|
Article |
Metodologías activas en la enseñanza de la ecuación de
la recta y sus elementos para estudiantes de bachillerato
Ricardo Florencio Yoza
Lucas[*]
Franklin Iván Chaguancalle
Torres*
Catalina Vitelia Vargas
Pérez*
John Fernando Granados Romero*
Abstract
The research was developed with the purpose of
evaluating the effectiveness of active methodologies in the teaching of the
equation of the straight line and its elements for high school students, given
the recurrent problem of low academic performance and lack of motivation in the
learning of abstract mathematical concepts. For this purpose, a
quasi-experimental design with a mixed approach was implemented, where the
performance of an experimental group, exposed to innovative pedagogical
strategies through the PhET platform, was compared
with a control group that received traditional instruction. The sample included
first year high school students from a specific educational unit, using data
collection instruments such as questionnaires with Likert scales, semi-structured
interviews and systematic observations, whose results were analyzed using
statistical and qualitative software. The results showed that the
implementation of active methodologies, focused on collaborative learning and
contextualized problem solving, significantly enhanced conceptual
understanding, practical application skills and intrinsic motivation towards
mathematics in the experimental group, while the control group showed
limitations in the transfer of knowledge to real situations and less autonomy
in solving exercises. However, persistent challenges related to personalized
feedback and the availability of technological resources were identified,
factors that require attention to optimize the effectiveness of these
methodologies. Consequently, the study concluded that the integration of
interactive digital tools and student-centered pedagogical strategies
constitutes a promising approach to transform the teaching of complex
mathematical content, although its scalability depends on continuous teacher
training, the adequacy of educational infrastructure and the design of didactic
materials aligned with the specific needs of the student context.
Keywords: Active methodologies; equation of the straight line;
mathematics teaching; meaningful learning; baccalaureate.
Resumen
La investigación se desarrolló con el propósito de
evaluar la efectividad de las metodologías activas en la enseñanza de la
ecuación de la recta y sus elementos para estudiantes de bachillerato, ante la
problemática recurrente de bajo rendimiento académico y desmotivación en el
aprendizaje de conceptos matemáticos abstractos. Para ello, se implementó un
diseño cuasiexperimental con enfoque mixto, donde se comparó el desempeño de un
grupo experimental, expuesto a estrategias pedagógicas innovadoras mediante la
plataforma PhET, con un grupo de control que recibió instrucción tradicional.
La muestra incluyó estudiantes de segundo año de bachillerato de una unidad
educativa específica, utilizando instrumentos de recolección de datos como
cuestionarios con escalas Likert, entrevistas semiestructuradas y observaciones
sistemáticas, cuyos resultados se analizaron mediante software estadístico y
cualitativo. Los resultados evidenciaron que la implementación de metodologías
activas, centradas en el aprendizaje colaborativo y la resolución de problemas
contextualizados, potenció significativamente la comprensión conceptual, la
capacidad de aplicación práctica y la motivación intrínseca hacia las
matemáticas en el grupo experimental, mientras que el grupo de control mostró
limitaciones en la transferencia de conocimientos a situaciones reales y menor
autonomía en la resolución de ejercicios. Sin embargo, se identificaron
desafíos persistentes relacionados con la retroalimentación personalizada y la
disponibilidad de recursos tecnológicos, factores que requieren atención para
optimizar la eficacia de estas metodologías. En consecuencia, el estudio
concluyó que la integración de herramientas digitales interactivas y
estrategias pedagógicas centradas en el estudiante constituye un enfoque
promisorio para transformar la enseñanza de contenidos matemáticos complejos,
aunque su escalabilidad depende de la formación docente continua, la adecuación
de infraestructura educativa y el diseño de materiales didácticos alineados con
las necesidades específicas del contexto estudiantil.
Palabras clave: Metodologías
activas; ecuación de la recta; enseñanza de matemáticas; aprendizaje
significativo; bachillerato
Introduction
This article focuses on the implementation of active
methodologies in teaching the equation of a straight line and its elements to
second-year students of the General Unified Baccalaureate at the Alfredo Portaluppi Velásquez Public School during the 2024-2025
academic year. The main objective of this research is to facilitate the
understanding of these mathematical concepts through a didactic approach that
integrates active methodologies. This approach will allow for a comparison of
the impact of these methodologies with traditional teaching, with the aim of
evaluating their effect on mathematics education and meaningful learning among
students.
Globally, UNESCO recognizes the importance of promoting greater global
awareness and strengthening mathematics education to address the obstacles that
hinder the achievement of sustainable development (UNESCO, 2023). In this
context, it is recognized that mathematics education not only enhances critical
and analytical thinking, but also provides the tools necessary to address
complex problems and foster innovation in the search for sustainable solutions.
Mathematical education in developing countries requires special attention
for three fundamental reasons: first, a nation's economic strength depends on
the ability of its education system to produce mathematically literate workers
and consumers; second, the mathematics learning deficit in these countries is
considerable and shows few signs of diminishing; and finally, widespread
negative attitudes toward mathematics, coupled with an expectation of failure,
represent a significant barrier to progress (ProFuturo,
2024).
In this context, the results of the PISA 2022 assessment, conducted by
the Organization for Economic Cooperation and Development (OECD), highlight the
severity of the learning crisis facing adolescents in Latin America and the
Caribbean. With unprecedented participation from countries in the region, the
report reveals that 75% of 15-year-olds fail to demonstrate basic mathematical
skills. Furthermore, learning trends show no significant improvement in most of
the countries assessed (Inter-American Development Bank (IDB) and World Bank,
2024).
In Ecuador, mathematics education faces significant challenges. Despite
the progress reported in the Ser Estudiante 2023 assessment, which showed
improvement in various curricular areas, the alarming reality that 7 out of 10
students fail to reach the minimum level of competency highlights the urgency
of reforming current educational strategies (Enríquez, 2024). Therefore, it is
essential to implement innovative and effective methodologies that not only
foster interest in this discipline but also develop critical and
problem-solving skills in students, thus ensuring a more comprehensive and
quality education.
Traditionally, knowledge of the equation of a line and its elements has
been transmitted through narrative processes, where the teacher communicates
the content in a unidirectional manner. However, recent research suggests that
this strategy may not be the most appropriate for achieving meaningful and
lasting learning (Armijos and Serrano, 2024; Hidrobo, 2023). In this context,
there is a need to implement active methodologies, such as the use of
interactive simulations, project-based learning, and the flipped classroom,
which have proven to be effective tools for optimizing academic performance and
student participation (Mendez, 2023; Zumárraga et al., 2023). In addition, the
use of active methodologies at the secondary level has begun to attract the
attention of researchers. Studies such as that by Vale and Barbosa (2023)
suggest that incorporating these strategies can significantly enhance
intellectual performance, motivation, and interest in learning mathematics
among young people.
Internationally, various studies have explored the implementation of
active methodologies in teaching linear equations and related mathematical
concepts. For example, in Vietnam, Nguyen et al. (2022) conducted an
experimental study based on activity theory to teach linear equations in the
plane, using a pretest-posttest design with a control group in the Mo Cay
district of Ben Tre province. The results showed that students in the
experimental group, who received instruction through activity theory, achieved
better learning outcomes than those in the control group, who were taught using
traditional methods. In addition, there was a significant decrease in the
number of low-performing students, although the number of outstanding students
did not increase significantly.
Another study by Rodríguez et al. (2023) examined the application of
active methodologies in the classroom, focusing on secondary school students.
The findings indicate that these teaching strategies optimize school
performance, self-esteem, motivation, and student satisfaction. The results
indicate that students take on greater responsibility when they actively
participate in their learning.
Hidrobo's (2023) study investigates the impact of motivation on teaching
the equation of a straight line to tenth-grade students. This article
highlights the effectiveness of active methodologies, which encourage student
participation and achievement, facilitating the understanding of complex
mathematical concepts through skills such as group work and the use of
technological tools. Progress in motivation and academic performance is
observed, underscoring the need to transform education to make it more effective
and attractive to high school students.
In Ecuador, the study by Armijos and Serrano (2024) investigates
differentiated teaching of the equation of a straight line to high school
students in the Amazon region. This study emphasizes the importance of
designing active teaching strategies that encourage student participation and
responsibility, thereby improving their understanding of the subject. The
results suggest that the use of integrative approaches can be a valuable tool
for facilitating meaningful teaching in mathematics.
In this context, Candela and Rodríguez (2023) analyzed the impact of
active methodologies, particularly problem-based learning, on mathematics
learning in general basic education and upper basic sublevel students at the
Eloy Alfaro Educational Unit in Chone during the 2023 school year. The
inductive method was used to develop the research. The results revealed that
the application of active methodologies in curriculum programs and in the
classroom generates cognitive, social, and motivational advantages for
students, highlighting that these methodologies promote the construction of
meaningful knowledge, the development of relevant skills such as problem
solving and critical thinking, as well as greater motivation and commitment,
collaborative learning, and practical application of knowledge.
Among these methodologies, the PhET platform is presented as an
innovative teaching resource that allows students to interact with mathematical
concepts through visual and manipulative simulations. These simulations
facilitate understanding of the relationship between the elements of the
equation of a line, such as slope and intersection, as well as its graphical
representation, which can contribute to better use of the content (Yanchapaxi,
2023; Ávila, 2024; Gani et al., 2020).
The study conducted by Yanchapaxi (2023) examined the implementation of
active methodologies and the use of PhET simulations in physics teaching,
highlighting their impact on the educational process. The research shows that
these resources improve conceptual understanding and encourage collaboration
among students. It suggests that the integration of analytical components in
teaching allows for progress monitoring and the development of pedagogical
skills, which promotes a more efficient and practical educational environment.
This methodology can be equally applicable in teaching the straight line and
its elements at the high school level.
According to Ávila (2024), in his article “Use of the PhET virtual
simulator as a tool for distance learning in mathematics,” the validity of the
PhET simulator in the teaching of mathematical concepts, including the equation
of the line, is examined. The active techniques provided by digital instruments
promote more revealing and interactive instruction in distance learning
environments, emphasizing that these skills not only improve conceptual acuity
but also promote performance and interaction among students.
The PhET simulator allows the user to interact practically in the
student's relationship with technology, based on mathematics and science
situations. It has been the subject of several research studies to validate its
effectiveness in the phases of the teaching-learning process. This simulator,
accessible from mobile devices and computers, can be easily downloaded and used
without an internet connection. Its use is of vital importance in the study of
the straight line and its elements, as it facilitates reciprocal, practical,
motivating, participatory, effective, quality, and meaningful learning,
allowing students to develop their thinking in a creative way.
The integration of the PhET platform into the teaching of the equation of
a line not only seeks to optimize academic performance, but also promotes a
more dynamic and motivating teaching environment for high school students. This
content is fundamental in the education of General Unified High School
students, as it lays the foundation for the development of more advanced
mathematical skills and their application in various disciplines. The linear
equation, which represents the relationships between variables in a Cartesian
plane, is an essential concept in mathematics and has practical applications in
everyday life, underscoring the importance of its teaching.
This research focuses on the implementation of a methodological proposal
based on active methodologies, with the aim of evaluating its impact on the
teaching of the equation of a straight line and its elements to second-year
General Unified High School students at the Alfredo Portaluppi Velásquez Public
School during the 2024-2025 school year. Using a quasi-experimental approach,
the influence of these techniques on students' academic performance and
motivation is investigated, as well as the evaluation of the most effective
pedagogical strategies to facilitate the understanding of the equation of a
straight line.
The research problem is formulated around three key questions: How do
active methodologies influence high school students? How can the teaching of
the equation of a straight line be improved through the implementation of
active methodologies? Will the use of the PhET platform in the learning process
motivate high school students? Based on these questions, specific objectives
are established that seek to determine the impact of active methodologies on
academic performance, balance the most relevant skills, and evaluate the
methodological proposal considering its effect on teaching.
The specific objectives of this article focus on identifying the most
effective active strategies to optimize the understanding of the equation of a
straight line, assessing their influence on teaching and student motivation,
and implementing the methodological proposal in two selected classrooms. By
achieving these goals, we hope to contribute to the improvement of mathematics
education, transforming the teaching experience and sparking students' interest
in this field of knowledge. This article not only seeks to validate the
effectiveness of active methodologies, but also to enrich the pedagogical
process in related content.
Materials and methods
This research study was developed using a
quasi-experimental design at a descriptive level, employing a mixed approach
and framed within a cross-sectional time period. The design and level selected
allowed for the evaluation of the effectiveness of active methodologies in
teaching the equation of a straight line and its elements, without random
assignment of participants, since a pre-formed sample was used. On the other
hand, authors such as Osmanović and Maksimović (2022) emphasize that
contemporary educational research requires the integration of quantitative and
qualitative methodologies. and qualitative methodologies. In this context, the
mixed approach prioritized the numerical analysis of data collected through
standardized instruments to understand students' perceptions and experiences
through surveys and observations, while the cross-sectional time period allowed
for the collection of information without longitudinal follow-up of
participants.
The population consisted of 200 students belonging
to the Alfredo Portaluppi Velásquez Public School, considered a heterogeneous
group in terms of academic skills and references, from which a sample of 60
11th-grade students was selected through non-probabilistic convenience
sampling, which facilitated data collection in a specific educational setting.
Various data collection instruments were used to
obtain information, including surveys that provided detailed and in-depth
information about their learning experiences, and direct observation to record
student behavior during class sessions where active methodologies were
implemented.
According to Kusmaryono et al. (2022), the Likert
scale is widely used in educational research to measure students' perceptions,
attitudes, and experiences regarding programs or interventions. Therefore, data
collection was carried out using a survey consisting of 10 questions with this
type of scale. This instrument was applied at two key moments in the study: at
the beginning to establish a baseline of knowledge and attitudes before the
implementation of active methodologies, and after the intervention to evaluate
the changes produced, which enabled a rigorous comparison of the impact of the
strategies implemented.
For this quasi-experimental study, specific
dimensions were established to comprehensively examine students' perceptions of
different aspects of the teaching-learning process of the linear equation.
These dimensions include: conceptual understanding, teaching methodology,
motivation and relevance, practical application, and assessment and feedback.
Each of these dimensions is operationalized through specific indicators
expressed in the form of items in the questionnaire.
The quantitative data were analyzed using Minitab
statistical software, a precise and easy-to-use tool that offers general
statistical applications and is widely used in the field of education,
complemented by Microsoft Excel for data organization and visualization, while
Atlas.Ti, a specialized software that facilitated the coding and analysis of
the observations made, was used for the analysis of the qualitative
information.
The study was developed through seven sequential
methodological phases that ensured the scientific rigor of the research. First,
formal authorization was obtained from the competent authority of the
educational institution to carry out the project, which allowed legitimate
access to the field of study and the participants. Subsequently, the pretest
instrument was administered to the selected students in order to assess their
prior knowledge and attitudes toward the equation of the line before any pedagogical
intervention. In the third phase, both the traditional class with the control
group and the intervention with the pedagogical strategy based on active
methodologies with the experimental group were implemented, maintaining
controlled conditions to ensure the internal validity of the study. After the
intervention, the post-test instrument was administered to both groups to
assess the changes in student learning and attitudes as a result of the
different methodologies used. The fifth phase consisted of validating and
determining the reliability of the instrument using appropriate statistical
techniques, which ensured the robustness of the data collected. Next, the
results obtained were tabulated, analyzed, and quantitatively interpreted,
presenting them in tables and statistical graphs that facilitated their
understanding and comparison. Finally, conclusions were drawn based on the
empirical evidence collected, which responded to the objectives initially set
and provided significant knowledge about the effectiveness of active
methodologies in teaching the equation of a straight line (Ilbay Cando and
Veloz León, 2023).
In terms of ethical aspects, the research on active
methodologies in teaching the equation of a straight line was conducted under
strict principles of academic and scientific integrity to ensure a
comprehensive study process in all its phases, from planning to the
dissemination of results. The criteria of those involved were respected,
ensuring that their participation was voluntary and that they could freely
express their opinions and experiences without any coercion or undue influence.
Likewise, the personal data of the participating students was kept anonymous by
encoding the information collected and removing any direct identifiers from the
instruments used, thus protecting their privacy and confidentiality in
accordance with current ethical standards in educational research.
Results
The study included 60 valid observations during the
application of the survey to students, distributed between the control group
and the experimental group, and its validation was carried out using Cronbach's
alpha, obtaining a coefficient of 0.9071. This implies excellent reliability
and suggests high homogeneity and consistency among the items when measuring
the underlying construct.
The diagnostic survey applied to the control group
consisted of ten statements designed to explore students' perceptions of
learning the equation of a straight line using a traditional methodology. The
questions were: Q1. “I understand the basic concepts of the equation of a
straight line well”; Q2. “The traditional methodology has helped me learn about
the equation of a straight line”; Q3. “The teacher's explanations are clear and
understandable”; Q4. “I feel motivated to learn about the equation of a straight
line”; Q5. “Class activities are useful for understanding the equation of a
straight line”; Q6. “I can apply the equation of a line to real-life problems”;
P7. “I feel that I can do exercises related to the equation of a line without
difficulty”; P8. “The assessments adequately reflect my understanding of the
subject”; P9. “I receive sufficient feedback on my performance in the subject”;
and P10. “I would recommend this methodology to other students for learning
about the equation of a line.”
For the analysis, a five-level Likert scale was used
with the following coding: “Strongly disagree” (1), “Disagree” (2), ‘Neutral’
(3), “Agree” (4), and “Strongly agree” (5), allowing for the quantification of
students' attitudes and perceptions regarding each of the statements.
The following tables show the frequencies of the
results obtained in the diagnostic survey. Table 1 presents the results
observed in the control group, while Table 2 presents those of the experimental
group.
Table 1. Results obtained from the diagnostic survey of students in the
control group
|
|
P1 |
P2 |
P3 |
P4 |
P5 |
P6 |
P7 |
P8 |
P9 |
P10 |
|
Totalmente de acuerdo |
5 |
5 |
12 |
7 |
9 |
5 |
5 |
6 |
5 |
10 |
|
De acuerdo |
7 |
12 |
10 |
9 |
12 |
6 |
6 |
10 |
10 |
14 |
|
Neutral |
16 |
10 |
7 |
9 |
7 |
15 |
12 |
12 |
11 |
5 |
|
En desacuerdo |
0 |
2 |
0 |
3 |
1 |
1 |
5 |
0 |
2 |
0 |
|
Totalmente en desacuerdo |
2 |
1 |
1 |
2 |
1 |
3 |
2 |
2 |
2 |
1 |
|
N= |
30 |
30 |
30 |
30 |
30 |
30 |
30 |
30 |
30 |
30 |
Note: Prepared by the authors using Minitab
software.
The results in Table 1 show that, in general,
student responses are distributed heterogeneously, with a slight tendency
toward neutrality and agreement on most items. The statements with the highest
concentration of responses in the “Strongly agree” and “Agree” categories were
question 3 and question 10, suggesting a favorable perception of the teaching
work and the methodology used, even though it is traditional.
On the other hand, question 6 shows an increase in
neutral and disagree responses, which could indicate difficulties in linking
the content to practical situations. Questions 1 and 5 show some dispersion,
but with a significant core of positive responses, which can be interpreted as
an acceptable understanding of the concepts and usefulness of the activities.
Finally, there is a low frequency in the disagree
and strongly disagree categories for most items, indicating that there is no
strong rejection of the methodology, although there is a need for improvement
in key areas such as the practical application of knowledge and feedback.
The analysis by dimensions of the control group
reveals significant patterns in students' perceptions of the teaching of the
linear equation.
In the Conceptual Understanding dimension, only 40%
of students claim to understand the basic concepts well, with a majority
(53.3%) maintaining a neutral position, suggesting moderate but not
consolidated understanding.
In terms of Teaching Methodology, there is notable
strength in the teacher's explanations, with 73.3% agreeing or strongly
agreeing, while the traditional methodology is positively rated by 56.7% of
participants.
Regarding Motivation and Relevance, the perceived
usefulness of classroom activities stands out in particular (70% agree),
although personal motivation shows more moderate results (53.3% agree). The
dimension of Practical Application emerges as the weakest area, with only 36.7%
agreeing both on the application to real problems and on the resolution of
exercises without difficulty, which points to a gap between theoretical
understanding and practical application.
In terms of Evaluation and Feedback, approximately
half of the students express satisfaction, while it is noteworthy that, despite
the difficulties identified in other dimensions, the recommendation of the
methodology achieves a resounding 80% approval rating.
Table 2. Results obtained from the diagnostic survey of students in the
experimental group
|
|
P1 |
P2 |
P3 |
P4 |
P5 |
P6 |
P7 |
P8 |
P9 |
P10 |
|
Totalmente de acuerdo |
4 |
2 |
10 |
6 |
9 |
5 |
3 |
2 |
2 |
7 |
|
De acuerdo |
8 |
14 |
6 |
8 |
12 |
8 |
9 |
14 |
7 |
12 |
|
Neutral |
12 |
11 |
10 |
10 |
6 |
11 |
10 |
8 |
16 |
9 |
|
En desacuerdo |
5 |
2 |
3 |
4 |
1 |
4 |
7 |
5 |
3 |
1 |
|
Totalmente en desacuerdo |
1 |
1 |
1 |
2 |
2 |
2 |
1 |
1 |
2 |
1 |
|
N= |
30 |
30 |
30 |
30 |
30 |
30 |
30 |
30 |
30 |
30 |
Note: Prepared by the authors using Minitab
software.
The data compiled in Table 2 for the experimental
group reflect diverse perceptions regarding learning the equation of a straight
line. The responses show a tendency to fall between the categories of “Agree”
and “Neutral” for most items, suggesting a moderately favorable assessment of
the methodology used.
In key questions such as questions 3, 5, and 10,
there is a higher concentration of responses at the high levels of agreement,
indicating that students recognize strengths in the way the sessions were
conducted.
However, in items such as questions 7, 8, and 9, the
number of responses in disagreement and strongly disagree is more notable,
highlighting certain weaknesses in the perception of support and the
development of autonomous skills.
Overall, the distribution suggests that, although
the experience was rated relatively positively, there are specific aspects that
require attention in order to optimize the teaching-learning process within
this group.
The analysis by dimensions of the experimental group
reveals significant trends in student perception of the teaching of the
equation of the straight line. In the Conceptual Understanding dimension, a
balanced distribution is observed, with 40% of students stating that they
understand the basic concepts, but an equal proportion remain neutral,
reflecting a partial understanding of the fundamentals. With regard to Teaching
Methodology, the teacher's explanations receive a favorable rating (53.33%
agree), while the traditional methodology shows a similar level of acceptance.
In terms of Motivation and Relevance, the usefulness
of classroom activities stands out very positively, rated favorably by 70% of
participants, constituting the highest-rated aspect of the entire survey,
although personal motivation to learn shows more moderate results (46.67%).
The Practical Application dimension presents
significant challenges, with only 43.34% considering that they can apply the
equation to real problems and 40% feeling confident in performing exercises
without difficulty.
Finally, in the Evaluation and Feedback dimension,
while evaluations are considered adequate by 53.34%, feedback emerges as the
weakest area with only 30% satisfaction, which constitutes a clear opportunity
for improvement, despite which 63.33% would recommend the methodology to other
students.
The diagram illustrated in Figure 1 shows the
distribution of responses from the experimental group in relation to ten
statements about learning the equation of a straight line.
Figure 1. Consolidated graph of the survey conducted with
students from both groups
Overall, there is a positive trend toward the
methodology implemented, with levels of agreement (blue) and total agreement
(green) standing out in most items. For example, in questions 2 and 10, the
levels of agreement are particularly high, suggesting that students perceive
benefits in the methodology used and would be willing to recommend it. Question
3, which assesses the clarity of the teacher, also shows a good level of
approval, demonstrating the teacher's effectiveness in conveying content. However,
in items such as questions 7, 8, and 9, which address autonomy in solving
exercises, perceptions of assessments, and feedback received, there is greater
dispersion in the responses, with increases in the categories of disagreement
and total disagreement, revealing areas for improvement in pedagogical support
and the consolidation of practical skills.
Finally, in question 6, related to the applicability
of knowledge in real contexts, a neutral response predominates, which could
indicate a lack of clarity on how to transfer learning to situations outside
the classroom. Overall, the graph suggests a mostly positive assessment,
although with specific aspects that require reinforcement to improve the
educational experience.
The exit survey administered to students included
the following questions: Q1. How would you rate your understanding of the basic
concepts related to the equation of a line at the end of the course?; Q2. To
what extent do you think the methodology used facilitated your understanding of
the equation of a line during the course?; Q3. Were the explanations provided
by the teacher clear and understandable to you throughout the course?; Q4. How
would you rate your motivation to learn about the equation of a straight line
at the end of the course?; Q5. Do you consider that the activities carried out
in class were effective in facilitating your understanding of the equation of a
straight line?; P6. Do you believe that the equation of a straight line can be
effectively applied to real-life situations and problems after completing the
course?; P7. Do you consider that you can perform exercises related to the
equation of a straight line without difficulty at the end of the course?; P8.
Do the assessments used in this course adequately reflect your understanding of
the concepts and content of the subject?; P9. Do you think you received
sufficient feedback on your performance in this subject during the course?; and
P10. Would you recommend this methodology to other students as an effective way
to learn about the equation of a straight line?
Table 3. Results obtained from the exit survey of students in the control
group
|
|
P1 |
P2 |
P3 |
P4 |
P5 |
P6 |
P7 |
P8 |
P9 |
P10 |
|
Totalmente de acuerdo |
2 |
1 |
3 |
1 |
4 |
1 |
3 |
1 |
2 |
4 |
|
De acuerdo |
4 |
7 |
8 |
9 |
8 |
6 |
4 |
10 |
10 |
8 |
|
Neutral |
9 |
9 |
7 |
7 |
6 |
10 |
9 |
5 |
5 |
6 |
|
En desacuerdo |
3 |
3 |
1 |
3 |
2 |
3 |
2 |
2 |
3 |
2 |
|
Totalmente en desacuerdo |
3 |
1 |
2 |
1 |
1 |
1 |
3 |
3 |
1 |
1 |
|
N= |
21 |
21 |
21 |
21 |
21 |
21 |
21 |
21 |
21 |
21 |
Note: Prepared by the authors using Minitab
software.
Table 3 shows the frequencies of the results
obtained in the exit survey conducted with the control group, while Table 4
presents the results observed in the experimental group.
In general, the results of the survey applied to the
control group show a majority trend toward neutral and agree responses, with a
lower frequency of responses at the extremes of strongly agree and strongly
disagree. In questions Q1, Q2, Q3, Q4, Q6, and Q7, the highest concentration of
responses is in the Neutral option, suggesting a moderate or undefined
perception among students regarding their understanding and experience with the
equation of a straight line.
This was especially true for P6 (real-life
application) and P7 (performing exercises without difficulty), where 10 and 9
students, respectively, responded with neutral, reflecting doubts about their
practical mastery of the subject.
For questions P5, P8, P9, and P10, there is a slight
predominance of responses in the Agree category, highlighting that students
consider class activities, assessments, feedback, and the recommended
methodology to be acceptably effective, although not overwhelmingly so.
The low frequency of responses in Strongly Agree for
almost all questions indicates a lack of strong positive conviction on the part
of the control group, while responses in Disagree and Strongly Disagree remain
relatively constant but low, suggesting that, although there was no high
satisfaction, there was also no overwhelming rejection. Overall, the control
group reflects an intermediate position regarding the learning of the linear
equation, with a slight inclination toward acceptance of the methodology applied,
but with ample areas of opportunity to strengthen both conceptual clarity and
motivation and practical application.
The analysis by dimensions of the control group
shows different trends in students' perceptions of the teaching of the equation
of a straight line, since in the dimension of conceptual understanding, only
42.9% expressed some level of agreement with their mastery of basic concepts
(P1), while 42.9% remained neutral, indicating partial understanding at the end
of the course. with regard to teaching methodology, 42.9% considered that the
methodology applied facilitated their learning (P2), although the clarity of
the teacher's explanations (P3) stands out, with 52.4% expressing agreement,
making it one of the most highly rated aspects; in terms of motivation and
relevance, 47.6% felt motivated to learn (P4) and 57.1% considered the
classroom activities to be effective (P5), highlighting the importance of
classroom dynamics; However, in terms of practical application, only 33.3% felt
they could apply the equation to real-life situations (P6) and the same
percentage felt they could solve exercises without difficulty (P7), while 47.6%
remained neutral on both items, highlighting limitations in knowledge transfer.
Finally, in terms of evaluation and feedback, 52.4% considered the evaluations
to be adequate (P8) and 57.1% rated the feedback received positively (P9),
while the recommendation of the methodology (P10) reached 57.1% agreement,
suggesting a generally favorable perception despite the areas for improvement
identified.
Table 4. Results obtained from the exit survey of students in the experimental
group
|
|
P1 |
P2 |
P3 |
P4 |
P5 |
P6 |
P7 |
P8 |
P9 |
P10 |
|
Totalmente de acuerdo |
5 |
6 |
5 |
7 |
7 |
9 |
9 |
8 |
6 |
11 |
|
De acuerdo |
10 |
8 |
11 |
10 |
8 |
9 |
8 |
9 |
8 |
4 |
|
Neutral |
7 |
7 |
5 |
4 |
6 |
3 |
4 |
6 |
7 |
4 |
|
En desacuerdo |
1 |
2 |
1 |
1 |
0 |
0 |
1 |
0 |
1 |
3 |
|
Totalmente en desacuerdo |
1 |
1 |
2 |
2 |
3 |
3 |
2 |
1 |
2 |
2 |
|
N= |
24 |
24 |
24 |
24 |
24 |
24 |
24 |
24 |
24 |
24 |
Note: Prepared by the authors using Minitab
software.
The statistical analysis of the survey applied to 24
students in the experimental group reveals favorable results in general terms.
Positive ratings (combining “Strongly agree” and “Agree”) exceed 58% in all
questions, suggesting a positive reception of the methodology implemented to
teach the equation of the line.
The analysis by dimensions of the experimental group
shows consistently positive results in student perception of the teaching of
the equation of a straight line, since in the dimension of conceptual
understanding, 62.5% agreed with their mastery of the basic concepts (P1) and a
remarkable 70.83% considered that they can perform exercises without difficulty
(P7), reflecting a solid theoretical and practical assimilation; With regard to
teaching methodology, 58.33% rated the methodology applied positively (P2) and
66.67% highlighted the clarity of the teacher's explanations (P3), suggesting
an effective synergy between pedagogical innovation and teacher communication.
In terms of motivation and relevance, 70.83% expressed high motivation to learn
(P4) and 62.5% considered the classroom activities to be effective (P5),
underscoring the success of the interactive strategies implemented.
The practical application dimension stood out
notably, with 75% of students recognizing the usefulness of the equation of the
line in real contexts (P6), demonstrating a successful transfer of knowledge to
everyday situations. In terms of assessment and feedback, 70.83% considered the
assessments to be adequate (P8), while feedback (P9) achieved a 58.33% approval
rating, the latter being the area with the greatest room for improvement.
Finally, the overall assessment of the methodology is reflected in the fact
that 62.5% of students would recommend it to others (P10), consolidating a
favorable perception that, despite opportunities for refinement in feedback,
validates the effectiveness of the experimental pedagogical approach.
The implementation of active methodologies through
digital platforms such as PhET significantly improves
conceptual understanding and practical application of the equation of a
straight line, as the experimental group showed a 25% increase in the ability
to solve exercises without difficulty (mean = 3.83) and 75% agreement in the
application to real problems (P6), results superior to the control group that
received traditional classes, where only 33.3% expressed confidence in solving
exercises (mean = 3.23) and 47.6% remained neutral in practical application.
The use of interactive tools such as PhET encourages greater motivation and active participation
in learning mathematical concepts, evidenced by the fact that 70.83% of the
experimental group expressed high motivation (P4) and 62.5% recommended the
methodology (P10), while in the control group only 47.6% felt motivated and
57.1% considered traditional activities to be effective, highlighting the
potential of technologies to create dynamic and contextualized learning
environments.
Although active methodologies demonstrate advantages
in the development of cognitive and practical skills, challenges remain in
personalized feedback and geometric communication, as in both groups less than
60% of students considered the feedback received to be adequate (P9),
suggesting the need to complement digital platforms with individualized
monitoring strategies to strengthen autonomy and mathematical expression.
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Universidad
de Guayaquil,
ricardo.yozalu@ug.edu.ec
https://orcid.org/0009-0008-7643-9278
Universidad de Guayaquil, franklin.chaguancalleto@ugedu.ec
https://orcid.org/0003-6155-1279
Universidad de Guayaquil, catalina.vargasp@ug.edu.ec
https://orcid.org/ 0000-0002-8555-4496
Universidad de Guayaquil, John.granadosr@ug.edu.ec