## 2 Materials and Method
### The Systematic Task-Based Assessment Method (STBAM)
#### 2.3.1 Formalization of ST, the Direct Modulus of the Systematic Task-Based Assessment Method
Accordingly, for the \(j\)-th student, we will let the symbol \(l(j)^{i}_{k}\) with \(k=1,2,\ldots,\pi(L,i)\), stand for an attainment indicator associated with the \(L^{i}_{k}\) attribute required to accomplish \(d_{i}\) for \(i=1,2,\ldots,m\). Similarly, \(p(j)^{i}_{k}\) for \(k=1,2,\ldots,\pi(P,i)\) will represent the corresponding success indicator for the \(P^{i}_{k}\) attribute linked to performing direction \(d_{i}\). In turn, \(a(j)^{i}_{k}\), where \(k=1,2,\ldots,\pi(A,i)\), will represent the success indicator of the \(A^{i}_{k}\) attribute while performing direction \(d_{i}\). Therefore, formally, for direction \(d_{i}\), the structure of the \(ST\) protocol also includes the indicator matrices \(IL(j)^{i}\), \(IP(j)^{i}\), and \(IA(j)^{i}\) with \((1\times\pi(L,i))\), \((1\times\pi(P,i))\), and \((1\times\pi(A,i))\) entries one-to-one, namely,  
\[IL(j)^{i}=\left( \begin{array}{c}l(j)^{i}_{1}\\ l(j)^{i}_{2}\\ \vdots\\ l(j)^{i}_{\pi(L,i)-1}\\ l(j)^{i}_{\pi(L,i)}\\ \end{array} \right)\ IP(j)^{i}=\left( \begin{array}{c}p(j)^{i}_{1}\\ p(j)^{i}_{1}\\ \vdots\\ p(j)^{i}_{\pi(P,i)-1}\\ p(j)^{i}_{\pi(P,i)}\\ \end{array} \right)\ IA(j)^{i}=\left( \begin{array}{c}a(j)^{i}_{1}\\ a(j)^{i}_{2}\\ \vdots\\ a(j)^{i}_{\pi(A,i)-1}\\ a(j)^{i}_{\pi(A,i)}\\ \end{array} \right). \tag{47}\]  
Each one of the indicators \(l(j)^{i}_{k}\), \(p(j)^{i}_{k}\) and \(a(j)^{i}_{k}\) can bear one of the binary values \(\checkmark\) or \(X\). The \((\checkmark)\) pointer will appear whenever the student's performance exhibits a particular manifestation of the general attributes Learning (\(L^{i}_{k}\)), Procedure (\(P^{i}_{k}\)), or Attitude (\(A^{i}_{k}\)). Correspondingly, if one of these attributes is not displayed, then the \((X)\) marker will show up. An instruction with indicators graded (\(\checkmark\)) in one or more competence-bearing attributes ensures that the student not only \"knows\" something (\(L\)) but can also \"do it\" (\(P\)) and embraces the task with the right disposition (\(A\)). This approach guarantees holistic development, promoting learning beyond memorization. In what follows, the symbolic statement (\(l(j)_{k}^{i}\rightarrow\checkmark\)) will signal that the \(j\)-th student indicator \(l(j)_{k}^{i}\) is linked to a success \(\checkmark\) pointer. Similarly, (\(p(j)_{k}^{i}\rightarrow\checkmark\)) will signify that \(p(j)_{k}^{i}\) bears an achievement \(\checkmark\) pointer, and (\(a(j)_{k}^{i}\rightarrow\checkmark\)) will be interpreted similarly. The total number of indicators linked to direction \(d_{i}\) that display a binary pointer \(\checkmark\) for student \(x_{j}\) will be denoted using the \(z(j)_{L}^{i}\) symbol and defined through  
\[z(j)_{L}^{i}=\sum_{k=1}^{\pi(L,i)}I\left(l(j)_{k}^{i}\rightarrow\checkmark\right), \tag{48}\]  
with \(l(l(j)_{k}^{i}\rightarrow\checkmark)\) being the indicator function of the statement that \(l(j)_{k}^{i}\) ties to a binary value \(\checkmark\), namely,  
\[I(l(j)_{k}^{i}\rightarrow\checkmark)= \begin{cases}1&if\quad l(j)_{k}^{i}\text { ties to a binary pointer value }\checkmark\\ 0&if\quad l(j)_{k}^{i}\text{ ties to a binary pointer value }X\end{cases} . \tag{49}\]  
This pool rates the student's commitment to adequately completing the tasks specified in direction \(d_{i}\) according to the \(L\) attribute. Many indicators marked \(\checkmark\) indicate a successful display of learning competence. Correspondingly, the total number of \(p(j)_{k}^{i}\) indicators associated to \(d_{i}\) that display a binary pointer \(\checkmark\) will be denoted using the \(z(j)_{P}^{i}\) symbol and given by  
\[z(j)_{P}^{i}=\sum_{k=1}^{\pi(P,i)}I\left(p(j)_{k}^{i}\rightarrow\checkmark \right), \tag{50}\]  
where \(I(p(j)_{k}^{i}\rightarrow\checkmark)\) stands for the indicator function of the statement that \(p(j)_{k}^{i}\) takes a binary value \(\checkmark\), namely,  
\[I(p(j)_{k}^{i}\rightarrow\checkmark)= \begin{cases}1&if\quad p(j)_{k}^{i}\text { associates to a binary pointer value }\checkmark\\ 0&if\quad p(j)_{k}^{i}\text{ associates to a binary pointer value }X\end{cases} .