## 2 Materials and Method
### The Systematic Task-Based Assessment Method (STBAM)
#### 2.3.2 The ST-FIS, a Fuzzy Inference System-Based Modulus for the STBAM Scheme  
A Fuzzy Inference System (FIS) in the ST settings refers to a framework or method used to evaluate a student's performance by applying human-like reasoning based on vague or subjective inputs. An FIS is constructed with key elements and utilizes fundamental processes to make inferences, aiming to produce as output \(E\) the result of the evaluation process, which reflects the student's performance in a clear and precise manner. FIS in the ST setting allows uncertainty to be handled by combining the L, P, and A to produce a fair and balanced performance assessment. Key Elements of the Mathematical Representation of an ST-Fuzzy Inference System (ST-FIS)  
1. Fuzzy sets  
Fuzzy sets represent the foundation of the ST-FIS. A fuzzy set is defined by its membership function, which expresses the degree to which a value belongs to it. Let \(F\) stand for a fuzzy set concurring in the ST-FIS, and let \(\mu_{F}(z)\) be its own membership function, which maps a crisp input \(z\) belonging to a domain \(D\) to a number in the interval \([0,1]\), namely  
\[\mu_{F}(z):D\longrightarrow[0,1]. \tag{69}\]  
This mapping reflects how strongly \(z\) belongs to the fuzzy set \(F\). 2. Input variables  
Generally, an FIS considers a set  
\[Z=\{z_{1},z_{2},\ldots,z_{k},\ldots,z_{n-1},z_{n}\}, \tag{70}\]  
of input values \(z_{k}\), where each one represents a crisp value in a domain \(D\subset R\). For each input \(z_{k}\), there is an associated pool  
\[F=\bigg{\{}B_{1}^{F},\ldots,B_{2}^{F},\ldots,B_{m(F)-1}^{F},B_{m(F)}^{F}\bigg{\}}, \tag{71}\]where \(F\) is a fuzzy set, and \(B_{k}^{F}\) represents the \(k^{th}\) linguistic term with \(k=1,2,\ldots,m(F)\), which qualitatively interprets the input \(z_{k}\). These fuzzy sets collectively partition the range of \(z_{k}\) into overlapping regions, enabling smooth transitions between categories. Each linguistic term \(B_{k}^{F}\), for \(k=1,2,\ldots,m(F)\) entails a fuzzy set defined by its membership function \(\mu_{B_{k}^{F}}(z_{r})\). Particularly, for the ST settings, given a direction \(d_{i}\) for \(i=1,2,\ldots,m\), adapting an FIS takes up as input variables \(F\to L\), which encompass the learning of theoretical background, \(F\to P\), leading to the display of practical skills; it also takes up \(F\to A\), covering the student's disposition to complete tasks. These input variables are defined in the domains \(DL\subset R\), \(DP\subset R\), and \(DA\subset R\), respectively. Moreover, for the input variable \(L\), we associate linguistic terms  
\[L=\bigg{\{}B_{1}^{L},B_{2}^{L},\ldots,B_{m(L)-1}^{L},B_{m(L)}^{L}\bigg{\}}. \tag{72}\]  
Correspondingly, for \(P\), we arrange  
\[P=\bigg{\{}B_{1}^{P},B_{2}^{P},\ldots,B_{m(P)-1}^{P},B_{m(P)}^{P}\bigg{\}}. \tag{73}\]  
And for the input variable \(A\),  
\[A=\bigg{\{}B_{1}^{A},B_{2}^{A},\ldots,B_{m(A)-1}^{A},B_{m(A)}^{A}\bigg{\}}. \tag{74}\]  
3. Output variables  
Likewise, adapting an FIS for the ST protocol involves an output variable \(Y\) which corresponds to an Enlightenment or Proficiency skill demonstrated by the student. We have  
\[E=\bigg{\{}B_{1}^{E},B_{2}^{E},\ldots,B_{m(E)-1}^{E},B_{m(E)}^{E}\bigg{\}}. \tag{75}\]  
4. Rule base  
In the FIS joining the ST scheme, the rule base is composed of a set of \"IF-THEN\" rules that connect the input variables \((L,P,A)\) with the output variable \(E\) in linguistic terms. The ongoing FIS has a set of fuzzy rules that allow it to generate a comprehensive assessment of the student's performance. The rules capture the diversity of possible combinations between the inputs, adapting to different types of students and contexts. Each rule incorporates the three dimensions (cognitive, practical and attitudinal), ensuring that learning is not assessed in isolation. These rules reflect real-life situations that occur during the development of any activity, where performance depends on a balanced interaction among Knowledge, Procedure and Attitude.