## 2 Materials and Method
### Summative Evaluation
#### 2.1.1 Formalization of the Summative Assessment Protocol  
When the S method of evaluation, based on traditional logic, is used to characterize \(E\) it is necessary to establish, initially, the different sections or components of the learning to be evaluated. For this explanation, we will consider that the symbol \(s_{i}\), represents each section and that we have a number \(m\) of them, that is, we will define the matrix \(A\) (\(m\times 1\)) of these assignment sections formally,  
\[A=\left( \begin{array}{c}s_{1}\\ s_{2}\\ \vdots\\ s_{m-1}\\ s_{m}\end{array} \right)^{T}. \tag{1}\]Each assignment section \(s_{i}\) for, \(i=1,2,\ldots,m\) will have an associated weight which is symbolized by \(w_{i}\). The weight determines the relative importance that the development of the learning section \(s_{i}\) bears in determining the value of the global evaluation \(es^{j}\) of the \(j\)-th student's performance. Then, we will also have to consider a matrix of weights \(W\) with the same number of entries as the matrix \(A\) of leaning sections. Let us say  
\[W=\left( \begin{array}{c}w_{1}\\ w_{2}\\ \vdots\\ w_{(m-1)}\\ w_{m}\end{array} \right)^{T}, \tag{2}\]  
where the following condition must hold:  
\[\sum_{i=1}^{m}w_{i}=1. \tag{3}\]  
In addition, for \(j=1,2,\ldots,n\) and \(i=1,2,\ldots,m\), we will use the symbol \(p_{ji}\) to represent the number, expressed as a percentage, standing for the score or grade assigned by the teacher to the \(j\)-th student \(x_{j}\), for the performance shown while carrying out the activities contemplated in the assignment section \(s_{i}\). Therefore, with \(G\), we associate a global matrix of scores \(P\), by which we mean  
\[P=\left( \begin{array}{cccc}p_{11}&p_{12}&\ldots&p_{1m}\\ p_{21}&p_{22}&\ldots&p_{2m}\\ \vdots&\vdots&\vdots&\vdots\\ p_{n1}&p_{n2}&\ldots&p_{nm}\end{array} \right). \tag{4}\]  
Particularly, for the \(j\)-th student \(x_{j}\) we count on a matrix \(P(j,A)\) of a \((1\times m)\) dimension, which represents the evaluations obtained for performance in carrying out the activities associated with the learning section \(s_{k}\). To be precise,  
\[P(j,A)=\left( \begin{array}{c}p_{j1}\\ p_{j2}\\ \vdots\\ p_{j(m-1)}\\ p_{jm}\end{array} \right)^{T}. \tag{5}\]  
Consider now a group \(G\) which includes a number \(n\) of students. Let us represent through \(es(j,A,W)\), for \(j=1,2,\ldots,n\), the total assessment or grade that the Summative Assessment (SA) method produces in correspondence to a student \(x_{j}\) for the displayed performance. Let us symbolize by \(ES\) the \((n\times 1)\) matrix of grades that are assigned to the \(G\) group, that is,  
\[ES=\left( \begin{array}{c}es(1,A,W)\\ es(2,A,W)\\ \vdots\\ es(n-1,A,W)\\ \end{array} \right)^{T}. \tag{6}\]  
Therefore, the global assessment matrix \(ES\) of the group \(G\) is obtained by multiplying the matrices \(P\) and \(W\), by which we mean \[ES= \begin{pmatrix}p_{11}&p_{12}&\ldots&p_{1m}\\ p_{21}&p_{22}&\ldots&p_{2m}\\ \vdots&\vdots&\vdots&\vdots\\ p_{n1}&p_{n2}&\ldots&p_{nm}\\ \end{pmatrix}\begin{pmatrix}w_{1}\\ w_{2}\\ \vdots\\ w_{m}\\ \end{pmatrix} . \tag{7}\]  
Specifically, the teacher's grade \(es(j,A,W)\) assigned through the summative scheme is given by the scalar product of \(P(j,A)\) and \(W\), that is,  
\[es(j,A,W)=\sum_{i=1}^{m}p_{ji}w_{i}. \tag{8}\]  
To introduce a criterion to assess the consistency of the mathematical scheme backing the SA method, we will assume that independently of the teacher's grading, the student's actual abilities determine a matrix \(PO(j,A)\) of objective scores, namely  
\[PO(j,A)=\left( \begin{array}{c}po_{j1}\\ p_{0j}\\ \vdots\\ po_{j(m-1)}\\ p_{jm}\\ \end{array} \right)^{T}.