In this paper, we propose an oriented-graph - design of the database generated in a virtual educational platform with the records of students access to the learning resources. This theoretical model lets us compute a visualization index for students and for the available material in the platform in a total and a partial way, by the 1-norm of the adjacency matrix of the graph. These coe cients let us construct a classi cation system that it could be useful for determining di erent levels for students and for the material. Then, we can propose di erent sca olding for the learning activities and e ective learning outcomes for a meaningful learning experience depending on the interaction between the students and the resources.
Current educational context is characterized by the diversity of the instruction, students, and multimedia supports. Moreover, there exist a large number of hybrid courses in which the blended, online and face-to-face learning are present at the same time. In this framework, a thought about the material provided by the teacher is an essential action in order to nd learning patterns and improve the learning and teaching processes.
Learning analytics techniques may provide very useful tools that help the
teachers to optimize their work (see [
Copyright © 2018 for this paper by its authors. Copying permitted for private and academic purposes
Although learning analytics give the teachers methods and tools of gathering
information on how learners are interacting with learning resources, usually there
is a gap between this information and the pedagogical actions that help to the
teachers in their learning designs ([
Learning designs describe frameworks that can be used to help the teachers in
the design and the choice of learning resources, learning tasks or activities, and
learning supports in order to create a meaningful learning experience for the
students, especially with the use of Information and Communication Technologies,
(see [
In the conceptual framework that links learning analytics and learning design,
we have four dimensions on which we can work: temporal analytics, comparative
analytics, cohort dynamics and tool speci c analytics . The third one, the cohort
dynamics, is a category that helps us to propose di erent learning outcomes
depending on the di erent interaction patterns manifested in a course between
students and resources. In this context, the study of whether a student has
accessed or not to a speci c resource is one of the most requested questions
by di erent educational agents as desired information about a course (see [
One of the usual available tools in an educational platform is the ow visualizations of the number of visits that each resource has had trough the time. These graphics are often performed with the absolute frequencies of the visits, so its interpretation has to be done carefully because it could be deceptive and confused. For example, we can have a resource that has the highest number of visits and, however, it has been seen only by a minority number of the students. In Figure 1 we can see a typical ow of the visits of the resources by the students in a course in Moodle. We have information about the number of visits per day but we do not know what resources have been visited or the students that have accessed to the material. Moreover, it is necessary to take into account that the interpretation of the obtained results can imply di erent learning insights depending on the context of implementation. So, simple tools of an overview of the results, for students and resources, are needed in order to get a good understanding of this type of cohort dynamics.
This is the motivation for this work: to obtain more speci c information about the visualization of the educational resources by the students, in an easy way that let us create a simple tool. We propose a simple mathematical model that allows us to analyze the interaction between them and, nally, to obtain a classication system for both of them. This approach is based on the consideration of the visualization index. Its computation and the subsequent classi cation let us provide an overview of the results of the analyses that it is easily understandable for non- experts in learning analytics. In addition, this index can be applied for a global course, and in a partial form (for students or for resources).The proposed model is based on an oriented graph having as nodes both, the students and the resources. Since the most of the learning management systems for education have implemented models with social network analysis, it would make easier its integration in the usual educational platforms.
This work is organized as follows: In Section 2, we list the goals of this article. The proposed model to analyze the data is developed in Section 3, together with a simple example that helps to understand the essential ideas. Finally, the conclusions and references are given. 2
The main goals of this work are: 1. To de ne a measurement index of visualization that let us analyze what happens or what happened in a course regarding the interaction between the students and the provided resources. 2. To de ne a classi cation system for resources and students that depends on the developed metric. 3. To obtain a simple procedure that gives us a general overview of the situation. 3
This section describes the design of the mathematical model that let us compute the visualization indices of a course. 3.1
Model of the database
We have used a graph approach to model the data. A graph database is a
database that can be structured in graph form so that the nodes of the graph
contain the information, and the edges contain properties and/or de ne relations
between the information contained in the nodes. One of the main strengths of
this kind of databases is the capability to give answers in short time for questions
regarding relations (see [
Remark 1. We will state the following notation 1. The set X = fx1; : : : ; xlg is going to be the set of nodes that represent the students. 2. The set Y = fy1; : : : ; ysg contains the resources that the teacher provides through the virtual platform 3. The database D that we can download from the virtual platform, usually in .csv format. This database contains the set of all registers Rk. Each register represents a case in which each student visits one speci c resource. We can now attach a graph structure to D. In order to do so, we have to de ne the set of nodes, N , and the set of arrows, A. The graph, G = (N; A), is going to be composed by two layers of nodes in such a way that all the arrows have their source in one layer and target in the other one. 1. Layer 1: the nodes are the elements of the set X. 2. Layer 2: the nodes are the elements of the set Y . 3. We have an edge xi ! yj if and only if the student xi has visited the resource yj . From now on this relation will be expressed as yj 2 xi.
Example 1. In this example, we suppose that we have a face-to-face course with six students and ve resources (four of them about contents of the course, and y5 that is a learning task). So, the layer 1 has six nodes and the layer 2 has ve nodes. We suppose that we have obtained the database D that is shown in Table 1. The look of the corresponding graph of D and the ltered data are in Figure In the sequel, B will denote the set f0; 1g, and the set of matrices with entries in B with l rows and s columns will be denoted by Bl s.
De nition 1. The adjacency matrix of D is de ned as the adjacency matrix, A 2 B(l+s) (l+s), of the associated graph: 2.
User Resource x1 y1 x2 y1 x3 y1 x1 y2 x3 y3 x3 y2 x2 y2 x3 y5 x5 y5 x4 y1 yO1Y bf x1 y2 E O Y x2 y3 O x3 y4 x4 <yO5 x5 x6
where for i = 1; : : : ; l , j = 1; : : : ; s
x1 : : : xl y1 : : : ys ys cij = 0 0
1 C = (cij )CC
C C C C C
A 0 1 if yj 2 xi 0 if yj 2= xi (1) (2) (3) Remark 2. Note that we are only interested in the block C that have been dened. So, the study of the adjacency matrix is reduced to the study of the matrix C. For this reason, in this paper, we will use both, the letter A and the letter C, to make reference to such adjacency matrix if there is no possible confussion. Example 2. The adjacency matrix associated to the vector database of Example 1 is 0 0 0 0 0 0 0 1 1 0 0 0 1 B 0 0 0 0 0 0 1 1 0 0 0 C BB 0 0 0 0 0 0 1 1 1 0 1 CC BB 0 0 0 0 0 0 1 0 0 0 0 CC
BB 0 0 0 0 0 0 0 0 0 0 1 CC A = BB 0 0 0 0 0 0 0 0 0 0 0 CC
BB 0 0 0 0 0 0 0 0 0 0 0 CC BB 0 0 0 0 0 0 0 0 0 0 0 CC BB 0 0 0 0 0 0 0 0 0 0 0 CC B@ 0 0 0 0 0 0 0 0 0 0 0 CA
0 0 0 0 0 0 0 0 0 0 0 where the matrix C is highlighted in bold characters. 3.2
Visualization indices Once we have obtained the adjacency matrix associated to D, we can compute the total visualization index of the course by the following formula De nition 2. The total visualization index, D, is the rate of resources that have been visited, at least, one time.
D := k A k1 ; l s k A k1 being the 1-norm of the matrix A, that it is, the number of 10s in the matrix A. Ths visualization index is a real positive number less or equal than one, D 2 [0; 1].
De nition 3. The partial row-visualization index, xi , is the rate of resources that have been visited by the student xi at least one time, k ai k1 being the1-norm of the i-th row of the matrix A, that it is, the number of 10s in the i-th row of the matrix A. As more resources the student visit, as higher the index is.
De nition 4. The partial column-visualization index, yj , is the rate of students that have visited the resource yj at least one time, (5) (6) k a j k1 being the1-norm of the j-th column of the matrix A, that it is, the number of 10s in the j-th column of the matrix A. As more students visit the resource, as higher the index is.
Like the visualization index, the partial visualization indices are positive real numbers less or equal than one.
We highlight that the value of the visualization index has a di erent interpretation depending on the type of the course we are teaching. For instance, we have to take into account that the index has di erent consequences if we deal with an in-person course or an online course. This fact is very important in the prescriptive design stage. The ideal scenario would get the indices in di erent checkpoints that let us analyze the dynamic of the course.
Example 3. The total visualization index for D given in Example 1 is D = As we can see in the partial row-visualization indices shown in Figure 3, the student x6 has not visited any resource. On the other hand, the student x3 is the one that more di erent resources has visited.
The partial column-visualization indices are included in Figure 4. In this case, the resource y1 has a high visualization index. It could mean that the content of the resource has not been well understood because of most of the students have revisited it. Another possibility is that this resource is directly related to the assessment. In the case of y4, the interpretation is the inverse. Finally, at the stage computation, the task has been visited by less than half of the students. Although we can have a global insight of the use of the resources with the values of the indices mentioned above, it would be interesting to have a visualization tool that let us label the resources or the students depending on the value that the index has. This system provides us with a ranking of the most visited resources and the most visiting students. The system is done by constructing classi cation intervals, each of them carrying with a label, in such a way that we assign to every student or resource the label corresponding to the interval to which the visualization index belongs.
We develop the procedure only for the resources since for the other case the procedure can be developed in a similar way.
De nition 5. Let I be the set of partial column-visualization indices of all resources. Let C be the set fvmin; I1; I2; I3; I4; vmaxg where vmin and vmax are the minimal and maximal values that the visualization indices take in the database D, and I1; I2; I3, I4 are the intervals between Q1; Q2; Q3, the quartiles of the set I. Then, we de ne the following map : Y
! C Learning analytic techniques, together with strategies for learning design, may provide very useful tools that let the teachers optimize their sequences of the didactical resources.One of the current challenges that the techniques and systems of learning analytics face are to help to assess the success and adequacy of a concrete educational resource, taking into account the pedagogical and local context of the course in which these procedures are being implemented. In this work, we propose a mathematical model that can be useful to solve this need.
Our future work is related to the use of the visualization index to obtain information about an e ective instructional course by sca olding. In addition, we are working in a visualization tool, based on the classi cation system developed in this work, that allows the users to optimize the learning resource design of a course, and purpose prescriptive actions based on learning design for the students.