## 3 The Method: Broader Terms Curriculum Mapping
### Step 2: Processing of Broader Terms  
This paper uses natural language processing (NLP, [36; 37]) to convert broader terms assigned to courses into quantitative data, i.e., into frequencies of words. It will be assumed that these words--these tokens as they are called in the NPL literature [36]--extracted from broader terms, still carry conceptual meaning and can still be used to characterize courses and the program-degree. For this reason, in this paper, _token_ and _key_ (program or course) _concept_, K, are used as synonyms.  
NLP applies a sequence of processing functions to an original set of broader terms. Tokenization, the first of these functions, identifies words in broader terms that are included in a corpus; in a dictionary of tokens. Recalling Section 3.1's example of obtaining broader terms for Mathematics (C1:MATH)--\"Function of a real variable\" and \"differential calculus\" were the resulting broader terms--and, considering the corpus of English words, after tokenization the following set of tokens { function, of, a, real, variable, differential, calculus} characterizes the Mathematics course.  
But the above set includes tokens (i.e., \"of\" and \"a\") that add no value to the course characterization; therefore, these tokens, known as stop-words, as well as any punctuation signs and numerals, should be removed. Moreover, words written with capital letters and different conjugations of the same word should be replaced by an adequate \"stem-word\" (in a process known as stemming, [36]).  
Denoting the stemming and the purging of meaningless tokens as normalization, if a study program has \(N\) courses, after tokenization and normalization of course \(Ci\) broader terms (\(i\in\{1,2,\ldots,N\}\)), a multiset \(\mathcal{K}_{i}\) (allowing multiple instances of the same token) of \(m_{i}\) tokens \(\mathcal{K}_{i,k}\) is obtained (with \(k=1,2,\ldots,m_{i}\)). For the program-degree as a whole, a set \(\mathbf{K}\) (no repetitions) with a total of \(M=\left|\bigcup_{i=1}^{N}\mathcal{K}_{i}\right|\) tokens is obtained.  
With K\(j\), the \(j\)th token in set \(\mathbf{K}\), and the frequency of this token in course C\(i\) is found from  
\[b_{i,j}=\sum_{k=1}^{m_{i}}\delta_{i,k}(\text{K}j)\, \tag{1}\]  
with  
\[\delta_{i,k}(\text{K}_{j})=\begin{cases}0,&\mathcal{K}_{i,k}\
eq\text{K}j\\ 1,&\mathcal{K}_{i,k}=\text{K}j\end{cases}. \tag{2}\]  
Equation (1) represents the elements of an \(N\times M\) matrix \(\mathbf{B}\) of token frequencies per course.  
Considering the broader terms for the five courses in Table 1 (column three), after tokenization and normalization the resulting course-token matrix is (Reference [19] provides the R programming code [38] used in this section):  
\[\mathbf{B}_{\text{SC}}=\begin{pmatrix}\text{K1:manag}&\text{K2:calculus}&\text{K3:control}&\text{K4: Kenenergi}&\text{K5:linear}&\text{K4:logistic}&(...)\\ 0&2&0&0&2&0&(...)&\text{C1:MATH}\\ 0&1&0&0&0&0&(...)&\text{C2:PHY}\\ 2&0&2&1&1&2&(...)&\text{C3:LOGOP}\\ 3&0&1&2&0&1&(...)&\text{C4:ENER}\\ 1&0&0&0&0&0&(...)&\text{C5:FIN}\end{pmatrix}\, \tag{3}\]  
where, given the large number of identified tokens (70), only the columns for the six most frequent are shown.  
Observe how this matrix attaches quantitative information to courses based on token frequency. Observe, for instance, the link that emerges between courses C1:MATH and C2:PHY via token K2:calculus. Matrix \(\mathbf{B}_{\text{SC}}\) shows that this token is found twice among the tokens associated with course C1:MATH, and once among those associated with course C2:PHY (see also the underlined words in Table 1).  
This ability to describe a study program quantitatively is an important breakthrough and a way to bridge the gap created by tacit understanding and unclear LO-statements. But, at the same time, notice how unpractical the analysis of the data in the matrix format is.  
To achieve a clearer understanding of the quantitative data emerging from NLP, an alternative to matrix or tabular representations of data is essential.