## 2 Materials and Method
### The Rubric
#### 2.2.1 Mathematical Setup for the Rubric  
There is no single universal formula for all rubrics, as they vary according to the context and evaluation criteria. We offer here a possible adaptation of a qualitative heading. This method is usually represented by a table which we will refer to concisely using the symbol \(T(R)\) (see Table A2). Usually, in the rows of \(T(R)\), the criteria are accommodated. These indicate the specific aspects of the task to be assessed. They are placed starting from the second cell of the first column of \(T(R)\). We will assume that we have a number \(m\) of them, that is, \(i=1,2,\ldots,m\). In this way, we introduce the matrix \(H\) of the criteria:  
\[H=\left( \begin{array}{c}h_{1}\\ h_{2}\\ \vdots\\ h_{m-1}\\ h_{m}\end{array} \right)^{T}. \tag{15}\]  
In what follows, the symbol \(ph(j)_{i}\) will be used to notate the performance of student \(x_{j}\) relative to achieving the tasks linked to criterion \(h_{i}\). Typically, a simple rubric \(R\) includes relative weights assigned to its criteria. These determine the degree of importance in assessing each student's overall performance. Let\(\rho_{i}\) be the relative weight of the criterion \(h_{i}\), \(i=1,2,\ldots,m\), in the global assessment. By this, we mean that \(\rho_{i}\) represents the percentage of the overall student's assessment which is fixed by the criterion \(h_{i}\). Let us represent through P the matrix of \((m\times 1)\) of weights \(\rho_{i}\), \(i=1,2,\ldots,m\). This consists of several entries that are equal to the matrix \(H\) of criteria, let us say  
\[P=\left( \begin{array}{c}\rho_{1}\\ \rho_{2}\\ \vdots\\ \rho_{m-1}\\ \rho_{m}\end{array} \right)^{T}, \tag{16}\]  
where the condition  
\[\sum_{i=1}^{m}\rho_{i}=1. \tag{17}\]  
must be met. Suppose \(T(R)\) represents the table of a simple rubric \(R\) expressed in its general form, and that is obtained by considering a number m of evaluation criteria \(h_{i}\) according to Equation (15). We will use the symbol \(c_{is}\), to represent the cell in \(T(R)\) that is determined by the intersection of the \(i\)-th row and the \(s\)-th column of \(T(R)\). From the second cell of the first line of \(T(R)\), the general identifiers of the possible levels of performance exhibited by students when carrying out the activities stipulated in the different criteria appear. These generally refer to the various qualities that the expected achievement can have for each criterion. A linguistic descriptor will identify each level of performance, denoted by a symbol \(l_{s}\), where \(s=1,2,\ldots,k\). Then, the formalization of \(R\) must include a matrix \(PL\) given by  
\[PL=\left( \begin{array}{c}l_{s}\\ l_{2}\\.\\ l_{k-1}\\ l_{k}\end{array} \right)^{T}. \tag{18}\]  
Additionally, we assign to each level of performance \(l_{s}\) a rating \(\alpha(l_{s})\) according to the hierarchical ordering put by inequality  
\[\alpha(l_{1})<\alpha(l_{2})<\ldots<\alpha(l_{(k-1)})<\alpha(l_{k}). \tag{19}\]  
Usually, the Rubric formulation sets for  
\[\alpha(l_{s})=s. \tag{20}\]  
Then, symbolically, for a simple rubric, we will have a matrix \(POL\) of ordered level indicators, namely,  
\[POL=\left( \begin{array}{c}\alpha(l_{1})\\ \alpha(l_{2})\\ \vdots\\ \alpha(l_{(k-1)})\\ \alpha(l_{k})\end{array} \right)^{T}. \tag{21}\]  
In addition, for \(j=1,2,\ldots,n\) and \(i=1,2,\ldots,m\), we will represent using \(\phi(j)_{is}\) the numerical ranking that, according to the performance ordering of Inequality (19), was obtained by student \(x_{j}\) for achievement shown in carrying out the activities contemplated in the criterion \(h_{i}\). Then, we have  
\[\vartheta(j)_{is}=I(ph(j)_{i}\to l_{s})\alpha(l_{s}). \tag{22}\]  
where through the expression \((ph(j)_{i}\to l_{s})\), we point to the condition the \(j\)-th student has obtained an assessment \(l_{s}\) for work displayed in the activities aligning with the criterion \(h_{i}\), and \(I(ph(j)_{i}\to l_{s})\) is the indicator function of the condition \((ph(j)_{i}\to l_{s})\), by which we mean  
\[I(ph(j)_{i}\to l_{s})=