**Contents**



## **About the Editors**

#### **Francisco D. Fern´andez Mart´ın**

Interests: program evaluation; plurilingual education; peer learning; entrepreneurship education; bullying behavior; service learning; civic engagement; inclusive education; counseling; elderly education; special educational needs

Special Issues, Collections and Topics in MDPI journals Special Issue in Sustainability: Interactive Learning Environments in Student's Lifelong Learning Process: Framework for Sustainable Development Goals of the 2030 Agenda

#### **Jos ´e Mar´ıa Romero Rodr´ıguez**

Interests: educational technology; teacher training; higher education; sustainability education; social networks; educational innovation; active methodologies; Internet risks; mobile learning.

Special Issues, Collections and Topics in MDPI journals

Special Issue in Sustainability: Sustainable Development Goals (SDGs): The Challenges of the 2020-2030s for Quality Education

Special Issue in Education Sciences: Educational Research and Innovation in the First Global Catastrophe of the 21st Century: Committed to Education

Special Issue in Sustainability: Education for Sustainable Development: Commitments and Challenges——Selected Papers from CIEI 2021

#### **Gerardo G ´omez Garc´ıa**

Interests: E-Learning; Innovation; ICT; Digital Literacy; Flip Teaching; Fake News; Informational Literacy

#### **Magdalena Ramos Navas-Parejo**

Interests: active methodologies; information and communication technologies (ICT) in education; reading promotion for educational inclusion

Special Issues, Collections and Topics in MDPI journals

Special Issue in Sustainability: Interactive Learning Environments in Student's Lifelong Learning Process: Framework for Sustainable Development Goals of the 2030 Agenda

Special Issue in Mathematics: Emerging Technologies in Learning of Mathematics Education

## **Preface to "Active Methodologies for the Promotion of Mathematical Learning"**

For years now, particularly with the rise of information and communication technologies (ICT) in education, new teaching methods and innovations have been emerging in classrooms at all stages of education. This has led teachers to rethink the educational model and teaching tools in order to respond to the way students learn in the 21st century.

Currently, the new methodological currents tend to want to abandon the master class to give way to the type of active and autonomous learning methodologies that are on the rise which provide a better response to the educational demands of today. The suitability of the method will depend on the educational context in which it is applied, so teachers must be familiar with the different types of methodologies and be able to discern the most appropriate one, such that it fits the needs of their classrooms and allows for the fulfilment of the objectives set. The teaching–learning process requires scenarios that promote meaningful learning.

In this sense, research in educational innovation seeks to improve certain aspects and is inclined towards the inclusion of active or emerging methodologies, which achieve greater involvement of students in their learning. These are educational practices in which students play a greater role, learning through experimentation and interaction with their peers. The role of the teacher becomes that of a learning guide, so that the planning part of teaching becomes more relevant than the expository part. To this must be added the need to use ICT in a coherent way.

It can be deduced that active methodologies involve a set of methods and strategies in which the students are the central axis of the didactic process, achieving their active participation, which generates significant and effective learning.

### **Francisco D. Fern´andez-Mart´ın, Jos´e-Mar´ıa Romero-Rodr´ıguez, Gerardo G ´omez-Garc´ıa, and Magdalena Ramos Navas-Parejo**

*Editors*

## *Article* **E-Learning in the Teaching of Mathematics: An Educational Experience in Adult High School**

#### **Antonio-José Moreno-Guerrero, Inmaculada Aznar-Díaz, Pilar Cáceres-Reche and Santiago Alonso-García \***

Department of Didactics and School Organization, University of Granada, 18071 Granada, Spain; ajmoreno@ugr.es (A.-J.M.-G.); iaznar@ugr.es (I.A.-D.); caceres@ugr.es (P.C.-R.) **\*** Correspondence: salonsog@ugr.es

Received: 29 April 2020; Accepted: 21 May 2020; Published: 22 May 2020

**Abstract:** Currently, the e-learning method, due to the period of confinement that is occurring due to COVID-19, has increased its use and application in the teaching and learning processes. The main objective of this research is to identify the effectiveness of the e-learning method in the teaching of mathematics with adults who are in high school, in contrast to the traditional expository method. The study developed is quantitative, descriptive and correlational. The research design is quasi-experimental, with a control group and an experimental group. The results show that the use of the e-learning method has a positive influence on motivation, autonomy, participation, mathematical concepts, results and grades. It can be concluded that the e-learning method leads to improvement in adult students who are studying the mathematical subject in the educational stage of high school, provided that it is compared with the expository method. Therefore, this method is considered effective for its implementation in adults.

**Keywords:** emerging methodology; educational innovation; e-learning; educational experimentation; adults; students

#### **1. Introduction**

Technological development is a reality today [1]. This fact is reflected in our society [2], specifically in the labour, social and educational fields [3]. This technological advance facilitates, strengthens and speeds up the performance of daily tasks [4].

In the educational field, technological progress is reflected in the development of the so-called information and communication technologies (ICT) [5]. ICTs directly influence the development of teaching and learning processes [6], since they promote innovative pedagogical actions, as well as generate new learning spaces [7]. These pedagogical events enhance the transformation of the classroom as we know it [8], since they allow for the elimination of spatial-temporal barriers [9], as well as access to a large amount of information [10], with different formats [11]. It has also promoted the improvement of students' motivation, autonomy, involvement and attitude towards educational content [12–14].

Among the pedagogical actions based on ICTs is e-learning, which is defined as the pedagogical act that takes place online, thanks to the use of the Internet and technological devices, whether mobile or not, with synchronous or asynchronous connection, and from anywhere [15]. Therefore, the e-learning method becomes a pedagogical tool that facilitates access to learning for the whole of society [16].

The method of not is of recent creation [17], since its beginnings date back to 1993, when it began to be used more assiduously, having a greater impact in the field of education [18]. Prior to that date, distance learning was widely used.

This method of teaching is currently on the rise due to COVID-19 [19]. Its flexibility in terms of location, time, effort and costs [20], makes it the most appropriate option for training and evaluating students [21].

It should be borne in mind that two types of resources are required to develop the e-learning method: digital and technological [22]. Among the digital resources are educational videos, teaching platforms, videoconferences, podcasts, social networks, among many other resources [23]. While technological resources can be the desktop computer, tablet, smartphone, among others [24].

The use of e-learning by the members involved in the teaching and learning process becomes a challenge [17], because an average level of digital competence is required to apply it with guarantees [25]. Therefore, teachers and students need to be trained in the use of the various technological and digital resources [26].

This teaching method has a number of characteristics that make it different from other teaching methods [27]. Some authors see it as an evolution of distance education [28,29]. For others, it is a new teaching modality that differs substantially from face-to-face teaching [30].

Be that as it may, e-learning has a number of characteristics, among which are promoting dialogue and group activities, enhancing students' interpersonal relations [31]; encouraging collaboration among students themselves, achieving joint goals in the elaboration of different tasks [29]; facilitating communication, both synchronous and asynchronous [32]; enabling learning to take place from any location, provided that a technological device is available [33] to encourage the acquisition of digital competence in students [34]; enable adaptation to the individual pace of students [35]; enhance motivation, as the student can develop his or her own learning style [36]; promote the acquisition of learning to learn competence [37]; be adapted to the circumstances of each individual, both personal and occupational [38]; provide access to an unlimited amount of learning resources [39]; facilitate teacher monitoring of student activity [40]; and promote student familiarisation with the use of technological and digital resources [41].

It should also be noted that the e-learning method is a special case of distance learning [42]. There are several reasons for this [43]. In distance learning, email is used to receive the contents of the subject, not having a virtual medium [44]. In addition, a large number of theoretical contents are presented, which are not interactive and whose sequencing is closed [45]. Additionally, contact with the teacher is sporadic, which acts as a mere transmitter of content. In this case, the student is a passive receiver, who usually has a feeling of loneliness [46].

In other words, the teaching-learning process can take place 24 hours per day, every day of the year [47], allowing students to be trained while they are on the move or in a place other than their usual one [48], promoting a change in the teacher-learner mentality, and with it the philosophy of learning, in which the student organizes his or her training process and the teacher guides that action [49], and allowing unlimited access to network resources [50]. Therefore, the use of e-learning totally changes the perspective we had of teaching until now [51].

However, the e-learning method can generate a spatial and temporal gap [52], so it is necessary to personalize the educational experience of the students, trying to keep the learners motivated and committed [53]. Moreover, in developing countries, the use of ICT is not as widespread as in developed countries, leading to a lack of acceptance of technological resources and, therefore, of e-learning, not having the desired effect on educational learning [54,55].

Mathematics in the field of social sciences is considered a necessary instrument to be able to decipher the closest environment and represent various facts, be they social, scientific and technical that occur in today's world [56]. Mathematics facilitate the understanding of various phenomena, be it social reality itself, economic aspects or historical facts, among others [57]. In this case, mathematics becomes an adequate tool to acquire knowledge, reflect on social aspects, and represent facts from the environment [58]. In other words, mathematics tries to convert all these facts into knowledge and information [59]. In addition, the language used in the mathematical field allows the phenomena that occur to be explained in detail and precisely [60].

It should be borne in mind that mathematics is instrumental, and is the basis for acquiring knowledge from other subjects, or in other fields, such as sociology or political science [61].

In addition, mathematics develops the student's intellect, promoting competencies that will allow him to function personally and socially [62]. It also promotes creativity, the development of autonomy, the improvement of self-esteem and entrepreneurship [63].

In the field of mathematics, there are educational actions in which e-learning has been developed as a teaching method [64,65]. One of the ideas is that applied in the MCIEC model (motivation, context, interactivity, evaluation and connectivity), which entails greater student involvement. This model allows the student to increase his or her ability to make an effort to understand mathematical content, thanks to increased interest, motivation and adaptation to the context [64]. The development of the e-learning method presents improvements if it is applied with an appropriate teaching and learning method. An example of this is the development of the e-learning method associated with the GeoGebra resource, which is integrated into the Moodle platform, improving aspects related to assessment, motivation and student interest. It also promotes learning in a more meaningful way and adapts assessment to students' needs [65]. Another similar case is that of the Working Memory Capacity (WMC) method, developed in the e-learning method. This method leads to an improvement in students' abilities to acquire various mathematical concepts. In this case, it improves students' academic performance. This is due to the increase of their involvement and motivation in mathematical contents [66]. Another case is the development of the e-learning method, associated with the Edmodo application, in the field of mathematics. This training process increases participation in learning. This involvement increases the memorization, comprehension, application, analysis, evaluation and creation of mathematical contents. It also increases students' attitude and acceptance of mathematical content [67]. The use of e-learning in the development of mathematics increases the commitment of students themselves, improving performance. It also increases interest, and thus, acquired results. It also improves the acquisition of mathematical content [68]. Another example is pedagogical action, in which e-learning is used with the individualized e-learning environment called UZWEBMAT. This combination promotes individualized attention of students. Moreover, it is adapted to the learning style of the students, improving their comprehension skills. It also increases their responsibility for learning and is reflected in motivation and academic performance [69]. In many cases, student learning, and therefore student outcomes, can be affected by poor connectivity, inflexible scheduling, and inadequate devices [70].

#### **2. Justification and Research Objectives**

The use of ICTs today, coupled with the global crisis being experienced by COVID-19, makes e-learning a necessary teaching method. This implies the application of new didactic strategies and pedagogical approaches [71].

This study presents a teaching method based on e-learning for adult students who study high school in the distance mode. In addition, it shows the pedagogical actions developed during the first quarter of the 2019–2020 school year. A contrast is also established with the traditional expository method developed with the students of the night school. All of this was done in the subject of mathematics applied to the social sciences.

The aim of this research is to give continuity to the application of the e-learning method in the teaching of mathematics, with the intention of contrasting the results obtained in other studies with similar characteristics [63–70].

The main objective of this research is to identify the effectiveness of the e-learning method in teaching mathematics to adults who are in high school, in contrast to the traditional expository method. The following specific objectives are established from this objective:


#### **3. Method of Investigation**

#### *3.1. Research Design and Data Analysis*

The study developed is quantitative, descriptive and correlational [72]. The research design is quasi-experimental, with a control group (GC) and an experimental group (Ge), that is, non-equivalent groups. In this case, the research process developed in other previous studies has been followed, where active teaching methods have been applied [73,74]. Unlike the investigations mentioned above, this study tries to know how an active teaching method influences, in this case, the e-learning method in the development of the development of the subject of mathematics. For this, a contrast is established with the exhibition method. The students are divided into two groups: the control group, made up of night school freshmen; and the experimental group, made up of distance school freshmen. In both groups the subject of mathematics applied to social sciences has been developed. In the control group the traditional expository method has been applied. In the experimental group the e-learning teaching method has been developed. The distribution of the students has not been random, because the groups have been formed by the head of studies, according to the registration requested by the students. The criteria for the distribution of the student body is based on the principles of equity and equality. In other words, the management team distributed the groups bearing in mind several criteria, including the length of time the students have been out of official studies and the grades of the last year enrolled. With respect to years of non-study, it established three criteria: (a) more than 10 years not enrolled in official studies; (b) between 10 and 5 years not enrolled in official studies; (c) less than five years not enrolled in official studies. With regard to the qualification, it established four criteria: (a) no subjects passed in the last year enrolled; (b) between 0 and 3 subjects passed; (c) between 4 and 6 subjects passed; (d) all subjects passed. Based on these criteria, it made an even distribution. These criteria are set out in the School Education Project. The information was collected at the end of the first quarter, that is, after the pedagogical intervention, through the application of a post-test (Table 1).


**Table 1.** Composition of the groups.

The Statistical Package for the Social Sciences (SPSS) v25 (IBM Corp., Armonk, NY, USA) was used to analyse the data collected. The statistics used are mean (M) and standard deviation, in addition to skewness (Skw) and kurtosis (Kme) statistics. Additionally, Student's t-test (tn1<sup>+</sup>n2−2) has been used to compare the means between the established groups. Finally, Cohen's d-test and the biserial correlation (rxy) have been applied, in order to know the effect size and the out-of-association. The significance level applied in the study was *p* < 0.05.

#### *3.2. Participants*

The sample applied in this research consists of 132 students. The sampling technique applied is for convenience. This is due to the ease of access to the students. In studies focused on the application of pedagogical methods, the sample size is not a determining factor [75,76].

The students are studying the first year of the adult baccalaureate, specifically the humanities and social sciences, at an adult education centre in Southern Spain. There is a total of 39.39% men and 60.61% women, with an age range between 18 and 33 years old (M = 23.3; SD = 1.89), where 40.15% have work, and 35.61% have family responsibilities.

The research was conducted in the first quarter of the 2019–2020 school year. Previously, permission was requested from both the school management and the students themselves. Both were informed of the objectives of the research. Neither the school nor the students refused to participate.

#### *3.3. Instrument*

The instrument used is an ad hoc questionnaire that has had as reference the questionnaires 77 and 78, which consists of 30 items (Appendix A). These are distributed in different dimensions: Socioeducational (five items), oriented to know the socio-educational aspects of the sample; motivation (two items), autonomy (two items); collaboration (two items); participation (two items); resolution (two items); class time (two items), in which the aim is to identify the attitudes, motivations and interests of the student in the application of the teaching method; concepts (two items), scientific data (two items), graphics (two items), results (two items), decision (two items), ratings (three items), which focus on the learning acquired in the subject of mathematics. In addition, teacher-ratings have been taken into account, obtaining the values of the grades established by the teacher. The questionnaire uses a Likert scale, composed of four items (1: None, 2: Few, 3: Enough and 4: Completely).

This questionnaire has been subjected to various statistical tests, for its validation and reliability. At first, the Delphi method was used, with qualitative validity, by eight experts, whose ratings were positive (M = 4.66; SD = 0.16; min = 1; max = 6). Then, the statisticians of Kappa de Fleiss and W de Kendall were used, whose results were adequate (K = 0.89; W = 0.87). Subsequently, it was validated through exploratory factor analysis with varimax rotation, whose data (Bartlett = 2981.09; *p* < 0.001; Kaiser-Meyer-Olkin = 0.89) are adequate. It was finalized using Cronbach's alpha (0.91), McDonald's omega method (0.89), compound reliability (0.85) and mean variance extracted (0.84), showing adequate metrics. Taking into account the statistical tests, the instrument is considered as valid and reliable. The internal consistency of each of the dimensions is: Socio-educational (0.941); motivation (0.884); autonomy (0.861); collaboration (0.952); participation (0.891); resolution (0.948); class time (0.923); concepts (0.891); scientific data (0.901); graphics (0.912); results (0.884); decision (0.896); and ratings (0.911).

#### *3.4. Dimensions and Study Variables*

The study focuses research on attitudes and mathematical development. Both aspects have marked the distribution and composition of the dimensions of this study [77,78].

In addition, the dependent and independent variables have been established. The dependent variables are associated with the dimensions indicated for this study. The teaching method developed during this research is established as the independent variable. In order to facilitate the understanding of the results achieved, each of the dimensions is analysed:


#### *3.5. Methodological Procedure*

The research process developed began with the validation and reliability of the instrument used. Subsequently, the selection of the sample and the application for permits were made. In this case, the pedagogical proposal was presented to the selected school, which agreed to participate. The centre, itself, requested information on the results achieved in the research.

Then, the pedagogical proposals were developed. On the one hand, the traditional exposition method (Gc), in which the teacher presented the theoretical contents, followed the sequence of the textbook and proposed tasks. On the other hand, there is the e-learning method (Ge), which will be explained in more detail in the next point.

At the end of the first quarter, data was collected using Google Form, which is a Google Drive tool. In other words, the data was collected on the last day of class, in the auditorium of the educational centre, which has a capacity for 300 people. To do this, the students used their own mobile devices. In the cases that they did not have, the centre gave them one to fill out the questionnaire. Indicate that the data collection was carried out at the same time, specifically at 18:10. This data was downloaded in Excel format and transcribed into the format of the selected statistical program. Finally, the various statistical tests were carried out and the results obtained were analysed.

#### *3.6. Pedagogic Procedure*

The pedagogical proposal developed with the experimental group is based on the e-learning method. For this purpose, the teacher has made use of the Moodle platform and e-mail. In addition, every week, a schedule was established, consisting of one hour of group attention and two hours of individualized attention. The three hours could be developed in a face-to-face way in the educational centre. It should be noted that these hours were not compulsory. Only those students who considered it necessary came to the centre, and on a voluntary basis. It should be noted that during the study procedure, hardly any students attended the centre to answer questions. The group that developed the expository method, had an hour of tutoring with the teacher of the subject, to solve doubts individually, or attend to the concerns of the students. During this period, the teacher also attended to the student through a virtual platform and by e-mail.

The Moodle platform contained all the content to be dealt with in the subject during the first term, distributed by didactic units. In this case, four didactic units were established for the first quarter. Each one of the didactic units of the Moodle platform was structured in different sections:


have been varied, having different types: short answer, long answer, assumptions, problem solving and autocomplete, relate columns and operations, among others. In this case, all the tools available in Moodle have been used;


The evaluation methods and instruments used have been:


On the other hand, the pedagogical proposal developed by the control group was based on the presentation of theoretical contents by the teacher. In addition, activities have been developed, both from the textbook and from cards given by the teacher. As a method and instrument of evaluation, the following have been applied:


#### **4. Results**

The data presented in Table 2, after the descriptive statistical analysis, show diversity of response among students who attend both the night school and the distance school. According to the data provided by the asymmetry and kurtosis statisticians, the response distribution is considered normal. This is because the values are between ±1.96, according to [79]. The students in the control group show a mean response that is around 2. Some dimensions are slightly below and others are slightly above. In the control group the dimension with the highest rating is resolution. In contrast, the dimension with the lowest rating is decision. The students in the experimental group show a response tendency that is around 2.5 points. The least valued dimension in the experimental group is decision. The most valued dimension in the experimental group is teacher-ratings. According to the statistic that shows the standard deviation, an even trend of response is observed in the students. This is presented in all the dimensions of the study, both in the control group and in the experimental group. Kurtosis is platykurtic in all study dimensions, both in the control group and in the experimental group.


**Table 2.** Results obtained for the dimensions of study in GC and EG of high school students.

a. Established grade group (None: 1–4.9; Few: 5–5.9; Enough: 6–8.9; Completely: 9–10).

The means presented by the control group and the experimental group show relevant differences. In the control group, there is diversity of means between the study dimensions. The resolution and teacher-ratings dimensions stand out from the total mean. On the other hand, the decision dimension is much lower than the mean. In the experimental group, these differences are more pressing. In this case, the dimensions motivation, autonomy, participation, resolution, concepts, results, ratings and teacher ratings are located above the mean. On the other hand, the collaboration, class time, scientific-data, graphics and decision dimensions are located far below the totalised mean. Furthermore, even ratings are observed, both in the control group and in the experimental group, in the collaboration, class time, scientific-data, graphics and decision dimensions (Figure 1).

To identify the value of independence between the expository-traditional method and the e-learning method, Student's t statistical test has been used. The values present higher averages in favour of the experimental group, although it is not significant in all cases. The dimensions motivation, autonomy, participation, concepts, results, ratings and teacher-ratings show a significant relationship. In all the dimensions where there is a relationship of significance, the force of association is average, if the values of the biserial correlation are taken into account. The size of the effect is low in class time and graphics, and very low in the rest of the dimensions (Table 3).

**Figure 1.** Comparison between control group and experimental group.



\*\*. The correlation is significant at the level 0.01. a. Established grade group (None: 1–4.9; Few: 5–5.9; Enough: 6–8.9; Completely: 9–10).

#### **5. Discussion**

The rise of information and communication technologies, related to the current situation of confinement caused by COVID-19, makes the e-learning method relevant in recent times, thus promoting innovative educational practices [1–6].

The e-learning teaching method breaks with the classic stereotypes of teaching and learning processes, since it modifies the spaces and time of training, allowing the development of the pedagogical act in any place and at any time. This can be achieved if technological devices and digital resources are available, as well as internet access [22–27].

In the present research, the influence of e-learning in the field of mathematics has been analysed, in contrast to the traditional expository method, in adult students who are studying for high school. As shown in the results obtained, there are significant differences between the values achieved in the control group and the experimental group. These differences have always been in favour of the e-learning method.

In the group where the expository method has been developed, the lowest values have been produced in the decision. In this case, students have difficulties in making decisions by themselves when solving the proposed mathematical problems. On the other hand, the most valued dimension is the resolution, that is, the carrying out of activities in class. This may be due to the fact that the teacher, present in the expository method, can respond to the needs that the students may have during the development of the different practices.

In the group where the e-learning method is developed the dimension with the highest score is teacher-rating. This shows that the students' grades are increasing, being in line with [58]. On the other hand, the less valued dimension, as in the control group, is decision. In other words, neither the expository-traditional method nor the e-learning method allows the student's decision-making to improve when it comes to solving a problem on their own.

Both in the control group and in the experimental group, students have shown a tendency to respond evenly. This shows that the students agree on the teaching methods applied. This does not mean that there are equal values in all the study dimensions. In the control group the means of the dimensions have not been equal to each other. Examples of this are the dimensions resolution and teacher-ratings, which are above the totalised mean. That is to say, for the students who have developed the expository method, in these dimensions, they show a better evaluation. On the other hand, the decision dimension is much lower than the average.

Something similar occurs in the experimental group. The averages thrown between the different dimensions are not equal to each other. In this case, the contrasts are more relevant. For example, the dimensions motivation, autonomy, participation, resolution, concepts, results, ratings and teacher ratings are above the total average. On the other hand, the collaboration, class time, scientific-data, graphics and decision dimensions are much lower than the total average.

If the means of the control group and the experimental group are compared, there are dimensions in which there are no significant differences. This is the case of the collaboration, class time, scientific-data, graphics and decision dimensions, which present evenly distributed means, although always with higher values in the experimental group.

Where there are significant differences, in favour of the e-learning method, are in the dimensions of motivation [36,59], autonomy [35], participation [60], concepts [55], results, ratings and teacher ratings [58]. In other words, the e-learning method favours these aspects in the pedagogical act.

In the dimension where there is a greater contrast, when comparing the expository-traditional method with the e-learning method, it is in autonomy. This may be mainly due to the fact that the e-learning method favours self-regulation of learning [39].

If this study is compared with other studies in which e-learning has been developed, improvements in students can be observed. On the one hand, there is an improvement in motivation, autonomy, participation, concepts, results and grades. All these aspects are reflected in other studies, in which the e-learning method is associated with a clearly defined and structured pedagogical approach. In the studies analysed, student effort, which has an impact on their qualifications, is due to increased motivation and interest. In other words, the pedagogical approach influences whether the student can be more or less motivated. In addition, the fact that the student is more motivated leads to an increase in participation, which will lead to improvements in the acquisition of mathematical concepts. It will also influence the resolution of various activities. All of this is ultimately reflected in the grades, which increase. Therefore, it can be indicated that there is an improvement in students' academic performance. Furthermore, it should be taken into account that the e-learning method will favour the autonomy of the student, adapting to his or her learning style, which implies more individualised attention to the teaching and learning process. What is clear from all this research is that the e-learning method is associated with a clearly defined pedagogical process, as shown in this research [64–70].

#### **6. Conclusions**

In general, it can be indicated that the dimensions of motivation, autonomy, participation, concepts, results, self-evaluation and teacher qualification have proved to be significant. That is to say, according to the study group, differences are observed in the evaluations given by the students. It should be

borne in mind that these differences may be motivated by the application of the teaching method applied. In one group the expository method has been developed and in the other the e-learning method. The most valued dimensions have been those of the group in which the e-learning method has been developed. This can be due to several reasons. One of them is the applied method, since the e-learning method makes the student the guide of his/her own learning. That is, they have more weight in the teaching and learning process, while the teacher is a guide. This aspect can have a direct influence on motivation, autonomy and participation. This fact, in turn, can lead to a better acquisition of mathematical concepts and results, given that being motivated and having more autonomy in learning, allows the student to increase his or her participation, and in his or her view, to present more interest in the contents being developed. Finally, the improvement in the concepts and results generates an improvement in the qualification of the students, and therefore, an improvement in the self-evaluation of the didactic actions developed. The rest of the dimensions, such as collaboration, resolution, class time, scientific data, graphics and decision, no differences were observed. This may be due to the method itself. In this case, both the e-learning method and the expository method, due to their didactic processes, do not require greater collaboration among students, nor in the feeling of class time. The other dimensions may be due to the fact that neither the expository method nor the e-learning method lead to an increase in the understanding and development of scientific data, the development of graphs or decision-making.

It can be concluded that the e-learning method is an improvement for adult students who are studying mathematics in the educational stage of high school, provided that it is compared with the expository method. In this case, the improvements occur in motivation, autonomy, participation, concepts, results, ratings and teacher-ratings. Therefore, the use of the e-learning method would be effective for its implementation with adults who study mathematics in high school.

The prospective of the research is based on two aspects. On the one hand, the aim is to present the scientific community with new data on the application of innovative teaching methods. In this case, the e-learning method is compared with the traditional expository method for teaching mathematics to adults studying in secondary schools. On the other hand, the aim is to publicise the educational practice developed in this research, so that other teachers, in similar circumstances, can develop it.

The limitations of the study are several. On the one hand, the study sample presents some specific socio-educational characteristics, so one must be cautious when extrapolating the data to other populations. The access to the sample has been for convenience, due to the fact that the educational groups are established by the educational centres themselves. This has prevented the application of other sampling techniques. Finally, the fact of not applying a pretest and posttest study process makes it impossible to be categorical in ensuring that the e-learning method directly influences the dimensions, since there may be other elements that may have been include in the development of the study. Therefore, the results obtained should be treated with caution.

As a future line of research, it is presented to develop this didactic method in other educational stages and in other educational subjects.

**Author Contributions:** Conceptualization: I.A.-D. and P.C.-R.; methodology: P.C.-R.; software: A.-.J.M.-G.; formal analysis: A.-J.M.-G.; investigation: I.A.-D., P.C.-R., A.-J.M.-G. and S.A.-G.; data curation: P.C.-R. and A.-J.M.-G.; writing—original draft preparation: I.A.-D., P.C.-R., A.-J.M.-G. and S.A.-G.; writing—review and editing: I.A.-D., P.C.-R., A.-J.M.-G. and S.A.-G.; visualization: I.A.-D.; supervision: S.A.-G. All authors have read and agreed to the published version of the manuscript.

**Funding:** This study has been financed by the "Study and analysis of technological resources and innovation in teacher training in the field of Higher Education and its applicability to the development of the Santander Region (Colombia)", in the Framework Cooperation Agreement for the strengthening of research and education, signed between the Corporación Escuela Tecnológica del Oriente, the Secretariat of Education of Santander and the AreA HUM/672 Research Group of the University of Granada. Code: ISPRS-2017-7202. Period: 2017–2021.

**Acknowledgments:** We acknowledge the researchers of the research group AREA (HUM-672), which belongs to the Ministry of Education and Science of the Junta de Andalucía and is registered in the Department of Didactics and School Organization of the Faculty of Education Sciences of the University of Granada.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A**


#### **Table A1.** The instrument used is an ad hoc questionnaire.


**Table A1.** *Cont.*

#### **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **B-Learning in Basic Vocational Training Students for the Development of the Module of Applied Sciences I**

#### **Francisco-Javier Hinojo-Lucena, Juan-Manuel Trujillo-Torres, José-Antonio Marín-Marín and Carmen Rodríguez-Jiménez \***

Department of Didactics and School Organization, University of Granada, 18071 Granada, Spain; fhinojo@ugr.es (F.-J.H.-L.); jttorres@ugr.es (J.-M.T.-T.); jmarin@ugr.es (J.-A.M.-M.)

**\*** Correspondence: carmenrj@ugr.es; Tel.: +34-627-199-357

Received: 9 May 2020; Accepted: 18 June 2020; Published: 5 July 2020

**Abstract:** Information and communication technologies are a step forward in education, as they have given rise to innovative methodologies, such as blended learning. This type of training can be applied at any stage or educational typology such as basic vocational training. The main objective of this article is to know the degree of effectiveness of this methodology in this stage, specifically in an applied science module. For this purpose, a quasi-experimental design has been applied with a control group and an experimental group with a total of 147 participants. The results show how those students who have worked through b-learning have experienced better results in all the dimensions of the study. In conclusion, the implementation of this methodology in basic vocational training brings benefits, such as motivation and autonomy in the teaching–learning processes of all students.

**Keywords:** b-learning; ICT; vocational training

#### **1. Introduction**

The technological field continues advancing to this day. This fact is reflected in the social field [1], especially in the labour, social and educational fields [2]. This technological and digital boom has made it easier to carry out multiple tasks in the domestic and personal sphere, becoming a tool that facilitates people's daily lives.

This progress is also reflected in the educational field [3]. In this case, the use of technologies in the educational field, as in other fields of study, is called information and communication technologies (ICT) [4]. The use of ICT generates benefits for student training [5], because it facilitates the application of innovative teaching and learning processes, thereby promoting new learning spaces. This leads to new educational proposals [6], transforming educational events as we know them today [7]. This is because ICT promotes the elimination of spatial-temporal barriers [8] and facilitates access to an enormous amount of resources [9] in different kinds of support and media. All of this leads to improvements in student development [10], mainly due to the increase in motivation, autonomy, predisposition and attitude of the students in the treatment of the proposed pedagogical contents [11–13].

Among the different pedagogical methods that the incursion of ICT in the educational field has brought about is b-learning [14], also known as blended learning, combined learning or hybrid learning [15]. When defining it, the authors agree in not considering this teaching method only as a mixture of classroom learning and online learning [16], but rather as a methodological basis that uses the best of the classroom learning teaching process, and the best of e-learning [17]. In order for the b-learning teaching process to be developed with a minimum of guarantees, the characteristics of those involved in the pedagogical act must be analysed [18], identifying the students' learning styles [19], and the changing the role of the teachers and instructors [20], with the student being the organiser of his or her own learning [21], while the teacher must become a guide [22], developing an open

methodology [23], attending to the students individually [24], and promoting a more autonomous educational act in the student himself [25], with the intention of facilitating the expansion of the students' knowledge [26]. The manner and proportion of combining classroom and virtual teaching will depend on the needs and characteristics of the environment where the teaching and learning process takes place [27].

Among the main characteristics of b-learning is the fact that the teacher acts as both an online tutor and a traditional teacher [28]; promotes personal links between teacher and student [29] and between the students themselves [30]; encourages the combination of technology, learning development and teacher support [31]; facilitates the application of other teaching methods, complementary to b-learning [32]; allows for the development of synchronous and asynchronous communication [33]; focuses more on the curricular elements that the student should develop than on the environment in which he or she is developing [34]; develops ubiquitous learning [35]; eliminates spatial-temporal barriers [36]; favours the development of the digital competence of teachers and students [37]; promotes digital literacy [38]; adapts to the pace, style and pedagogical development of the learner [39]; facilitates attention to diversity [40]; provides access to a wealth of digital resources [41]; and generates a shift in roles between teachers and learners [42].

The use of b-learning, if not properly applied, can lead mainly to technological dependence for teachers and learners [43] and an increased workload for teachers [44].

In order for the b-learning method to be implemented with a number of guarantees, it first requires a great deal of effort and dedication on the part of the teacher [45], as well as instructional design of the teaching and learning process [46]. The key competence of learning to learn should also be enhanced [47], new roles for teachers and students should be clearly established [48], curricular flexibility should be encouraged [49], different learning styles for students should be addressed [50] and cooperative and collaborative work should be promoted [51].

The b-learning method, in the teaching and learning processes, generates improvements in motivation [52], in academic performance [53], in the relationship between teacher and student [54], in learning autonomy [55] and in collaboration [56].

At the same time, the expository method consists of the presentation of a topic in a structured way with the intention of providing information organised, according to criteria appropriate to the intended purpose. This methodology is fundamentally centred on the verbal presentation by the teacher of the contents of the subject under study. This method is also often referred to as a "master class", to refer to a subject taught by a teacher on special occasions [57]. This method is basically focused on the unidirectional communication of the teacher with the student. The teacher teaches by showing the content to be learned, exposing them, so that the student learns through attentive listening and note taking, and the subsequent completion of tasks [58]. Among the advantages of this method is the saving of time and it also means that the teacher is able to attend to big groups, among other considerations [59]. Among the disadvantages are the low participation of the student, little feedback, difficulty attending individually to the student, not facilitating autonomous learning, a passive position for the teacher, the students receiving such a large quantity of information that they do not have time to assimilate it and exceeding their capacity for attention [60].

In other words, the differences between the b-learning method and the expository method lie mainly in the role of the student in the teaching and learning process. In the b-learning method, the role is active. While in the expository method, the role is passive [28–42,57–60].

In the subject of mathematics, educational experiences based on the b-learning method have been developed [61]. Research shows how the pedagogical process allowed teachers to experience the social and cognitive development of students, through synchronous and asynchronous discussions with their peers and facilitators [62]. In addition, this method improves learning outcomes and attitudes towards learning mathematical content [63].

Ultimately, the application of active teaching methods can generate benefits in the teaching and learning processes. These benefits directly influence the students themselves. An example of this is the b-learning method, which leads to an increase in, among other aspects, motivation, academic performance, the relationship between teachers and between students, learning autonomy and collaboration.

This fact is also reflected in subjects such as mathematics, where students' attitudes improve substantially.

#### **2. Justification and Objectives**

The application of information and communication technologies in education has led to the emergence of new methods of teaching. These methods include b-learning [64]. The research presented below is based on analysing the contrast between the b-learning teaching method and the expository method. To this end, this research has focused on students of Basic Vocational Training. All students enrolled at this stage of education are at risk of social exclusion. This is due to their socio-educational characteristics. That is to say, being students with a lack of motivation, with behavioural problems, without study habits and with difficulties in the acquisition of new content.

The pedagogical act has been carried out in the subject of Applied Sciences I, in which they develop pedagogical actions aimed at the acquisition of theoretical and practical competences at a professional, personal and social level. That is, it focuses on science and math.

In the expository method the teacher has had an active role and is always exposing the theoretical contents while the students have a passive role and do not intervene during the class. In b-learning, teachers have developed their pedagogical actions by using a virtual platform. In point 3.6. of the present manuscript, both teaching methods are explained in detail.

It is important to indicate that both pedagogical processes could be observed by the researchers themselves, since they continuously supervised the pedagogical actions, making sure that the established pedagogical processes were adequately developed.

These two methods have been applied in the teaching–learning processes of the educational reality of the participants in the research, so we wanted to measure the influence of both separately, and how this affects the qualifications of the students. This research also tries to present more studies on the use of b-learning in mathematics related subjects [61–63], specifically for students who present adverse socio-educational characteristics, as in the case of students in Basic Vocational Training.

Therefore, the main objective of the present study is to identify the degree of effectiveness of the b-learning method in the module of Applied Sciences I, for Basic Vocational Training students, in comparison with the expository method, and in different areas of socio-pedagogical development. From this general objective, the following specific objectives are developed: (i) to define the level of motivation of the students, both in the control group and in the experimental group; (ii) to specify the level of interaction (teacher–student, student–student, student–content), both in the control group and in the experimental group; (iii) to investigate the level of autonomy of the students, both in the control group and in the experimental group; (iv) to identify the level of collaboration of the students, both in the control group and in the experimental group; (v) to identify the level of deepening of didactic content, both in the control group and in the experimental group; (vi) to discover the level of problem solving in the didactic activities proposed, both in the control group and in the experimental group; (vii) to analyse the perception of the class time developed, both in the control group and in the experimental group; (viii) to specify the influence of the teaching method through the grades, both in the control group and in the experimental group; (ix) to identify the contrast of averages in the different dimensions of study between the group that applies the expository method and the b-learning method.

#### **3. Research Method**

#### *3.1. Research Design and Data Analysis*

The present research is quantitative, descriptive and correlational [65], applying a quasi-experimental design, by means of control group (Gc) and experimental group (Ge). The study follows the structure and model of previous research [66–69]. The control group has experienced the exposure method. On the other hand, the experimental group has followed the b-learning method. The distribution of students is not random, since the groups were already defined from the beginning of the course. The criteria for the distribution of the students were established in the previous academic year, in a meeting between the different Heads of Studies of the secondary education centres, in the presence of the Education Inspector. At this meeting, the distribution criteria are based on the principle of equity. The information was collected at the end of the educational experience, which took place in January of the 2019/2020 academic year, by means of a post-test (Table 1).


**Table 1.** Groups' composition.

The analysis of the data collected has been carried out using the Statistical Package for the Social Sciences (SPSS) programme, version 25. The statistics used are the mean (*M*), standard deviation (*SD*), skewness (*S*kw) and kurtosis (*K*me). In addition, Student's t-test (tn1 + n2-2) was used to compare the group means. Finally, Cohen's d test and the biserial correlation (rxy) have been applied to identify the effect size and the association force. The significance level applied in the study was *p* < 0.05.

#### *3.2. Participants*

The sample used in this study is composed of 147 students. The sampling technique used is a convenience sample, due to the ease of access to the population. In relation to the total number of the sample, the authors [70,71] establish that, in the application of pedagogical methods, the size of the sample is not a determining factor.

The students that make up the two groups that are part of the study are studying the module of Applied Sciences I, of the Basic Vocational Training. The students in this educational stage present specific characteristics, among which the following stand out: having reached fifteen years of age and not exceeding seventeen years of age; having studied the first cycle of Obligatory Secondary Education (ESO), or exceptionally, the second year of ESO; and having been proposed by the teaching team, after acceptance by the parents. These students usually have had a previous negative experience during their stage in ESO, not reaching the necessary competences, with high levels of absenteeism, low academic performance, lack of motivation and no study habits [72].

The Basic Vocational Training student body is made up of 61% men and 39% women, aged between 15 and 17, with an average age of 16.3 years and a standard deviation of 0.432. The students were composed of three professional families or groups of training cycles with common characteristics, on the one hand, the professional family of electricity and electronics (2 groups), the professional family of physical activities and sports (2 groups) and the professional family of personal image (2 groups).

The research was carried out in the first month of the second quarter, in the academic year 2019–2020. It is important to highlight that three teachers participated, so they were trained in b-learning and the Moodle platform, as well as in the expository method. In order to proceed with the research, the corresponding permits were requested and collected, informing all the parties involved of the objectives of the study. There was collaboration at all times between all the people involved in the study.

#### *3.3. Instrument*

The instrument used was an ad hoc questionnaire, following the structure of other questionnaires that collected data on active teaching methodologies [66–69]. The qualifications established by the teacher have also been taken into account and it is described in the pedagogical procedure.

The questionnaire is composed of nine dimensions (socio-educational, motivation, interactions, autonomy, collaboration, deepening of content, problem solving, class time and ratings), with 35 items with answer format based on Likert scale (from 1 = None to 4 = Completely).

The instrument was tested for validity and reliability. The Delphi qualitative validity method was applied (*M* = 4.46; *DS*= 0.21; min = 1; max= 5); the Kappa statistic by Fleiss and W by Kendall (*K* = 0.88; *W* = 0.86); the exploratory factorial analysis with varimax rotation (Bartlett = 2. 771.01; *p* < 0.001; Kaiser-Meyer-Olkin = 0.89); Cronbach's alpha (0.89); McDonald's omega method (0.88); the reliability of the compound (0.87); and the mean variance extracted (0.85). Bearing in mind all these values, it is considered a valid and reliable instrument.

#### *3.4. Dimensions and Study Variables*

The dimensions used in this study are based on other research [66–69], whose items have been considered as independent variables: socio-educational; motivation; interactions; autonomy; collaboration; deepening of content; problem solving; class time; rating; and teacher ratings. On the other hand, the pedagogical method developed has been considered as a dependent variable. All these variables have been measured through the questionnaire.

#### *3.5. Methods*

The methodological procedure applied in the investigation started with the validation and reliability of the questionnaire. Subsequently, the study population and the research sample were selected, requesting at that time all of the corresponding permits, both from the educational centres and from the trainees themselves.

The methodological procedures to be developed were then determined and specified. On the one hand, the pedagogical acts of the expository method (Gc) and the b-learning method (Ge) were established.

Then, data was collected using a form previously developed with the Google Form tool. Finally, they were downloaded in an Excel table, transcribed to the statistical program used, and the statistical tests and analyses were carried out.

#### *3.6. Pedagogical Procedure*

In order to develop the b-learning method with the Basic Vocational Training students, each student was assigned a computer at the educational centre, so that they could access a Moodle platform specifically assembled for the development of the module, by means of a username and password. The access to the platform could be done from any place and at any time, as long as they had a device with Internet access. Those students who, due to different circumstances, could not access the platform from their homes, were provided with a corner with computers for their use in the libraries of the educational centres. In this case, the tutor also had access to the platform 24 h a day, being able to enter it outside school hours to correct or solve doubts about activities. The pedagogical development was divided into two clearly defined lines: the virtual sessions and the face-to-face sessions:

For the virtual period, the students had to read the theoretical contents prepared for their study or knowledge, carry out activities to consolidate the acquired contents and ask for the necessary help through the forums, the tutor or a classmate.

For the face-to-face period, the time was dedicated to consolidate the theoretical contents, to carry out cooperative and collaborative activities, to solve the individual difficulties that the students presented before certain types of activities and tasks, as well as to develop activities to attend to the transversal elements, such as reading comprehension, oral and written communication, audio-visual communication, ICT and values education.

The qualification criteria of the Basic Vocational Training students who have developed the b-learning method were:


In the expository method, the teacher has made the theoretical presentation of the contents during the development of the sessions. This theoretical presentation followed what was established in the didactic program as well as what was indicated in the textbook used for the level of the students. On certain occasions, the teacher has used the digital blackboard to show certain contents in an interactive way. In addition, during the development of the classes, the teacher established a series of tasks, both in the development of the session and at home. The time structure of each session was distributed as follows: 10 min to remember the contents worked during the previous session; 10 min to solve the doubts that the students could have or to correct the exercises developed at home; 25 min for the theoretical exposition of the contents; and 10 min for the accomplishment of tasks. The qualification criteria have been:


#### **4. Results**

The data presented in Table 2, related to the descriptive analysis, shows diversity of scores between the control group and the experimental group. In this case, students in Basic Vocational Training who have developed the educational experience through b-learning present better averages in all the dimensions studied. Although, if we analyse it in detail, we can see that the average of the experimental group is in an intermediate zone, so the scores reached are not very high. These are around 2.5 points. On the other hand, in the control group the averages are relatively low, given that they are in all cases below 2. In the experimental group, the most valued dimension is class time, while the least valued dimension is resolution. In the control group, the dimension with the highest score is the relationship between students. On the other hand, the least valued dimensions are resolution and teacher ratings. If the values of the standard deviation are taken into account, a trend of grouped response is shown, not having dispersion in any of the dimensions. With respect to kurtosis, most of them are platicurtic, although there are also, to a lesser extent, leptokurtic and mesocurtic types. If the values reached in asymmetry and kurtosis are taken into account, it can be established that the distribution of the sample is normal. This is because the values are between ± 1.96, as marked by [73].


**Table 2.** Results obtained for the dimensions of study in CG and EG of students in Basic Vocational Training.

<sup>a</sup> Established grade group (None: 1–4.9; Few: 5–5.9; Enough: 6–8.9; Completely: 9–10).

The comparison of means shows that the total mean of the experimental group is 2.5, i.e., in the intermediate zone. This indicates that the scores have not been high, but rather average. In contrast, in the control group, the idealised mean is at 1.7, which marks a low response trend. If the totalised means of the control group and the experimental group are compared, a considerable distance is observed, so that the application of one teaching method or another influences the dimensions studied. In the control group, the student–student mean stands out from the idealised mean. The same occurs with class time in the experimental group (Figure 1).

**Figure 1.** Comparison between control group and experimental group.

By means of the Student's t-test, the relationship of significance, in each of the study dimensions, of the b-learning method in relation to the expository method has been identified. In this case, the statistical values show the relation of significance in all the study dimensions. The greatest difference in means is presented in class time, with up to one point of difference. On the other hand, the dimension with less mean difference is teacher–student. If we take into account the strength of association, we can see that the relationships are medium and low-medium. The dimensions class time, collaboration, ratings, teacher-ratings and motivation have a medium relationship strength. The dimensions with a medium-low relationship are located the rest of the dimensions. According to Cohen's d, the effect size is very low in all dimensions, except in class time, ratings and teacher-ratings, where the effect size is low (Table 3).


**Table 3.** Study of the value of independence between control group and experimental group.

\*\* The correlation is significant in level 0.01. <sup>a</sup> Established grade group (None: 1–4.9; Few: 5–5.9; Enough: 6–8.9; Completely: 9–10).

#### **5. Discussion**

The results achieved have been able to show that the b-learning method is an effective teaching and learning process, compared to the expository method. In this case, it is effective with students of Basic Vocational Training, in the module of Applied Sciences I. In other words, it is effective for students who are at risk of social exclusion.

The inclusion of ICT in the educational field is enabling innovative teaching and learning processes to be applied, thus favouring the academic development of students [1–5].

If we analyse each of the groups in detail, we can see that, as in the control group, the results achieved are relatively low in all the study dimensions. This may be due to the characteristics of the students in Basic Vocational Training already indicated by [68], where the students present a poor academic background in previous educational stages. Resolution and teacher ratings are among the least valued dimensions. That is, they present difficulties in the resolution of the various pedagogical actions, and therefore, this is reflected in the qualifications established by the teacher.

An example of this is the b-learning teaching method, considered as a didactic process that mixes the best of the expository method with the best of the e-learning method, allowing, among other aspects, to adapt to the rhythms and learning styles of the students, as well as to provoke a change in the roles of the agents involved in the pedagogical act [14–20].

In the present study, we have analysed how the b-learning method influences the students of Basic Vocational Training, specifically in the module of Applied Sciences I. To this end, a contrast has been established with the expository method. According to the results obtained, it is observed, in general terms, that there are better averages in the experimental group, which has developed the b-learning method, with respect to the control group, which has received a teaching based on the expository method.

Additionally, in the group that has developed the b-learning method, the ratings are higher than those offered by the control group, although it does not present very high ratings. Rather, the scores are average. Even so, there is a contrast between the groups, so it can be considered that the b-learning method favours the academic development of the students. The most valued dimension has been class time. This may be because the method applied may be new to students. On the other hand, the less valued dimension is resolution. In this case, the same happens as in the control group, so the educational base of these students is affected for the development of any educational process they develop, although unlike the control group, the grades are not so affected.

In both groups, the response trend is grouped, so that students maintain the same line of assessment, according to the teaching method applied. That is to say, they agree when giving their opinion about one teaching process or another.

In this case, it can be indicated that the b-learning method, in comparison with the expository method generates an improvement in Basic Vocational Training students in motivation [52]; in the relationship between the teacher and the student [54]; in the relationship between the students and the didactic content [49]; in the relationship between students [51]; in autonomy [55], in collaboration [56]; in the deepening of content [50]; in the resolution of pedagogical acts; in the perception of the class time; in self-evaluation [59]; and in the grades of the module studied [53].

In other words, this study confirms that which has already been established by other authors in relation to the b-learning method and the expository method. With the b-learning method, a positive attitude is produced in the student, since it generates an active process in the formative process. On the other hand, the expository method generates a passive act in the teaching and learning process in the students themselves. The difference of the student's role in these methods provokes significant differences in motivation, in the relationship established between the agents involved in the training process, in the self-management in the collaboration, in the deepening of content, in resolution, in the classtime and in the academic performance.

#### **6. Conclusions**

It can be concluded that the b-learning method is effective in the teaching and learning processes of the students of Basic Vocational Training in the module of Applied Sciences I, in comparison with the expository method, having a direct influence on the feeling of the students' own class time. This research shows how important it is to introduce this type of innovative method in the vocational training stage, since it has important advantages for students in many aspects of their learning processes.

The prospective of this study is on two different levels. On the one hand, it tries to provide data to the scientific community on the use of the b-learning method in Basic Vocational Training students. On the other hand, it tries to offer an effective teaching and learning process for teachers working with these types of students.

The limitations of the study are the focus on the selection of the sample, which has been for convenience. In addition, the study population presents specific characteristics, so one must be cautious when extrapolating the results. For future lines of research, it is proposed to analyse this teaching and learning process in the second year of Basic Vocational Training and in other modules.

**Author Contributions:** Conceptualisation, F.-J.H.-L. and J.-M.T.-T.; methodology, J.-A.M.-M.; software, J.-A.M.-M.; validation, C.R.-J.; formal analysis, F.-J.H.-L.; investigation, C.R.-J.; resources, J.-M.T.-T.; data curation, J.-A.M.-M.; writing—original draft preparation, C.R.-J.; writing—review and editing, C.R.-J.; visualisation, J.-M.T.-T.; supervision, F.-J.H.-L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Ministry of Education and Vocational Training of the Government of Spain (project reference: FPU18/01595).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

1. Hinojo, F.J.; Aznar, I.; Romero, J.M.; Marín, J.A. Influencia del aula invertida en el rendimiento académico. Una revisión sistemática. *Campus Virtuales* **2019**, *8*, 9–18.


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Analysis of Factors Influencing Students' Access to Mathematics Education in the Form of MOOC**

**Dalibor Gonda 1,\*, Viliam Duriš ˇ 2, Gabriela Pavloviˇcová <sup>2</sup> and Anna Tirpáková 2,3**


Received: 2 July 2020; Accepted: 23 July 2020; Published: 27 July 2020

**Abstract:** Restricting the movement of students because of COVID-19 requires expanding the offer of online education. Online education should reflect the principles of pedagogical constructivism to ensure the development of students' cognitive and social competencies. The paper describes the preparatory course of mathematics, realized in the form of MOOC. This course was created and implemented based on the principles of pedagogical constructivism. The analysis of the respondents' approach to MOOC revealed a difference between bachelor and master students in the use of MOOC. Bachelors found a strong correlation between their approach to MOOCs and the way they are educated in secondary schools. The results of the research point to the need of more emphasis should be placed on advancing the learner's skills in navigating and analysing information. The questionnaire filled in by the participants also monitored the students' access to learning. The results of the experiment confirmed the connection between the preferred approach to learning and students' activities within the MOOC.

**Keywords:** constructivism; mathematics learning; MOOC; new teaching techniques; students' access to MOOC

#### **1. Introduction**

Recently, many pedagogical experts have questioned traditional teaching methods such as lectures and testing [1] (pp. 167–202), [2] (pp. 3–17). According to Mascolo [2], the basis of pupil-centred education is constructivism. Constructivism is based on the European genetic epistemology of Jean Piaget and American cognitive psychology. Constructivist epistemology includes cognitive constructivism and social constructivism [3] (pp. 241–250). Cognitive constructivism pursues the individual development of knowledge through interaction with the environment, and social constructivism refers to the dialogue of students with each other and the teacher and to the social context in which learning takes place [4] (pp. 61–86). According to Lave and Wenger, an important part of constructivism is social constructivism, which focuses on cultural and social learning conditions, on social interaction in learning [3,5] (pp. 241–250). Pedagogical constructivism is a combination of cognitive and social constructivism and demands that teaching should use authentic problem solving, creative thinking and group work [5]. Medová and Bakusová [6] (pp. 142–150) stressed the role of real-life problems in constructivist mathematics education. Astin [7] also focused on group work in his research.

Recently, we have seen a massive increase in the offer of various online courses, even for university students. There are countries where online courses have become an integral part of teaching, especially at universities. In the USA, for example, more than 30% of university students attend at least one online

course [8]. Using state-of-the-art computer technologies, online courses offer students a wide range of engaging and interactive learning environments that have demonstrated support for satisfaction, motivation and persistence among participants [9] (pp. 435–447), [10] (pp. 24–32), [11] (pp. 221–231), [12] (pp. 306–331).

Online courses encourage students to be independent, to develop the skills of personal reflection and abstract conceptualization [13] (pp. 227–243), [14] (pp. 309–328). For more success in using online courses [15] (pp. 1–28) suggests more emphasis should be placed on advancing the learner's skills in navigating and analysing information.

Another, relatively new, but especially effective element in online learning are the so-called massive open online courses or MOOC. The term "massive open online course," or MOOC, was first used to describe a course on learning theory taught by George Siemens and Stephen Downes at the University of Manitoba in 2008. According to Downes, the idea was to "invite the rest of the world to join the 25 students who were taking the course for credit" [16].

MOOC is based on the principle of sharing and freedom. According to Jeffrey [17], this is "self-service learning and crowdsourced teaching".

MOOC courses meet with several positive responses in the professional community. For example, Friedman [18] (pp. 175–186) considers the MOOC a breakthrough in higher education, and Mozoué [19] sees them as an alternative to full-time education and making education accessible to a wider range of society. There are also doubts about their contribution to higher education. Several higher education analysts are sceptical and express their doubts as to whether the MOOC is an adequate alternative to classical higher education or online education, especially in terms of teaching and access to students [8,20] (pp. 7–26), [21] (pp. 87–110). They also point out that the use of MOOC requires participants to be able to work independently and thus have the necessary level of critical literacy and the ability to navigate the course. Therefore, according to Kop, Fournier, & Mak [22] (pp. 74–93), more experienced and independent students are more successful in this environment. It also happens that many participants are struggling with a lack of instructional support at the MOOC and do not complete their courses.

There are currently several empirical studies that evaluate not only the MOOC teaching strategy but also the results of MOOC-related learning. According to Toven-Lindsey [23] (pp. 1–12) and Rhoads and Lozano [21] (pp. 87–110), there are considerable differences in pedagogical approaches, most courses still use elements that are common in traditional classes, including lectures, multi-choice assessments, and discussions about current groups.

Currently, there is already an offer of mass open online courses (MOOC) in Slovakia, but such offer is limited—only in some universities are MOOC offered for selected courses, as a means of supporting the quality of education. This is even though external students make up approximately one quarter of university students. According to [24] (pp. 451–460), due to many online learning opportunities, including MOOC courses, it is necessary to analyse their quality and to improve the effectiveness of education using analytical methods. One of the first Slovak universities involved in the MOOC project since 2013 is the Slovak University of Technology. Slovakia was also involved in the project BizMOOC—Knowledge Alliance to enable a European-wide exploitation of the potential of MOOCs for the world of business Programme: Erasmus+. It was found in the project that one of the major obstacles to using MOOC is the language barrier (see www.bizmooc.eu). The above findings showed the need to create MOOC in the national language in Slovakia. The aim of our research was to create a preparatory MOOC of mathematics that would consider the principles of pedagogical constructivism and to conduct research on the behaviour of students in using this course. In this way, we wanted to find out whether not only the cognitive but also the social component of the student's personality in relation to his/her approach to learning develops within the MOOC. This would determine whether MOOC can be a suitable alternative to full-time education.

#### **2. Materials and Methods**

#### *2.1. Objective*

For several years, many Slovak universities have introduced the so-called "tutoring mathematics" in the form of various mathematics courses. These courses are intended primarily for students admitted to the 1st year of higher education and are usually organized in full-time form lasting from 3 to 5 days. It was this situation that motivated us to create a pilot preparatory e-learning course in mathematics. In our case, we chose a relatively under-used MOOC model in Slovakia, where students can not only educate themselves but also discuss and present their problem-solving procedures. In addition to the above, we were motivated by idea of Giroux [25], according to which students should not only be educated, but also be active participants in the learning process. Our research team has set a goal to develop a pilot mathematics preparatory course in the form of MOOC and examine its use by students. At the same time, we investigated whether, in addition to the development of cognitive competencies, participants in the course also develop social competencies. In the case of the development of both competencies, the MOOC in the proposed form could be a suitable alternative to the full-time form of education.

As courses of a similar type were not available in Slovakia so far, we developed the course ourselves and offered it to the students and we not only observed the extent to which the students used the course, but we also considered it necessary to find out the students' reactions to the product. We were therefore interested in the extent to which students will use the different parts of the MOOC and what attitude they will take to it.

The MOOC course lasted one month and was made available to students admitted to the first year of undergraduate and graduate study at the technical faculty of a selected university in Slovakia. There were separate MOOC modules for each stage of the study with respect to the achieved education.

During enrolment in the first year (both undergraduate and graduate), students were acquainted with a preparatory course in mathematics in the form of the MOOC containing the "mathematical minimum" needed to master mathematics in the given field of study for which they were admitted. Created MOOC and possibilities of its use were introduced to students by MOOC authors themselves. At the same time, each student received access data to the portal. The access data were anonymous for the research team, only used to monitor the activities of individual students within the MOOC. These data were also used in the final questionnaire. Student activity data served as data for statistical evaluation of MOOC rate and usage. Our MOOC consisted of the following modules: Module 1—algebraic equations and inequalities, Module 2—non-algebraic equations and inequalities, Module 3—functions, Module 4—elemental geometry. Each module was given a week within the MOOC. MOOC was created and launched on the website: https://www.mooc.km.fpv.ukf.sk/, which we were developing for a long time. This training system works both in Slovak and English and the course materials for individual modules were gradually made available. The study materials were divided into two parts. The first part consisted of theoretical bases of the studied problems such as definitions of basic terms (8 pdf files) and assignments of tasks in text form (8 pdf files) and the second part consisted of sample examples in audio-visual form. The video sequences included instructions for solving basic sample examples for individual modules (32 video sequences). Solutions of various problem tasks supporting the construction of new computational strategies for students were the subject of webinars. Every Friday, the webinar was held twice (at 10:00 and 17:00), which was focused on problematic issues related to topics provided to MOOC participants in each week. The webinar was always led by a member of the author team. The aim of the webinar was to support the ability of students to create their own solutions of given tasks with creative use of already acquired theoretical knowledge and skills with solving standard tasks. The principles of pedagogical constructivism were consistently applied to webinars. The heuristic general didactic method was used, in which the teacher acted as a moderator of the participants' discussions. Each registered participant automatically became a member of the MOOC discussion forum without teacher participation. At the same time participants

could create different discussion subgroups—these subgroups could be created by the participants. Another possibility through MOOC that we created was the possibility to address the teacher in the form of a question or by requesting to check the correctness of the task. From the questions asked to the teacher, we gradually created the content in the "Frequently Asked Questions" menu. For each of the topics covered, exercises were also available for download with the option to send suggested solutions to the teacher for review.

#### *2.2. Sample*

We were interested in the extent to which students will use the individual parts of our MOOC course and what attitude they will take towards it. The research took place in September before the beginning of the winter term of the academic year 2018/2019 at a selected university of the Slovak Republic. Respondents of the research were engineering fields of study students, namely 48 undergraduate students and 35 graduate students. The respondents were between 19 and 26 years of age. Participation in MOOC was voluntary, which was also reflected in students' lower interest from compared commonly used full-time form.

#### *2.3. Information Collection Tools*

The data necessary for the evaluation of the research were obtained by monitoring the activity of students involved in the MOOC preparatory course of Mathematics. We monitored the number of views of each video sequence and the number of downloads of study materials. An important source of data was the content and form of discussions among students within the discussion group. The administrator was able to track the overall activity of each MOOC member, so it was possible to determine the priorities of each MOOC member when choosing the options offered within the course. After completing the MOOC course, respondents completed a questionnaire. All students who participated in the MOOC were able to complete the questionnaire, regardless of whether they completed the course or not.

#### **3. Results and Discussion**

The basis for evaluating the suitability and usability of MOOC as a preparatory course in mathematics was a questionnaire developed and used by Aharony and Bar-Ilan [26] (pp. 146–152). Just like the authors of the questionnaire, in our case we also observed 5 areas in the questionnaire:


The 4th part of the questionnaire—Learning Strategies (LS), which reflects the student's approach to learning and education, was very important for us. According to [26] there is the deep learning versus the surface learning approach; terms that are based on the early work of Marton and Säljö [27]. Deep learners tend to seek for their 'inner self' through the learning process [28,29].

Contrarily, surface learners learn only important and essential facts, applying minimum study efforts [28]. A surface learning approach is associated with students who study only superficial details [30]. They are concerned with the time needed to accomplish the learning task; therefore, they try to choose the quickest way to accomplish their learning assignment, without asking further questions and without fully understanding the text meanings. Surface learners usually memorize facts; thus, meta-cognitive skills are mostly not involved in their learning process [28].

Cognitive appraisals of threat and challenge refer to "dispositions to appraise ongoing relation-ships with the environment consistently in one way or another" [31] (p. 138). Cognitive appraisal addresses the person's evaluation of events for his or her well-being [32].

For the reasons stated above and also in accordance with [26], we divided the fourth area of the questionnaire into two parts (areas), namely: learning strategies: deep learning (LS-D), consisting of questions 1, 3, 6, 8, 12, 13 and surface learning (LS-S), which consisted of questions 2, 4, 5, 7, 10, 11, 14. For the same reasons, we divided the fifth questionnaire into two parts: threat perception (CAQ-T), consisting of questions 1, 2, 3, 4, 6, 7, and challenge perception (CAQ-CH), which consisted of questions 5, 8, 9.

Part of our research was also tracking the activities of students who attended the MOOC course. We divided the activities into two areas: cognitive constructivism (MOOC-CC) and social constructivism (MOOC-SC).

Subsequently, we identified research questions:


To find answers to individual research questions (Q1–Q5) we analysed the results obtained by the questionnaire method as well as by monitoring the respondents' activities. There was an answer to each question on the 5-point Likert scale, where 1 means "absolutely disagree" and 5 means "totally agree". The results obtained in our research by the questionnaire method in both groups of students are illustrated in the following figures (Figures 1–6).

**Figure 1.** Answers of undergraduate and graduate students in PU (average values).

**Figure 2.** Answers of undergraduate and graduate students in PEOU (average values).

**Figure 3.** Answers of undergraduate and graduate students in LS-D.

**Figure 4.** Answers of undergraduate and graduate students in LS-S.

**Figure 5.** Answers of undergraduate and graduate students in CAQ-CH.

In Figures 1–6 we can see that there are differences between the answers of students of undergraduate and graduate study to questions in individual areas of the questionnaire. We wondered if the differences are statistically significant.

The statistical significance of the differences between the two groups of students in the answers to the questions was verified in each area of the questionnaire (PU, PEOU, LS-D, LS-S, CAQ-CH and CAQ-T) based on calculated values, called total score. As the assumption of a normal distribution of observed traits was not met, we used the non-parametric Wilcoxon two-sample test to verify the Q1 research hypothesis [32].

**Figure 6.** Answers of undergraduate and graduate students in CAQ-T.

In our case, the first selective file consists of undergraduate students and second file consists of graduate students. The results of the selected area of the questionnaire (total score) of both groups of students represent the realization of two mutually independent random samples from continuous distributions. We conducted Wilcoxon's two-sample test in STATISTICA.

The results obtained using Wilcoxon's two-sample test were summarized in the following Table 1.


**Table 1.** The results of Wilcoxon's two-sample test.

Note: Values exceeding the critical value are indicated \* in the table.

Since the calculated probability value *p* < 0.05, in three cases—in the LS-D, CAQ-CH and CAQ-T areas, the hypothesis *H*<sup>0</sup> is rejected in all three cases at the significance level *p* = 0.01 and we can say that among the undergraduate and graduate groups is a significant difference in the answers to the questionnaires in the LS-D, CAQ-CH and CAQ-T.

Based on the results obtained in the statistical analysis of PU, PEOU and LS-S questionnaire, the hypothesis *H*<sup>0</sup> cannot be rejected, i.e., the observed differences are not statistically significant.

#### *Analysis of Student Activity within MOOC*

Figures 7 and 8 show the average MOOC visit values for each activity. The activities of undergraduate and graduate students were evaluated separately. This division was based on the results of the questionnaire in the field of LS, where it turned out that undergraduate and graduate students have different approaches to education. While graduate students prefer deep learning, undergraduate students prefer surface learning. Therefore, when answering other research questions, we evaluated the individual parts for undergraduate and graduate students separately.

We divided the activities in the created MOOC course into two parts. The first part was called "Cognitive Constructivism", which included those activities where there was no cooperation with other course participants or with the teacher. They were theory, video, and exercises to practice. The average utilization of the individual activities is shown in Figure 7.

**Figure 7.** Average visit in options under 'Cognitive constructivism'.

In both groups of students, the most attention was paid to video sequences. In all activities graduate students were more active in all activities. The biggest difference was in the use of the offer of the necessary theoretical knowledge on the individual topics covered within our MOOC.

The second part consisted of activities in which the participants cooperated with each other and possibly with the teacher. We called this part "Social Constructivism" and it included activities: a webinar, a question to the teacher, a discussion forum, frequently asked questions.

Based on the results shown in Figure 8, graduate students were more active in the second part activities, and even more than in the first part. Only in the activity "Question to the teacher" were undergraduate students more active. We interpret this because of the fact that undergraduate students come from a high school (secondary school) environment where the teacher has a dominant position in pupils learning [33].

**Figure 8.** Average visit in options under 'Social constructivism'.

Again, we can see that there are differences between both groups of students (undergraduate and graduate) in both activity areas. We were interested in finding out whether these differences as well as the links between the monitored areas in both groups of students are statistically significant. We used the statistical method—Spearman order correlation coefficient, which expresses the degree of dependence between X and Y.

In our case, we calculated the following Spearman correlation coefficient values for both groups of students (Tables 2 and 3).


**Table 2.** Spearman correlation coefficient (undergraduate students).

*\* p* < 0.05.

**Table 3.** Spearman correlation coefficient (graduate students).


\* *p* < 0.05.

We observe in undergraduate students a high degree of bonding between PU and LS-S (*R* = 0.71) and between PEOU and LS-S (*R* = 0.75). It is the relationship between surface learning and attitudes to the use of MOOC. So, we can state that the more undergraduate students prefer a superficial approach to learning, the more they consider MOOC to be a useful and easy to use tool. Based on the results, a significant degree of binding was also observed between LS-S and MOOC-CC (*R* = 0.49), i.e., between surface learning and cognitive constructivism activities in MOOC. This can also be interpreted as suggesting that undergraduate students with a superficial approach to education tend to use activities that do not interact with other course participants. Further evaluation revealed a significant correlation (*R* = 0.5) when using video in MOOC, which undergraduate students considered the easiest way to obtain information. We can say that undergraduate students approach the use of MOOC rather than a suitable tool to help them master the curriculum with minimal effort. Other connections between observed areas in undergraduate students were not statistically significant.

A significant degree of linkage between several fields of study can be observed in the graduate students. In particular, we observe a high degree of binding between LS-D and PU (*R* = 0.7), PEOU (*R* = 0.82) as well as CAQ-CH (*R* = 0.69). Based on the correlation coefficient values given above, it can be said that the more graduate students prefer a profound approach to learning, the more they perceive the MOOC as a cognitive challenge and consider it a useful and very usable tool for learning. In other words, based on the results we can see that the graduate students prefer deep learning. Equally

significant is the degree of binding between LS-D and MOOC-CC (*R* = 0.58) and also between LS-D and MOOC-SC *(R* = 0.70), i.e., between deep learning and cognitive and social constructivism activities in our MOOC course,

Unlike undergraduate students, where these links were not confirmed at all. Based on the calculated values of correlation coefficients, we observe significant degrees of binding in the MOOC-CC between the use of tasks and exercises (*R* = 0.61) and theory (*R* = 0.53). The use of video had only a slight degree of custody (*R* = 0.37) regarding given access to education compared to undergraduate students. However, a significant degree of binding has been shown between the use of video and the perception of the usefulness of MOOC (*R* = 0.57) in graduate students. This means that the more graduate students perceive the usefulness of MOOC, the more they use video sequences in the MOOC course. In the MOOC activities included in social constructivism, there were significant degrees of links between deep learning and webinar (*R* = 0.63), discussion forum (*R* = 0.64) and frequently asked questions (*R* = 0.53). Based on the calculated correlation coefficient values, we got a zero link between LS-S and the activity *question to the teacher*, which was often used by undergraduate students. Based on the above results, we can conclude that in the case of graduate students with a perception of MOOC as a challenge for their education, a positive attitude towards MOOC from the point of view of its usefulness and ease of use also strengthens.

The results of our research correspond to those of [26], which also confirmed the differences between undergraduate and graduate students in LS and CAQ. The behaviour of students within our MOOC was also confirmed by the findings of Kop, Fournier, & Mak [22], that more experienced and independent students are more successful in the MOOC environment. Based on the above, it can be concluded that more frequent use of MOOCs in higher education institutions requires more space should be devoted to activities falling under the principles of social constructivism in secondary schools. In acquiring new knowledge based on the principles of cognitive constructivism, we recommend using the heuristic method in secondary schools, where the teacher acts as a moderator of the pupil's learning. In this way, critical literacy of pupils will be strengthened, which, according to Lewin [34], is one of the important prerequisites for successful mastery of MOOC.

#### **4. Conclusions**

Based on the results of the research, it can be stated that students' access to education, as well as the perception of new situations in terms of threat or challenge depends on the level of study. Master students prefer deep learning, so they care more about understanding the subject matter and the value of their knowledge, they are willing to devote more time to study. Bachelor students, on the other hand, prefer surface learning, i.e., the acquisition of the necessary knowledge without a deeper understanding and with the least effort. It was also shown that with the growth of the preference for surface learning among bachelor students and with the growth of the preference for deep learning among master students, there is also a growing perception of the usefulness and applicability of MOOCs. An important element of the research was the students' approach to the activities in our MOOC course, which were divided into cognitive and social constructivism activities. Significant differences between bachelor and master students emerged in this area. Bachelor students showed a positive correlation between their superficial approach to education and the use of MOOC activities included in cognitive constructivism. They used videos the most for their education, which we can consider the easiest way to get information. In the quantitative evaluation of attendance at individual activities of the course, bachelor students more often used the possibility of asking questions to the teacher than masters. On the contrary, master students showed a positive correlation between their perception of MOOC as a challenge and the use of activities of both cognitive and social constructivism in our course. Significant degrees of connection were also shown in the masters between their in-depth approach to education and the use of all activities of our MOOC course with a higher preference of those that were included in social constructivism. Compared to bachelor students, they made much more use of the discussion forum, webinar, theory on the topic as well as tasks and examples. We believe that differences in behaviour and attitude towards MOOC among undergraduate and graduate students reflect strongly on the way and methodology of education at secondary schools in Slovakia. Undergraduate students attended the MOOC course before they started university (before the semester), so we can attribute their behaviour to high (secondary) school students with very little or no e-learning experience. On the other hand, awareness, and responsibility for the study in terms of its need for the future and self-assertion in life is still low for 19-year-olds. Another aspect may also be the overall atmosphere in society and the speed of time when young people have higher demands on their surroundings, but not towards themselves. On the other hand, graduate students already have a bachelor's degree in higher education and know that they are required to be independent and responsible in their studies. They also have more experience with e-learning and are more open to communicating with their classmates and educators.

Based on our findings, we can state that MOOCs created on the basis of pedagogical constructivism have the potential to be a full-fledged alternative to full-time education. However, future MOOC participants need to be prepared to prefer deep learning.

**Author Contributions:** Conceptualization: D.G.; Methodology: D.G., V.D.; Software: V. ˇ D.; Validation: A.T.; ˇ Formal Analysis: A.T., G.P.; Data Curation: A.T., D.G.; Writing—Original Draft Preparation: D.G.; Writing—Review & Editing: G.P. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Measuring Arithmetic Word Problem Complexity through Reading Comprehension and Learning Analytics**

### **Maria T. Sanz 1,\*, Emilia López-Iñesta 1,\*, Daniel Garcia-Costa <sup>2</sup> and Francisco Grimaldo <sup>2</sup>**


Received: 28 July 2020; Accepted: 4 September 2020; Published: 10 September 2020

**Abstract:** Numerous studies have addressed the relationship between performance in mathematics problem-solving and reading comprehension in students of all educational levels. This work presents a new proposal to measure the complexity of arithmetic word problems through the student reading comprehension of the problem statement and the use of learning analytics. The procedure to quantify this reading comprehension comprises two phases: (a) the division of the statement into propositions and (b) the computation of the time dedicated to read each proposition through a technological environment that records the interactions of the students while solving the problem. We validated our approach by selecting a collection of problems containing mathematical concepts related to fractions and their different meanings, such as fractional numbers over a natural number, basic mathematical operations with a natural whole or fractional whole and the fraction as an operator. The main results indicate that a student's reading time is an excellent proxy to determine the complexity of both propositions and the complete statement. Finally, we used this time to build a logistic regression model that predicts the success of students in solving arithmetic word problems.

**Keywords:** learning; reading comprehension; complexity; problem-solving; arithmetic word problems; fraction operator; technological environment

#### **1. Introduction**

Previous work has studied the relationship between performance in mathematics problem-solving and reading comprehension in students of all educational levels [1–3]. Authors such as Pólya [4] and Puig and Cerdán [5] have shown that reading and understanding the statement are key phases of the problem-solving process. The National Council of Teachers of Mathematics (NCTM) [6] determined that, in solving a mathematical problem, many of the necessary skills present in all areas of the educational curriculum are required, such as reading, reflection and understanding. The latest PISA report [7] indeed highlights that a solid reading competence is fundamental for academic achievement in all subjects of the educational system (including mathematics), while being a prerequisite for successful participation in most adult life [8–10].

Our research is framed within the context of arithmetic word problems (from now on AWPs or AWP in singular) and focuses on how to measure the complexity of the statements involved. To this end, we computed the reading comprehension of students through a technological environment and use learning analytics to predict student performance in solving this sort of problems.

#### *1.1. Complexity of Arithmetic Word Problems*

AWPs are texts or statements describing real-life situations in which unknown quantities need to be determined from other amounts that are known [5,11,12]. AWPs are some of the first problem-solving activities in the elementary school mathematics curriculum, and as such, they deserve special care and attention.

The complexity of AWPs has been conceptualized through research on the resolution of verbal problems and on the difficulties they present for schoolchildren. Daroczy et al. [13] showed that these difficulties can be caused by either one or a combination of linguistic and numerical complexity. Linguistic complexity refers to the linguistic and morphological aspects of the statement (e.g., how words are combined to form the text). Numerical complexity, in turn, refers to the numerical factors of the statement (e.g., both the quantities and the relationships between them).

According to Castro et al. [14], the complexity of AWPs can be measured following four main approaches:


To the author's knowledge, none of the previous approaches has yet measured the complexity of AWPs through the students' reading comprehension of the statement itself. This aspect makes our research a novel and original contribution to the state of the art of mathematical problem-solving.

#### *1.2. Measuring the Complexity of AWP Statements through a Technological Environment*

To measure the complexity of an AWP, we split its statement into propositions, as follows from the partial semantic approach defined above. Our unit of analysis is thus a proposition, which contains a verb and a quantity associated with the related action. Figure 1 shows an example of an AWP statement [20] divided into three propositions.


**Figure 1.** Example of an arithmetic word problem (AWP) statement ([20]) divided into propositions.

Propositions can be classified into levels to facilitate comparability and to determine their complexity. Hunt [21] names these constructions T-units or minimal terminable units of language. T-units each consist of a main clause plus the subordinate clauses it may include, and they can be organized on a number of levels: declarative sentences represent level 0; level 1 adds a subordination to sentences of level 0; level 2 adds a subordination to those of level 1 and so on. The higher the level, the more complex the sentence will be. This way, Proposition 1 in Figure 1 belongs to level 0, and Propositions 2 and 3 are of level 1, since they are respectively subordinate by the terms "of them" and "now".

We measure the complexity of each proposition by obtaining the time per word that students spent while reading the corresponding segment of the AWP statement. The time per word is computed through a technological learning environment able to control which information is displayed at any time and to register the interaction of students with the content. This novel approach is more powerful than to control reading from printed texts.

The use of intelligent tutors or technological learning environments (e.g., Moodle, Edmodo or Bakpax) has increased in recent years across all educational stages [22]. However, these environments have not yet been used to measure reading comprehension and the complexity of AWPs. These tools can usually be accessed through mobile devices and smart screens and allow one to register student–computer, student–teacher or student–content interactions [23–26], thereby giving rise to the so-called learning analytics research field. This field deals with applying data analytics to education and it is defined as the area of investigation in charge of measuring, compiling and analyzing data sets obtained through the use of computer-assisted learning platforms that track and record student digital interactions [27,28].

Technological environments and learning analytics are a cutting-edge approach to detect patterns on student strategies when solving a learning task. They are also helpful in understanding study habits, the use of teaching materials or the time dedicated to the proposed activities [29], sometimes supplemented by information on attendance, participation or motivation [30].

This work focuses on the analysis of the student–computer and student–content interactions obtained through the Read and Learn (R&L) technological environment [24,31]. R&L is a research tool to carry out experiments that analyze the strategies of students when they first have to read a text or problem statement and then answer a series of questions in a digital context.

#### *1.3. Predicting Student Performance When Solving AWPs*

Mathematical models have been extensively used to try to predict the probability of correctly solving a learning task. These models are commonly used to build a personalized route that guides students through an adapted teaching–learning process [32].

Logistic and Bayesian knowledge tracing models stand out among the statistical prediction models used for this purpose. The former have been used to predict the probability of success from the students' previous skills and the difficulty of the task [33]. The latter use hidden Markov models to estimate latent parameters and predict student success [32].

Following previous work on the matter [26,34], this work presents a binary logistic regression model to predict student performance from the complexity of an AWP measured by the reading comprehension of its statement.

The remainder of the paper is organized as follows. Section 2 describes the materials and methods used to measure the complexity of AWPs, the features of the R&L technological environment, a validation experiment for a sample population and the tested hypotheses. Section 3 presents the experimental results that determined the feasibility of our approach for assessing the complexity of mathematical problems through reading comprehension. Section 4 shows how to build a logistic model to predict student performance from the complexity computed for an AWP. Finally, discussion and conclusions are drawn in Section 5 in the context of the state-of-the-art literature.

#### **2. Material and Methods**

#### *2.1. Procedure for Measuring the Complexity of AWPs*

The complexity of an AWP can be derived from the complexity of all the propositions that form its statement. To estimate the complexity of a proposition, we compute the reading time per word for a group of students using the R&L technological environment. The reading time of proposition *j* in task *i* (*Tij* in Equation (1)) thus comprises the time spent by each student (*tijs*) in the group (of size *n*) and the number of words in the proposition (*k*).

$$T\_{ij} = \left\{ \frac{\mathfrak{t}\_{ijs}}{k} ; \, s \in \{1, \ldots, n\} \right\} \tag{1}$$

The total complexity of an AWP can in turn be measured by averaging the previous reading times per student for all propositions (Equation (2)), where *m* represents the number of propositions in the statement.

$$T\_i = \left\{ \frac{1}{m} \sum\_{j=1}^{m} \frac{t\_{ijs}}{k} ; \; s \in \{1, \dots, n\} \right\} \tag{2}$$

#### *2.2. Instrument*

R&L is a technological environment in which to design research experiments on reading comprehension in text and image-related learning tasks. It is a web tool that can be accessed through mobile devices, computers and smart screens using any browser on any operating system.

Experiments in R&L can include enriched texts with a list of questions and answers. A number of configuration settings are available, such as the possibility of accessing the statement at any time or only under certain conditions, the effect of alternatively hiding and showing parts of texts by clicking on them (Figure 2), the use of open-ended or multiple-choice questions, the number of attempts allowed to complete the task or the definition of feedback to be given after answering the questions.


**Figure 2.** An example of an experimental setting in R&L wherein the texts of both questions and answer options are totally hidden.

R&L records all user interactions with the statements, questions and response options along with timestamps, which allows tracking the access history to the learning content with a level of precision of milliseconds. Any user action is registered, such as displaying a hidden proposition, moving the focus from the statement to the questions and vice versa. This way, we can determine aspects such as: what part of the statement the student is focused on, which point in time a certain proposition is read, how long a student remains in the same proposition, how many times a proposition is consulted and in which order students access the statement, the questions and the answer options.

R&L is able to digest these learning data flows and compute the variables of interest from the previously recorded data (e.g., the time reading a proposition or answering a question). Data can then be exported in CSV so it can be further used in any preferred data analysis software (e.g., R or SPSS). For more details on R&L the interested reader can check out the literature [24] and keep up with our website about data analytics and technological tools in education https://go.uv.es/grimo/datte.

#### *2.3. Experimental Design*

To test our proposal we have conducted a descriptive quantitative study involving a group of 70 students, 26 girls and 44 boys, aged between 15 and 16 years old.

At the time of the study, the students belonged to two public secondary schools in Spain selected by a convenience non-probability sampling. One school is located in an upper-middle socioeconomic area of a town of twelve thousand inhabitants. The other one is located in a multicultural suburb with medium-low socioeconomic status in a city of eight hundred thousand inhabitants.

Informed consent was obtained from schools, teachers and students before the start of the experiment. Anonymity of the data was guaranteed by just collecting the year of birth, gender, course and a dummy school code for each student. Any combination of data with a frequency of less than 5 observations was considered subject to statistical secrecy and it was removed to prevent de-anonymization.

The experiment was run individually using the school's computer room. Students were introduced to the R&L technological environment before starting the session. Following fair and ethical practices, participants were made aware that they were involved in a research study. They were clearly informed about the aims of the study and that their performance would not be considered in their grades.

Participants were asked to solve a couple of AWPs presented as two tasks with their corresponding statement and five answer options. The statements were designed taking in to account the mathematical and the grammatical complexity. We built two isomorphic tasks [35] dealing with mathematical introductory concepts related fractional numbers over a natural number, basic mathematical operations with a fractional whole and the fraction as an operator. In addition, we classify the propositions of the statements into levels as defined by Hunt [21], which allows the measured reading comprehension to be compared.

Tasks were written in Spanish since all participants were native Spanish speakers. For the sake of readability, we also show the translation of the statement into English as follows:


Both tasks have an equal mathematical structure, expressed in terms of the relationships between the variables and quantities involved. This means that they are solved by applying the same rules, procedures, and algorithms. The question is placed at the end of the statement following the pattern *axb* = ? where *a* and *b* are known quantities. Note that the semantic relationship between the variables and the unknown quantity, the lack of data in the question and the absence of irrelevant data is equivalent in both statements. The tasks can be classified as two AWPs of multiplicative comparison according to Puig and Cerdán [5]. This sort of problems use a scalar function (*I*) to link two extensive quantities (*E*) of the same type of magnitude (*ExI* = *E*, the Schwartz relation [36]). For example, the scalar function in task 1 is "two-thirds of," while the two extensive quantities are "thirty candies" and the unknown quantity of "strawberry candies".

The proposed AWPs use the fraction (i.e., two-thirds) as an operator [37] that transforms an initial quantity (i.e., thirty candies or one-half of a pizza) into a final quantity (e.g., strawberry candies or a fraction of the pizza). This transformation is associated with the scalar function and the multiplication operator, as shown in Figure 3. The tasks are consistent [38] since they can be solved by directly translating the key terms in the statement (e.g., are or is) into the operation to be performed, in this case a multiplication.

**Figure 3.** Use of the fraction as an operator in the proposed AWPs.

We can determined the grammatical complexity of the tasks by dividing the statement into propositions and analyzing their syntax. Each statement is composed of three propositions, as shown in Table 1. The first two relate to the informative part of the statement and the third one is the question. We configured the tasks in R&L so that just one proposition could be displayed at a time while the rest of them remained hidden (see the different colored segments in Figure 4).

**Table 1.** Propositions of tasks 1 and 2.


The length of the informative parts is the same in both statements (i.e., 3 + 6 words for P11 + P12 and P21 + P22 as from the original text in Spanish). The number of words in the question part differs (i.e., 5 to 7 words for P13 and P23 as shown in Table 2) due to the introduction of rational numbers that change the Spanish quantifier "cuántos" by "qué porción de," although it keeps the same length in English.



**Table 2.** Syntax of the propositions from the original text in Spanish.

The grammatical complexity of each proposition is also represented by the number of nous, verbs, numerals, prepositions and conjunctions in Table 2. The type of sentences can be categorized into

levels as defined by Hunt [21]. Propositions P11 and P21 are declarative sentences of level 0. The rest of propositions are level 1 since they include a subordination to the previous sentences by the terms "of them" (P12), "of it" (P22), "candies" (P13) and "of the pizza" (P23) respectively.

#### *2.4. Research Hypotheses*

We pose the following hypotheses in line with previous work on the mathematical concepts dealt with by our study:


Hypotheses 1 and 3 were tested by comparing the average reading times of propositions of the same level. Regarding H1, an increase in complexity from P11 to P21 was due to the mere presence of fractional instead of natural numbers. By comparing the complexity of P12 and P22 we checked the effect of reformulating the whole (H3) from a natural number (i.e., thirty candies) to a fractional one (i.e., one-half of a pizza).

To test H2, we compared the average reading times of level 1 subordinate propositions with that of proposition P21. Propositions P12 and P22 include the syntagms "of them are" and "of them it is" that refer to the use of the fraction as an operator (from now on, we refer only to syntagms "of them are" in order to improve readability). We take proposition P21 as the reference level 0 declarative sentence since it also uses a rational number (i.e., one-half of a pizza), but it does so as a fractional quantity.

#### **3. Analysis and Results**

Reading times were rather dispersed in our group of students, as shown by the high standard deviations in Table 3 (values are expressed in seconds per word or s/word). The Kolmogorov–Smirnov test confirmed that the times recorded did not follow a normal distribution (*p*-value < 0.05) for the propositions (*Tij*) or the complete statement (*Ti*). Therefore, we use the median as a good representative of each set of times. We did not use the mean in our analysis, since it is affected by outliers in the obtained asymmetric distributions. For example, see how most of the students read faster than the average reading time (empty circle) in the box-plots shown in Figure 5.

We checked for differences in the reading times due to the socioeconomic context and the gender of students. Differences between school were not statistically significant following the non-parametric Wilcoxon signed-rank test for paired samples (*p*-values > 0.05). Reading times were also not statistically different between boys and girls (*p*-values > 0.05). We can then use the data obtained for the whole group to study the complexity of the statements.


**Table 3.** Reading times (s/word) for each proposition (*Tij*) and task (*Ti*).

**Figure 5.** Distribution of reading times for each proposition (*Tij*) and task (*Ti*).

By comparing the reading times in Table 3 we can test our hypotheses as follows:


Student performance was rather good when solving the two proposed tasks. The success rate was 94.3% for task 1 and 62.9% for task 2. The median reading time of all propositions in task 2 was longer than that of task 1 (7.21 s/word and 3.67 s/word respectively) and the distribution was more sparse (e.g., compare *T*<sup>2</sup> and *T*<sup>1</sup> in Figure 5). The previous results confirm that solving task 2 was more complicated than solving task 1.

#### **4. Predicting Student Success from the Proposed Complexity Measure**

We use a binary logistic regression model to predict the student success when solving an AWP. The model estimates the probability of succeeding (or failing) in completing a task from the complexity of its statement, measured as the reading time per word. The data obtained in our study were used to train a model for each task, as described by Equation (3), where *Tij* is the time taken by students to read each proposition (*j*) of the problem (*i*).

$$P(\text{success} = 1) = 1/(1 + e^{-\left(b\_0 + \sum\_{j=1}^{n} b\_j T\_{ij}\right)})\tag{3}$$

We discarded outliers from our data and kept the results of 58 students to build the model for task 1 and of 57 students for task 2. We trained the models with a random sample of 50 students and validated them with the remaining eight students (task 1) and seven students (task 2). Table 4 shows the relation between the reading time per proposition and the success of students from direct observation of the data. Faster reading times led to better performance in task 1 (indirect relation), whereas slower students were the best performers in task 2 (direct relation). These results are in line with the complexity of the statements analyzed above.

**Table 4.** Relation between the reading time per proposition and student success.


The model built for task 1 is shown in Equation (4). It explains between 0.142 (Cox and Snell *R*<sup>2</sup> value) and 0.424 (Nagelkerke *R*<sup>2</sup> value) of the dependent variable. It gives an accuracy of 98.3% when calibrating on the train set and it correctly predicts the success of the eight students in the validation set. The sign of the coefficients obtained for each proposition (*bj*) reproduces the indirect relation previously found between the reading time and the probability of successfully solving task 1 (see Table 4).

$$P(\text{success} = 1) = 1/\left(1 + e^{-(7.302 - 0.063 \cdot T\_{11} - 0.788 \cdot T\_{12} - 0.269 \cdot T\_{13})}\right) \tag{4}$$

We analyzed the odds ratio (OR) to understand the magnitude of the effect, that is, how much the probability of success changes as a result of increasing by one second the reading time of a proposition, the rest being constant. An OR greater than one indicates an increase in the probability while an OR less than one implies a decrease. Taking more time to read proposition P12 (i.e., higher values of *T*12) lowers the probability of success since *OR* = 0.455. Increasing the reading time for propositions P11 and P13 does not affect the student's success that much since OR remains near to one (*OR* = 0.939 and *OR* = 0.764 respectively).

The model built for task 2 (see Equation (5)) is more limited since it explains between 0.056 (Cox and Snell *R*<sup>2</sup> value) and 0.175 (Nagelkerke *R*<sup>2</sup> value) of the dependent variable. It gives an accuracy of 65.4% when calibrating on the train set and it correctly predicts the success of four students in the validation set. All coefficients are positive and confirm the direct relation found in Table 4. They are also close to zero, which makes OR rather close to one. For example, increasing the reading time of proposition P22 slightly raises the probability of success (*OR* = 1.117); the time taken to read propositions P21 and P23 does not have any significant effect on student success (*OR* = 1.009 and *OR* = 1.059 respectively).

$$P(\text{success} = 1) = 1/\left(1 + e^{-\left(-0.896 + 0.009 \cdot T\_{21} + 0.111 \cdot T\_{22} + 0.057 \cdot T\_{23}\right)}\right) \tag{5}$$

Far from being contradictory, the models represent the different complexities of the two statements. The overall reading time for task 1 was half the overall time for task 2 (e.g., see *T*<sup>1</sup> and *T*<sup>2</sup> in Table 3). Students having reading comprehension problems in task 1 thus showed higher probabilities of failure. On the contrary, task 2 appeared as a more complex AWP whose successful resolution could benefit from investing more time in reading its propositions.

#### **5. Discussion and Conclusions**

We have presented a novel proposal to measure the complexity of an AWP through the student reading comprehension of its statement. The approach allowed us to predict the students' success from their reading times when solving the task. The students' reading time has demonstrated to be a good proxy to determine the complexity of AWPs and it can become an essential tool for the design of problem statements. By analyzing the statement propositions, one can adjust the level of complexity of the task to focus on certain student profiles.

The paper also introduces the use of the R&L technological environment to compute the complexity of a problem statement, without the need to use traditional paper-and-pencil questionnaires. In addition to that, R&L enables the collection of extensive data on student interactions and opens the way for more data-driven research on the topic.

The results obtained confirm that our procedure for measuring the complexity of AWPs is consistent with previous findings [14]. The two tasks under study can be classified as multiplicative comparison problems according to the semantic approach [5], whose difficulty lies in the introduction of fractional versus natural numbers [39–41].

We identified the complexity of the syntagms "of are" or "of them it is" (or its equivalent "son de" in Spanish), which is related to the multiplication operator and to the concept of "fraction of" or "part of" [37]. These ideas begin to be developed in the school curriculum from the fourth year of primary education. The complexity of this concept, though, increases when it is applied to a fraction. These results may be linked to the design of tasks for current textbooks, where the concept of natural number is introduced through graphic support and considering the whole as a discrete quantity. However, when this concept is introduced over a fraction in the sixth year of primary education, the visual representation is usually removed and the whole becomes a continuum. That results in the mathematical concept being taught through a rote rule, which associates this expression with the multiplication of fractions and leads to possible errors in later courses, as shown by researchers at the Rational Number Project (http://www.cehd.umn.edu/ci/rationalnumberproject/) and the National Assessment of Educational Progress (https://nces.ed.gov/nationsreportcard/). Our work confirmed this issue with a sample group of students of the last year from compulsory secondary education.

The complexity of the statement propositions has been used to build binary logistic regression models that predict the probability of success in solving AWPs. The models confirmed that the propositions that most affect probability are those that involved a more difficult mathematical concept. In our study, these propositions are the ones that deal with the fraction as an operator over both a natural and a rational number.

It is worth noting that our approach also proposes the segmentation of the statement into propositions, whose complexity can be measured and compared following the classification into levels by Hunt [21]. In our study, first level propositions are declarative alphanumeric sentences where the numerical values are either natural numbers or fractions. Second level propositions introduce a subordinate clause through the syntagms "of them are" or "of it is". This fact goes far beyond evaluating the complexity by the success rate [44] and allows comparing the complexity of mathematical concepts within and across AWPs.

This work opens up a line of research on using technological environments and data analytics to determine the complexities of AWPs by measuring the level of understanding of each the statements and dealing with the mathematical concepts that make them more difficult to solve. Next steps include the design of a longitudinal study by students' age that analyzes the evolution of the concepts and the possible blockages that occur. Future work will also help to define an index that allows creating AWPs statements with prefixed complexities by weighting the propositions in the statement according to their level following the classification by Hunt [21].

These sorts of metrics and tools can be implemented by intelligent tutors designed to teach maths through problem-solving. They can help to track personalized teaching–learning paths for each student while using reading comprehension as one of the key drivers for predicting students' skills [26]. Despite the benefits provided by technological environments, the development of digital teaching competence continues to be a challenge for the education system [45,46]. However, the introduction of emerging tools and data analytics is progressively providing teachers and researchers with new experimental scenarios to study, for example, the possible impact of the use of feedback oriented to success when students interact with a given statement [31]. As Alonso et al. pointed out [22], the development of good teaching practices that integrate technology in the classroom can help teachers to start applying digital learning tools effectively and to improve their digital competence.

**Author Contributions:** Conceptualization, M.T.S. and E.L.-I.; data curation, M.T.S., E.L.-I., D.G.-C. and F.G.; writing—original draft preparation, M.T.S. and E.L.-I.; writing—review and editing, M.T.S., E.L.-I., D.G.-C. and F.G.; supervision, M.T.S., E.L.-I. and F.G.; project administration, F.G.; funding acquisition, F.G. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was partially supported by the Spanish Ministry of Science, Innovation and Universities (MCIU), the Spanish State Research Agency (AEI) and the European Regional Development Fund (ERDF) under project RTI2018-095820-B-I00 and the projects UV-SFPIE-PID19-1098335, UV-SFPIE-PID19-1095187.

**Acknowledgments:** The authors would like to thank the support of the Capgemini-University of Valencia Chair for Innovation in Software Development

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Learning Mathematics with Emerging Methodologies—The Escape Room as a Case Study**

#### **Arturo Fuentes-Cabrera 1, María Elena Parra-González 2,\*, Jesús López-Belmonte <sup>1</sup> and Adrián Segura-Robles <sup>2</sup>**


Received: 22 July 2020; Accepted: 10 September 2020; Published: 15 September 2020

**Abstract:** Nowadays, different methodologies are booming in the field of education, and active gamification-based methodologies such as the Escape Room are an example of these methodologies, which is the base of this research. The purpose of this research is to analyze the effectiveness of the use of an Escape Room as an active methodology to learn mathematics. A quantitative research method was performed through an experimental design. Two study groups were set up. With the control group, a traditional training methodology was used, and with the experimental group, an innovative one was used through an Escape room experience. A total of 62 students of the 3rd level of Secondary Education from an educational center in Ceuta (Spain) participated. Results show how the experience developed through the escape room improved achievement, motivation and autonomy in a significant way. It has also reduced learning anxiety significantly. It is concluded that the use of the Escape room in Mathematics improves learning achievement, anxiety, motivation and autonomy, with gender being a variable to be taken into account, especially in motivation and autonomy. Therefore, the escape room has a greater potential than a traditional methodology in Mathematics.

**Keywords:** active methodology; educational innovation; escape room; gamification; methodological contrast; mathematics; secondary education

#### **1. Introduction**

In recent times, traditional education, or more precisely, traditional teaching methods, gave place to a non-central place, and innovation is increasingly taking center stage [1]. Traditional teaching in mathematics is understood as the teacher being the main figure of the educational act, which is a situation more focused on teaching. On the other hand, there is talk of new didactic approaches, when they focus on the student, and on learning as the main action of the educational act. This is causing new innovative teaching modalities to appear that produce a higher incidence in students, such as flipped learning, gamification or problem-based learning [2]. The new didactic environments and approaches allow a greater involvement of the students in the daily life of the classroom and, above all, in the participation of their training process [3]. All this promotes profound changes in education, as well as in people's lives and in their daily actions [4]. Furthermore, this is supported by a total increase in the use of technology [5], as the main support for educational innovation today [6]. For this reason, education approaches the idea of a digital society in which students live daily [7].

This transformation takes place within the framework of constant adaptation of teaching to the digital society and to new ways of life [8]. For all these reasons, fundamental concepts such as that of active methodology derive, as the main source of new forms of teaching, transmission of knowledge and new forms of participation in the educational process of all the agents involved in it. In this new paradigm, the student takes the leading role to become responsible for their training process, always guided by the teacher and always trying to achieve the objectives and strengthen the content [9]. All this tends to have, as its main source, the construction of self-knowledge by the students, and undoubtedly produces new opportunities and means available to students [10]. In this sense, their motivation is increased, causing students a greater interest and a better attitude towards learning [11].

Today's students live in a world surrounded by technologies, as well as with great stimulation caused by this. Therefore, all these actions are aimed at concluding a favorable process for the development of students, with increasingly virtual environments [12]. These innovative learning environments produce a ubiquity in spaces and times dedicated to teaching [13]. Similarly, new methods are generated for a better understanding of the contents [14].

#### *1.1. Gamification as an Innovate Tool*

When we talk about gamification, we are referring to one of the active methodologies that, in recent years, has been most developed by teachers and students from all over the world [15]. For this reason, it has earned a place among the most widely used and studied methodologies [16]. As its name suggests, it bases its main focus on the application of the game phenomenon and on converting spaces reserved for formal learning into recreational spaces in which to continue learning [17,18]. Gamification involves the use of tools and elements typical of the game in the classroom, bringing along different benefits as pupils feel more motivated, active in class, and even want to take part in their learning processes in an active way, as within the game structure they feel less pressure in class. [19]. This methodology focuses on facilitating the effort of the students and making their task more enjoyable. For this reason, it focuses on the game as a tool to achieve high levels of learning and achievement of objectives [20]. How could it be otherwise? The game has also undergone a transformation over the years, passing by the classics to the most modern ones, with a great load of virtual reality, an aspect that has been transferred to education [21]. The use of the game, as an educational tool, has had a high success rate, taking into account what has been studied in different investigations, considering it to be an effective and adequate method to put it into practice with students of any age [22,23] and even any educational stage in which their training is developed [12,24].

Gamification allows teachers to develop their own learning program for their students, based on formal knowledge structures, with the certainty of achieving more than acceptable academic indicators in factors as important as the ability to solve problems [25]. In addition, the interactive process that occurs between the agents involved in the educational process [26] and the collaboration between these same agents [27] are benefited. It also increases factors that are influential in the maturational development of the student, such as motivation [28], a positive attitude towards learning [29], interest in knowing their own training [30], the autonomy of the student [31], commitment to the educational act [32], dedication to teaching by teachers and learning by students [33,34], as well as attraction, enjoyment, absence of negative feelings and the satisfaction of facing the task [35,36].

The success of gamification is supported by focusing its method on a system of rewards that make the student increase his attitude and predisposition towards teaching, having a very positive impact on the psychosocial indicators [37] that we have mentioned and which, inescapably, will produce an increase in performance that is obtained from the dedication of the students [38].

In the field of mathematics, recent studies on the use of active and gamified methodologies show very beneficial results for teachers and students who have put it into practice, with all the aforementioned areas and others, such as effective resolution of practices, being developed and promoted [39,40].

#### *1.2. Using the Scape Room as a Gamification Tool*

Within gamification, there are many ways to put it into practice. One of them is the Escape Room [41]. This is considered a training modality that is based on the resolution of challenges, tests or

problems posed to students by teachers, which give rise to various situations in which students must have adequate knowledge to solve the practice of learning [42]. In an Escape Room, people—students in this case—are locked in a room and they are given several enigma and challenges, which they have to solve to be able to find the way out [43]. Therefore, this method is supported by the game design, with the students having to solve a series of tests, knowing how to self-manage their own knowledge, individually, and collectively to share their knowledge. This causes an increase in the participation of students when solving the aforementioned challenges or problems [44]. The method consists of "locking" the classrooms or spaces ready for practice, where they have to carry out various tasks and activities to solve puzzles and to be able to leave the place in the shortest possible time [45].

Several studies about the application of the Escape Room in the classroom offer us very favorable results in terms of its application in different contexts [46–48]. Thus, its application improves all the indicators previously exposed in relation to gamification. In this line, it produces a high motivation index in students [49], improves their activation and participation in the teaching and learning process [50], produces greater satisfaction for learning and attraction to it [51] and supposes a greater assimilation and reinforcement of the contents [52]. All this, as it cannot be otherwise, leads to a better and greater acquisition of content, positively impacting the student's grades and academic performance [53]. All this, produced by the learning environment, is very favorable for the attitudes of the students, and for their collaborative and personal practice [54].

In the area of mathematics, there are some investigations and training actions that lead to the success of students' learning through the use of the Escape Room. Despite typical problems, a lack of attention to instructions, as occurs in traditional teaching, turns out to be beneficial in its application in the field of mathematics, since it increases competitiveness for learning, motivation and student interest [55].

In addition, the implementation of activities related to the Escape Room in the field of mathematics, promotes the autonomy of students, and facilitates learning, increasing teamwork and the ability of students to resolve conflicts or challenges posed by the teacher [56]. All this favors collaborative work, as well as the autonomy of the students, enabling them to face future learning [57].

#### *1.3. Justification and Objectives*

Gamification implies the use of tools, design, and elements of games in classrooms [19]. This teaching methodology eases the students' effort, making it more enjoyable. The implementation of this innovative methodology fosters students' participation and motivation towards their learning processes [11]. Students take the leading role and become more responsible in their learning process, which is guided by teachers [9].

As it has been seen in the previous section, the use of Escape Rooms in education fosters better results in students' motivation [48], activation and participation in their learning processes [49], and satisfaction for learning and attraction to it [50].

This study was carried out at the third level of the Secondary Education stage of the Spanish educational system, due to the lack of motivation students have towards learning and practicing mathematics, as revealed in the scientific literature [58,59].

To check if the lack of motivation was caused by the methodology carried out by the teacher, the main aim of this experimentation is to analyze the effectiveness of the use of educational Escape Rooms in mathematics lessons, as compared to the implementation of a traditional methodology focused on teacher's lectures and presentations without the use of innovative materials and resources. In summary, the following dimensions were measured: learning achievement (as a number obtained in final evaluation of the subject); learning anxiety (example: I have felt nervous during classes); motivation (example: Does the methodology applied affect your motivation with regard to mathematical content?); and autonomy (example: To what extent has the methodology applied in the field of mathematics contributed to their autonomy?).

Bearing in mind this general purpose of this study, the specific aims are:


In addition to the objectives described, the following research questions arise:


#### *1.4. Intervention Description*

Due to the lack of motivation and active participation of secondary students during mathematics lessons, teachers sought and searched for an innovative methodology that could foster this motivation and participation when learning mathematics. As it has been highlighted in the introduction of this study, this methodology or strategy has been proven to be useful in education, and that is the reason for having chosen it for this experimentation.

In the implementation of the didactic unit carried out, mathematics contents have been worked on and they have been practiced. Within this didactic unit, different challenges and enigmas were designed. Different elements were taken into consideration, following another study about this methodology [57]. These elements were important to design the Escape Room experience, and they were:


All of these elements were merged into one story to engage the students. The story told by the teacher had a theme centered on action and suspense to encourage the motivation of the students. All this was done with the purpose that the students were immersed in the formative action in an active way.

#### **2. Materials and Methods**

#### *2.1. Research Design and Data Analysis*

Experimental design through a descriptive and correlational analysis was carried out, which was based on the quantitative perspective following experts within this field [60,61]. The students were classified into two different groups to be analyzed. On one hand, the control group followed traditional teaching methodology. On the other hand, an experimental group followed an Educational Escape Room as a methodology for practicing the mathematics contents. Methodology was defined as an independent variable; other dimensions and the effectiveness of methodologies were selected as dependent variables to be evaluated. Stratified sampling was used to select participants. Stratified sampling is a technique where the researcher divides the entire population into different subgroups. Then, it randomly selects the final subjects from the different strata proportionally. Both groups share a course, work area, content and teachers, so it is established that there is no prior significant difference in both groups.

Statistical Package for the Social Sciences (SPSS) v25 program was used for statistics analysis. For this analysis, the descriptive statistics mean (M) and standard deviation (SD) were used. The measurement of the effect size has been obtained by biserial correlation (*r*) established by Cohen [62]. In addition, a *p* < 0.05 is established in the study as a statistically significant difference. The value of the effect size of Pearson *r* correlation varies between −1 (a perfect negative correlation) to +1 (a perfect positive correlation). In this case, after verifying that the data do not follow a normal distribution, non-parametric statistics are used. Specifically, the Mann–Whitney U test is used to compare two groups with no normal distribution [63].

#### *2.2. Participants*

The participants were 62 students from secondary education who took part in this experiment. Recently it has been determined by studies of relevant impact that the sample size in these type of investigations does not condition the performance of these experiments [64].

The selection of students was done carrying out an intentional sampling, thanks to the ease of access to the students. They are enrolled in an educational center of the Autonomous City of Ceuta (Spain). One of the workers in this center detected the need of this research after working with these students.

These students were specifically selected from the third level of the Secondary Education stage of the Spanish educational system (*n* = 62; Mage = 15 years; SD = 1.62). The composition of both groups, control and experimental ones, is specified in Table 1.


**Table 1.** Study groups by sex.

#### *2.3. Instrument*

Data collection was acquired by an ad hoc questionnaire. The design of this tool was done following other validated instruments found within the scientific literature [65,66]. The questionnaire has 32 items in total, which are divided into 20 different dimensions. A Likert scale type is followed with a range of five points (from 1 = Strongly disagree to 5 = Strongly agree).

The instrument was validated first in a qualitative manner and afterwards in a quantitative way. In the first phase, a Delphi method was carried out to do the qualitative validation. Within this procedure, 8 experts in active methodologies in education of different universities were involved. Reviewers rated each item based on its transparency and relevance on a scale of 1 to 6, recommended indicators in the literature [67]. The questionnaire was highly valued by these experts (M = 4.87; SD = 0.21; min = 1; max = 6), and the recommendations given were followed. Kappa de Fleiss and W de Kendall were applied to achieve the indexes of concordance and relevance of observations granted, this showing positive results (K = 0.87; W = 0.89). Afterwards, for the validation, an exploratory and confirmatory factor analysis by the principal components' method with varimax rotation was done quantitatively. The results show an appropriate factorial structure to the initial theoretical approach,

and the correlations among factors are positive. The tests determined the dependence between the delimited variables (Bartlett's test of sphericity = 2647.21; *p* < 0.001) and the adequacy of the sample (Kaiser–Meyer–Olkin = 0.86).

More statistical analyses were used to measure the reliability of the questionnaire, such as Cronbach's alpha (α), compound reliability (CR) and average variance extracted (AVE), confirming the internal consistency of all the results of the questionnaire.

#### *2.4. Procedure*

The experiment was carried out in several phases. First of all, the ad hoc instrument was designed, and validated. Then, the selection of students who took part in the research was done. Family consent had to be asked to develop this study. In this research, ethical principles of confidentiality were respected. Thus, students were in a random manner divided into two groups with the same number of them, one being established as a control group and the other as an experimental group.

Data collection took place before and after the teaching procedure, with two months' time between them. Teachers taught contents during 10 sessions, through a traditional method with the control group and an innovative method using an Escape Room with the experimental group. The sessions lasted 55 min. The contents taught were related to solving problems using systems of linear equations with two unknowns.

Participants were divided into two groups, which happened to be the group classes where they are enrolled. The control group followed a traditional methodology learning process. In the traditional teaching methodology, the role of the teacher focuses on the presentation of the contents and on the completion of the exercises on the blackboard. All participation in the teaching process has the teacher. The student focuses attention on the actions carried out by the teacher. Therefore, students play a passive role. This hinders and prevents the interaction of educational agents to carry out learning and solve problems in a collaborative way. The students did the problems using systems of equations individually in their own class notebooks. Meanwhile the experimental group followed a learning methodology based on the Escape Room. This fostered the interaction of students with their peers, teamwork, motivation, and participation of students to learn and practice contents.

The Escape room was designed by mathematics teachers with the help of the researchers, who are experts in the field. There were two computers in the room, and a tablet, so the students could surf the net to try to solve the codes that needed its use. There were 5 enigmas hidden in the room, which were based on mathematics problems that needed to be solved, and when one enigma was solved, it led to the other one, because it had a track for the next one. The mathematical problems were problems to practice mathematics in an innovative way. An example of the tasks performed in the Escape Room is as follows: The narrator tells a story to set the students in a haunted house where the doors have been mysteriously closed and a strange noise has been heard. The shadow of a ghost peeks out and tells them that in order to get out of the haunted house they have to solve various problems. The ghost in each test provides students with a card with exercises (Figure 1) that they will have to solve in a satisfactory way to continue advancing in the story and to reach the exit. All the tests, depending on their complexity, have a certain time for the students to solve them. Once the time had elapsed, the students received a clue or puzzle to find the next test.

$$\begin{cases} \frac{\chi + \mathfrak{P}\chi}{\mathfrak{Q}} = \mathfrak{S} \\ 4 - \frac{\mathfrak{D}\chi - \mathfrak{p}}{\mathfrak{Q}} = 1 \end{cases} \quad \begin{cases} \frac{\chi + 1}{\mathfrak{Q}} + \frac{\mathfrak{p} - 1}{\mathfrak{Q}} = 0 \\ \frac{\chi + \mathfrak{D}\chi}{\mathfrak{P}} - \frac{\chi + \mathfrak{p} + \mathfrak{Q}}{\mathfrak{Q}} = 0 \end{cases}$$

**Figure 1.** Example of tasks performed.

Teachers were only assessors or guides in the experience, as the students, in groups, had to solve the enigmas with cooperative work. Each group had a color, and the enigmas had the same colors themselves, so each group had to find the tracks of their colors. The enigmas were hidden around the classroom. In order to find them easily, they had to pay attention to the details told in the story. Each time an enigma was solved, the students got a badge, which could be exchanged for a reward afterwards. The rewards could gain extra points in the attitudinal section of the subject in the participation block or new clues to find the enigmas. This was done with the experimental group, meanwhile the control group had the same number of mathematical problems, which was five, displayed on the board in the class, and they had to solve them. These problems were to practice mathematics, so they needed no specific explanation as the processes of those problems were previously explained and they had to practice them. To complete the study, the data were collected and analyzed.

#### **3. Results**

All dimensions have had a significant improvement after applying the Escape Room as a pedagogical tool (Table 2). Regarding learning, we find significant values of Learning Achievement (U = 339,000; Z = −2073; *p* = 0.038) with a large effect size (d = 0.525). Related to Learning Anxiety, we get similar results (U = 339,000; Z = −5754; *p* = 0.000) with a large effect size (d = 0.528). Motivation and Autonomy show significant differences too (U = 654,000; Z = 2480; *p* = 0.013 and U = 654,500; Z = 2461; *p* = 0.014).


**Table 2.** Mann–Whitney U test for control and experimental groups differences.

\* Cohen's *d* small < 0.20, Medium < 0.50, large > 0.50.

Effects sizes can be considered large d = 0.653 and d = 0.655. In all analyzed dimensions (Achievement: RC <sup>=</sup> 36.06; RE = 26.94; Motivation: Rc = 25.90; RE = 26.94 and Autonomy Rc = 37.11; RE = 25.85), the experimental group obtains higher values than the control group. In anxiety, where these values are different, values are lower in the case of the experimental group (Rc = 44.61; RE = 18.31). Based on the results, the experience developed has caused significant effects on all the dimensions analyzed.

From another perspective (Table 3), we seek to see if the gender variable could affect some of the changes produced during the experience. The gender variable only influences motivation (U = 52,000; Z = −2620; *p* = 0.008) and autonomy (U = 33,500; Z = −3365; *p* = 0.001) in the experimental group. Women are the most motivated and show the highest level of anxiety in general. This effect does not occur in the control group, where there are no differences by gender. Women are the most motivated and show the highest level of anxiety in the control group, although these differences do not become significant.


**Table 3.** Mann–Whitney U test for control and experimental groups differences by gender.

\* Cohen's *d* small < 0.20, Medium < 0.50, large > 0.50.

Finally, it is relevant to know how the different dimensions analyzed are correlated, both for the control group and for the experimental group (Table 4). In the control group, where the Escape Room was not carried out, the different dimensions do not seem to maintain positive or negative correlations (*p* > 0.05). In the experimental group, positive and significant correlations appear. The positive relationship between motivation and achievement stands out (*r* = 0.364, *p* < 0.05), as does autonomy and achievement (*r* = 0.404, *p* < 0.05), and motivation and autonomy (*r* = 0.684, *p* < 0.01).


**Table 4.** Correlation between dimensions for each group.

\* Significance with values less than 0.05. \*\* Significance with values less than 0.01.

#### **4. Discussion**

The research carried out has made it possible to achieve the proposed general purpose of analyzing the effectiveness of the use of educational Escape Rooms in mathematics lessons, this compared to the implementation of a traditional methodology in the 3rd year of Secondary Education. The results obtained in this study have confirmed the potential of the literature on teaching innovation and the use of active methodologies in teaching and learning processes [1,2]. In this case, the research has focused on the use of the Escape Room as a methodology to gamify a subject as traditional and rigorous as mathematics.

Despite the acceptance and use of gamification by teachers in general [16], this work acquires its reason for being in the scarcity of studies concerning the use of the Escape Room in the area of Mathematics. Therefore, with the intention of increasing the impact of the literature on the state of the question, this study has been carried out.

This work is based on the analysis of previous research reported on gamification and, specifically, Escape Rooms. It has been conducted in order to analyze the dimensions that offer benefit to the scientific literature on this active methodology. Therefore, this study acquires its potential in presenting some findings on a training approach that has been scarcely studied in the subject of mathematics.

Studies on the implementation of the gamification and Escape Rooms reveal great benefits both in the teaching process [20,22], and in various indicators related to learning [25,27,55]. The latter have to do with an improvement in problem solving [26], in the interaction and collaboration between the people involved [27,28], in activation and attitude [53,57], in motivation [29,52,66] in satisfaction with the environment generated and the task to be carried out [54], and autonomy of students [33,68,69], as well as the results and academic performance achieved [38,64], among the indicators more outstandingly reflected in impact studies.

In particular, this research has been articulated in the analysis of four dimensions, such as learning achievement, learning anxiety, motivation and autonomy. The results show that these dimensions have experienced a significant improvement after the application of the Escape Room as a pedagogical tool in the Mathematics subject. These findings are associated with previous studies on the use of this approach that reveal an improvement in the achievement of students in their formative action [38,56], in the control of learning anxiety [12], in motivation [29,41,52] and in student autonomy [31,56,57].

At a higher level of specificity, the gender variable has only influenced students' motivation and autonomy, as other studies reveal [68,69]. It has been found that women are more motivated and show a higher level of learning anxiety, in analogy with other works [70]. However, this effect produced is not found in the control group, where there are no differences by gender.

At the correlational level, in the control group the correlations are absent. In contrast, in the experimental group, there are correlations between Achievement-Autonomy, Motivation-Autonomy and Motivation-Achievement. This last correlation stands out, which generates a positive effect on students by increasing their motivation for learning and homework due to the achievement attained. These results are in congruence with the specific literature on the analyzed art [12,70]

#### **5. Conclusions**

This work shows the analysis of an escape room experience with students. The results obtained allow us to know the students' assessment of the intervention with the escape room and inform about the effect on learning process.

Based on these findings, use of the Escape Room in the Mathematics course has contributed to the improvement in the different dimensions studied, such as learning achievement, learning anxiety, motivation and autonomy (with gender being a variable to take into account), and in a general way, in motivation and autonomy. Therefore, the use of this active gamified methodology is positioned as a didactic approach with greater potential than traditional methodologies.

The potential of Escape Room education has been demonstrated. This resource is effective in both increasing motivation and promoting active learning. As in other studies, there is agreement on the idea that, through this type of game, it is possible to facilitate the learning of a specific topic in a motivating and efficient way.

#### **6. Prospective and Limitations**

The prospective of this research focuses on the promotion of the Escape Room and the promotion of gamified practices in the field of mathematics. As revealed in this study, the use of this active methodology benefits several important indicators in the training process. For this reason, the present work encourages the teaching community to use the Escape Room as a methodological alternative in the area of Mathematics. Furthermore, this research contributes to establishing the bases of this active methodology in this specific field, with the purpose of serving as a support for future research by other members of the scientific community.

The limitations of this study are focused on the particularities of the participants, who are involved in a specific context. Furthermore, this study lacks a prior analysis of the groups that have been subjected to experimentation and control, as they were classes already organized by the educational center, which assured its composition was based on heterogeneity. Although the effects are positive, we must take into account the area of work within mathematics. In this case, the positive results were obtained in the work in the areas of algebra, logic and geometry. Therefore, as a future line of research, it is intended to articulate a teaching network to apply the Escape Room in the Mathematics subject with the intention of obtaining a representative sample that allows the results to be generalized to the entire student population. In addition, another of the possible lines of action may focus on the application of this active methodology in different educational stages.

**Author Contributions:** Conceptualization, J.L.-B., and A.F.-C.; methodology, M.E.P.-G.; software, A.S.-R.; validation, A.S.-R.; formal analysis, M.E.P.-G.; investigation, M.E.P.-G., J.L.-B., A.F.-C., and A.S.-R.; data curation, A.S.-R.; writing—original draft preparation, J.L.-B. and A.F.-C.; writing—review and editing, J.L.-B. and M.E.P.-G.; visualization, A.F.-C.; supervision, M.E.P.-G. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Influence of Musical Learning in the Acquisition of Mathematical Skills in Primary School**

#### **Aurelio Chao-Fernández 1, Dorinda Mato-Vázquez <sup>2</sup> and Rocío Chao-Fernández 1,\***


Received: 29 September 2020; Accepted: 4 November 2020; Published: 10 November 2020

**Abstract:** In this research we analyze the influence of musical activities in the acquisition of mathematical knowledge and skills of a sample, 50 students from both a public and a private school in A Coruña (Spain), at a cognitive level. Based on a quantitative study with a quasi-experimental design, we evaluated students' knowledge acquisition; we worked with musical activities related to mathematics in the experimental group (EG), and with traditional mathematical activities in the control group (CG). We used a questionnaire that the teachers completed before and after putting the activities into practice, after collecting—writing daily field notes—the mathematical knowledge acquired by the students. The results indicate that there are significant differences between the pretest and the posttest, between CG and EG, but there are no differences between public and concerted schools. In short, it is concluded that music represents an excellent tool in mathematical learning.

**Keywords:** musical activities; mathematics; learning-teaching; preschool

#### **1. Introduction**

The close relationship between music and mathematics has been known since ancient times, Pythagoras (582 BC) being considered as the first to establish connections between the two disciplines when joining them in his school, so that music and arithmetic were studied together until the Middle Ages [1].

However, most of the works developed in this area were specific music treatises formulated by mathematicians (Descartes, Mersenne, Euler, D'Alembert, etc.), or compositions created through mathematics (Bartók and the golden reason, Stochastic music by Xenakis, Halffter's Fibonacciana, etc.), being necessary to wait until the end of the 20th century for it to be envisioned as an emerging study area by Milton Babbitt, David Lewin, and especially John Clough [2].

Recently An, Ma and Capraro [3] or Johnson and Edelson [4] have argued the connection between both disciplines claiming that musical notes, scales and tuning are related to various areas of mathematics, from proportions and integers to geometry and trigonometry. That is, most of the theories are built on the relationships and common structures of both components [5], while studies aimed at teaching and learning of mathematics and music at the same time are scarce [6–9]; even more unusual are those reporting the benefits related to teaching them jointly. This is despite the fact that it has been demonstrated, through neuroimaging, that musicians and mathematicians activate common brain areas [10]. In this line, Winner, Goldstein and Vincent-Lancrin [11] made visible an investigation carried out by Graziano, Peterson and Shaw in which they stated that the improvements in the area of mathematics were greater among students who were also studying piano, especially in subjects related to spatial learning.

Undoubtedly, at a neurological level, music provokes different responses in the areas of the brain that affect both cognitive and emotional levels, since it activates imagination and creativity [12], both very necessary to approach mathematical learning from an affective framework [13], building the foundation from which the processes of cognition act: perception, attention, memory, intelligence, thought and language [3,14].

According to studies, the interdisciplinary approach of teaching music with mathematics positively affects cognitive development and skills related to both disciplines and student's academic results [1,15,16]. Some of the reasons may be that learning is more attractive when the contents are addressed together, which provides emotional security and confidence [17,18]. Benefits are significantly increased if interdisciplinary work begins in the first stage of the education system.

In spite of the above, the contributions, experiences and teaching materials that work between subjects at school in an interdisciplinary way [19] are solely incipient, except for the occasional song to facilitate the processes of memorization of, for example, the multiplication tables. This trend also occurs in Spain, which does not come as a surprise, since an absolute disconnection among the same course subjects is usual [10]. Although the Organic Law 2/2006 of Education of May 3 [20] organizes the curriculum of Infant Education in Spain in three areas of knowledge, indicating that the work must be done in an interdisciplinary manner, it is quite different in real-world scenarios [21].

There are two types of schools depending on their ownership: public and private since 1985, according to the Organic Law Regulating the Right to Education [22]. There is plenty of research that analyzes whether there are differences between both types, the results being torn between those that do find different results [23,24] and those that conclude that there is no significant difference in relation to the ownership of the center [25–27] and claim other inputs to be more relevant.

The edition of PISA [28] goes one step further by adding two parameters: the appreciation of the students (their satisfaction is greater when they study in charter and private centers than in public centers) and the performance in the evaluations (which is lower in public schools).

Calero and Waisgrais [29], Meunier [30] and Salinas and Santín [31] among others justify the differences in academic performance in public centers due to the lower quality of resources and the greater number of students of immigrant origin.

From this referential framework, we have asked ourselves the following research question: How does music and, more specifically, music teaching, influence students' mathematical learning?

The presented work is part of a larger study that analyzes the effects, cognitively, of musical activities in math classes, and how these scaffolds influence student learning depending on certain variables such as the course, sex, ownership of the center, professions and studies of parents [32].

Particularly, the objective of this research is to analyze the influence caused by musical activities, cognitively, in the acquisition of mathematical knowledge in the last course of Pre-school Education, taking into account the ownership of the center.

#### **2. Materials and Method**

To analyze the influence on the acquisition of mathematical knowledge, a quantitative methodology has been used, based on a quasi-experimental design with a pretest and a posttest and two reference groups: an Experimental Group (EG), which received the stimulus or treatment; and the Control Group (CG), which only served as a comparison since it did not receive such treatment.

#### *2.1. Context and Participants*

The study involved a sample of 50 children between 5 and 6 years old, 25 girls (G) and 25 boys (B), 24 from a Public School (PS) and 26 from a Charter school (CS) of the 3rd year of Early Childhood Education in the province of A Coruña (Spain) and their respective teachers in the school year 2018–2019 (see Table 1). Besides, in this research, centers with similar socio-economic characteristic were selected to avoid possible influences.


**Table 1.** Distribution of the sample.

#### *2.2. Procedure*

The research carried out refers to a pretest and posttest design. The purpose is to evaluate the mathematical learning before and after the intervention of the teachers to be able to subsequently make a statistical comparison between the students' learning and the possible variations depending on the two groups into which the sample was divided (the CG formed by 26 students and the EG by 24), and the ownership of the center to which they belonged (in the CG, 12 students belong to PS and 14 to CS, and in the EG, 12 to PS and 12 to CS).

In order to obtain the data, the same contents were worked on in both groups, although, to verify if there were differences in the academic results of the students, preselected musical resources were used in certain activities within the classroom—related with mathematics contents—with the experimental group (EG), while with the control group (CG) mathematics content was worked with traditional didactics.

The mathematical contents developed with the strategy of the musical activities worked in the classroom, and with the respective teachers, deal with "Properties of the objects" (PO), "Basic operations with concrete elements" (BO) and "Space-time relations" (STR) and have been elaborated from Decree 330/2009, currently in force in the Autonomous Community of Galicia (Spain).

#### *2.3. Evaluation*

The evaluation technique that was used was direct observation; that is to say, the progress of the students during the accomplishment of the programmed activities was contemplated, focusing mainly on the acquired learning, as well as on the rhythm and characteristics of this acquisition. That is why in this work we collected, through field notes, relevant descriptive and reflective information by the teacher-tutors in the day-to-day process. This systematic collection resulted in a class diary of each student that allowed to cover a questionnaire designed by the researchers that would make up the pretest and the posttest.

The questionnaire consisted of 10 items with six response options, ranging from 0 (nothing) to 5 (much), to evaluate some fundamental contents of the mathematics based on LOE [20]. It was then endorsed by a system of inter-rater validation formed by four experts (teachers and professors specialized in mathematics and music). In this way, the most relevant items were selected for their relevance (they should be related to the object of study) and clarity (easily understandable).

The items of the questionnaire are the following:


The distribution of the items of the questionnaire regarding the contents worked in the classroom is as follows: PO *(I*1, *I*2, *I*3, *I*4), BO *(I*5, *I*6, *I*7), and STR *(I*8, *I*9, *I*10).

After the pretest was carried out by the teacher-tutors (in the Spanish educational framework it is the Infant teacher/tutor in charge of teaching music) and the analysis of the data by the researchers, an activity plan was prepared related to the mathematical contents mentioned above (PO, BO and STR), which would be developed three days a week, 30 min per session, for two months with the EG, while in the CG the class sessions related to the same contents were held in a traditional way (cards and textbook). The participating teachers are part of an interdisciplinary research group led by the researchers and had already collaborated in past courses on other innovative experiences with other students. Therefore, the training they had, prior to the start-up of this project, was forged jointly in the whole group, which led to the design of the proposal being developed by all.

At the end of the 8 weeks, and after the achievement of the activities, the teachers—with all the data from the daily observations—covered the posttest test in both groups (Experimental and Control) to determine if the treatment had brought any change in the acquisition of mathematics knowledge. Finally, the tabulation of the data, the statistical analysis and the discussion of the results by the researchers was carried out using the statistical package SPSS v.23.0. Averages and standard deviations were used as descriptive statistics, the Cronbach's alpha is used to find the reliability coefficient, the Wilcoxon T test, appropriate for related variables, and the Student's *t*-test for independent variables. The reliability coefficient obtained by Cronbach's alpha (internal consistency) is satisfactory with values of 0.803 in the pretest and 0.821 in the posttest; for the control group 0.811 and 0.892 for the experimental group, as well as 0.796 for the Public School and 7.01 for the Concerted School.

The development of work with the EG was carried out by formulating a logical sequence of activities structured in sessions using musical instruments, songs, choreographies, working duration, height and intensity, in addition to any situation of interaction with schoolchildren that would respond to their questions.

Each of the sessions consisted of three phases: (1) in the general assembly, an introductory activity was carried out in order to check the previous knowledge, present the contents and stimulate the students. We tried to incite them to action based on what they knew; (2) the development activities were carried out, in which the children demonstrated what they were learning in the previous phases, and (3) finally, the relaxation phase is the highlight in which relaxation tasks were carried out, but without losing connection with the central theme of the program.

Taking Noll [33] into account, all the proposed activities had a playful nature, since at these ages the engine of emotional, intellectual and social developments are games. In addition, they influence knowledge structures and relationships with the environment.

It should be noted that the teacher made an effort to promote a climate of security and trust. At all times, the teacher worried about helping the students when developing the activities, reminding them of the collective rules and guiding those who were blocked by providing them with new patterns of action.

Since music is attractive to students, even more so at early ages, we have proposed a series of musical activities that motivate them to progressively relate them to mathematical concepts. These examples show an example of how the quality of the sound "Duration" has been worked from music and mathematics in an interdisciplinary manner.

#### *2.4. Musical Games*

*The musical race* Objectives:


Work group: the whole class. Resources: the song *The musical race.* Time: 15 min.

Description: We listen to the song The musical race. After the first time, we ask students about figures: did all they run at the same speed? Which one was the fastest? Which one was the slowest? Then, we listen to the song again, this time imitating the figures that are part of the race.

In this activity, through the listening, students have to get to learn, and understand the duration of musical figures (half note, quarter note, eighth note, quarter note rest). Afterwards we check whether that knowledge has been acquired through questions, and finally, through movement, students try to link them to the mathematical concept "Fast-slow", through the experience of a race where they imitate the slower and faster musical figures.

*Simon Machine* Objectives:


Work group: Groups formed of 4 to 5 children. Resources: Simon machine. Time: 30 min.

Description: Randomly, the machine illuminates and emits its own sound as it lights. After waiting, the student must repeat the sequence offered by the machine, in the correct order, using theirs visual and audible memory. If the student succeeds, the machine will respond with a longer sequence, and so on. If the student fails, the student must assign his turn to the next classmate. The different levels of difficulty increase the speed of the sequence to be repeated.

In this activity, in addition to trying to involve the student in interdisciplinary work-through the "Fast-slow" sound sequence-it is intended to improve memory, attention and concentration, increasing the difficulty progressively.

*Musical domino* Objectives:


Work group: the whole class. Resources: adapted domino. Time: 20–25 min.

Description: The domino consists of the following figures: whole, half, quarter, eighth notes and quarter rest, in two colors (red and black). Each student takes 6 tiles that are placed face down. The one that gets the tile with two half notes begins.

The students will have to join the tiles, for example, black red-black red, and they will then clap taking into account the time that each figure occupies; in this case, the student will count to one. When it comes to the silence tile, the student will not clap, but will just wait for the necessary time. The first to finish the tiles wins.

This activity includes the musical figure known as the whole note which implies more difficulty than previous tasks since it is necessary having understood the concept of duration related to quantity. *Let's dance!* Objectives:


Work group: the whole class. Resources: music player. Time: 15 min.

Description: There will be a brainstorm of the steps that students want to introduce in the choreography. Once the steps have been proposed, a selection is made and they will be adapted to a song known to children. This was one of the choreographies:

First: We all jump once.

Second: We turn to our right twice.

Third: We move the arms up in circles 3 times.

Fourth: We move the hip to the right and left 4 times.

Fifth: We cover our nose and crouch 5 times.

Repeat as many times as necessary as the children want.

This activity was one of the most attractive for students, since in addition to having fun, they consolidated the musical and mathematical knowledge that the class had been working on, and other activities were added such as laterality, numbering, geometric figures, etc.

#### **3. Analysis and Results**

#### *3.1. Averages and Standard Deviations of the Pretest and Posttest by Groups and Ownership of the Center in the Respective Items*

The averages and standard deviation show that, with respect to the initial and final tests, there are differences in the academic performance of the Experimental Group, compared to the results obtained by the participants of the Control Group according to the ownership and regarding *I1* (Table 2).



Note: M = Mean value; SD = Standard Deviation.

It is stated that the performance in CS has experienced an improvement in the EG (went from 2.41 to 4.43), while in PS (went from 2.50 to 3.89) an improvement is also reflected, although less than in the CS.

The starting scores were higher in the PS. However, in the posttest score the scores obtained by the CS are higher (Table 2).

As it can be seen in Table 2, the scores obtained in the *I*<sup>2</sup> (to order objects by size) reveal that the EG, starting from very low marks in the *pretest*, has improved more than the CG. If we compare by ownership, we see that in the CG, the students of the PS had lower averages in the *pretest* and higher in the *posttest*. On the other hand, in the EG the scores of the PS sample were and continue to be higher.

The values, in general terms, indicate that the participating sample has a positive attitude towards the use of musical activities in the teaching–learning process of mathematics as shown in the results of Table 2. In this sense, there are differences between CG and EG, in favor of the latter in all cases. The students of the CS stand out again, although all of them experience an increase in their grade (Table 2). While the CG averages are homogeneous in both centers and in both tests, in the EG the differences were greater in the students of the PS at the beginning of the project, but the students of the CS showed superiority in the second test.

Concerning averages and standard deviations of the *I*4, we obtain higher values in all cases when the questionnaire is passed a second time. Differences are shown between the Control and Experimental groups—in favor of the first in the pretest and in favor of the second in the posttest—in both centers (Table 2).

In the case of *I*5, the EG stands out again with respect to the CG in the posttest result, which analyzes the grouping of elements according to the quantity. Starting from the averages in the initial tests—in which the CS stood out slightly in both groups—the CS CG stood out in the posttest, although in the EG there is a small inclination towards the PS. In addition, if the results are assessed before and after the application of the activities, it is necessary to highlight that the results of the pretest were low in general, while those of the EG posttest are very high, with the PS students reaching scores very close to 5 (Table 2).

In *I*6, the average scores obtained by the students according to the interdisciplinary method reflect differences between the CG and the EG, in favor of the latter in all cases. Comparing the results to the type of center being analyzed, the students of the CS stand out again, although all of them experience a increase in the grade (Table 2). While the CG averages are homogeneous in both centers and in both tests, in the EG the differences between the schools were higher in those belonging to the PS at the beginning of the project, but those of the CS showed superiority in the second test. The obtained data are shown in Table 2.

Another aspect included in the instrument used refers to associating the numerical name with the quantity of elements embodied in the responses of item 7 with an average close to the response value 5 (=4.89 and 4.83) equivalent to "Totally agree" in the GE of the PS and the CS.

Based on Table 2, the mean and standard deviations of the scores obtained by group and center typology express that the CG of the CS experienced a slight improvement and no change is perceived in the PS students. Additionally, the EG improved meaningfully and in a special way in PS and CS, since the scores of the initial questionnaire were higher than in the CG than in the EG for both centers. Therefore, it can be inferred that musical activities have been of great benefit to students.

Regarding the question of whether they identify the morning, afternoon or evening, which is shown in Table 2, the analysis show that students quantify it as very positive with values that approach the response 5 (x = 4.86 and 4.33) relative to the option "Totally agree". These results confirm that the perception of the participating students about the intimate relationship between music and mathematics follows the same line of the previous Items regarding ownership and the two groups. While the initial values were homogeneous in PS and CS, in the posttest the EG gets distanced from the CG with noteworthy differences; especially those of the PS are on the verge of the maximum score, although it is true that they started from higher values than those of the CS (Table 2).

Regarding *I*9, dissimilar results were found, which contrast with the other values of the study. Such data contrast with the previous values; reflected in Table 2, that in the EC the values in the posttest have decreased in relation to the pretest, both in the CG and in the EG. However, in PS the

results increased somewhat in both groups. The reasons given by the teachers of the CS to explain this were convincing, alluding to a very negative family event that had happened to one of the students, which affected the rest of the students.

In reference to *I*10, which assesses the knowledge of the sample on the fact of recognizing before, now and after, it should be indicated that considering the *I*<sup>10</sup> content, the averages and standard deviations of the scores obtained show again that the EG improved in all centers, so we can affirm that the effects of the program of recreational musical activities applied to assess the mastery of the students are noteworthy and positive. In Table 2, we can confirm that the EG started from lower values than the CG in both schools; however, the posttest shows better grades in the EG, mostly in the students of the PS.

#### *3.2. Control Group in Each of the Items*

Another aspect analyzed is the comparison of the pretest and posttest between the means obtained by the Experimental Group and the Control Group in each of the items. These results lead us to corroborate that the effects of the program of musical activities focused on mathematics are positive and significative on the performance of the students in all the items that were applied to assess the mastery of basic concepts of mathematics (Figure 1). The initial values of the EG are lower than those of the CG in all items except *I9*, although in *I1*, *I5*, *I8* and *I10* there are few differences between one group and the other. In *I9*, the average mark in the EG pretest is somewhat higher than in the CG. Once the posttest was administered and analyzed, we could see from the obtained results that the EG improved in all the items, especially highlighting the *I1*, *I5*, *I7* over the CG. The *I9* item does not offer high scores in any case for reasons of internal nature, which the teachers kindly explained.

**Figure 1.** Group interactions by item.

#### *3.3. Comparisons of the Pretest and Posttest between the Means Obtained by the Experimental Group and the Control Group in Each of the Contents*

The obtained results, in relation to the worked contents and the groups that make up the study sample, offer the following data: the EG presents worse qualifications in the pretest than the CG in all cases; however, by applying the second questionnaire, the CG increased its performance values on a regular basis and the EG had an important improvement.

The interaction of groups and measurements for performance by content in the pretest—both in the CG and in the EG—show that BO (*I5*, *I6* and *I7*) is the highest, followed of PO (*I1*, *I2*, *I3* and *I4*) and finally STR (*I8*, *I9* and *I10*). Likewise, the posttest also shows that both the EG and the CG remain the highest BO, followed by PO and then STR.

#### *3.4. Comparisons of the Global Means of the EG and CG*

It follows that according to the global results, the total averages of the CG (2.67) and the EG (2.36) show higher values in the GC in the pretest. However, the scores of both groups in the posttest differ quite in favor of the EG, with 4.35 for the EG and 2.99 for the CG.

We can therefore affirm that the applied measures had a positive impact, which is in line with other studies that have shown the effectiveness of music to stimulate cognitive functioning [34,35].

#### *3.5. Comparisons of Means for Related Samples*

The results obtained in the pretest and posttest were analyzed to check if there are statistically significant differences, through the Wilcoxon test for related samples. Through this analysis, we intend to contrast the null hypothesis of non-existence of significant differences between the means of the subjects before and after applying the proposal of musical activities in the mathematics class.

Observing the results of Table 3, we can verify that the statistic with its associated *p*-value (0.000) is less than the level of significance (0.05), which allows us to conclude that, between the means obtained in the pretest and the posttest, there are statistically significant differences, favorable to the posttest.



<sup>a</sup> Wilcoxon signed rank test; <sup>b</sup> Based on negative ranges.

#### *3.6. Comparison of Means for Independent Samples*

Next, we proceed to apply the Student's *t*-test for independent samples, given the parametric nature of the distribution, in order to determine if there is a significant difference between the results of the Public School and the Concerted School, on the one hand, and the Control Group and the Experimental Group on the other.

In both cases, the first assumption of normality is verified, based on the Shapiro-Wilk test, applicable for *n* < 30, since the data have a normal distribution. Additionally, the second assumption, consisting of determining the existence of equality of variances, using Levene's test, resulted in the variances being equal.

When analyzing the data (Table 4), it is observed that there is no significant difference between the mean of the two groups, PS and CS, as *p*-value = 0.470 > 0.05.

**Table 4.** Equality of means between the Public School (PS) and the Concerted School (CS).


However, there are significant differences between the CG and the EG, since *p*-value = 0.000 is less than 0.05, which is why differences are shown in favor of the EG (Table 5).

**Table 5.** Equality of means between the Control Group (GC) and Experimental Group (GE).


#### **4. Discussion and Conclusions**

The results obtained in this research have allowed us to identify the influence of musical activities (musical instruments, songs, choreographies, working duration, height and intensity, in addition to any situation of interaction with schoolchildren that answered their questions) in the acquisition of mathematical knowledge of students in Early Childhood Education. We can therefore affirm that the objective we set ourselves at the beginning of the study has been fulfilled and corroborate what authors such as Hall [36] are right to point out about the benefits of interdisciplinary work between mathematics and music. Taking into account the results according to the variables group and ownership, we can affirm that the musical teaching has promoted advantages in the mathematical learning of the students.

We can emphasize that, as for the previous mathematical knowledge that the schoolchildren possessed, the sample is homogeneous in both cases with respect to *I*<sup>1</sup> "To recognize circles, triangles and squares", To *I*<sup>6</sup> "To create compositions with Cuisenaire rods", to *I*<sup>8</sup> "To identify morning, afternoon and evening" and *I*<sup>2</sup> "To order objects by size".

However, differences between groups can be seen in *I*<sup>10</sup> "To recognize before-now-after" and in *I*<sup>4</sup> "To arrange objects by their height". In relation to ownership, there are differences in *I*<sup>9</sup> "To use different measuring units", *I*<sup>7</sup> "A To Associate the numerical name with the number of elements" and *I*<sup>5</sup> "To group items by quantity".

Focusing on the two groups (CG and EG) and the 10 items of the questionnaire, we achieved that the EG showed greater success compared to the first application of the questionnaire, substantially improving compared to the CG, which remained stable or increased its performance minimally. Specifically, the improvement highlights in items 1, 5, 7 and 8, being 8 the one in which a greater difference was evidenced. In items 2, 4, 6, 9 and 10, the CG had better qualifications in the pretest than the EG, however, in the posttest the EG stood out.

Based on the ownership and the items, in general, we appreciate positive effects of the program of musical activities on the performance of the students in the 10 items. In the case of *I*9, the average of the posttest performance in PS and CS of the CG decreased, and it increased minimally in the EG, but it was due to a valid and consistent justification, according to the teachers.

With respect to the two groups and contents, the results indicate improvement in the EG with respect to the CG in the three contents: objects properties, basic operations with concrete elements, and space-time relationships.

Likewise, the non-parametric tests reaffirm the aforementioned, giving results that indicate significant differences between the results of the test prior to the implementation of the proposal and the one after it Wilcoxon's T). Regarding the CG and the EG, it is perceived through the Student's *t*-test that the EG (who received the math class through musical activities) presents significant differences with respect to the CG, in favor of the former. Finally, it should be noted that the Student's *t*-test, in relation to ownership, shows that there are no significant differences between public and concerted schools. This shows that the differences are not due to the type of school, or the students, but are linked to the fact of working on mathematics through musical activities.

Consequently, it can be concluded that the activities applied had a positive effect, in line with other studies that have shown the effectiveness of rhythmic patterns, intensities, durations, heights, speeds, symmetry, etc., as effective musical mechanisms in the acquisition of mathematical skills [10]. The present investigation echoes these findings and extends them by showing evidence of such benefits in childhood education, age at which studies are not frequent, as the previous authors indicate, in addition to warning that further research is necessary.

It becomes evident after the implementation of this experience the fact that the application of musical activities as a resource in mathematical learning represents an excellent alternative for teachers of Early Childhood Education, who seek to meet the learning needs of children in a fundamental stage for their integral development. In this way, the proposed objective has been achieved, which is an important contribution to the progress of the interdisciplinary work of both subjects.

In short, the basic knowledge of mathematics was achieved through guided and planned musical experiences, which allowed students to be stimulated in a pleasant and conductive environment, since the activities were carried out with high levels of motivation, harmonizing all their dimensions, both physical and emotional. Therefore, we can affirm that for these teachers music represented an excellent alternative, since it had a positive impact on the performance and motivation of the children in these classrooms. Consequently, we confirm that learning should occur in an interdisciplinary and pleasant context. In addition, it must be adapted to the needs that schoolchildren have to explore and get to know their surroundings. Finally, it becomes clear that professionals trained in mathematics didactics and musical didactics are necessary to undertake this challenge [37].

**Author Contributions:** Conceptualization, R.C.-F. and D.M.-V.; Methodology, software investigation and resources, R.C.-F., D.M.-V. and A.C.-F.; Formal analysis, A.C.-F., R.C.-F. and D.M.-V.; Writing of the manuscript, R.C.-F., D.M.-V. and A.C.-F.; Supervision project, A.C.-F., R.C.-F. and D.M.-V. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** This project is related to the research stay of Rocío Chao-Fernández at the University of A Coruna where she investigated the interdisciplinary work of music and mathematics at school, for which she has obtained funding from the University of A Coruna.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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