Edited by: Tecnológico Superior Corporativo
Edwards Deming
January - March Vol. 6 - 2 - 2023
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e-ISSN: 2576-0971
Received: December 12, 2022
Approved: May 02, 2023
Page 26-41
Inverted learning model for the online high school
mathematics class
Modelo de aprendizaje invertido para la clase online de
Matemáticas del bachillerato
María Alejandra Saucedo James
*
Mayra Jacinta Moncayo Carpio
*
Carlos Andres Acosta Vergara
*
ABSTRACT
In these times of covid-19, the Ecuadorian educational
system has been challenged to sustain the teaching-learning
processes in the online modality. This has demanded
initiative and creativity on the part of teachers and students
in order to meet the expected results. The present work
proposes the inverted learning model designed for the
development of online Mathematics classes, which includes
students and teachers of the first year of high school of the
Vinces Educational Unit, Vinces canton, province of Los
Ríos. In order to obtain the results, an observation guide
was applied to the virtual classes, a questionnaire to 177
students and an interview to teachers as data collection
instruments. These made it possible to show the current
reality of the teaching method, as well as the limitations
faced by educators and students in making possible the
online adaptation of the planning of the development of the
contents of the subject of Mathematics.
Keywords: Inverted classroom, inverted learning, online
classroom, virtual classroom, Baccalaureate, Mathematics.
RESUMEN
En estos tiempos de covid-19, al sistema educativo
ecuatoriano se le ha planteado el desafío de sostener los
procesos de enseñanza-aprendizaje en la modalidad online.
* Ingeniera en Sistemas, Universidad Técnica de Babahoyo, E-mail:
msaucedoj@utb.edu.ec ORCID: 0000-0001-9595-5552
* Licenciada en Educación, Universidad Técnica de Babahoyo
mmoncayoc@utb.edu.ec, ORCID: 0000-0002-7655-1576
* Ingeniero Comercial, Universidad Técnica de Babahoyo
cacostav@utb.edu.ec, ORCID: 0000-0001-7755-4923
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Este ha demandado iniciativa y creatividad por parte de
maestros y estudiantes a fin de cumplir con los resultados
esperados. El presente trabajo propone el modelo de
aprendizaje invertido diseñado para el desarrollo de las
clases online de Matemáticas, que incluye a educandos y
docentes del primer año bachillerato de la Unidad
Educativa Vinces, cantón Vinces provincia de Los Ríos. Para
la obtención de resultados se aplicó una guía de
observación a las clases virtuales, un cuestionario a 177
estudiantes y una entrevista a docentes como
instrumentos de recolección de datos. Estos permitieron
evidenciar la realidad actual del método de enseñanza,
además de las limitaciones que afrontan los educadores y
educandos en posibilitar la adaptación online de la
planificación del desarrollo de los contenidos de la
asignatura de Matemáticas.
Palabras clave: Aula invertida, aprendizaje invertido,
clase en línea, clase virtual, Bachillerato, Matemática.
INTRODUCTION
Currently, information and communication technologies (ICT) have modified the social
encounters between individuals worldwide, giving rise to the emergence of new
paradigms that propose a diversity of knowledge and skills focused on the variety of
existing technologies. In this context, education is not at the margin of this, the presence
of platforms for teaching and learning, have facilitated the substantial change between
face-to-face and online in terms of academic activities, which have allowed to continue
with the educational processes in these times of pandemic, avoiding the permanent
paralysis of school actions.
The process of adaptation from face-to-face to online has revealed shortcomings in the
use of technology, especially for teachers, who have had to relearn new ways of
presenting didactic content. In this scenario, there is a need for motivated teachers and
students, willing to receive new information. This will encourage the use of ICT in
education, facilitating access to content and teaching-learning methods, in full harmony
with the individuals who handle it. The interaction between the subject and the
technology to educate requires new views on how to proceed and act in a context which
is mediated by a computer.
Thinking about online education requires first of all to look at the current pedagogical
models and dialectically deny their best contributions in order to build an educational
option in new Learning Management Systems (LMS). There are several proposals:
Blended-learning (Bartolomé and Sandals, 1998), E-learning (Elliott Masie, 1999),
Inverted Classroom (Lage, Platt and Treglia, 2000), among others that have transcended
and by their nature propose diverse pedagogical actions. Therefore, the inverted
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classroom offers a direct coincidence between what is developed in face-to-face classes
and what is intended to be achieved in the future, being essential its study and theoretical
systematization.
According to the authors Abío et al., (2017) Llanos and Bravo (2017, p. 6) the flipped
classroom learning model or flipped classroom consists of turning the class around, the
student outside the classroom, seeks from his own autonomy the acquisition of
knowledge by accessing the contents, mainly those of audiovisual type developed and
selected by the teacher according to the educational needs, the technological tools that
allow appropriating and expanding knowledge. Gallardo et al. expresses that this model
allows the teacher to dynamize and modernize the teaching-learning processes,
facilitating the use of classroom time aiming at achieving greater depth in the study of
disciplinary contents. (p. 96)
The flipped learning model is based on the need to match different types of learning, the
use of multimedia tools is considered as an important contribution to teaching and a
highly influential tool for acquiring knowledge, in which students actively contribute to
the elaboration of knowledge and value their own learning in a very significant way (Park
and Choi, 2017, p. 192).. In addition, Mendoza (2017) highlights that, the intentional
material taught by the teacher refers to the evaluation of the contents to be taught and
the resources to be used inside and outside the classroom, for the better understanding
of concepts, procedures, skills, abilities, aptitudes and values.
Fúneme (2019, p. 161) refers that the learning model aims to study the conditioning
factors of the teaching and learning process of Mathematics, in order to adequately focus
the efforts of teachers in the educational field. The central objective of mathematics
teaching is the cognitive development of students, which is correlated with the form of
instruction and interaction of the teacher in the classroom. (Holguín et al., 2016, p. 287)..
Mathematics teachers should consider the interests and needs that allow them to create
an active environment of motivation, enthusiasm and attention to their class for problem
solving in a mathematical context. (Pochulu and Font, 2016, p. 15), students should
assume the acquisition of knowledge, directed by the educator, strengthening class
discussion or reflection. The teacher points out Oviedo (2018, p. 5)., for his part must
assume strategies to generate student participation in class; therefore, it is necessary to
make known the evaluation strategy to be implemented to determine the impact
generated in the population of students who acquire knowledge through the classes
(Vega et al., 2015, p. 11)The evaluation is a fundamental dimension for the fulfillment of
the virtual educational process. (Jimenez, 2019).
Moncada Cerón (2013) defines the concept of model as a conceptual pattern on which
is built from reality, providing elements that direct actions (p. 39). Relating this definition
to the educational level, Tünnermann (2008)conceptualizes the term educational model
as the concretion of each of the elements an institution possesses (teaching, research,
extension, linkage and services) in order to make its educational project a reality. (p.
15).
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For the development of this research, a literature review was carried out in different
scientific databases on the variables of study: flipped learning, online class and its related
terms based on the glossaries of the Social Sciences. Likewise, search chains with
exclusion criteria were applied in Spanish, English and Portuguese to improve the
accuracy of the results related to the research topic.
From the search carried out, a total of 15 research precedents related to the variables,
descriptors and scope of the proposed topic were compiled, namely: Núñez Paz and
Rodríguez (2020); Fúneme (2019); Madrid et al.;Guerrero (2019); Rahmadani et al.
(2020); Lai & Hwang (2016); Wei et al. (2020); Cevikbas and Kaiser (2020); Seitan et al.
(2020); Kumar et al. (2017); Clark (2015); Urban (2020); Cedillo et al. (2021); Lo et al.
(2018); TARAZI, (2016).
With the literature review of the different studies and the consideration of systemic
summaries related to the inverted learning model and the online mathematics class in
the baccalaureate, it has been possible to identify that the topics studied are the
following:
The inverted classroom as a strategy to improve performance, academics and attitudes
in mathematics learning.
Transformation of mathematics teaching through the application of the inverted
classroom.
The inverted classroom, impact and effectiveness in motivation and learning for
Mathematics versus traditional teaching models.
Proposal for a didactic intervention with an inverse methodological approach to learn
inequalities.
Once the conclusive summaries of the study background have been exposed, it can be
seen that the tendency is to produce inverted classroom technologies in order to
motivate and intervene, but there is a model thinking of inverted learning as a process
for learning to teach mathematics online at the baccalaureate level, therein lies the
importance of the research.
After the introduction, this paper is organized in the following sections: literature review
on the flipped classroom learning model, the online mathematics classroom, a
description of the research design, the modeling, discussion of the results obtained and
the corresponding conclusion.
MATERIALS AND METHODS
The analysis offered does not dispense with a subject that contextualizes it, in the
particular case of this study we choose the subject Mathematics, which within the
common core of the curriculum is essential in the learning of students and its presence
in secondary education levels. In accordance with everything reviewed concerning the
subject, the central scientific question posed for this article is: how to model the learning
invested in the online classes of the subject of Mathematics in students of the first year
of high school?
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This research is assumed as explanatory from a high preponderance of modeling of the
processes of inverted learning applied to Mathematics classes through the face-to-face
modality, which seeks to discover the current situation on the methodology or strategies
used by teachers to achieve compliance with the teaching in online session and achieve
learning in their students; all this from a model thinking of inverted learning which
facilitates its future transformation perspective in the use of teaching of the
Baccalaureate.
For the development of this research, 320 students in the first year of high school in the
specialties of Science and Computer Science Technician and 2 teachers of the subject of
Mathematics were considered as the study population. The random probabilistic
sampling was calculated by means of the online data processing sheet, obtaining a sample
of 177 students to whom the scientific research instruments were applied.
For the collection of research data, the observation of online classes of the subject of
Mathematics was used, which allowed obtaining information for the realization of a
documentary analysis on how teaching-learning activities are developed, determining the
role of teacher and student in the inverted learning model in the online modality.
Likewise, an interview was conducted with the teachers who teach the subject in the
first year of high school, with the purpose of determining the strategies used in the
development of their classes, in order to reach the contents and achieve assimilation.
Finally, a survey was applied to students to identify their perceptions about the learning
model used, as well as the didactic resources and assertiveness of the teacher when
interacting with the online Mathematics class.
All the instruments offered (questionnaire, observation and interview guide) facilitate
the triangulation of data for a systemic analysis, modeling the process of teaching-learning
Mathematics, through the logical procedures of thinking that facilitate the use of
modeling as a direct scientific method of research.
RESULTS
Table 1. Availability of technological equipment for the development of the online class.
Description
Frequency
Percentage
Totally agree
66
37,29%
Agreed
74
41,81%
Neutral
11
6,21%
Disagreement
22
12,43%
Strongly disagree
4
2,26%
Total
177
100,00%
It can be seen that more than 70% of the students have technological equipment for the
development of the virtual session, 20% do not have this resource. In the courses
observed, a great majority of them connect to the online class, being highlighted by the
teachers, those who do not have technological resources, look for options to connect.
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Table 2. Digital resources sent by the teacher to review the class topic
Description
Frequency
Totally agree
81
Agreed
76
Neutral
0
Disagreement
17
Strongly disagree
3
Total
177
It is evident that more than 80% of the students receive didactic material from the
professor, while 12% indicate that they do not have access to it. In the observations
made, it is evident that the teacher sends the material at the end of the class online, but
in his opinion not all students can receive it due to different difficulties regarding
accessibility to a technological device.
Table 3. Review of class material by the student at home
Description
Frequency
Percentage
Totally agree
26
14,69%
Agreed
93
52,54%
Neutral
50
28,25%
Disagreement
6
3,39%
Strongly disagree
2
1,13%
Total
177
100,00%
It was determined that more than 60% of respondents review the material of the subject
of study, about 30% do not review it. As for the development of the classes, no activities
are generated to strengthen the review of these materials, as stated by teachers, this is
due to the short time allotted for the development of the virtual class.
Table 4. You actively participate in the online class (ask questions or contribute ideas).
Description
Frequency
Percentage
Totally agree
42
23,73%
Agreed
24
13,56%
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Neutral
44
24,86%
Disagreement
64
36,16%
Strongly disagree
3
1,69%
Total
177
100,00%
It is evident that 60% of respondents do not participate in the online session, while 30%
do. It was determined that the development of the class is established on the basis of
the explanation of the subject, expressing the teachers that the short duration of the
class does not allow the participation of all students.
Table 5. In the online class you learn by working as a team and collaboratively using forums,
networks, chat
Description
Frequency
Totally agree
24
Agreed
1
Neutral
5
Disagreement
96
Strongly disagree
51
Total
177
More than 80% of the students stated that they do not perform collaborative work in
the online class, while 30% reflect a positive response. It was possible to observe the
assignment of group activities to be done at home, however, the teacher does not know
digital educational tools to develop this online activity.
Table 6. In the online class, your teacher answers your questions or doubts about the class
topic.
Description
Frequency
Percentage
Totally agree
54
30,51%
Agreed
82
46,33%
Neutral
31
17,51%
Disagreement
9
5,08%
Strongly disagree
1
0,56%
Total
177
100,00%
Seventy percent of the students state that their concerns about the class topic are
answered, while 20 % say that this is not done. It was verified that the teacher motivates
the students to express their doubts, but according to his criteria many do not do it,
due to different situations (connectivity, embarrassment, lack of interest).
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Table 7. Is evaluated in the online class by the teacher to verify their learning online form.
Description
Frequency
Percentage
Totally agree
1
0,56%
Agreed
19
10,73%
Neutral
45
25,42%
Disagreement
71
40,11%
Strongly disagree
41
23,16%
Total
177
100,00%
Eighty percent of respondents claim not to be evaluated in the online class, while 10%
reflect a positive response, it is observed that the teacher explains in great detail the
subject, but no LMS platform is applied for the evaluative part in a synchronous manner,
according to their criteria this is due to the lack of expertise in the use of these learning
tools.
Table 8. The teacher reinforces the explanation of the class topic.
Description
Frequency
Percentage
Totally agree
18
10,17%
Agreed
0
0,00%
Neutral
0
0,00%
Disagreement
85
48,02%
Strongly disagree
74
41,81%
Total
177
100,00%
It is emphasized that 90% of the respondents agree that there is no reinforcement of
the class topic, while 10% affirm that there is. In the courses observed, it was found
that there is no generalized feedback of the contents, teachers report that the student
does not express concerns about the explanation provided.
Table 9. You would like to receive a much more engaging and fun learning experience in
online classes
Description
Frequency
Totally agree
40
Agreed
60
Neutral
52
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Disagreement
22
Strongly disagree
3
Total
177
Fifty-eight percent of the students want fun learning, while 42% indicate that they do
not. As for the observations made, it was evident that the development of the online
class is limited to the transmission of information, without prioritizing interaction.
According to the teachers' criteria, this is due to a lack of knowledge of a specific
methodology for teaching through a computer.
Table 10. The Inverted Classroom learning model will improve your mathematics learning in
the online classroom.
Description
Frequency
Percentage
Totally agree
41
23,16%
Agreed
71
40,11%
Neutral
45
25,42%
Disagreement
19
10,73%
Strongly disagree
1
0,56%
Total
177
100,00%
Of those surveyed, 63% consider that the invested learning improves their training in
Mathematics, while 36% give an unfavorable answer. In the observations, the lack of a
model designed for online education is evident; in the teacher's opinion, students do not
achieve the academic level through this modality of study.
DISCUSSION
The educational process in the context of the covid-19 pandemic, has been developed
in the online modality, by public and private institutions, generating a number of
questions and challenges due to the same activities on which they have tried to establish.
According to the results obtained, it was found that students and teachers have
technological equipment (PCs, tablets, smartphones) to connect to the online class.
However, teachers express the complexity of coupling to this modality, due to the lack
of knowledge of methods, techniques, strategies, use and application of digital tools that
allow them to make the most of this learning space.
In the same analysis of results, students expressed their desire to receive dynamic and
interactive learning that motivates them to participate and be active in the construction
of online knowledge. From this, the need arises to offer adaptations to the acting learning
model, singularized in its inverted character for the online modality, from the
assumptions of the conception of this in the face-to-face (Jon Bergmann and Aaron Sams,
2007).
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The modeling of the online teaching-learning of the mathematics class, in its conception
of inverted learning, requires a simplified representation of this process, in subsystems
and components that constitute it.
The online inverted class, arises in the processes of its conception, therefore the authors
consider that the first subsystem has to be established in: the orientation of the
autonomous online activity, which has as purpose to project the individual actions of the
student with the digital learning objects by internet, previous to the interaction with the
teacher and other students in the synchronous class schedule, originating in a particular
way the interrelation of the students with the materials oriented by the teacher to learn
before the asynchronous class.
In the analysis carried out, it is emphasized that the digital resources used by the teacher
are sent to the students at the end of the asynchronous class, and not prior to it as
proposed in the model, in which this first subsystem composed of three components is
proposed (See Illustration 1Subsystem 1.-Orientation of online self-employed activity).
In the first component, digital resources (videos, podcasts, presentations, blogs, among
others offered) are received through the EVEA (Virtual Teaching-Learning
Environments) according to the educational institution. Subsequently, the contents
shared by the teacher should be understood individually through the reading,
systematization and analysis of the digital resources offered. This will facilitate the
processes of interpretation of the learning objects, from the synthesis of the contents
and their assimilation that prepares them in advance for the development of the online
session.
Illustration 1Subsystem 1.-Orientation of online self-employed activity
The teacher in the application of the inverted learning model, during the orientation of
the autonomous online activity, must define the objective of the class, select or create
the digital didactic resources and prepare the interactive activity for the students. Once
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the first phase has been executed, a second subsystem is proposed, which is composed
as an online synchronous learning process, where the main objective of the subsystem
is to systematize the theoretical and practical contents of the Mathematics class in a
synchronous way, through the interaction of the teacher with the students and the
online learning objects (see Figure 2: Subsystem 2). Illustration 2Learning Feedback
Subsystem 2.).
The referred subsystem is established on the logic of the internal relationship of the
components of the teaching-learning process, namely, objectives, contents, methods,
resources and evaluation. The first component is called: Orientation of learning
objectives and contents, in which the guiding phase of the activity takes place on the
basis of what has been previously learned by the student and facilitates the asynchronous
online pedagogical interaction, where the learner exemplifies what has been learned,
proposes unresolved questions during his individual understanding and interpretation, in
addition the teacher explains and argues to homogenize the previous knowledge and the
content of mathematics treated, which as a third component will allow revealing the
cognitive feedback of the content and the learning method used.
Illustration 2Learning Feedback Subsystem 2.
The role of feedback seeks to improve cognitive skills and consolidate knowledge in
students. In the online mathematics class currently being developed, there is a lack of
practice activities, reinforcement and evaluation of what has been learned. Therefore, it
is emphasized that the third subsystem will consist of an online feedback and
consolidation process, its objective is to strengthen the practical knowledge of
Mathematics through practice as a key point in the evaluation of learning.
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On the other hand, one of the elements of greater attention is the evaluation, this
responds to the need to obtain the progress of how much the student is learning. The
first component is called autonomous practical exercise, this proposes that the student
consolidates the knowledge through the development of practical activities of
mathematical exercises, as well as digital support material, if any of them find difficulties
in this activity to reinforce what they have learned, which, as a third component of
evaluation, allows to demonstrate the assimilation of the contents taught by the teacher.
(See
Illustration 3Online evaluation and consolidation)
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Illustration 3Online evaluation and consolidation
The inverted classroom learning model for online modality, from the conception of the
proposed subsystems: orientation of the autonomous online activity, learning feedback,
evaluation and online consolidation, then, the internal relationships that arise in the
epistemological analysis are established.
• The essential relationship between the content, the analysis made by the student
of the subject through technological resources and facilitates understanding.
Therefore, a quality of asynchronous internalization of the autonomous activity
measured by the technology and the Mathematics content is revealed, which
allows the student to be prepared to act in the following synchronous didactic
space.
• The communicative relationship between the teacher-student action mediated by
educational technology and the study contents simultaneously reveals the
synchronous communication process of learning, which is characterized by the
achievement of the stated goals.
• The internal relationship between the independent work done by the student, the
reinforcement of learning based on the materials offered, the self-assessment
based on interacting with technological resources, is where the asynchronous
individual assessment of learning achievements is revealed.
• Therefore, the internal logic of the subsystems, their particular relationship
approach, the singularization of the synchronous-asynchronous correlation of
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inverted teaching-learning and the achievements of the online mathematics class
mediated by educational technologies are unveiled.
In summary, inferences must be made on the integrative quality generated from the
direct interaction of the proposed subsystems, where the processual character of the
online Mathematics class is revealed, from the adaptation of the inverted classroom
learning model for its conception, development and improvement, in the subsystems
offered in this article. (See Illustration 4. Design of the Inverted Classroom Learning
Model aa)
Illustration 4. Design of the Inverted Classroom Learning Model a
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