## 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
\tag{59}\]  
As we elaborated for the summative and the rubric methods, we consider that independently of the teacher's assessment \(est(j,z,D)\) given by Equation (58), the student's pool of abilities determines an objective performance at achieving the different activities contributed into the directions composing the \(ST\) construct. According to Equation (58), we represent the associated objective performance ratings through the symbols \(zo(j)_{L}\), \(zo(j)_{P}\) and \(zo(j)_{A}\), which, recalling Equation (54), can be expressed by  
\[zo(j)_{L}=\sum_{i=1}^{m}zo(j)_{L}^{i},\ zo(j)_{P}=\sum_{i=1}^{m}zo(j)_{P}^{i}\ \text{and}\ zo(j)_{A}=\sum_{i=1}^{m}zo(j)_{A}^{i}, \tag{60}\]  
respectively, in similar way as we set for Equation (57), we consider the vector \(zo(j,D)\) of objective grading inputs  
\[zo(j,D)=(zo(j)_{L},zo(j)_{P},zo(j)_{A}). \tag{61}\]  
Therefore, based on Equation (58), we can consider an objective grade \(est(j,zo,D)\) to be given by  
\[est(j,zo,D)=\frac{zo(j)_{L}+zo(j)_{P}+zo(j)_{A}}{TI}, \tag{62}\]  
where \(TI\) is given by Equation (56). Therefore, the direct arrangement of the ST scheme entails an absolute deviation \(\Delta est(j,z,zo,D)\), formally given by  
\[\Delta est(j,z,zo,D)=|est(j,zo,D)-est(j,z,D)|. \tag{63}\]  
Moreover, the direct \(est(j,z,D)\) assessments provide a device to estimate what would be the uncertainty conferred to the grade assigned to the \(j\)-th student, as mainly determined by the teacher's subjectivity while assigning the achievement pointers (\(\bigvee\)) to the involved \(l(j)_{k}^{i},p(j)_{k}^{i}\), and \(a(j)_{A}^{i}\) indicators. Recalling the simulation procedures we arranged for the SA and R methods, we now introduce a labeling index \(q=1,2,\ldots,\theta(m)\), and as set by Equation (54), we can consider simulated replicate values \(z(q,j)_{L}z(q,j)_{P}\) and \(z(q,j)_{A}\), given by  
\[z(q,j)_{L}=\sum_{i=1}^{m}z(q,j)_{L}^{i},\ z(q,j)_{P}=\sum_{i=1}^{m}z(q,j)_{P}^ {i},\ z(q,j)_{A}=\sum_{i=1}^{m}z(q,j)_{A}^{i}, \tag{64}\]  
where \(z(q,j)_{L}^{i},z(q,j)_{P}^{i}\), and \(z(q,j)_{A}^{i}\) are simulated versions of \(z(j)_{L}^{i},z(j)_{P}^{i}\), and \(z(j)_{A}^{i}\), (cf. Equations (48)-(52)), which along with Equation (57) yield the simulated rating triplet  
\[z_{q}(j,D)=(z(q,j)_{L},z(q,j)_{P},z(q,j)_{A}). \tag{65}\]  
The last equation concurrently determines the simulated grade \(est(j,z_{q},D)\):  
\[est(j,z_{q},D)=\frac{z(q,j)_{L}+z(q,j)_{P}+z(q,j)_{A}}{TI}. \tag{66}\]  
Then, agreeing with Equations (62) and (66), we can consider replicating absolute deviations \(\Delta est\big{(}j,z_{q},zo,D\big{)}\), given by  
\[\Delta est\big{(}j,z_{q},zo,D\big{)}=\Big{|}\overline{est}(j,zo,D)-est(j,z_{q },D))\Big{|}. \tag{67}\]  
Then, we can introduce the Mean Absolute Deviation \(\Delta\overline{est}(j,z_{q},zo,D)\), given by  
\[\Delta\overline{est}(j,z_{q},zo,D)=\frac{\sum_{q=1}^{\theta(m)}\Big{|} \overline{est}(j,zo,D)-est(j,z_{q},D))\Big{|}}{\theta(m)}. \tag{68}\]  
We in turn set the convention that \(eso(j,zo,D)\) lies in the interval \((0,1]\). Therefore for the considered simulation runs, we set \(\overline{est}(j,zo,D)=1/2\). Then, we can offer \(\Delta\overline{est}(j,z_{q},zo,D)\) as a proxy for the average rating error for \(est(j,z,D)\) relative to \(\overline{est}(j,zo,D)\)Simulation Procedures for the ST Device  
The generation of replicates of student grades assigned using the ST scheme is based on the random assignment of binary indicators to the evaluation attributes Learning (L), Procedure (P), and Attitude (A). For each direction in an activity, a set of specific indicators is generated per attribute, and each of them is assigned a value of 0 or 1, using the function \(randi([0,\,1])\) in MATLAB 2018b. A value of 1 indicates that the student showed evidence of the attribute being assessed, while a value of 0 indicates no evidence of displaying the referred trait. This procedure is repeated a defined number \(\theta(m)=100\) of times to generate an equal number of possible performance configurations \(z_{q}(j,D)\), (cf. Equation (65)).