## 3 DMAIR Description
### Learning and Teaching (LT)
#### 3.2.2 LT Component Structure  
In order to create the desired learning paths formed of the learning blocks, the most important structures that will be used in this approach are the ontology of concepts (\(O\)), the learning block (\(B\)), the block relationship (\(BR\)), the block matrix (\(BM\)) and the learning path (\(BLP\)), presented as follows:  
* \(O\) is an ontology of concepts, which is a structured set of semantic tags related by a hierarchic relationship. It can be represented as a graph or as a symbolic logic structure. In this paper, an ontology is a directed graph of concepts: \[O=\{V,E\}\] (1) where
* \(V=\{v_{i}|i=\overline{1,nV}\}\) is the set of concepts that can be determined using an automated method of extracting concepts from a corpus of text (e.g., using Natural Language Processing);
* \(E=\{(v_{i},v_{j})|i,j=\overline{1,nV}\}\) is the set of edges, where an edge defines a hierarchical relationship between two concepts;
* \(B\) is the learning block, which is the most basic unit of learning. A block is a quintuple of elements, codified as follows: \[B_{i}= \begin{array}{c}<\!\!I,O,N,S,P\!\!>/\!<\!nI_{i},nO_{i},nN_{i},nS_{i},P \!\!>,\\ i=\overline{1,nB}\end{array} \] (2) where
* \(I_{i}=\{kwi_{j}|j=\overline{1,nI_{i}}\}\) is the set of keywords that define the prerequisite concepts, i.e., the concepts needed to be known in order to follow the current learning block;
* \(O_{i}=\{kwo_{j}|j=\overline{1,nO_{i}}\}\) is the set of keywords that define the concepts learnt in the current learning block;
* \(N_{i}=\{kwn|j=\overline{1,nN_{i}}\}\) is the set of semantic tags that reflects the upper level in a digital ontology of concepts that describe the learning block;
* \(S_{i}=\{kws_{j}|j=\overline{1,nS_{i}}\}\) is the set of semantic tags that reflects the lower level in a digital ontology of concepts that describe the learning block;
* \(P\) is the ensemble of learning processes, methods, instruments, and elements needed in the learning process run in the current learning block. A graphical representation of a learning block is shown in Figure 2. We can observe that the elements of \(I_{i}\) and \(O_{i}\) sets are represented as sockets that can be connected with similar ones. Also, a block is characterized by a level in a correspondent ontology of concepts that is formed based on the set of blocks and delimited by an upper level (concepts that are more general than the ones studied in the current block) and by a lower level (concepts that are more particular or detailed than the one studied in the current block). The colours are expressed in order to differentiate easier between the elements of the block. * \(BR\) is the block relationship established between two blocks, which can be differentiated into two types: * linear, \(B_{i}\leftrightarrow B_{j}\), which is established between two blocks situated on the same ontology level, based on the keywords matching. The connection can be made if the output keywords of the first block are fully or partially matched with the input keywords of the second block: \[ \begin{array}{c}B_{i}\leftrightarrow B_{j}\to O_{i}\cap I_{j}>0,\\ i,j=\overline{1,nB}\end{array} \] (3) * leveled, \(B_{i}\updownarrow B_{j}\), which is established between two blocks situated on neighboring levels related to a specific ontology, based on the sets of semantic tags. The connection can be made if the current semantic tag of the first block is connected in the ontology \(O\) with the semantic tag of the second block: \[ \begin{array}{c}B_{i}\updownarrow B_{j}\rightarrow\exists(kws_{k},kwu_{l} ),\\ i,j=\overline{1,nB},k=\overline{1,nS_{i}},l=\overline{1,nN_{j}}\end{array} \] (4)
* \(BM\) is the bidimensional array or matrix that contains a specific configuration of a set of learning blocks. A bidimensional array is a structure formed of learning blocks that may be connected based on the four edges of a block: \[