Mathematical competencies with applications of mirror classes in regular basic education

International Journal of Evaluation and Research in Education (IJERE)
Vol. 12, No. 4, December 2023, pp. 2230~2245
ISSN: 2252-8822, DOI: 10.11591/ijere.v12i4.25577  2230
Journal homepage: http://ijere.iaescore.com
Mathematical competencies with applications of mirror classes
in regular basic education
Maruja Dionisia Baldeón De La Cruz1
, Melba Rita Vasquez Tomás1
, Judith Soledad Yangali Vicente1
,
Jhon Holguin-Alvarez2
1
Postgraduate School, Universidad Privada Norbert Wiener, Lima, Perú
2
Department of Investigation, Faculty of Law and Humanities, Universidad César Vallejo, Lima, Perú
Article Info ABSTRACT
Article history:
Received Sep 27, 2022
Revised Sep 19, 2023
Accepted Oct 10, 2023
The problem about learning in mathematics lies in the lack of application of
teaching strategies for the resolution of the calculation in mathematical
problems, problems of movement, mathematical regularity, equivalence, and
uncertainty; in populations with difficulties to learn collaboratively. Here the
application of mirror classes in populations with interactive problems is
demonstrated, thus contributing to strengthening their knowledge from
cooperative information management, the use of analysis skills based on co-
evaluation. In this sense, the purpose of the study was to evaluate the mirror
class, as a pedagogical strategy, in order to optimize the competence of
solution of quantity problems in basic school students of two educational
institutions in Lima, Peru. The study was conducted under the action
research approach where the diagnostic and exit test, question guide and
field diary were the instruments used for data collection. It was concluded
that the mirror class as a pedagogical strategy favors the optimization of
competition and solves quantity problems in basic education students. So,
they also developed concrete knowledge, to assign them in cognition more
enduringly as a form of social learning.
Keywords:
Mathematical competence
Mirror class
Pedagogical strategy
Virtual education
This is an open access article under the CC BY-SA license.
Corresponding Author:
Maruja Dionisia Baldeón De La Cruz
Research Vice President, Universidad Privada Norbert Wiener
Av. República de Chile N°. 432, Santa Beatriz, Jesús María, Lima, Perú
Email: maruja.baldeon@uwiener.edu.pe
1. INTRODUCTION
Mathematical competencies promote the development of mathematical processes such as thinking,
reasoning, argumentation, interpretation, among others, in students [1]. However, national and international
standardized evaluations show worrying results regarding mathematics learning in Peru. Thus, the regional
comparative and explanatory study (ERCE 2019) conducted in Latin American and Caribbean countries
showed that 61% of Peruvian students in sixth grade of primary school are below the minimum level of the
competencies established in the area of mathematics [2]. Likewise, the program for international student
assessment (PISA) in 2018 reported that the academic performance of Peruvian students, in general, is low,
ranking 64th out of 79 countries [3].
Mathematical competence is defined as the ability to understand the relevance of mathematics in the
world and thus use it to exercise the role of constructive, committed and reflective citizens [4]. In this regard,
D’Amore, Godino, and Pinilla [5] pointed out that achieving mathematical competencies is a challenge and
requires considering four didactic requirements: i) Epistemological requirement, which is the theoretical
reference that guides the teacher in the teaching and learning process; ii) Cognitive requirement, which is the
theoretical construction of a learning object by the student; iii) Communicative requirement, which consists

Int J Eval & Res Educ ISSN: 2252-8822 
Mathematical competencies with applications of mirror classes in … (Maruja Dionisia Baldeón De La Cruz)
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of discursive interaction in a learning situation; and iv) Socio-cultural requirement, which refers to the
relationships that develop among all the elements that are part of the classroom. In Peru, the national
curriculum proposes in the area of mathematics, the approach focused on problem solving. From this
approach, four competencies are considered for the development of mathematical thinking: i) Solution of
quantity problems; ii) Solution of form, movement and location problems; iii) Solution of regularity,
equivalence and change problems; and iv) Solution of data management and uncertainty problems.
The results of the research contribute to demonstrate the effects of developing mirror classes in the
quantity problem solving competence in more than half of the participants, developing significant levels of
achievement. At the same time, it is shown that, in motivation, teacher accompaniment, as well as teaching
strategies are elements of the session that improve the beliefs of the students, as well as their own parents,
and the apprehension that resources applied strategies generate greater collaboration and intrinsic motivation
to learn. Finally, teachers begin to accept that the reflection of knowledge in others as a strategy of the zone
of proximal development, allows progressively staggering increasingly positive performances in learning
mathematics.
Given this, the hypothesis is raised that the effects of the mirror class can cause improvements in the
way students learn, associating information to solve problems collaboratively, with stronger motivational
interrelationships for the development of mathematics, both in learning, as well as in the perception of self-
efficacy of teachers. This will also influence the parental expectation towards the children about the solution
they arrive at, changing the traditional classes for classes with potentially cognitive interrelationships. It is
consistent with the review of the literature that, the position that the interrelationships can be more
cooperative if models of proximal learning are provided through classes that reflect the development of the
skills of other more expert subjects over those who are apprentices or have poor school performance.
The mathematical competency: solution of quantity problems demands greater use of time in class
with students because it addresses several thematic fields. This competency encourages students to solve and
formulate problems that allow them to comprehensively construct the notions of number, number systems,
their operations and properties. It also requires students to use strategies and develop logical reasoning, based
on the mobilization of the following skills: i) Translation of quantities into numerical expressions;
ii) Communication of understanding of numbers and operations; iii) Use of estimation and calculation
strategies and procedures; and iv) Argument of statements about numerical relationships and operations [6].
In that sense, competencies differ from abilities in that they are developed in the long term while skills are
acquired in a specific learning experience [7].
The translation of a verbal statement into a symbolic representation requires understanding the
variables and relationships established within the verbal statement, in addition to the syntactic characteristics
of the symbolic representation [8], being the verbal-to-graphic representation one of the most difficult for
students [9]. On the other hand, Font, Godino, and D’Amore [10] refer that comprehension is an ability that
the student evidences in practice. This allows students to develop a more flexible understanding by making
several representations [11]. Likewise, they contribute to organize students' thinking and make mathematical
ideas more concrete [12]. Hence, other research [13] points out that representations should be considered as
instruments that allow mathematical competence to be generated. In this sense, going through several
equivalent representations improves the learning of mathematics [14].
On the other hand, Alsina et al. [15] explained that the strategies used in problem solving acquire an
important role because they favor the learning of mathematical concepts, consolidating comprehension,
expression and their application. Reciprocally, the problems done in class should encourage students to
express their ideas or explain how they have solved them [16]. In this regard, communication allows students
to formulate questions and argue in and with mathematics, thus contributing to the development of
mathematical competences [1]. Hence, the current teaching of mathematics requires the implementation of
new pedagogical strategies that favor the understanding the mathematical ideas of students. Therefore, they
contribute to the development of competencies in the context of virtual classes as a result of the COVID-19
pandemic [17].
In this sense, mirror classes promote interaction between students and teachers where knowledge is
shared through a virtual platform between two national or international institutions [18]. In this regard, the
research literature has so far revealed that the cognitive mechanisms of students increase as much as their
knowledge when they use the expertise of their peers to improve their cognitive or procedural skills. This
means that this conception is based on socio-cultural and imitative perspectives of learning, since the mirror
class method has revealed certain empirical glimpses in research. It shows approaches towards its realization
for the improvement of the abilities described. Mirror class is a strategy of pedagogical internationalization
that allows strengthening the competence of students by establishing a connection between collaborative
learning, the theory of research competences, the application of information and communication technologies
(ICT) and identifying the communicative tools in virtual learning environments [19].

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Nowadays, with the dominance of technologies in learning at the secondary or university level [19]
have found that mirrored learning (oriented to mirror classes), generates positive effects on the participants of
these types of classes by introducing telephony and artificial media in learning. In essence, they found that
learning among Swedes and Finns was much higher in the face-to-face and screen-based interaction
strategies, which allowed significant improvement of this ability in the classroom. A study aimed at
understanding the method steps of the teacher [20] found that teachers are more predisposed to generate
command strategies in classrooms where learning exchanges are carried out by the same teacher but in an
experiential way, as opposed to those who only work under information and communication systems in the
classrooms in which they participate, since there is greater co-occurrence of knowledge generation in
classrooms where they work in a participatory (experiential) way.
On the other hand, this strategy seems to have appeared in other researches, which with similar
structure have oriented their research work towards mirror activities [21] who with an addition of inverted
classroom, found improvements in students of an internship on surgery, and which, under constructivist
strategies of collaboration allowed finding more evidence of self-learning in students who carry out this
experience. In the school field, Shimpi, Akhtar, and Moore [22] reported that imitation as a learning
replication strategy in young children, who imitated the learning acts of others who were unknown to them.
Although an attempt was made to control the interaction variable between them, first sketches were generated
about autonomous learning and imitation of strategies among totally unknown infants. Although the age of
the participants should be taken into account, since in the initial stage the cognitive capacities influence the
quality of the denial of the interaction as to make a mirror learning.
In the pedagogical context, we can find that the imitative effects in learning also depend on the
quality of the learning itself, from the ecologist theory, this quality by which neurons perform imitation is
enduring with respect to the goals of the recipients of such learning. Less quality in the strategy (durability,
exemplarism, balance, and motivation), can affect the reception of learning in subjects with less expertise,
denoting that learning by mirror method depends both on the relationship: subject-expert/subject-inexpert,
and on the mediation of the teacher to regulate the factors of the quality of the imitated learning [23]. Current
evidence has determined that in addition to the acquisition of learning and the development of skills, mirror
learning may include the acquisition of goals, attitudes, and other aspects such as confidence, self-belief,
about the visionary achievement of learning in their educational mirroring interactions [24], [25].
In view of that, we could focus on the Vygostkian perspective, alluding to the use of the proximal
potential with the acquired potential among the participants of an experience, although experimental evidence
differs in the inclusion of the interaction variable in mirror learning [22], [26]. Although the situationally of
this learning can also occur through sociocultural processes of teaching from the more expert to the more
inexperienced, the inclusion of the interrelation variable in the method with refractory or mirror classrooms is
necessary. This model also belongs to a certain extent to Chevallard’s approach in 1985, regarding the
didactic transposition among learners [27], since there is an origin that must be transferred to other
participants so that they can take advantage of the substantial aspects of knowledge, in order to use the
borrowed sources as didactic elements in the teaching of knowledge to others. In this sense, we also focus
these contributions on the co-constructive method analyzed [28] who argue that learning is both negotiated
and bounded, in interactive segments, which we predict as convenient among the peers performing the
learning. For this reason, the perspectives complement each other, from their socio-biological aspect to the
socio-cultural aspect described during the history of mankind.
From the Vygotskian perspective, the study contributes to the development of mathematics, based
on proximal and potential development, it contributes to the achievement of skills to develop problems,
exemplifying mathematical classes with students with greater knowledge and skills, with others with low
potential. to make it. When carrying out student exchanges, the development of low-level skills is promoted,
supporting them in the experience of those who, if they have the potential to achieve it, therefore, mirror
classes based on student exchange are applied. For all the explanation, this study has the aim to evaluate the
mirror class as a pedagogical strategy, in order to optimize the competence of solution of quantity problems
in basic school students of two educational institutions of Lima, Peru.
2. RESEARCH METHOD
2.1. Participants
The aim of the study was to evaluate the mirror class as a pedagogical strategy to optimize the
problem-solving competence in quantity in basic school students in two educational institutions in Lima,
Peru. In this sense, this research was developed under the qualitative approach, with an action research
design. Under this design, four phases were assumed: planning of an action plan to optimize the quantity
problem-solving competence, action to execute the action plan, observation to collect evidence from the

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implementation of the plan, and reflection on the actions recorded. The sample consisted of 40 students of
both sexes whose ages ranged between 8 and 10 years old, chosen from a total of 64 corresponding to the
third grade of basic education from two public educational institutions from different districts of Lima, Peru,
selected by means of purposive sampling.
2.2. Instruments and procedure
For data collection, two semi-structured interview guides were used as instruments for both the
parents and the teachers who taught the mirror classes; a diagnostic test and an exit test were also used, in
addition to the field diary. The first instrument used was the diagnostic test, which was developed by the
students individually and online. It consisted of five multiplicative problems in order to collect information
on the current situation of the level of development of the competency solve quantity problems. The
problems were oriented to the skills that make up the aforementioned competency: translation of quantities
into numerical expressions, communication of understanding of numbers and operations, use of estimation
and calculation strategies and procedures, and argument of statements about numerical relationships and
operations. Scores were classified into the following achievement levels: beginning (0-10), process (11-14),
and satisfactory (15-20). Subsequently, 12 mirror classes were developed, focused on multiplicative problem
solving (multiplication and division problems), conducted virtually due to the context of the COVID-19
pandemic. The duration of each class was 150 minutes. In the development of the classes, the use of
unstructured concrete material and teamwork were prioritized as strategies. At the end of the 12 mirror
classes, the students were evaluated with an exit test consisting of five multiplicative problems, to
corroborate the progress in the development of the solution of quantity problems competency. Similar to the
pedagogical test, the questions were oriented to the skills of the competency Solve quantity problems. The
achievement levels considered were beginning (0-10), process (11-14), and satisfactory (15-20).
On the other hand, the two semi-structured interview guides were applied, consisting of 8 questions
each, addressed to the 2 teachers and 8 parents of both classrooms, on their perception of the mirror class as a
pedagogical strategy in the achievement of the solution of quantity problems competence in the students. The
field diary was used to record everything observed in each mirror class. The research followed the four
phases [29], namely planning, execution, observation, and reflection.
2.2.1. Planning phase
Coordination meetings were held among the responsible teachers, in these meetings an action plan
was established to optimize the solution of quantity problems competence, specifically in the resolution of
multiplicative problems. For this purpose, first, a review of the curricular planning of each classroom was
considered in order to determine the possible subtopics that could be included in the resolution of
multiplicative problems. Secondly, an agreement matrix was organized to establish the pertinent guidelines
for the development of the mirror classes (topic and subtopics, date and time of the mirror classes, virtual
learning tool to be used in the meetings, materials and evaluation instrument). Subsequently, the mirror
classes referred to multiplicative problems and the evaluation instruments (rubric) were designed. Finally, the
diagnostic evaluation was applied to third grade students of the selected educational institutions, both located
in different districts of Lima-Peru, in order to identify the level of achievement of the Solution of quantity
problems competence.
2.2.2. Action plan execution phase
The 12 mirror classes referred to multiplicative problems were carried out and the evaluation
instruments (rubric) were applied. In the development of the mirror classes, the stipulations of the agreement
matrix had to be considered, in addition to complying with the following recommendations: all meetings are
synchronous, teachers must be connected virtually in all sessions, innovative and relevant activities must be
generated, students will work in teams to solve, create and argue the resolution of multiplicative problems,
the means of communication will be kept open between students and; between teachers and students, all
materials will be shared with students before the mirror classes (videos, worksheets, and practices) in order to
address queries or doubts. At the end of the 12 mirror classes, an exit evaluation was applied to the students
to identify the level of achievement reached in the competency solve quantity problems.
2.2.3. Evidence observation phase
The actions carried out in the plan implementation phase were observed and evidence was collected
using diagnostic and exit tests, interview guides and the field diary. The preliminary tests were developed
with performance tests, instead, the interviews were carried out personally with those parents who agreed to
the evaluation. The teacher’s diary was a clear tool for the teacher’s auxiliary record and diary according to
the development of their classes.

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2.2.4. Reflection on actions phase
Weekly meetings were developed between the teachers of both educational institutions to evaluate
the results obtained in the previous phase, so that improvements and aspects to be strengthened in the
research were identified. The meetings consisted of questioning and proposal stages. In the first stage, it was
allowed to focus on the problems that arose, as well as the solutions that did not work in the development in
class. On the other hand, the proposals made it possible to complete cyclical learning processes in the
classroom, progressively improving those that were deficient.
3. RESULTS AND DISCUSSION
3.1. Interpretation and analysis
The analysis of the results was based on the data obtained from the diagnostic test and the exit test,
in addition to the development of interviews with parents, teachers, and the analysis of the field diary applied
in 12 activities to third grade students in two educational institutions in Lima, Peru. To this end, we began
with the transcription and organization of the information and the theoretical and data triangulation. It
allowed us to argue the study from different expert authors and the definition of the Peruvian Ministry of
National Education (Ministerio de Educación Nacional del Perú) in relation to the solution of quantity
problems competence and its capabilities in the area of mathematics, in addition to the responses of the
participants that allowed us to know their perception in relation to each of the categories of the study.
The information given by the participants of instrument 1, teacher interview, instrument 2 and the
observation record card of the field diary were processed in the Atlas.ti software version 8.4.24, in which
each one was enlisted according to the categories of the study; solution of quantity problems competence and
mirror classes. According to the data, memos were created, associated with the theory that underlies the
study, and finally semantic networks were constructed that demonstrate the relationships between each of the
categories, subcategories and memos, in order to understand and interpret what is the perception of teachers,
students and parents about the implementation of mirror classes as a pedagogical strategy for the
development of skills, such as: translating quantities into numerical expressions; communicating
understanding about numbers and operations; using estimation and calculation strategies and procedures;
arguing statements about numerical relationships and operations.
3.2. Data coding
In the coding process, the selective coding method [30] was considered, which allows starting the
processing from the definition of a list of codes previously established from the method design of the
research and the categories and subcategories of the study. In this case, four indicators were reported on the
evaluated competence. Thus, the semantic network of codes of the research is proposed, which are made up
of the categories, solution of quantity problems competence and mirror classes as shown in Figure 1.
Figure 1. Research codes and code groups
Thus, between the category solution of quantity problems competence and mirror classes; an
association relationship was evidenced with the subcategory motivation, two relationships of belonging with
the subcategories teaching strategies and teaching support and a causality relationship with the subcategory

2235
evaluation. Although the evaluation has been more relational with the arguments, the teaching
accompaniment with the translation of the quantities, as well as the teaching strategies seem to come together
better with the development of numerical estimation. Finally, motivation has been associated in a better
predisposition with the forms of communication of students when learning mathematics.
3.3. Category analysis
3.3.1. Solution of quantity problems competence
The purpose of the competency solves quantity problems is for the student to solve and formulate
problems that make it possible to understand numerical systems, their operations and properties through the
implementation of different resources [31]. Thus, for this category, four subcategories have been established
for the analysis, which have been established according to the skills that the child must develop to achieve
this competence: translating quantities into numerical expressions, communicating understanding about
numbers and operations, using estimation and calculation strategies, and arguing statements about numerical
relationships and operations. The analysis associated with each of the subcategories is presented, based on
the instruments applied to teachers, parents and elementary school students from educational institutions in
Lima, Peru.
a. Translation of quantities into numerical expressions
In the analysis of this subcategory, the contribution of the mirror classes in the development of this
ability in the students was evidenced. Their application allowed the creation of problems based on a given
situation or requirement. Likewise, it allowed the resolution of multiplicative problems from the translation
of the verbal statement of the problem to a concrete, graphic and later symbolic representation, using
unstructured didactic materials, the guidance of the teacher and the accompaniment of the parent. In this
regard, several researchers [32]–[34] stated that the creation and resolution of problems are essential tasks for
the development of competencies and therefore of mathematical thinking. On the other hand, the opportunity
to provide them with autonomy in the development of their activities strengthened the teaching process,
teamwork and the development of student learning through the virtual sessions as shown in Figure 2.
Figure 2. Network of codes and quotations of the translation of quantities into numerical expressions ability
It can be seen from the teachers’ answers to the question: Do you consider that your students can
translate actions of repeating and distributing quantities, to multiplication and division expressions with
natural numbers, when solving problems from the mirror classes? Why?
(348:418)-D1: These mirror classes have particularly helped my children a lot ...
(820:878)-D1: It has helped them a lot because now they understand better.
(592:687)-D2: They themselves were discovering things in these classes, being in the virtual classes.
(1459:1576)-D2: Yes, I consider that they can translate statements about repeating and distributing
into multiplication and division expressions.

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Now, with respect to the question addressed to the parents, do you consider that your child can create
problems on multiplication and division and solve them from the mirror classes? Why? It was evidenced:
(215:266)-P1: Yes, in multiplication and division they can also create ...
(416:495)-P2: Yes, they have learned to create because it has been more didactic among classmates
(575:630)-P3: Yes, he learned to create problems by himself, he does it and solves it...
(691:742)-P4: They could create it with the same examples they were given ...
(873:960)-P5: They created their multiplication and division problems following the examples and
solved them ...
(1098:1283)-P6: When she reasons and listens carefully to the class is where she herself sometimes
tells me mom this is how it is, we add, then we subtract, then we multiply by the amount ...
(1337:1418)-P7: They have advanced but they are still reinforcing multiplication and division ...
(2401:2476)-P8: Yes, a little more because they helped each other among classmates to do the
problems.
On the other hand, in the analysis of the field diary applied to the students it was found:
(1644:1903)-DC1: The students actively participated in the resolution of the multiplication problems
by representing them with concrete material such as the Mac kinder box, the rulers or base ten, then
graphically and finally through multiplication ...
(2581:2740)-DC2: One of the problems was worked in teams and then one member of each team
explained how they solved it. The teacher gave pertinent feedback ...
(4201:4286)-DC3: They represented the problems graphically with drawings, diagrams or number
lines ...
(9188:9377)-DC4: They used the drawings, the table of points, the table of direct proportionality,
the number line, the decomposition, as they found easier to solve the problems.
b. Translation of quantities into numerical expressions
In the analysis of this subcategory, the contribution of mirror classes in the development of the
ability communication of understanding of numbers and operations, through representations in a concrete,
graphic and symbolic way, from the use of didactic and ludic strategies such as the use of unstructured
didactic material and the application of virtual games, was evidenced. In this perspective, [35] point out that
representations favor the understanding of mathematical concepts. They contribute to organize students'
thinking and make mathematical ideas more concrete [12]. On the other hand, ICT implemented in the
teaching and learning process enhance the understanding of mathematical content [36]. For their part, parents
recognize the added value of mirror classes in their children's learning, in addition to the development of
other skills such as teamwork, as shown in Figure 3.
Figure 3. Network of codes and quotations of the communication of understanding of numbers and
operations ability

2237
It can be seen from the teachers’ answers to the question: Do you think that your students acquired
greater understanding of the meaning of multiplication and division from the mirror classes? Why?
(2666:2842)-D1: Yes, because after the mirror classes we have been analyzing some problems and
they already realize when it is multiplication, when it is necessary to divide and they have even
realized the importance of knowing multiplication ...
(3340:3458)-D2: Yes, because in the development of the mirror classes the children participated and
answered the questions, besides that in their practices they solved and created multiplicative
problems."
Now, with respect to the question addressed to the parents: Do you think that your child acquired greater
understanding of the meaning of multiplication and division from the mirror classes? Why? It was evidenced:
(2480:2568)-P3: Yes, my son learned, now he solves by himself ...
(2572:2762)-P4: Yes, I think so because he has been able to solve his own problems. When they said
multiplication, he said double, triple, he did it by himself or also with his small caps and with the
beans ...
(2907:2975)-P5: I was looking at her so she already had those hints that she could do ....
(3035:3157)-P6: The divisions were a little difficult for her because it was her first time, but she had
already made progress in multiplication ...
(3366:3410)-P7: They have helped her, she did not understand almost anything ...
(1337:1418)-P7: They have made progress but they are still in reinforcement with multiplication and
division ...
(2401:2476)-P8: My child made progress with multiplication and division.
(2743:2860)-DC 1: Most students were very attentive and participative since it was a topic that
generated much interest in them ...
(4288:4382)-DC 2: Then they explained the process of the resolution and received feedback in a
timely manner ...
(5314:5482)-DC 3: Students found double and triple relationships between the factors of the
multiplications from graphs and problems posed. In addition, they completed them ..."
Here the motivation or attention to the task converge, the relationship seems to have achieved a
better understanding of the problems with semiotic representation of quantities. Likewise, it has been found
that shared evaluation can support the generation of new responses in cognitively less favored students. In
this sense, assessment can promote a sense of learning and mathematical learning because of students’ own
motivations.
c. Use of estimation and calculation strategies and procedures
It was found that estimation and calculation procedures were worked on through problem solving. In
this context, the strategies used by the students were mostly oriented to the use of didactic resources such as
unstructured materials, among which the Mackinder box and homemade objects from their environment such
as small caps, chip-taps, marbles and seeds stand out in the development of mathematical activities. In this
regard, Torrecilla, Carrasco, and Cerezo [37] pointed out the importance of the use of educational material in
the teaching and learning of mathematics, since it contributes to the development of logical thinking due to
the manipulation that allows for greater understanding of concepts. In this perspective, strategies play an
important role because they favor the learning of mathematical concepts by consolidating understanding [15].
On the other hand, it allowed parents to identify the motivation involved in student learning, the
implementation of playful, participatory strategies that encourage teamwork as shown in Figure 4.
It can be seen from the teachers’ answers to the question: “What strategies used in the teaching and
learning process of multiplication and division in the mirror classes allowed their students to acquire greater
understanding?”
(4075:4236)-D1: With the materials and graphics, the children who did not understand
multiplication and division very well were able to understand ...
(4506:4527)-D2: Use of the Mackinder box, the seeds or objects that the children used to solve the
multiplicative problems. In addition, the graphic representation ...

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Now, with respect to the question addressed to parents, “What strategies or activities used in the teaching and
learning process of multiplication and division in the mirror classes allowed their child to acquire greater
understanding?” It was evidenced:
(5801:5815)-P3: Mackinder box …
(2572:2762)-P3: We had a bottle with small plates and with that he used to solve the multiplication
problems, for division it was through pure multiplication ...
(6395:6403)-P4: Mackinder ...
(6423:6587)-P4: Small plates, my child used to do it with beans and also those little division and
multiplication games for them to answer, with roulette they also did that a lot ...
(6964:6980)-P5: Little Mackinder box ...
(7275:7336)-P6: They used the Mackinder box for multiplication and division ...
(7409:7436)-P7: I have not used any purchased material, but I have used the small caps and seeds
we have at home ...
(7963:8006)-P8: Used seeds, soda caps.
(3261:3325)-DC1: They justified the reason for the representations they used ...
(3148:3260)-DC2: The students represented multiplications in different ways, using the Mackinder
box, in different situations.
d. Argument of statements about numerical relationships and operations
In the analysis of this subcategory, the ability of most students to argue and explain the strategies
and procedures applied in the development of multiplicative problems, that is, multiplication and division,
was evidenced, which allows highlighting the contribution of the mirror classes. In this perspective, the
problems solved in class should encourage students to express their ideas or explain how they have solved
them, otherwise it makes no sense to consider the competence to communicate as an objective to be achieved
[16]. In this regard, Solar et al. [1] points out that oral or written communication contributes to give meaning
to ideas and to share them with others. Hence, classroom is the appropriate space for students to develop
communicative and argumentative practices [38]. However, some parents pointed out that in the development
of the classes their children, due to their shyness and insecurity to express themselves in front of others,
presented limitations to share their knowledge, as shown in Figure 5. This could also be demonstrated in
internationalization classes as in other evidence found [39].
Figure 4. Network of codes and quotations of the use of estimation and calculation strategies and procedure’s
ability

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Figure 5. Network of codes and quotations of the argument of statements about numerical relationships and
operations ability
It can be seen from the teachers’ answers to the question: “Do you consider that your students can
make arguments or explanations about multiplication and division problems? Why?”
(4506:4527: 5432:5660)-D1: Most of the children do ... some are shy and do not give you the
answer, they just tell you that they have already done it, but when you take them by teaspoonful,
they explain how they have been doing it, but others are more confident.
(5945:6053: 6470:6487)-D2: Yes, when they understand and solve the problem, they can explain the
process they have followed to solve it ... definitely yes.
Now, with respect to the question addressed to the parents, “Do you consider that your child can make
arguments or explanations about multiplication and division problems? Why?” It was evidenced:
(8343:8433)-P1: In that regard he did roughly because they are shy to speak in public ...
(8820:8860)-P2: He would need a little more to get around ...
(9161:9238)-P3: Mi child explains to you how they managed to solve the problem ...
(9270:9307)-P4: He says and explains how he did it ...
(9506:9591)-P5: She is a little bit more shy to express herself in public, but at home she does it very
well ...
(9774:9844)-P6: Yes, she does, the only thing is that she has a little bit of stage fright ...
(9894: 9998)-P7: Yes, I can do it but if the teacher asks her because she doesn’t do it on her own,
she is very shy ...
(10054:10152)-P8: Yes, they can do it for their teacher, not for others. They are a little bit
suspicious, a little bit shy.
(3918:4198)-DC1: The students solved problems where they multiplied in a rectangular way, that is,
rows by columns and vice versa to find the product, thus discovering that the order of the factors
does not alter the product, relating it to the commutative property in multiplication ...
(4668:4827)-DC2: The students solved multiplications using the board of dots. This also allowed
them to strengthen their understanding of the commutative property.

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3.3.2. Mirror class
The category, mirror class is defined as a pedagogical strategy that promotes interaction between
students and teachers where knowledge is shared through a virtual platform between two national or
international institutions [18]. For this category, learning sessions have been indicated as a subcategory. For
its analysis, the following domains of analysis were established: motivation, teacher support, evaluation and
teaching strategies.
a. Learning sessions
The analysis of this subcategory allowed identifying, from the perception of teachers, parents and
students, those factors that are related to the development of each of the 12 activities proposed in the mirror
classes for the development of learning and skills of the competency solve quantity problems. The aspects,
such as motivation, teacher support, evaluation, and teaching strategies were identified as shown in
Figure 6 to Figure 9, respectively. Motivational components were analyzed under convergent restructuring,
and the citation code network for the analyzed database.
Figure 6. Network of codes and citations of the motivation domain
Figure 7. Network of codes and citations of the teaching accompaniment domain

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Figure 8. Network of codes and citations of the evaluation domain
Figure 9. Network of codes and citations of the teaching strategies domain
It can be seen from the following questions and answers to the teachers: “Do you consider that the
mirror classes contributed to make your students feel more motivated to learn about multiplication and
division operations? Why?”
(6600:6796)-D1: With the mirror classes they have become more interested in these multiplication
and division problems because we saw that they were two classroom groups which interacted even
in the work groups ...
(7460:7645)-D2: Yes, because the children interacted with classmates from another place and that
was new for them; also, working with different activities and strategies such as the use of didactic
material ...
In relation to the question: “Do you consider that the teacher support in the mirror classes has contributed to
the development of your students’ mathematical skills?” The teachers answered:

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(8399:8574)-D1: Yes, because the students were being observed at all times during the classes and
feedback was given as needed, not only in the classes themselves but also outside of them ...
(8871:8902)-D2: Yes, we have always been supporting the students to give them feedback when
necessary, so that they can continue advancing in their learning ...
On the other hand, in response to the question: “What improvements have you observed in your
students’ learning as a result of the mirror classes?” It was found:
(9218:9281)-D1: Students have acquired greater understanding of multiplication and division, they
have implemented the use of concrete material and its graphic representation in problem solving; in
addition, most of them are now more participative ...
(10006:10076)-D2: The children did not participate much now with the mirror classes I see that
they have become more participatory, they have not been so afraid to participate, they have realized
that when they participated if for A or B they made a mistake, they were given feedback and they
understood ...
Finally, in response to the question: “Do you consider that mirror classes are a good strategy for
your students’ learning?” We obtained:
(9218:9281)-D1: I think so, because it has allowed most of the children to learn comprehensively
about multiplication and division through various strategies, in addition to interacting and learning
from other children …
(10006:10076)-D2: Of course, it is, because they have learned and interacted with other children.
Now, with respect to the questions addressed to the parents: “Do you consider that the mirror classes helped
your child feel more motivated to learn about multiplication and division operations? Why?” It was found:
(10386:10487)-P1: Yes, she has been more motivated because when she interacts with her friends,
she wanted to find out one thing, another thing ...
(10769:10864)-P2: Yes, he has been more motivated because they had a little competition among his
classmates ...
(10868:10951)-P3: Yes, he always wanted to answer when the teachers asked ....
(11116:11195)-P4: I could see that when there were mirror classes, he was well motivated there.
(11199:11270)-P5: Yes, teacher, because he begins to let go and meet friends, he interacts ...
(11199:11270)-P6: Yes, he is more motivated ...
(12008:12067)-P7: She told me mom, I finished my homework and she was happy ...
(12071:12190)-P8: Yes, teacher, because as all the children have been participating in both
classrooms, she also wanted to be there.
On the other hand, in relation to the question: “Do you consider that the teacher accompaniment in
the mirror classes has contributed to the development of mathematical skills in your child? Why?” It was
evidenced:
(12388:12462)-P1: They learned more with the mirror classes, it seems to me that they have given
more mathematics ...
(12702:12798)-P2: Yes, they have developed very well because they have given the opportunity to
participate several times ...
(13194:13258)-P3: Yes, my son has been able to understand ...
(13194:13258)-P4: Yes, because I watched my son solve his homework by himself ...
(13415:13472)-P5: My child has started to interact and understand the classes more ...
(13633:13755)-P6: When she has needed her teacher, she has always been there ...
(2572:2762)-P7: I think so ...
(13980:14062)-P8: Yes, my little girl has improved in mathematics, she solves her homework and I
see that she understands.
On the other hand, in relation to the question: “What improvements have you observed in your
child’s learning as a result of the mirror classes?” It was identified:

2243
(14172:14220)-P1: She likes to participate, to give her opinion, she is more confident ...
(14374:14413)-P2: In trying to solve better more calmly ...
(15779:15922)-P3: Well in what I have seen that my little daughter has achieved is that she can
understand better, she can also let go, express herself and can solve problems ...
(16089:16204)-P4: She has improved in solving alone the tasks that are left for her because she
understands and I see that she likes math more ...
(16208:16407)-P5: In doing her homework, she finished her class and already had her homework
ready. The other thing they have learned is to express themselves, not 100%, I could say 70% about
emotions and to give opinions ...
(17177:17256)-P6: It is good that mirror classes are held, it would be good in other areas as well".
Finally, in relation to the question: “What is your opinion about mirror classes?” We obtained:
(17177:17256)-P1: It is good that mirror classes are held, it would be good in other areas as well.
(17729:17810)-P2: Yes, it is good, teacher, that they interact with other children and adapt to
changes ...
(17814: 17887)-P3: Yes, it is good, but in my opinion virtually is not the same as face-to-face ...
(18137:18245)-P4: In case of virtual classes, I think they should continue with mirror classes, this
way as they have done ...
(18249: 18297)-P5: I would like them to continue with mirror classes ...
(18413:18500)-P6: I think they should continue with mirror classes, my little daughter has improved
in mathematics ...
(18630:18726)-P7: Personally, I have seen it well ...
(18730:18830)-P8: The mirror classes are good, my little daughter could interact with other
children and solved her problems on her own …
(3918:4198)-DC1: The students represented multiplications in different ways, using rulers, in
various situations ...
(3148:3260)-DC2: Students solved division problems using the technique of division between one
and two digits in the quotient ...
(9063:9185)-DC3: Students solved combination problems from graphs, double-entry tables and
explained how they solved the combination in each case.
Otherwise, the results obtained in the diagnostic and exit test applied to the participating students are
presented in Table 1. As shown in the table, the results obtained in the exit test corroborate that the mirror
classes as a pedagogical strategy favored the optimization of the solution of quantity problems competence in
basic education students in Lima, Peru. The mirror classes have allowed students to reflect on ways of
solving mathematical problems expressed collaboratively.
Table 1. Achievement level of the solution of quantity problems competence
Achievement level Diagnostic test Exit test
A (achieved) 13 (33%) 38 (95%)
B (process) 24 (60%) 2 (5%)
C (beginning) 3 (7%) 0 (0%)
Total 40 (100%)
4. CONCLUSION
The application of the mirror class as a pedagogical strategy optimizes the development of the
solution of quantity problems competence in basic education students because it provides spaces for
interaction between students and teachers that generate an exchange of knowledge and reflection on the
pedagogical work among peers. Going through different ways of representing a problem allows for a better
understanding of mathematical concepts in students because these concepts become more concrete. In this
sense, the use of didactic materials favors the understanding of mathematical concepts by allowing the
student to manipulate them in a concrete way. Teachers should give students the opportunity to express their
ideas or explain how they have solved the problem, thus contributing to the development of the argument of
statements about numerical relationships and operations ability.

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2244
ACKNOWLEDGEMENTS
The research was financed by the Competitive Funds of the Norbert Wiener Private University
(UPNW/N°. 859-2021).
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BIOGRAPHIES OF AUTHORS
Maruja Dionisia Baldeón De La Cruz has a doctorate in Education, and is a
professor at the Norbert Wiener University Graduate School. She develops studies related to
the achievement of mathematical skills. She can be contacted at email:
maruja.baldeon@uwiener.edu.pe.
Melba Rita Vasquez Tomás is a doctor in Education, professor at the Norbert
Wiener University Graduate School. She develops studies related to the development of the
environment and educational improvement. She can be contacted at email:
melba.vasquez@uwiener.edu.pe.
Judith Soledad Yangali Vicente is a Ph.D. and postdoctoral fellow in education.
She is the Director of the research development area of the Norbert Wiener Private University.
She can be contacted at email: judith.yangali@uwiener.edu.pe.
Jhon Holguin-Alvarez is an advisor for research projects at the Faculty of Law
and Humanities of the César Vallejo University, Peru. Renacyt Researcher qualified by the
National Council of Science, Technology and Technological Innovation of Peru (Concytec);
He is a Doctor of Education and researches topics related to gamified learning, and the
development of humanistic skills, proactivity, and prosociality. He can be contacted at email:
jholguin@ucv.edu.pe or jhonholguinalvarez@gmail.com.

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