## 2 Materials and Method
### The Rubric
#### 2.2.1 Mathematical Setup for the Rubric
\begin{cases}1&if\qquad j-th\;student\;performed\;l_{s}\; in\;criterion\;h_{i}\\ \\ 0&if\quad j-th\;student\;did\;not\;perform\;l_{s}\;in\;criterion\;h_{i}\end{cases} . \tag{23}\]  
Moreover, in the general case, the cells \(cr_{is}\) of \(T(R)\) implicitly contain information associated with the evaluation criteria \(h_{i}\) and with the performance levels \(l_{s}\). In fact, the data carried by each cell \(cr_{is}\) can be conceived as a ternary association of the linguistic descriptors \(u_{is}\), \(l_{s}\) and the numerical hierarchy indicator \(\alpha(l_{s})\). The linguistic descriptor \(u_{is}\) specifies the characteristics or skills that the student must demonstrate to opt for the \(s\)-th performance level \(l_{s}\) while carrying out the provisions of the \(i\)-th criterion. Then, we will have an association  
\[cr_{is}\rightarrow(u_{is},l_{s},\vartheta(j)_{is}). \tag{24}\]  
Even more so, in the general case of a regular \(R\), in addition to the above triad which typifies the cells \(cr_{is}\) of \(T(R)\), for each evaluated student \(x_{j}\), with \(j=1,2,\ldots,n\), the rubric's formal set up includes a matrix \(A(j,H,PL)\) of a \((m\times k)\) dimension, with entries \(a(j)_{is}\) for \(j=1,2,\ldots,n\) and with \(i=1,2,\ldots,m\) and \(s=1,2,\ldots,k\), which are given by  
\[a(j)_{is}=\vartheta(j)_{is}\rho_{i}. \tag{25}\]  
We rely on the \(R_{i}A(j,H,PL)\) symbol to designate the \(i\)-th row of \(A(j,H,PL)\), that is  
\[R_{i}A(j,H,PL)=\Big{(}a(j)_{i1},\;\;\;a(j)_{2}\;\;\;\ldots,\;\;\;a(j)_{i(k-1)},\;\;\;a(j)_{ik}\Big{)}. \tag{26}\]  
Moreover, for the \(j\)-th student, we also let \(TA(j,H,PL)\) represent the total sum of the entries of \(A(j,H,PL)\), let us say  
\[TA(j,H,PL)=\sum_{i=1}^{m}\sum_{s=1}^{k}a(j)_{is}. \tag{27}\]  
Now, for \(ph(j)_{i}\) expressing the quality of accomplishment of the duties tied to the \(i\)-th criterion by the \(j\)-th student, the rubric's arrangement assigns one and only one performance level. We will denote this performance level using the symbol \(l_{s}(i)\). We also convene that \(\alpha(l_{s}(i))\) stands for the numerical hierarchy indicator of \(l_{s}(i)\). Thus, we will have that all the entries of \(R_{i}A(j,H,PL)\) but one will vanish. We denote such a non-vanishing entry through the symbol \(a(j)_{is(i)}\) which, using Equation (25), transforms into  
\[a(j)_{is(i)}= I(ph(j)_{i}\to l_{s}(i))\alpha(l_{s}(i))\rho_{i}. \tag{28}\]  
Provided that for \(i=I,2,\ldots,m\), we have  
\[\alpha(l_{s}(i))\in\{1,2,\ldots,k\}. \tag{29}\]Likewise, we let the symbol \(S(j,H,PL)\) represent the \(m\)-tuple  
\[S(j,H,PL)=\Big{(}a(j)_{1s(1)},a(j)_{2s(2)},\ldots,a(j)_{(k-1)s(k-1)},a(j)_{ks(k)} \Big{)}. \tag{30}\]  
Then, we ca define  
\[TS(j,H,PL)=\sum_{i=1}^{m}a(j)_{is(i)}. \tag{31}\]  
From Equations (27) and (31), we would have the equivalence  
\[TA(j,H,PL)=TS(j,H,PL). \tag{32}\]  
Therefore, we set the total direct quantitative grade \(er(j,H,PL)\) assigned by the teacher to the \(j\)-th student to be  
\[er(j,H,PL)=\frac{TS(j,H,PL)}{k}=\frac{TA(j,H,PL)}{k}. \tag{33}\]  
Consider a group \(G\) that includes a number \(n\) of students. Let \(er(j,H,PL)\) for \(j=1,2,\ldots,n\) be the global evaluation or grade that \(R\) assigns to a student \(x_{j}\) in a particular assignment. We will then have an \((n\times 1)\) matrix \(ER\) of global grades assigned to \(G\), namely,  
\[ER=\left( \begin{array}{c}er(1,H,PL)\\ er(2,H,PL)\\ \vdots\\ er(n-1,H,PL)\\ er(n,H,PL)\end{array} \right)^{T}. \tag{34}\]  
Additionally, \(er(j,H,PL)\) provides a criterion for assigning a qualitative assessment for the overall performance of the \(j\)-th student. This is symbolized by \(erQ(j,H,PL)\). Moreover, we could propose the following inference rule,  
\[erQ(j,H,PL)=\left\{ \begin{array}{ccl}\text{Deficient}&if&0\leq er(j,H,PL)< \frac{1}{k}\\ \text{Low}&if&\frac{1}{k}\leq er(j,H,PL)<\frac{1}{k}\\ \text{Acceptable}&if&\frac{2}{k}\leq er(j,H,PL)<\frac{3}{k}\\ \text{Good}&if&\frac{1}{k}\leq er(j,H,PL)<\frac{4}{k}\\ \text{Outstanding}&if&\frac{1}{k}\leq er(j,H,PL)<1\end{array} \right.