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Incidence of GeoGebra software in the
teaching-learning process on the derivative in
the Second Year of Unified General
Baccalaureate
Incidencia del software GeoGebra en el proceso de
enseñanza-aprendizaje de la derivada en el segundo año
de Bacherato General Unificado
José Luis Gallo-Calero
Ministerio de Educación del Ecuador, Quito, Ecuador
jose.galloc@educacion.gob.ec
https://orcid.org/0009-0001-0599-8805
Andrés Almeida-Flores
Ministerio de Educación del Ecuador, Quito, Ecuador
andres.almeidaf@educacion.gob.ec
https://orcid.org/0009-0004-2100-2723
Diego Zavala-Urquizo
Universidad Central del Ecuador, Quito, Ecuador
Facultad de Filosofía, Letras y Ciencias de la Educación, Carrera de Pedagogía de las
Ciencias Experimentales Matemática y Física
dzavala@uce.edu.ec
https://orcid.org/0000-0003-4883-922X
Edwin Vinicio Lozano
Universidad Central del Ecuador, Quito, Ecuador
Facultad de Filosofía, Letras y Ciencias de la Educación, Carrera de Pedagogía de las
Ciencias Experimentales Matemática y Física
elozano@uce.edu.ec
https://orcid.org/0000-0003-1167-4361
(Received on: 12/02/2025; Accepted on: 1/04/2025; Final version received on: 15/12/2025)
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Suggested citation: Gallo-Calero, J.L., Almeida-Flores, A., Zavala-Urquizo, D., y Lozano, E.V.
(2026). Incidence of GeoGebra software in the teaching-learning process on the derivative
in the Second Year of Unified General Baccalaureate. Revista Cátedra. 9(1), 71-89.
Abstract
This article presents a study on the use of the free software GeoGebra in the teaching and
learning process of mathematics, focusing on differentiation, with the aim of demonstrating
its impact on second-year students of the Unified General Baccalaureate at the "Juan
Wisneth" municipal school. This is particularly relevant given that in Ecuador, the
educational methodology is traditional and minimally oriented towards the digital realm.
The students were divided into two groups: the first group was introduced to the program
using a didactic guide, while the second group continued with the established academic
curriculum provided by the institution. The research is quasi-experimental with a
quantitative approach. Data was collected using three instruments: a diagnostic assessment
(before the intervention), a formative assessment (during the intervention), and a
summative assessment (at the end). Furthermore, this research is part of a socio-
educational project with a descriptive level of detail. This finding demonstrates that the use
of GeoGebra enhances student learning, as evidenced by higher scores among those who
used the software. Consequently, the impact of the digital age on mathematics, particularly
in the study of derivatives, encourages institutions to use free software for improved
learning.
Keywords
Mathematics, derivatives, software, GeoGebra, academic performance.
Resumen
Este artículo presenta el estudio sobre el uso del software libre denominado GeoGebra, en
el proceso de enseñanza-aprendizaje de la Matemática, centralizada en el campo de la
derivación, con la finalidad de evidenciar la incidencia de la misma en los estudiantes del
segundo año de Bachillerato General Unificado del colegio m, puesto
que, en Ecuador existe una metodología tradicional y mínimamente orientada al ámbito
digital dentro de la educación. Para esto, el estudiantado fue dividido en dos grupos: el
primero fue incluido al programa mediante una guía didáctica, mientras que el segundo
continuó con el pénsum académico establecido y otorgado por la institución. La
investigación es de tipo cuasiexperimental con enfoque cuantitativo. De igual manera, para
la recolección de datos se emplearon tres instrumentos: evaluación diagnóstica (antes de la
intervención), evaluación formativa (durante); y, evaluación sumativa (final). Asimismo, la
modalidad de investigación forma parte de un proyecto socioeducativo con un nivel de
profundidad descriptiva. Este hallazgo evidencia que la utilización de GeoGebra favorece en
la enseñanza-aprendizaje de los estudiantes al mostrar calificaciones más altas en aquellos
que utilizaron el software. Por consecuente, la implicación de la era digital en la Matemática,
específicamente al tratar el tema de la derivada, favorece a las instituciones el uso de
software libres para un mejor aprendizaje.
Palabras clave
Matemática, derivadas, software, GeoGebra, rendimiento académico.
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1. Introduction
In order to provide better education in both public and private schools, this article presents
and explains the impact of GeoGebra software as a technological, technical, and strategic
tool for teachers to generate effective, agile, and engaging learning experiences for students,
achieving favorable results in their understanding of various mathematical topics. This
research stems from postgraduate studies. To this end, a guide on the derivative was
developed, reviewed, and validated using the online GeoGebra software, known to the
students participating in this research as the "Didactic Guide to the Derivative."
Additionally, diagnostic, formative, and summative assessment tools were used to
quantitatively determine the acceptance or rejection of this educational resource.
In this context, teachers play a fundamental role in ensuring quality education through the
continuous updating of their knowledge and the strengthening of their digital skills. Mastery
of Information and Communication Technologies (ICTs) thus becomes an essential
condition for designing relevant and innovative learning experiences. This reflects the
characteristics of new generations of students, who develop their cognitive and social skills
in digital environments, demonstrating a high level of familiarity with the use of
technological tools for knowledge construction.
In this way, society can demand that teachers, students, and citizens in general have the
capacity to solve problems and face new challenges, offering timely solutions that
contribute to its development. In this sense, within the academic sphere:
New educational needs arise from the transformations taking place in
society; and it is here that the great challenges of the 21st century become
apparent. These impending changes are related to education, particularly
to the different teaching methods employed by teachers and the learning
situations that arise in the learning environment. (Olivo and Corrales,
2020, pp. 8-9).
Thus, the multiple needs faced by students, teachers, and the education system in general
become evident. One of these is the digital age, where the teacher must be a guide and the
student the primary builder of knowledge. However, the development of subjects through
a blackboard, a textbook, or a notebook is still prevalent, and there is no focus on innovating
new teaching strategies that are geared toward a more active and participatory
methodology.
This study was conducted at the Wisneth Municipal School, in the second year of the Unified
General Baccalaureate (BGU), with the aim of demonstrating the impact of the GeoGebra
program in the following contexts
1.1 Needs of contemporary education
ICTs have become essential tools for supporting teaching and learning processes. Therefore,
ICTs in education is definitive, and thus it is necessary to change methodological practices,
63). This change implies significant opportunities and challenges in the development of new
teaching and learning skills for teachers and students both inside and outside the classroom.
However, the implementation of digital competence in education depends heavily on the
resources available to the educational institution and how teachers use these resources. In
and student to build a bridge between intuitive ideas and formal mathematical concepts,
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providing an appropriate learning environment that involves knowledge, pedagogical
).
1.2 Needs for teacher training in Mathematics
Teaching mathematics has become the biggest challenge for some teachers, such as Álvarez
et al., who state that, according to the Ministry of Education's 2016 guidelines, this teaching
revolves around students being able to reason, think, relate, and apply mathematical
knowledge and premises to everyday life situations (Álvarez et al., 2020, p. 213). In other
words, learning mathematics becomes difficult due to the complexity, precision, and
abstraction of the content covered in class.
Similarly, according to Ayil, the creation of innovative virtual environments has become
necessary in current technological development so that students can actively participate in
their learning (Ayil, 2018, p. 36). Therefore, innovation in mathematics teaching must be
dynamic, ensuring that students have a more active role, where the resources used capture
their attention, motivating them and generating interest in acquiring knowledge and
mastering skills, thus transforming a large part of traditional teaching spaces.
1.3 Needs for the teaching of Mathematics
The difficulties involved in understanding concepts, analyzing, and solving mathematical
problems on a blackboard or in a notebook are numerous. Since it is difficult to grasp and,
above all, to master certain skills, the subject becomes tedious and boring. Holguín et al.
(2020) mention that "mathematics is considered one of the most complex subjects in the
academic curriculum, which is reflected in high failure rates. For this reason, new strategies
are being used to improve the teaching and learning method" (p. 72). One of the difficulties
in the teaching and learning process of defining and developing calculus, specifically the
topic of the derivative, is that there is no single way to represent it, as there are many
methods, such as graphical, algebraic, or numerical.
Based on the above, the aim of improving the teaching and learning processes of
mathematics, specifically in the area of the derivative, is framed within the implementation
of GeoGebra software as a teaching resource. To achieve this, a dynamic approach and
strategies were developed to capture students' attention, employing the words of Blázquez
et al., who state that motivation plays a significant role in prospective memorythe ability
to remember what needs to be done at the precise moment (Blázquez et al., 2008).
Consequently, if teachers aspire to achieve good results in the teaching and learning process
of derivatives, they must first awaken students' curiosity, interest, and motivation through
various didactic or technological resources, depending on their needs.
Finally, this application aims to contribute to overcoming the difficulties present in the
teaching and learning process, considering that the software is a beneficial tool for teachers,
students, and the entire educational community. Beyond achieving the understanding and
acquisition of a mathematical concept essential for students at higher levels of education,
the goal is to foster their interest and motivation, making the most of tools with which they
feel comfortable and which are new to them. This allows for more in-depth teaching,
optimizing time and enabling students to develop useful cognitive skills in both the school
and social environment.
2. Methodology
This research is based on the following methods, methodology, techniques, and
instruments:
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2.1 Research Approach
The research employed a quantitative approach. Hernández et al. state that the
measurement and statistical analysis, in order to establish patterns of behavior and test
set of processes organized sequentially
to verify certain assumptions, starting with a defined idea, progressing through additional
processes, and culminating in the presentation of the results report.
2.2 Level of research
The research focused on a descriptive level. On the one hand, Guevara et al. state that
on students is obtained.
On the other hand, Hernández and Mendoza maintain that the main function of the study is
to specify the characteristics, properties, and profiles of communities, groups, objects, or
any phenomenon (Hernández and Mendoza, 2018, p. 108). This scope allows for the
collection and measurement of data on the variables initially identified, with the possibility
of predicting an event in a rudimentary way, provided that the theoretical foundations and
background information are well established.
2.3 Type of research
The design of a research study is based on the steps, procedures, and strategies that must
be followed to address the research according to the model adopted for controlling
variables. Three types were used: documentary, field, and experimental, focusing on a
quasi-experimental design.
On the one hand, documentary research, according to Muñoz (2015), is that which deals
are generally theoretical, abstract, and
other hand, Hernández et al. state that field research consists of studies carried out in a
realistic situation, in which the researcher manipulates one or more independent variables
under carefully controlled conditions (Hernández et al., 2014, p. 150). Thus, this type of
research allows for the recording and control of data with the support of evaluations or
other data collection instruments, in order to facilitate information management.
The research
characteristic is to quantitatively verify the causality of one variable on another; this implies
is needed,
-
experimental type, one that manages the experimental and control groups. This design is
used when it is not possible to use subjects randomly, so they are already pre-selected.
2.4 Population and sample
According to Mejía, the population is the totality of elements or individuals that comprise
the study, delimited by the researcher according to the parameters established in the study
(Mejía, 2015, p. 95). Therefore, the research involved a population of 61 second-year
school. These students were divided into two sections: the experimental group of 30
students, belonging to the first section, to whom the proposed didactic guide was applied,
and the second, control group of 31 students who were not subjected to the same guide.
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The population coincides with the sample, since it is a specific educational institution where
the hypothesis is to be tested or rejected. For the aforementioned reasons, a non-
probabilistic convenience sampling method was used. Regarding the first point, Arias et al.
state that sampling is used when the population is very small or less than 100 individuals,
and the population is chosen directly based on shared characteristics or a biased judgment
on the part of the researcher (Arias et al., 2021, p. 115). As for the second point, convenience
convenient for the research, for the sample; this convenience arises because it is easier for
).
2.5 Research Technique
during the teaching and learning process provide information that allows for evaluation;
however, sometimes it is necessary to apply tests in order to evaluate specific elements and
technique allows us to measure the level of learning achieved by a student in a given content
area or topic to determine whether the teaching guide benefits or hinders student academic
performance.
2.6 Instrument and validity
The questionnaire was used as the instrument. Hernández and Mendoza define a
questionnaire as a data collection instrument used in scientific research, consisting of
questions administered to a sample or population (Hernández & Mendoza, 2018, p. 250). In
this application, three questionnaires were administered for diagnostic, formative, and
summative assessment. Each questionnaire consisted of 10 questions with structured
items.
instrument truly measures the variable it seeks to measure. It is achieved when it is
demonstrated that the instrument reflects the abstract concept through its empirical
-
designed and intended to have sound content, criteria, and construct. Based on the above,
the assessment instruments were reviewed and approved by three experts in the field.
2.7 Reliability
consi
aims to ensure consistency in the instruments' methodology and the population to which
they are applied, resulting in similar data or results. All students taking the assessments
should be on equal footing. To determine the reliability of the three assessments, pilot tests
were administered to 15 randomly selected third-year high school students. It is
recommended that pilot testing be conducted with a class of the same or higher grade level,
and that students have recently covered the topic of derivatives. Once the assessments were
administered, data tabulation and the calculation of Cronbach's alpha for each instrument
began. The following reliability results were obtained:
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Scale
Levels
Less than 0.200
Very low reliability
De 0.210 a 0.400
Low reliability
De 0.410 a 0.600
Regular reliability
De 0.610 a 0.800
Acceptable reliability
De 8.210 a 1.000
High reliability
Table 1. Cronbach's Alpha
Assessment instruments
Reliability coefficient
Levels
Diagnostic
0.891
Confiabilidad elevada
Training
0.954
Confiabilidad elevada
Summative
0.905
Confiabilidad elevada
Table 2. Results obtained from Cronbach's Alpha in the assessment instruments
Once the results of Cronbach's alpha were observed using the Kuder-Richardson method, it
was concluded that the three instruments have high reliability, according to the scale
proposed by Hernández and Mendoza, and can be applied to the students of the
experimental and control groups of the institution.
3. Results
Within the statistical analysis of the instruments applied to the students, the results were
tabulated and organized; the descriptive measures were analyzed in terms of frequency
distribution, percentages, arithmetic means, mean, mode, standard deviation and advanced.
3.1 Diagnostic assessment
Within this category, the type and level of students' knowledge were established before the
research process began. As Vera (2020) states, diagnostic t
beginning or end of the course to compare students' knowledge, that is, to understand the
before and after of the teaching-
were not issued, since this type of assessment serves to analyze students' responses and
their level of understanding and knowledge of the topic. Similarly, the following
nomenclature was used for the statistical analysis:
σ: Standard deviation
: Arithmetic mean.
n: Total number of data points
f: Sum of the frequencies.
: Sum of the product of the scores and the frequency.
The diagnostic instrument consisted of ten multiple-choice questions. The test was based
on prior knowledge from lower grades and the current grade. The test was administered in
person using a printed copy. The second-year students of the Unified General Baccalaureate
were divided into two groups. The first group, consisting of 30 students (experimental
group), will be referred to as the group that used the instructional guide; the second group,
consisting of 31 students (control group), did not use the GeoGebra program.
Below are the tables for both the experimental and control groups, showing scores, absolute
frequencies, and other data that allow for interpretation and provide insight into the
academic level at which the students in the experimental and control groups began the test.
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Ratings
Absolute
frequency
Product
xi2
fixi2
1
0
0
1
0
2
1
2
4
4
3
3
9
9
27
4
7
28
16
112
5
6
30
25
150
6
4
24
36
144
7
2
14
49
98
8
5
40
64
320
9
2
18
81
162
10
0
0
100
0
Total
30
165
1017
Table 3. Record of the diagnostic evaluation of the experimental group
Ratings
Absolute
frequency
Product
xi2
fixi2
1
0
0
1
0
2
0
0
4
0
3
4
12
9
36
4
5
20
16
80
5
8
40
25
200
6
3
18
36
108
7
6
42
49
294
8
3
24
64
192
9
2
18
81
162
10
0
0
100
0
Total
31
174
1017
Table 4. Record of the diagnostic evaluation of the control group
As shown in Table 3, a total of 30 students participated in the experimental group and were
evaluated on a scale of 1 to 10 points. No student obtained the maximum score; however, 9
students scored higher than 7, meaning that 30% of the students achieved the learning
objectives. This implies that 70% did not. These results were expected, given that this was
a diagnostic assessment and no intervention had yet been implemented with the group.
In Table 4, 31 students participated and were evaluated on a scale of 1 to 10 points. No
student obtained the maximum score; however, 35.48% of them scored 7 or higher.
Therefore, it is understood that 64.52% did not achieve the learning objectives. These
results are not very high; however, it should be noted that, as this is a diagnostic assessment,
few students are engaged.
3.1.1 Calculation of the arithmetic mean
Formula used in calculating the arithmetic mean of the experimental group with its
respective replacement:
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Equation 1
Formula used in calculating the arithmetic mean of the control group with its respective
substitution:
Ecuación 2
3.1.2 Calculation of the standard deviation
Formula used in the calculation of the standard deviation of the experimental group with its
respective replacement:
Equation 3
Formula used in calculating the standard deviation of the control group with its respective
substitution:
Equation 4
As shown in Figure 1, the control group obtained an average score of 5.61 out of 10, while
the experimental group obtained 5.50. These results are within the normal range, as they
were obtained at the beginning of the study before the intervention. Furthermore, the
standard deviation of the control group reflects that the scores are less dispersed compared
to those of the experimental group.
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Figure 1. Statistical data analysis of the diagnostic evaluation.
It can be said that both the experimental and control groups are in a very similar situation,
tending to be below average and mostly failing to achieve the required learning outcomes.
3.2 Formative assessment
Formative assessment, according to Mellado
and personal search for the evolution that each subject has experienced thanks to the
improvement of learning processes. Therefore, the test was developed with a structured
format and ten multiple-choice questions, each worth one point for a correct answer. The
topics covered included the definition of the derivative, derivatives of common functions,
and trigonometric derivatives. However, the instrument was administered virtually on the
CEVIM platform, Moodle, which is used by the municipal schools.
The data obtained from both the experimental and control groups are recorded in the
following tables, which include scores, absolute frequencies, and other data necessary to
interpret the data and understand the progress made by the groups during the intervention
of the teaching guide on derivatives using GeoGebra.
Ratings
Absolute
frequency
Product
xi2
fixi2
1
0
0
1
0
2
0
0
4
0
3
0
0
9
0
4
0
0
16
0
5
2
10
25
50
6
3
18
36
108
7
6
42
49
294
8
6
48
64
384
9
7
63
81
567
10
6
60
100
600
Total
30
241
2003
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Table 5. Formative assessment record of the experimental group
Ratings
Absolute
frequency
Product
xi2
fixi2
1
0
0
1
0
2
0
0
4
0
3
2
6
9
18
4
2
8
16
32
5
3
15
25
75
6
3
18
36
108
7
4
28
49
196
8
7
56
64
448
9
6
54
81
486
10
4
40
100
400
Total
31
225
1763
Table 6. Record of the formative assessment of the control group
On the one hand, Table 5 revealed encouraging data, as 25 students scored 7 or higher, with
only 5 students neither achieving nor mastering the learning objectives. Furthermore, 20%
of students obtained the maximum score of 10 points, and the most frequent score was 9,
with 7 students achieving 9 points. Therefore, it is evident that the experimental group has
made significant progress since the implementation of the teaching guide.
On the other hand, Table 6 shows that 21 students scored 7 or higher, while 10 students
have not yet achieved the learning objectives, with scores of 3 and 4. The most frequent
score was 8, achieved by 7 students. It should be mentioned that in the control group,
12.90% achieved the maximum score of 10. This demonstrates a very noticeable
improvement compared to the diagnostic evaluation.
3.2.1 Calculation of the arithmetic mean
Formula used in calculating the arithmetic mean of the experimental group with its
respective replacement:
Equation 5
Formula used in calculating the arithmetic mean of the control group with its respective
substitution:
Equation 6
3.2.2 Calculation of the standard deviation
Formula used in the calculation of the standard deviation of the experimental group with its
respective replacement:
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Equation 7
Formula used in calculating the standard deviation of the control group with its respective
substitution:
Equation 8
Analysis of the formative assessment data shows that the experimental group has an
average score of 8.03, while the control group has an average score of 7.26, both scores
being out of 10 points. In this respect, both groups achieved the learning objectives;
however, the standard deviation of the experimental group is 1.49, which is lower than that
of the control group, which stands at 2.05.
3.3 Summative assessment
through
the gathering of evidence with a fundamentally accrediting and operational function of
standardized, universal, and procedural. The instrument consisted of 10 structured
questions, and the topics were presented cumulatively. Among the topics reviewed were:
derivatives of common functions, trigonometric functions, derivatives using the chain rule,
and derivatives of the addition, subtraction, multiplication, and quotient of functions. It was
administered virtually for the reason mentioned above.
Ratingss
Absolute
frequency
Product
xi2
fixi2
1
0
0
1
0
2
0
0
4
0
3
0
0
9
0
4
1
4
16
16
5
3
15
25
75
6
6
36
36
216
7
5
35
49
245
8
5
40
64
320
9
4
36
81
324
10
6
60
100
600
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Total
30
226
1796
Table 7. Record of the summative evaluation of the experimental group
Ratings
Absolute
frequency
Product
xi2
fixi2
1
0
0
1
0
2
0
0
4
0
3
8
24
9
72
4
5
20
16
80
5
6
30
25
150
6
3
18
36
108
7
2
14
49
98
8
7
56
64
448
9
0
0
81
0
10
0
0
100
0
Total
31
162
956
Table 8. Record of the summative assessment of the control group
Table 7 shows the grades obtained by the students in the experimental group, in which
66.67% scored 7 or higher, with a total of 20 students achieving the learning objectives.
Thus, 33.33% of students scored between 4 and 6 out of 10. It is noteworthy that the highest
grade, 10, was the most frequent, and the median grade was 7 out of 10. Although the
average score in the summative assessment was lower than that in the formative
assessment, a high percentage of students still achieved the learning objectives.
Table 8 shows that within the control group, only 29% of students achieved the learning
objectives; consequently, 71% of students had grades below 7. Furthermore, the most
frequent grade was 3 out of 10, with a total of 8 students achieving this score, and the
average grade was 5 out of 10.
3.3.1 Calculation of the arithmetic mean
Formula used in calculating the arithmetic mean of the experimental group with its
respective replacement:
Equation 9
Formula used in calculating the arithmetic mean of the control group with its respective
substitution:
Equation 10
3.3.2 Calculation of the standard deviation
Formula used in the calculation of the standard deviation of the experimental group with its
respective replacement:
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Equation 11
Formula used in calculating the standard deviation of the control group with its respective
substitution:
Equation 12
The results show a significant difference in the average score between the two groups. The
experimental group had an average of 7.53, while the control group had an average of 5.23.
Therefore, the first group achieved the intended learning outcomes, while the second group
did not, as their score was below 7.
4. Analysis and discussion
In this section, the results collected in the study were analyzed and discussed. The
similarities and differences found between the experimental and control groups regarding
the teaching and learning of GeoGebra software were examined. Likewise, and due to the
circumstances of the country where this research was conducted, it was also discussed
whether or not the virtual environment affected the quality of instruction when subjected
to formative and summative assessments.
On the one hand, to test the hypothesis regarding the impact of GeoGebra software use (Hi)
and its lack thereof (Ho), it is necessary to extract the data from both assessments, including
both the arithmetic mean and the standard deviation for both groups. The following
mathematical language was used for this purpose:
Hi:
con
Equation 13
Ho:
Equation 14
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N°
Evaluations
Arithmetic mean
Standard deviation
1
Formativa
8.03
1.74
2
Sumativa
7.53
1.77
Overall average
7.78
1.755
Table 9. Statistical record of evaluations of the experimental group
N°
Evaluations
Arithmetic mean
Standard deviation
1
Formativa
7.26
2.05
2
Sumativa
5.23
1.88
Overall average
6.245
1.965
Table 10. Statistical record of control group evaluations
Table 9 shows an average score of 7.78 for both tests, which is higher than 7. It can be noted
that the students achieved the learning objectives with an average standard deviation of
1.77, demonstrating that the scores are not highly dispersed.
Meanwhile, Table 10 shows an average score of 6.245 for the two assessments, indicating
that the control group did not achieve the learning objectives, as their score was lower than
7 out of 10. Furthermore, they had a standard deviation of 1.965.
To determine critical values and rejection regions, it is taken into account that in the
calculation of the parametric Z test, the null hypothesis is rejected if:
Equation 15
Or also
Equation 16
Where Z_T is the theoretical value of Z for a significance level of 5%, =0.05; that is, the
research will have 95% reliability; otherwise, the research hypothesis is accepted with one
of the two alternatives. The corresponding mathematical language with its replacement is:
Equation 17
Once the theoretical bases have been detailed, the calculated parameterized Z test is found:
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Equation 18
Comparing the calculated Z value and the theoretical Z value, we understand that the former
is greater than the latter. That is to say:
Equation 19
Where Z_c=3.22 is outside the acceptance region of the null hypothesis, which leads us to
reject the null hypothesis Ho: (x_e )= (x_c ) and accept the research hypothesis Hi: (x_e )
(x_c ) with the alternative A_1:(x_e )> (x_C ). That is, in non-mathematical terms, the use of
GeoGebra software impacts the teaching-learning process of the Derivative in the second
On the other hand, since the tests were administered virtually, it is evident that the
experimental group, being in contact with the teaching guide, was not affected by the online
evaluation. However, the control group did not improve its results. This premise
demonstrates that teachers and students, when using GeoGebra software with the
accompanying online learning guide, develop a greater capacity to solve problems involving
derivatives and tackle new challenges. This is achieved not only through the GeoGebra
software itself, but also through engagement with the accompanying learning materials,
which help address social problems that may exist both nationally and internationally.
5. Conclusions
The use and application of the derivatives teaching guide using GeoGebra software
strengthened the understanding of formal mathematical concepts focused on derivatives.
Students who participated in the program achieved greater independent and collaborative
learning than those who did not. This resulted in the first group being more organized,
participative, and critical in their learning process.
The students in the experimental group showed considerable academic improvement
compared to the students who did not participate. This means the first group has greater
knowledge, which they can then apply in conversations with their classmates, in individual
tests on the same topic, and in everyday life. Promoting the use of the derivatives teaching
guide improves the teaching and learning process, meets the needs of contemporary
education by digitally integrating the educational environment with the free software.
Furthermore, it influences teacher training, as teachers transform the traditional learning
environment for students who learn primarily visually and interactively. Similarly, it fulfills
the need for teaching mathematics by offering different forms of representation, allowing
students to improve their ability to analyze and solve mathematical problems through both
in-person and digital learning experiences.
Bibliographic references
Álvarez, J., García, D., Erazo, C., & Erazo, J. (2020). GeoGebra as a mathematics teaching
strategy [GeoGebra como estrategia de enseñanza de la matemática]. Episteme
Koinonía, 3(6), 213226. https://doi.org/10.35381/e.k.v3i6.827
87
Licencia Creative Commons Atribución 4.0 Internacional (CC BY 4.0)
Revista Cátedra, 9(1), pp. 71-89, January-June 2026. e-ISSN: 2631-2875
https://doi.org/10.29166/catedra.v9i1.7849
Arias, J., & Covinos, M. (2021). Research design and methodology [Diseño y metodología de
la investigación]. Enfoques Consulting E.I.R.L.
Ayil, J. (2018). Virtual learning environment: A support tool for mathematics teaching
[Entorno virtual de aprendizaje: una herramienta de apoyo para la enseñanza de las
matemáticas]. Revista de Investigación en Tecnologías de la Información (RITI),
6(11), 3439. https://dialnet.unirioja.es/servlet/articulo?codigo=7107366
Blázquez, S., Ortega, T., Gatica, S., & Benegas, J. (2006). A conceptualization of limits for initial
learning of mathematical analysis at university level [Una conceptualización de límite
para el aprendizaje inicial de análisis matemático en la universidad]. Revista
Latinoamericana de Investigación en Matemática Educativa, 9(2), 189209.
https://www.redalyc.org/articulo.oa?id=33590202
Guevara, G., Verdesoto, A., & Castro, N. (2020). Educational research methodologies
(descriptive, experimental, participatory, and action research) [Metodologías de
investigación educativa (descriptivas, experimentales, participativas y de
investigación-acción)]. Recimundo, 4(3), 163173.
https://recimundo.com/index.php/es/article/view/860
Hernández, R., Fernández, C., & Baptista, M. (2014). Research methodology (6th ed.)
[Metodología de la investigación]. McGraw-Hill Education.
Hernández, R., & Mendoza, C. (2018). Research methodology: Quantitative, qualitative, and
mixed approaches [Metodología de la investigación: las rutas cuantitativa, cualitativa
y mixta]. McGraw-Hill Education.
Holguín, F., Holguín, E., & García, N. (2020). Gamification in mathematics teaching: A
systematic review [Gamificación en la enseñanza de las matemáticas: una revisión
sistemática]. Telos: Revista de Estudios Interdisciplinarios en Ciencias Sociales, 22(1),
6275. https://doi.org/10.36390/telos221.05
Mejía, E. (2005). Research techniques and instruments [Técnicas e instrumentos de
investigación]. Universidad Nacional Mayor de San Marcos.
Mellado, P., Sánchez, P., & Blanco, M. (2021). Trends in formative and summative student
assessment in Web of Science [Tendencias de la evaluación formativa y sumativa del
alumnado en Web of Science]. Alteridad, 16(2), 170185.
https://scielo.senescyt.gob.ec/pdf/alteridad/v16n2/1390-325X-alt-16-02-
00170.pdf
Muñoz, R. (2015). How to develop and supervise a thesis research project [Cómo elaborar y
asesorar una investigación de tesis]. Pearson.
Olivo, J., & Corrales, J. (2020). From virtual learning environments: Toward a new praxis in
mathematics teaching [De los entornos virtuales de aprendizaje: hacia una nueva
praxis en la enseñanza de la matemática]. Revista Andina de Educación, 3(1), 819.
https://doi.org/10.32719/26312816.2020.3.1.2
Parra, L., & Vázquez, M. (2017). Probability and non-probability sampling [Muestreo
probabilístico y no probabilístico]. Gestiopolis. https://www.gestiopolis.com/wp-
content/uploads/2017/02/muestreo-probabilistico-no-probabilistico-
guadalupe.pdf
Revelo, J., Lozano, E., & Bastidas, P. (2019). Teaching digital competence and its impact on
the mathematics teaching–learning process [La competencia digital docente y su
88
Licencia Creative Commons Atribución 4.0 Internacional (CC BY 4.0)
Revista Cátedra, 9(1), pp. 71-89, January-June 2026. e-ISSN: 2631-2875
https://doi.org/10.29166/catedra.v9i1.7849
impacto en el proceso de enseñanzaaprendizaje de la matemática]. Espirales, 3(28),
156175. https://dialnet.unirioja.es/servlet/articulo?codigo=8466473
Vera, F. (2020, August). The importance of the teaching–learning process and diagnostic
assessment [La importancia del proceso de enseñanza-aprendizaje y la evaluación
diagnóstica]. Atlante: Cuadernos de Educación y Desarrollo.
https://www.eumed.net/rev/atlante/2020/08/evaluacion-diagnostica.html
Authors
José Luis Gallo-Calero earned his Master's degree in Education, specializing in
Mathematics, from the Central University of Ecuador (Ecuador) in 2023. He obtained a
Specialist degree in Educational Quality Management from the Andean University Simón
Bolívar (Ecuador) in 2021. He earned his Bachelor's degree in Mathematics and Physics
Education from the Central University of Ecuador (Ecuador) in 2018. He currently teaches
in the Pedagogy of Experimental Sciences (Mathematics and Physics) program at the
Faculty of Philosophy, Letters, and Educational Sciences of the Central University of
Ecuador. He is the General Coordinator of the Higher Education Access Project (PAES). His
main research interests focus on innovation, strategies, and methodological guidelines. He
is a tenured teacher with the Ministry of Education of Ecuador.
Andrés Almeida-Flores: obtained his Master's degree in Education, specializing in
Mathematics, from the Central University of Ecuador (Ecuador) in 2023. He obtained his
Bachelor's degree in Mathematics and Physics Education from the Central University of
Ecuador (Ecuador) in 2018. He currently teaches in the Pedagogy of Experimental Sciences
(Mathematics and Physics) program at the Faculty of Philosophy, Letters, and Educational
Sciences of the Central University of Ecuador. He is a tutor for the Systematization of
Experiences in Rural Pedagogical Research and/or Intervention Practice. His main research
topics focus on innovation and the development of virtual tools for teaching mathematics.
He is a tenured teacher with the Ministry of Education of Ecuador.
Diego Zavala-Urquizo: He received his PhD in Education from Andrés Bello Catholic
University (Venezuela) in 2020. He received his Bachelor's degree in Basic Education from
Metropolitan University (Ecuador) in 2022. He received his Bachelor's degree in Business
Administration from Central University of Ecuador (Ecuador) in 2015. He received his
Master's degree in Systems Management from the Army Polytechnic School (Ecuador) in
2013. He received his Bachelor's degree in Business Administration from Central University
of Ecuador (Ecuador) in 2003.
He currently teaches in the Pedagogy of Experimental Sciences (Mathematics and Physics)
program at the Faculty of Philosophy, Letters, and Educational Sciences of Central
University of Ecuador. His main research interests focus on educational innovation and the
use of technology in the classroom.
Edwin Vinicio Lozano: He obtained his Master's degree in University Teaching and
Educational Administration from Indoamérica University (Ecuador) in 2004. He obtained
his Doctorate in Educational Psychology and Guidance from the Central University of
Ecuador (Ecuador) in 2000. He obtained his Bachelor's degree in Educational Sciences,
specializing in Educational Psychology and Guidance, from the Central University of
Ecuador (Ecuador) in 1997. He obtained his Primary Education Teaching Certificate from
the Alfredo Pérez Guerrero Higher Normal Institute (Ecuador) in 1992. He is a doctoral
candidate in Education at the National University of Rosario (Argentina).
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He currently teaches in the Pedagogy of Experimental Sciences (Mathematics and Physics)
program and at the Postgraduate Institute of the Faculty of Philosophy, Letters, and
Educational Sciences at the Central University of Ecuador. He is the coordinator of the
Psychopedagogical Area and the Coordinator of the Graduation Unit. His main research
topics focus on learning theories, psychopedagogy, educational innovation, and teaching
strategies and techniques.
Declaration of authorship-CRediT
José Luis Gallo-Calero: conceptualization, methodology, validation, formal analysis,
research, data curation and analysis, visualization, related concepts, final draft.
Andrés Almeida-Flores: related concepts, methodology, validation, research, organization
and integration of collected data, conclusions, supervision, first draft, revision, and editing.
Diego Zavala-Urquizo: related concepts, validation, formal analysis, research, organization
and integration of collected data, supervision, first draft, and editing.
Edwin Vinicio Lozano: related concepts, methodology, validation, research, organization
and integration of collected data, conclusions, supervision, first draft, revision, and editing.
Declaration of the use of artificial intelligence
The authors declare that they did not use Artificial Intelligence (AI) tools for any part of the
manuscript. No part of the scientific content, results, analyses, or interpretations was
generated by artificial intelligence. All material was reviewed and validated by the authors,
who are responsible for its accuracy and rigor.
