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Articl |
Innovation in mathematics instruction
to improve logical reasoning in High School Students
Innovación en la enseñanza de las matemáticas para
mejorar el razonamiento lógico en Estudiantes de Bachillerato
Luis Aurelio Logacho[*]
Lorena Salomé Andino Rueda *
Abstract
This study analyzes how innovative educational
strategies can improve logical-mathematical reasoning in first
and third year high school students at the Juan de Salinas Educational
Unit in Quito, Ecuador. The main objective is to strengthen problem-solving
skills, critical thinking and autonomy, overcoming rote learning and improving
performance in mathematics, an area with low results in international tests
such as PISA. The methodology used is based on the theories of Ausubel, Piaget
and Vygotsky, combined with cooperative learning and information and
communication technologies (ICT) such as GeoGebra. Interactive activities,
hands-on workshops and collaborative dynamics were implemented to connect
mathematical concepts with real-world situations.
Key words: Educational innovation, logical reasoning, meaningful
learning, ICT, mathematics, PISA tests.
Resumen
Este estudio analiza cómo estrategias educativas innovadoras pueden
mejorar el razonamiento lógico-matemático en estudiantes de primero y tercero
de bachillerato de la Unidad Educativa Juan de Salinas, en Quito, Ecuador. El
objetivo principal es fortalecer habilidades de resolución de problemas,
pensamiento crítico y autonomía, superando el aprendizaje memorístico y
mejorando el rendimiento en matemáticas, un área con resultados bajos en
pruebas internacionales como PISA. La metodología empleada se basa en teorías
de Ausubel, Piaget y Vygotsky, combinadas con aprendizaje cooperativo y
tecnologías de la información y comunicación (TIC) como GeoGebra. Se
implementaron actividades interactivas, talleres prácticos y dinámicas
colaborativas para conectar los conceptos matemáticos con situaciones del mundo
real.
Palabras
clave: Innovación educativa,
razonamiento lógico, aprendizaje significativo, TIC, matemáticas, pruebas PISA.
Introduction
Worldwide, and particularly in Ecuador, academic
performance in mathematics has been one of the major educational concerns due
to its direct impact on the development of critical skills necessary for
academic and professional success. Mathematics, as a fundamental discipline,
not only develops logical reasoning skills, but also fosters complex problem
solving and analytical thinking, essential skills in the 21st century work
environment. However, statistics from the National Institute for Educational Evaluation
(INEVAL) show that a significant proportion of Ecuadorian students fail to
achieve adequate levels of mathematical reasoning, which has a negative impact
on their overall performance and limits their access to higher education.
These deficiencies affect their preparation for
national exams and, therefore, their future academic and professional
opportunities. The problem of low mathematics achievement is not unique to
Ecuador. Globally, assessments such as the Program for International Student
Assessment (PISA) have shown that numerous education systems face difficulties
in equipping students with advanced, contextualized math skills that are
relevant to modern life. These gaps are particularly pronounced in education
systems with structural constraints, such as lack of adequate teacher training,
insufficient resources, and outdated teaching methods. This underscores the
urgent need to develop and implement innovative pedagogical strategies that
transform the way this crucial discipline is taught and learned.
In this context, this project was developed in the
Juan de Salinas Educational Unit, located in Sangolquí,
Ecuador, with the purpose of transforming traditional pedagogical practices.
The institution, aware of the challenges faced by its students, sought to
promote more active and meaningful learning, adapted to current needs.
The project
is based on a constructivist approach, which places the student at the center
of the learning process and encourages the active construction of knowledge
based on concrete experiences. The proposal incorporates technological tools
such as GeoGebra, which allows visualizing and solving complex mathematical
problems interactively, and active methodologies such as cooperative learning,
which emphasizes teamwork and joint problem solving.
This article
details the theoretical foundations of the project, based on educational
theories such as Piaget's and Vygotsky's constructivism and Ausubel's
meaningful learning. It also explores methodological strategies designed to
overcome traditional barriers in the teaching of mathematics, such as the lack
of connection with everyday life and the predominant use of rote approaches. In
addition, the results obtained during the implementation of the project are presented,
which show significant improvements in students' motivation, participation and
mathematical logical reasoning skills.
The importance of this project lies not only in the
progress achieved at the Juan de Salinas Educational Unit, but also in its
potential to serve as a replicable model in other educational contexts. By
addressing both structural limitations and pedagogical practices, this
initiative highlights the need to prioritize educational innovation, especially
in high-need contexts. In doing so, it seeks to ensure that students not only
acquire advanced mathematical competencies, but also develop confidence in their
abilities and a resilient mindset in the face of academic and professional
challenges,
Educational innovation refers to the implementation
of significant changes in the teaching and learning processes, with the
objective of improving educational quality, these changes may include new
methodologies, technological tools, pedagogical approaches and strategies that
encourage the active participation of students. In the context of mathematics,
educational innovation seeks to transform the way students interact with
mathematical concepts, promoting a more dynamic and applied learning (Area-Moreira,
2020, p. 15).
Constructivism, based on the theories of Jean Piaget
and Lev Vygotsky, emphasizes the importance of the environment and social
interaction in cognitive development (Coll, 1990, p. 67). Piaget emphasizes
that learning arises from the interaction of the individual with his or her
environment, which allows students to develop skills through active
exploration. For his part, Vygotsky highlights the role of the social context
and the use of mediators, such as language or technological tools, in learning.
Cooperative learning expands this perspective by organizing
work in heterogeneous groups, where each member contributes his or her skills
and experiences (Johnson & Johnson, 1999, p. 23). This dynamic promotes not
only the acquisition of knowledge, but also the development of interpersonal
skills such as communication, negotiation and leadership. In the mathematics
classroom, cooperative learning fosters joint problem solving, allowing
students to exchange ideas, clarify concepts and strengthen their understanding
(Torrego, 2018, p. 56).
The Finnish educational model has established itself
as an international benchmark thanks to its student-centered approach, the
personalization of learning and the use of active methodologies, such as
problem-based learning, these strategies encourage not only the acquisition of
knowledge, but also the development of critical and logical-mathematical
thinking. In this context, the integration of similar practices in the
baccalaureate can enhance logical reasoning through interdisciplinary
activities, collaborative projects and the use of educational technologies
(García, 2019, p. 78).
Mathematical didactics focuses on the design of
pedagogical strategies that facilitate deep learning of mathematical concepts,
while promoting the development of logical reasoning (Díaz Godino, 2004, p.
45). Through structured activities, such as problem solving and exploration of
mathematical patterns, students develop skills to analyze, abstract and connect
concepts, which is essential for their academic and personal formation (Rico,
2015, p. 78).
Logical reasoning is a fundamental cognitive skill
that allows students to analyze, synthesize, and evaluate information
critically. In mathematics, logical reasoning is essential for solving
problems, identifying patterns, and constructing valid arguments. Fostering
this skill not only improves performance in mathematics, but also prepares
students to face challenges in other areas of knowledge and in everyday life
(Johnson-Laird, 2010, p. 627).
Algorithm design is a powerful tool to improve
logical-mathematical reasoning in high school students. This practice fosters
sequential thinking by forcing students to break complex problems into orderly
and coherent steps (Rico, 2015, p. 56). In addition, it promotes critical
thinking by identifying patterns and abstracting key concepts. It also allows
them to apply problem-solving skills in practical and real contexts,
strengthening the relationship between mathematics and the world around them
(Sánchez, 2018, p. 34).
Meaningful learning, proposed by David Ausubel,
occurs when students relate new knowledge to previous concepts, thus building a
deep and lasting understanding. In mathematics, this approach is crucial for
students not only to memorize formulas, but to understand the underlying
principles and be able to apply them in different contexts (Ausubel, 1968, p.
48). The use of strategies that promote meaningful learning, such as solving
real problems and connecting to everyday situations, is critical to improving achievement.
Meaningful learning, proposed by David Ausubel,
holds that knowledge is assimilated more effectively when new concepts are
integrated into the learner's existing cognitive structures. This approach
contrasts with mechanical learning, in which data is memorized without deep
understanding or context.
In the mathematical field, this theory allows
students to relate abstract concepts, such as functions or equations, with
previous experiences or practical applications, facilitating a lasting
understanding and a greater ability to solve problems (Díaz Godino, 2004, p.
23). Mathematics teaching, based on meaningful learning, focuses on building
connections between ideas, promoting learning that is both functional and
transformative (Chamorro, 2010, p. 12).
Information and Communication Technologies (ICT)
ICTs have revolutionized education by offering tools
that facilitate access to information, collaboration and personalization of
learning, in mathematics, the use of educational software, simulations,
interactive platforms and digital resources can make learning more engaging and
effective. In addition, ICTs allow students to explore mathematical concepts in
a visual and dynamic way, which favors comprehension and logical reasoning
(UNESCO, 2017, p. 23).
The impact of ICT in mathematics education has been
widely recognized for its ability to transform learning from a static
experience to an interactive and dynamic one. Tools such as GeoGebra allow
students to experiment with mathematical concepts in real time, observe their effects,
and understand the relationships between different representations (Garcia,
2019, p. 78). These technologies not only facilitate the teaching of complex
topics, but also stimulate curiosity and interest, increasing student
participation and engagement (Puig et al., 2020, p. 89).
The PISA tests, developed by the Organisation
for Economic Co-operation and Development (OECD), assess students' competencies
in key areas such as mathematics, science and reading. In mathematics, PISA not
only measures students' ability to solve traditional problems, but also their
ability to apply mathematical concepts in real and complex contexts. An
innovative approach to teaching mathematics, focusing on logical reasoning and
meaningful learning, can contribute to improved results in these tests (OECD,
2019, p. 45).
The integration of educational innovation, logical
reasoning, meaningful learning, and ICT in mathematics education has the
potential to transform the learning experience of students,
by using innovative methodologies and technological tools, educators can create
learning environments that foster curiosity, creativity, and problem solving.
This not only improves mathematics achievement, but also prepares students for
the challenges of the 21st century (Hernández-Rojas, 2018, p. 12).
Materials and methods
The present study adopted a mixed methodological
approach, combining quantitative and qualitative research techniques to
comprehensively assess the impact of the project. This approach was chosen
because of its ability to capture both measurable outcomes and the subjective
perceptions of participants, providing a more holistic understanding of the
effects of the strategies implemented. The three main phases of the project,
detailed below, included an initial diagnosis, the design and implementation of
innovative pedagogical strategies, and a rigorous evaluation and monitoring
process.
Phase 1: Initial diagnosis. The initial phase of the
project focused on identifying the main weaknesses in the students'
mathematical logical reasoning and on understanding the existing pedagogical
practices in the Juan de Salinas Educational Unit. This stage included the
following activities:
Application of surveys and diagnostic tests:
Specific tests were designed and administered to measure the students' level of
mathematical logical reasoning. These tests evaluated skills such as problem
solving, critical thinking and understanding of abstract concepts. In addition,
surveys were administered to collect data on students' perceptions of
mathematics, including their levels of motivation, confidence and attitude
towards the subject.
Interviews with teachers: Structured and
semi-structured interviews were conducted with teachers to identify the
teaching methodologies used, common challenges faced in the classroom, and
teachers' training needs. These interviews revealed a strong reliance on
traditional methods, with an emphasis on memorization and mechanical problem
solving, and a low incorporation of technological tools.
Analysis of academic data: An exhaustive review of
the students' academic records and the results obtained in national tests such
as INEVAL was carried out. This analysis allowed us to identify trends and
correlations between current teaching methods and low performance in
mathematics.
Phase 2: Design and implementation of strategies.
Based on the findings of the initial diagnosis, innovative pedagogical
strategies were designed and implemented to address the critical areas
identified. This phase was characterized by a focus on teacher training,
reorganization of classroom practices and the introduction of active
methodologies and technological tools.
Teacher training workshops: Intensive workshops were
conducted to train teachers in the use of GeoGebra and other digital tools
designed to facilitate the teaching of complex mathematical concepts. In
addition, the workshops included training in active methodologies, such as
problem-based learning (PBL) and cooperative learning, to promote a more
dynamic and participatory approach in the classroom.
Reorganization of classroom practices: Traditional
classes were transformed into interactive learning environments, where students
actively participated in solving problems related to real-world situations. The
use of practical and experimental activities was promoted to connect
mathematical concepts with concrete applications, fostering a deeper and more
meaningful understanding.
Cooperative learning strategies: Students worked in
heterogeneous groups, which facilitated the exchange of ideas, the development
of social skills and collaborative problem solving. Group dynamics were
designed to encourage equal participation, ensuring that each team member
contributed to the achievement of common goals.
Phase 3: Evaluation and monitoring. The final phase
of the project focused on measuring the impact of the strategies implemented
and gathering feedback from participants to identify opportunities for
improvement. Activities carried out in this phase included:
Post-tests: Post-tests were designed and
administered to assess progress in students' mathematical logical reasoning
skills. These tests were aligned with the project objectives and allowed
comparison of the results with the initial data. The results showed a
significant improvement in students' performance, with notable increases in
their ability to solve problems and apply mathematical concepts in a practical
way.
Interviews
and focus groups: Interviews were conducted with students and teachers to
explore their perceptions of the project experience, identifying strengths and
areas for improvement of the strategies implemented. The focus groups provided
a space for open discussions, where participants shared their opinions and
suggestions on the impact of the project on their teaching and learning
processes.
Comparative data analysis: Post-test results were
compared with initial data to assess the degree of improvement achieved.
Qualitative data collected from interviews and surveys were also analyzed to
identify relevant patterns and trends. Overall, this methodological approach
made it possible not only to evaluate the effectiveness of pedagogical
strategies, but also to adapt and continuously improve teaching practices
according to the specific needs and contexts of students and teachers. The
findings of this research highlight the importance of an integrated approach
that combines quantitative and qualitative methods to address contemporary
educational challenges.
Results
PISA tests assess 15-year-old students to determine
their competencies in problem solving and the application of knowledge in
real-life situations. In the case of Ecuador, the results obtained in the
PISA-D 2017 tests reveal significant challenges in the area of mathematics.
This table reflects the academic performance in
mathematics during the last school years, allowing an analysis of the causes of
low performance and the need for innovative strategies to improve results.
Table 1. Table of averages in the
subject of mathematics.
|
School year |
course |
Average over 10
points |
No. of students |
|
2021
- 2022 |
PRMERO
BGU |
7.38 |
265 |
|
2022 - 2023 |
SECOND BGU |
7.52 |
240 |
|
2023
- 2024 |
THIRD
BGU |
9.08 |
177 |
Figure 2. Word cloud of the interview transcription document.
Impact on Learning and Motivation. The
preliminary results obtained in the implementation of the project were highly
encouraging, evidencing significant improvements both in academic performance
and in students' perception and motivation towards mathematics. These
improvements were observed through an exhaustive analysis of quantitative and
qualitative data collected during the different phases of the project,
highlighting the effectiveness of the pedagogical strategies adopted.
Impact
on Mathematical Logical Reasoning Skills. Post-tests designed to assess student progress revealed an average
improvement of 30% in mathematical logical reasoning skills compared to the
initial assessments. This progress was reflected in an increased ability to
solve complex problems and apply mathematical concepts in practical contexts.
Specifically, students showed stronger mastery in key areas such as:
●
Mathematical functions: Understanding and handling of linear and quadratic
functions, including the ability to represent them graphically and analyze
their properties.
●
Analytical geometry: Visualization and resolution of problems related to
points, straight lines and geometric figures in the Cartesian plane.
●
Problem solving: Development of systematic strategies to address
mathematical challenges in a logical and structured manner.
●
The use of
GeoGebra played a key role in this progress, as it allowed students to interact
dynamically with abstract concepts. The tool facilitated graphical
visualization and direct experimentation, transforming learning from a passive
to an active, exploratory process. For example, students were able to
manipulate parameters in real time to observe how they affected the properties
of functions, strengthening their conceptual understanding and their ability to
make connections between algebraic and graphical representations.
●
Increase in Motivation and Attitude
towards Mathematics. At the qualitative level, students reported a
positive change in their attitude towards mathematics, going from considering
it a difficult and unmotivating subject to perceiving it as an interesting and
accessible area. This change was captured through interviews and surveys, where
students highlighted the following aspects:
●
Interactivity and dynamism in
learning: The incorporation of technological tools and active
methodologies transformed traditional classes into more attractive and
participatory experiences.
●
Collaborative work: Group activities fostered an environment of mutual
support, where students learned from their peers and felt more confident in
facing complex problems.
●
Practical relevance: Activities linked to real-world situations helped
students understand the usefulness of mathematics in their daily lives and
future careers. In addition, many students noted that cooperative learning
allowed them to improve their social and communication skills, developing
greater confidence in themselves and their ability to solve problems. This
inclusive approach also benefited students who had traditionally shown
difficulties, who felt more integrated and supported in their learning process.
●
Teachers' Perception and Evaluation: Teachers involved in the project expressed
satisfaction with the tools and methodologies implemented, recognizing their
positive impact on student learning and motivation. During the interviews,
several teachers mentioned that GeoGebra not only facilitated the teaching of
complex concepts, but also allowed them to innovate in their pedagogical
practices. For example: Facilitation of explanations: The ability to
generate dynamic graphical representations improved the clarity of classroom
explanations. Adaptation to different learning styles: The tool made it
possible to cater to both visual learners and those who learn better through
direct practice. However, some teachers noted the need for more extensive
training to maximize the potential of educational technologies. While the
initial workshops provided a solid foundation, teachers indicated that ongoing
training would be key to address specific challenges and explore advanced
functionalities of the tools.
The findings so far show that the project has not
only improved students' academic performance, but also transformed their
perception and attitude towards mathematics. This impact, supported by the
integration of technologies such as GeoGebra and collaborative approaches,
highlights the potential of innovative pedagogical strategies to overcome
traditional barriers in teaching. Preliminary results also underscore the
importance of ongoing teacher training to ensure the long-term sustainability
and effectiveness of these initiatives. Overall, the project represents a
significant step towards creating more inclusive, dynamic and student-centered
educational environments.
By integrating the triangulation of sources, three
axes related to institutional perception were identified: trends, tensions and
absences, which synthesize the analysis of the training policy and allow
conclusions and discussions on the categories set out in the questions related
to conditions, institutional actions, foundations, key concepts and actors.
This project showed that pedagogical innovation can
transform mathematics teaching, increasing both academic performance and
student motivation. The combination of digital technologies and active
methodologies, such as cooperative learning, represents an effective way to
address current educational challenges.
To ensure the success and sustainability of these
initiatives, it is recommended:
Extend the scope of the project to other educational
levels and school contexts.
Establish continuous training programs for teachers,
focused on the use of technologies and active methodologies.
Conduct long-term research to evaluate the impact of
these strategies on students' academic performance and overall development.
The transformation of mathematics education is not
only possible, but also necessary to prepare future generations for the
challenges of the 21st century.
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