities with an insufficient score, that is, with a grade of achievement lower than 60%. The conclusion is that the study program had many deficiencies in teaching-learning strategies around the development of learning activities and possibly in evaluation techniques. It would be necessary to review, what was what happened? This question can be only answered once the study program completes.

Moreover, it is recommended to answer this question after having completed each one of the learning activities. The average grade of development of the competences acquired by the group was 58.37% (Equation (9)), which is too low. Recall that this study case considers simulated input values, it is obvious for other different input values the average grade of development of the competencies acquired by the group will change. It could be higher or lower than that obtained in this test.

In addition, if the competencies are aligned with the learning activities in a different way to that presented in Table 7 then the results of alignment presented in Table 8 would attain other values, so the weights shown in Figure 7 would modify, this in turn will produce a different final result than presented in Figures 10, 11 and 12. In order to explain these effects, we consider different weights to those presented in Figure 7, which are hypothetically generated, that is, these weights are not associated with an alignment matrix but we assume that they should be (Figure 14).

Then, as described, we can obtain the grades of development of competencies. The task completes through Equation (6), Equation (7), and Equation (8) by using hypothetical weights (Figure 12) and simulated input data (Tables 9, 10, and 11). Then we proceed to compare them with those shown in Figure 11 and Figure 12 through their distribution in frequency graphs (see Figure 14 and Figure 15, respectively). This proves that whenever the weights vij and uij change the results also change. Furthermore, we obtained the average grade of development of the acquired group's competencies using Equation (11). We found a value of 24.88%, which turns out to be much lower than a calculated one of 58.34% obtained for the simulated data study case. Recall that the simulated data is generated randomly, thus precluding any significance or relationship either with the student's behavior or with the learning activities.

Equation (12) allowed calculation of the number NH+ of students who acquired their competencies at the end of a learning activity H13 with a grade of development P greater than 60%, that is P>60% (30% of u13).

If we take the Ds values corresponding to the study case based on simulated data, we obtain NH+=40, which typify 80% of ns.

Likewise, if we aimed at calculating the number NH+ of students who acquired their competencies at the end of a learning activity H13 with a grade of development P greater than 60%, that is, P>60% (8.52% of u13), but taking Ds values corresponding to the hypothetical study case data, we would have obtained NH+=38, which represent 76% of ns. In both the simulated data case and the hypothetical study cases, the NH+ values are different.

Correspondingly, if the goal is obtaining the value of NH− with a grade of development P less than 60%, that is P<60%, then the result will be the complement of the value of NH+ obtained in both cases, but it is also possible to obtain NH− through Equation (13). This exercise can be illustrative whenever a comparison of the grade of development among learning activities Hij of the same Ui requires. Also, if the aim is to make comparisons among the grades of development at the end of the units the task could be achieved through using D and Equations (14) and (15). In addition, to know the grade of development of the competencies acquired, it is mandatory knowing which competences the student developed. This is only possible through obtaining the alignment matrix.

3.2 Study Case Based on Real Data

In the addressed education evaluation system, a study program can compose several units (Unit), and each one can associate several learning outcomes (RA) associated with their corresponding learning activity (AE). However, it is possible that each learning activity (AE) be divided into one or more sub-activities called “indicators”. Adapting data from Romero-Escobar in [³⁴], we can get a feel for an experimental study case.

The reference includes the results obtained in the study program corresponding to the subject "Operating Systems Management", which is pre-defined by the educational system in two units that comprise two learning outcomes (RA), and each one of them associating two activities to evaluate (AE) with 4 indicators each (Table 12). Percentage value of indicators is also pre-defined by the educational system.

In the Table 12, the scores of the 27 enrolled students identify each one by means of a letter. The letter 'E' stands for 'Competent', the letter 'S' signifies 'Sufficient' and the letter 'I' connotes 'Not Competent'. The official grading system does not consider other values for the score of the students. The meaning of 1.1.1 is: Unit 1, Result of learning 1 (RA 1) and Activity to evaluate 1 (AE 1), respectively. In addition, the report by [¹²], the evaluation of the activities (AE) was not carried out in such a way that the adopted techniques could be considered as a factor inducing evidence of the display of the indicators considered in the present study (knowledge, procedure, attitude).

In order to explain how the current education evaluation scheme works, we present in Figure 16 the symbolic representation of the elements of Table 12 describes as follows: the symbol Ui, for i=1…n, stands for the ith unit, using AEij, for j=1…m, we denote the jth activity of the unit Ui to evaluated, the symbol Fijz, for z=1,2,3,…, will stands for the zth indicator of the jth activity of the unit Ui to be evaluated, wij standing for the weight of the activity AEij, and finally, Iijz taken to symbolize the weight corresponds to Fijz. Wis is the percentage value corresponding to Ui

Using the listed notation convention, we go ahead and furnish the mathematical expressions that the education evaluation system takes into account at obtaining the results of Table 12. Alongside this, we elucidate what’s wrong with this evaluation scheme. Such a protocol assigns each evaluation result AEij of an activity of the sth student as follows:

where G={IijZifFijZ=E,0.775×IijZifFijZ=S,0.250×IijZifFijZ=I,

and where nz is the total number of indicators in AEij. In the Equation (16), we can be aware that the scheme of evaluation of the educational system considers only three qualitative values as rating, I, S, and E, the students who had a rating equal to ‘I’ will have only 25% of the value of IijZ.

Likewise, the students who had a rating equal to ‘S’ will credit for only 77.5% of the value of IijZ. It should be noted that the percentage represented by IijZ determines by the teachers based on their own criteria and can be arbitrarily modified during the subject’s program, if the teacher so wishes. In the other hand, If the teacher does not divide the activity to be evaluated (AE) into indicators then the official scheme evaluates the activity AEij as follows:

where the Fij value corresponding to rating to the activity AEij. For the unit, Wis, the educational system evaluates the sth student as follows:

The final evaluation, Ws, of the sth student is given by:

and the average rating of the group is obtained by:

which is W=8.62. Romero-Escobar [¹²] did not consider a formal relationship between competencies and learning activities to create the alignment matrix. Therefore, in contrast to the entries presented in Table 6, it is not possible to obtain the total number of attributes integrated into each activity to evaluate AEij. Neither could we include the weights corresponding to each activity AEij to as shown in Figure 6. Besides, this restriction extends to the values of each unit Ui. However, we can form a tree like diagram such as that appearing in Figure 6, provided we use as weights wij the ones taken from Figure 16.

The weights shown in Figure 17 are not the result of an alignment of competencies with the activities AEij to be evaluated. The official educational grading scheme relies on Equations (18) and (19) to obtain the rating after completing a unit and once the end the study program achieves (Figure 16), respectively. In other words, the rating of the unit Wis is the sum of ratings corresponding to each activity AEij and the Ws final rating is the sum of ratings corresponding to each unit Wis.

To acquire the comportments of the AEij values associating with the real data study case, we consider the Eij efficiencies at which the students performed the learning activity Hij, as pertaining proxy values, that is, we take Eij=AEij. Recall that our method estimates Eij values using a fuzzy inference system (Figure 4). Besides, since for the real data study case an alignment matrix is not available, we take the one corresponding to the simulated data study case as a possible surrogate (see Table 8). Given that, we only consider the first and second unit, both with the first and second corresponding activity Hij. Similarly, while analyzing the study case based on simulated data, the possibility of a random alignment was considered. It corresponds to the utmost subjective scenario compared to a consistent evaluation, so for this essay, taking as a base the random alignment associating with the case mentioned above renders convenient.

We calculate the total number of attributes integrated in the learning activity Hij through Equation (1) and the total number of attributes integrated into U_i through Equation (2) (Table 14). And its weights vij and ui obtained through Equations (4) and (5), respectively (Figure 18).

Let’s show in Figure 19 the distribution of Eij presented in a frequency graph. In panel (a) the distribution is concentrated in the highest value, 100%, and only three students presented evaluations lower than 100%. This means that the students performed the learning activity H11 skillfully and that the evaluation technique and the teaching-learning strategies were correct. In addition, more than 71% of the students developed their skills with a grade equal to 100% in the learning activities H12, H21 and H22. In general, evaluation techniques and teaching-learning strategies were the most appropriate. Values of Eij less than 77% correspond only to 11% of the total number of students.

Considering the weight values presented in Figure 18, we calculated through Equation (6) the grade Ds(Hij) of development of the competencies acquired by the sth student, while achieving the learning activity Hij (Figure 20).

In all the learning activities presented in Figure 20, a 71% of the total number of students developed their competencies with a grade equal to 100%. Furthermore, Figure 20 shows that an 18% of the total number of students developed the competencies with a grade lower than 100% but higher than 70%. A quite significant result for the educational system and for the curricular achievement of the teacher. If the evaluation techniques in [¹²] had considered the elements to evaluate presented in the simulated data study case, then, it could be said, that almost all the students acquired all the knowledge, that the procedures were achieved and that the attitude of the students was always positive throughout the study program. The results could always be biased towards the maximum grade of development using the qualifications of any of the groups or subjects presented in [¹²], due to the evaluation criteria that the educational system manages.

To calculate Ds(Ui) the grade of development of the competencies acquired by the sth student when ending all the learning activities of unit Ui, we use Equation (7) and the weight values presented in Figure 18 (see Figure 21).

In the same way, we obtain the Ds grade of development of competencies acquired by students throughout the study program (Figure 22) through the Equation (8).

Finally, we obtain the average percentage of the grade of development of the competencies acquired by the group through Equation (11), which is D=8.67. This value is scarcely differing to the one obtained by the educational evaluation system because the values of the weights ui are in almost equal proportion (see Figure 17 and Figure 18).