|
Article |
Estrategias Educativas Integradoras para el Precálculo
y la Física I: Comprensión del Concepto y la Representación Gráfica de
Relaciones
Víctor Manuel Barros[*]
Segundo Bienvenido Camatón Arízabal*
Gonzalo Rafael Naranjo Veintimilla*
Marco Vinicio Añazco Maldonado*
Abstract
This study analyzes the implementation of integrative
educational strategies between Precalculus and Physics I, focused on
understanding the concept of relationships and their graphical representation.
It addresses the issue of fragmented teaching in both disciplines, which
hinders the connection between mathematical models and physical phenomena. The
research was conducted using a mixed-methods, descriptive approach, with the
participation of first-year college students. Surveys, observations, and teaching
activities based on the interpretation of graphs and the use of technological
tools such as GeoGebra were applied. The results show a significant improvement
in students’ ability to interpret graphical representations, relate algebraic
functions to physical phenomena, and solve integrated problems. Likewise, an
increase in active participation and the development of critical thinking was
observed. It is concluded that the use of integrative strategies promotes
meaningful learning and strengthens interdisciplinary understanding,
contributing to a more solid foundation in the exact sciences.
Keywords: Precalculus, Physics I, graphical representation,
meaningful learning.
Resumen
El presente estudio analiza la
implementación de estrategias educativas integradoras entre el Precálculo y la
Física I, centradas en la comprensión del concepto de relación y su
representación gráfica. Se parte de la problemática de la enseñanza fragmentada
de ambas disciplinas, la cual dificulta la articulación entre modelos
matemáticos y fenómenos físicos. La investigación se desarrolló bajo un enfoque
mixto, de tipo descriptivo, con la participación de estudiantes de nivel
inicial universitario. Se aplicaron encuestas, observaciones y actividades
didácticas basadas en la interpretación de gráficas y el uso de herramientas
tecnológicas como GeoGebra. Los resultados evidencian una mejora significativa
en la capacidad de los estudiantes para interpretar representaciones gráficas,
relacionar funciones algebraicas con fenómenos físicos y resolver problemas
integrados. Asimismo, se observó un incremento en la participación activa y en
el desarrollo del pensamiento crítico. Se concluye que el uso de estrategias integradoras
favorece el aprendizaje significativo y fortalece la comprensión
interdisciplinaria, contribuyendo a una formación más sólida en las ciencias
exactas.
Palabras clave: Precálculo,
Física I, representación gráfica, aprendizaje significativo.
Introduction
In the field of higher education, particularly in
the exact sciences, one of the most persistent challenges is the difficulty
students face in connecting mathematical knowledge with its application in
physical contexts. This issue is most evident in the introductory courses of
Precalculus and Physics I, where students must not only understand abstract
concepts but also interpret and apply them in modeling real-world phenomena.
Various studies have pointed out that traditional teaching, characterized by disciplinary
fragmentation, significantly limits the development of deep and meaningful
learning (Redish, 2005; Hestenes, 1992).
Precalculus, as a fundamental foundation for the
study of calculus and other advanced areas of mathematics, introduces essential
concepts such as functions, relationships, and their graphical representations.
For its part, Physics I requires these same concepts to describe and analyze
phenomena such as motion, force, and energy. However, despite this close
epistemological relationship, in educational practice both disciplines are
often taught independently, leading students to develop a disjointed view of knowledge
(Bing & Redish, 2009). This disconnect hinders the understanding of Physics
as a quantitative science grounded in mathematical models.
One of the key elements linking Precalculus with
Physics is the concept of the relationship between variables, commonly
represented through functions and graphs. The interpretation of these
representations constitutes a fundamental skill for the analysis of physical
phenomena, especially in topics such as uniform linear motion and uniformly
accelerated motion. However, research has shown that many students experience
significant difficulties in interpreting graphs, particularly in the transition
between different representations (verbal, algebraic, and graphical)
(Leinhardt, Zaslavsky & Stein, 1990; McDermott, Rosenquist & van Zee,
1987).
In this context, graphical representation should not
be conceived solely as a visual resource, but as a cognitive tool that
facilitates the construction of meaning and the understanding of functional
relationships. According to Duval (2006), mathematical learning necessarily
involves the coordination of multiple representational registers, and
difficulty in this coordination is one of the main causes of failure to
understand mathematical concepts. This problem is exacerbated when students
fail to establish connections between mathematical representations and the
physical phenomena they describe.
From the perspective of meaningful learning,
proposed by Ausubel (1968), the acquisition of new knowledge depends on the
student’s ability to relate it to pre-existing cognitive structures. In this
sense, the integration of precalculus and physics content can foster the
construction of deeper meanings by allowing students to understand the utility
of mathematical concepts in explaining real-world phenomena. This integration
not only improves conceptual understanding but also contributes to the
development of critical thinking and problem-solving skills (Prince &
Felder, 2006).
Various studies have highlighted the importance of
implementing teaching strategies that promote interdisciplinarity in science
education. For example, Beichner et al. (2007) demonstrated that the use of
active methodologies in integrated learning environments significantly improves
academic performance in Physics. Likewise, studies in mathematics education
have shown that the use of multiple representations, including graphs, enhances
the understanding of functions and relationships (Ainsworth, 2006). This
evidence suggests that the integration of educational strategies centered on
graphical representation can be an effective way to improve learning in both
disciplines.
Within this framework, integrative educational
strategies emerge as a pedagogical alternative that seeks to overcome the
fragmentation of knowledge by promoting a holistic approach to learning. These
strategies involve the design of instructional activities that link content
from different disciplines, fostering the construction of meaningful
connections between them. In the case of Precalculus and Physics I, this
involves designing learning experiences in which students can explore,
interpret, and analyze graphs representing physical phenomena using
mathematical tools.
The use of digital technologies, such as graphical
visualization software, has also proven to be a valuable resource in this
process. Tools like GeoGebra allow students to interact with dynamic
representations, facilitating the understanding of the relationship between
variables and their behavior. According to Hohenwarter and Jones (2007), the
use of these technologies promotes more active and exploratory learning, which
contributes to better conceptual understanding.
Despite this evidence, the implementation of
integrative strategies in the classroom still faces various challenges,
including a lack of teacher training in interdisciplinary approaches and the
rigidity of traditional curricula. Therefore, it is necessary to research and
propose instructional models that guide the effective integration of
precalculus and physics content, with an emphasis on the use of graphical
representation as the central focus of learning.
In this regard, various studies in physics education
have shown that one of the main obstacles to student learning lies in the
inability to correctly interpret graphs representing physical relationships,
especially those related to motion. McDermott, Rosenquist, and van Zee (1987)
demonstrated that even college students have difficulty distinguishing between
a position-time graph and a velocity-time graph, reflecting a superficial
understanding of the underlying concepts. This issue highlights the need to strengthen
not only mathematical skills but also the ability to interpret representations
in physical contexts.
Similarly, the literature on mathematics education
has highlighted the importance of translation between different modes of
representation as an essential component of mathematical thinking. Duval (2006)
argues that understanding a concept does not depend solely on mastery of that
concept within a specific mode, but rather on the ability to translate it
between different modes, such as algebraic, graphical, and verbal. In this
sense, the teaching of precalculus cannot be limited to symbolic manipulation
but must integrate multiple forms of representation that allow students to
construct deeper meanings.
From a pedagogical perspective, active methodologies
have been widely recognized as effective strategies for promoting meaningful
learning in the sciences. Prince (2004) notes that approaches such as
problem-based learning and collaborative learning encourage active student
participation, which translates into greater retention and understanding of the
content. In the context of the integration of Pre-Calculus and Physics, these
methodologies allow for the design of learning experiences in which students not
only receive information but also construct knowledge through the resolution of
contextualized situations.
On the other hand, the incorporation of digital
technologies in the classroom has significantly transformed the possibilities
for teaching and learning in the sciences. Tools such as interactive simulators
and graphical representation software facilitate the visualization of complex
phenomena and allow students to experiment with different variables in real
time. According to Wieman, Adams, and Perkins (2008), the use of interactive
simulations in physics instruction improves conceptual understanding by allowing
students to actively explore the principles governing physical phenomena.
In this regard, the integration of technological
tools with teaching strategies focused on graphical representation can
significantly enhance learning in Pre-Calculus and Physics I. However, it is
essential that these tools be used in a pedagogically intentional manner—that
is, as means to facilitate conceptual understanding and not as ends in
themselves. The effectiveness of these strategies depends largely on the
teacher’s ability to design activities that promote reflection, analysis, and
critical interpretation of graphical representations.
Despite advances in educational research, there
remains a need for concrete teaching proposals that guide the effective
integration of mathematical and physical content in the classroom. In
particular, approaches are needed that allow the concept of relationship and
its graphical representation to be addressed as a central axis of learning,
facilitating the transition between mathematical thinking and the
interpretation of physical phenomena. This integration is especially relevant
in the early stages of higher education, where the foundations for learning
more complex disciplines are established.
In this context, the purpose of this study is to
design and analyze integrative educational strategies that promote
understanding of the concept of relationship and its graphical representation
among students in Pre-Calculus and Physics I. Specifically, it seeks to foster
the integration between both disciplines through the use of contextualized
teaching activities and the support of technological tools, with the aim of
strengthening the interpretation of physical phenomena based on mathematical
models. In this way, the study aims to contribute to the development of
meaningful, critical, and interdisciplinary learning that enables students to
understand the utility and applicability of the concepts addressed in their
academic training.
Materials and methods
This study was conducted using a mixed-methods
approach, of a descriptive nature with explanatory scope, aimed at analyzing
the effectiveness of integrative educational strategies in the understanding of
the concept of function and its graphical representation among students in
Pre-Calculus and Physics I courses. This approach allowed for the combination
of quantitative and qualitative data collection and analysis, with the aim of
obtaining a more comprehensive view of the phenomenon under study, in accordance
with the proposal by Creswell (2014), who notes that the integration of both
approaches fosters a deep understanding of educational processes. The research
design was non-experimental and cross-sectional, as variables were not
deliberately manipulated; rather, student behavior was observed in a natural
classroom setting following the implementation of a teaching intervention.
The population consisted of students enrolled in
Precalculus and Physics I courses at the beginning of higher education at a
public university, as well as the instructors responsible for teaching those
courses. A non-probabilistic convenience sample was selected , comprising
approximately 60 to 80 students and between 4 and 6 teachers, based on criteria
such as availability, voluntary participation, and access to the courses where
the teaching strategies were implemented. This type of sampling is common in applied
educational research, where the main objective is to analyze teaching-learning
processes in specific contexts (Hernández-Sampieri, Fernández & Baptista,
2014).
Regarding data collection techniques, a survey,
direct observation, and analysis of teaching activities carried out during the
intervention were employed. The survey was administered to both students and
teachers before and after the implementation of the strategies, with the aim of
identifying changes in the perception, understanding, and handling of concepts
related to functions, relationships, and their graphical representation in
physical contexts. The questionnaires were structured with closed-ended Likert-type
questions and some open-ended questions, which allowed for the collection of
quantifiable data while also gathering participants’ opinions and reflections.
To ensure content validity, the instruments were reviewed by experts in
mathematics and physics education, while their reliability was estimated using
Cronbach’s alpha coefficient.
Direct observation was conducted during class
sessions, using a structured guide that allowed for the recording of aspects
such as student participation, teacher-student interaction, the use of
graphical representations, and students’ ability to interpret relationships
between variables. This technique provided relevant qualitative information
about the teaching-learning process, complementing the data obtained through
the surveys. Additionally, an analysis was conducted of student work, such as
solved exercises, interpretations of graphs, and activities based on
contextualized problems, which allowed for an evaluation of the level of
understanding achieved.
The instructional intervention was designed based on
a set of integrative educational strategies focused on the use of graphical
representation as a connecting thread between Precalculus and Physics I. These
strategies included presenting problem situations contextualized in physical
phenomena, such as the analysis of uniform linear motion and uniformly
accelerated motion, in which students had to interpret and construct
position-time, velocity-time, and acceleration- -time graphs. In addition, the
use of technological tools was incorporated, specifically mathematical
visualization software such as GeoGebra, which allowed students to dynamically
explore the relationship between variables and observe changes in the graphs
based on different parameters.
The activities were conducted using an active
learning approach, promoting collaborative work in small groups, discussion of
results, and reflection on the processes followed. In this regard, students not
only solved exercises but also explained their reasoning, interpreted results,
and established connections between mathematical representations and physical
phenomena. The teacher’s role was that of a learning mediator, guiding the
construction of knowledge and facilitating interaction between students and the
content.
For the analysis of quantitative data, descriptive
statistical techniques were used, such as frequencies, percentages, and
measures of central tendency, in order to identify patterns in participants’
responses. Additionally, a comparison was made between the results obtained in
the initial phase (pre-test) and the final phase (post-intervention), which
allowed for an evaluation of the impact of the implemented strategies.
Regarding the qualitative data, a content analysis was applied, categorizing
the open-ended responses and recorded observations, with the aim of identifying
trends, recurring difficulties, and evidence of meaningful learning.
In terms of ethics, the research was conducted in
accordance with the principles of confidentiality, anonymity, and informed
consent. Participants were informed about the study’s objectives, and their
participation was entirely voluntary. Furthermore, it was ensured that the
collected data would be used exclusively for academic and research purposes.
Finally, it is important to note that, although the
study was conducted in a specific context, the results obtained can serve as a
reference for the implementation of similar strategies in other educational
settings, especially in those where there is a disconnect between the teaching
of mathematics and physics. In this way, the methodology adopted not only
allowed for the analysis of the effectiveness of integrative educational
strategies but also contributed to the development of a replicable teaching proposal
aimed at strengthening students’ understanding of the concept of the
relationship between s and their graphical representation as fundamental
elements in students’ scientific education.
Results
The results obtained in this study demonstrate a
positive impact of implementing integrative educational strategies on students’
understanding of the concept of relationship and its graphical representation
in Pre-Calculus and Physics I. The analysis was structured by comparing the
results of the initial assessment with those obtained after the instructional
intervention, considering both quantitative and qualitative data.
First, the data from the surveys administered to the
students reflect a significant improvement in the understanding of key concepts
related to the interpretation of graphs and the relationship between physical
and mathematical variables. Before the intervention, a high percentage of
students demonstrated difficulties in correctly interpreting position-time and
velocity-time graphs, as well as in establishing relationships between
algebraic and graphical representations. Subsequently, a notable increase in the
level of understanding was observed, suggesting the effectiveness of the
implemented strategies.
Table 1.
Level of conceptual understanding before and after
the intervention
|
Indicator assessed |
Before (%) |
After (%) |
|
Interpretation
of graphs (position-time) |
42% |
78% |
|
Interpretation
of graphs (speed-time) |
38% |
74% |
|
Relationship
between algebraic functions and graphs |
45% |
81% |
|
Understanding
the relationship between physical variables |
40% |
76% |
|
Integrated
problem-solving (Mathematics-Physics) |
35% |
72% |
The results in Table 1 show an average increase of
over 30% in all evaluated indicators, which demonstrates a substantial
improvement in students’ conceptual understanding. In particular, the progress
in the ability to relate algebraic expressions to their graphical
representations stands out, a fundamental aspect for modeling physical
phenomena.
Regarding the qualitative analysis, observations
made during class sessions identified significant changes in student
participation and reasoning. During the initial phase, there was a tendency to
solve exercises mechanically, without a deep understanding of the concepts.
However, following the implementation of the integrative strategies, students
began to explain their answers, justify their interpretations, and establish
connections between Pre-Calculus and Physics content.
Likewise, collaborative work fostered the discussion
of ideas and the joint construction of knowledge. Students demonstrated greater
confidence in interpreting graphs and relating them to concrete physical
situations, which was evident in group activities and in solving contextualized
problems. This finding aligns with the observations of Prince (2004), who
highlights that active learning promotes greater understanding and retention of
content.
Regarding the use of technological tools, it was
observed that the use of software such as GeoGebra facilitated the dynamic
visualization of relationships between variables, allowing students to
experiment with different scenarios and immediately observe changes in the
graphs. This interaction contributed to a better understanding of abstract
concepts, especially in the analysis of function behavior and its application
in Physics.
The bar graph clearly shows an increase in all
evaluated indicators, demonstrating an upward trend following the intervention.
The greatest differences are observed in the resolution of integrated problems
and in the interpretation of graphs, which confirms the effectiveness of
strategies centered on graphical representation as the core of learning.
On the other hand, students’ open-ended responses
reflected a positive perception of the methodology used. Many of them stated
that the use of graphs and digital tools allowed them to “better understand
physics” and “see what mathematics is for,” which indicates a shift in attitude
toward learning both disciplines. This aspect is relevant, as motivation and
the perception of the usefulness of knowledge are key factors in the
educational process.
In summary, the results obtained allow us to affirm
that the implementation of integrative educational strategies based on
graphical representation significantly contributes to the development of
conceptual understanding in Pre-Calculus and Physics I students. The observed
improvement is not limited to content mastery but also extends to higher-order
cognitive skills, such as the interpretation, analysis, and application of
knowledge in real-world contexts.
The results obtained in this study allow us to
conclude that the implementation of integrative educational strategies between
Precalculus and Physics I constitutes an effective pedagogical alternative to
overcome the fragmentation of knowledge that traditionally characterizes the
teaching of these disciplines. The articulation of content through the use of
graphical representation as a central axis fostered a deeper understanding of
the concept of the relationship between variables, allowing students to establish
meaningful connections between mathematical models and physical phenomena.
In this regard, it was evident that students
significantly improved their ability to interpret and construct graphs, as well
as to relate algebraic expressions to visual representations. This progress was
reflected not only in the quantitative results but also in the development of
higher-order cognitive skills, such as analysis, interpretation, and
reasoning—fundamental aspects of scientific education. The improvement observed
in solving integrated problems confirms that learning becomes more meaningful
when content is presented in a contextualized manner and linked to real-world
situations.
Likewise, the use of active methodologies, such as
collaborative learning and problem-based work, contributed significantly to
student participation and the collective construction of knowledge. Peer
interaction enriched the comprehension processes, creating spaces for
discussion and reflection that fostered meaningful learning. This finding
reinforces the importance of promoting dynamic learning environments in which
students take an active role in their educational process.
Furthermore, the incorporation of technological
tools, such as graphical representation software, facilitated the visualization
of abstract concepts and allowed students to interactively explore the
relationship between variables. This aspect proved key to strengthening
conceptual understanding, especially in topics where graphical interpretation
is fundamental, such as in the analysis of the motion of a " " in
Physics I. However, it is evident that the use of technology must be
accompanied by adequate pedagogical mediation, aimed at enhancing its
educational value.
Based on these findings, it can be stated that the
integration of educational strategies centered on graphical representation not
only improves academic performance but also contributes to the development of
an interdisciplinary view of knowledge. This perspective is essential in
training professionals capable of applying the concepts learned in real-world
contexts, moving beyond rote memorization and promoting critical and reflective
understanding.
Finally, the study highlights the need to rethink
teaching practices in the instruction of the exact sciences, incorporating
approaches that favor content integration and the use of active methodologies.
In this regard, it is recommended to strengthen teacher training in
interdisciplinary strategies and the pedagogical use of technological tools, as
well as to promote the revision of curricula to facilitate coordination between
subjects. In this way, it will contribute to the construction of educational
processes that are more coherent, meaningful, and aligned with the current
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Master of Education with a concentration in Mathematics Education
University of Guayaquil victor.barros@ug.edu.ec
https://orcid.org/0000-0001-8542-6454
Master’s Degree in Mathematics Education
University of Guayaquil segundo.camatona@ug.edu.ec
https://orcid.org/0000-0001-8327-2869
Master's Degree in Physics Education
University of Guayaquil gonzalo.naranjov@ug.edu.ec
https://orcid.org/0000-0002-2242-888X
Master's Degree in Operations Research
University of Guayaquil marco.anazcom@ug.edu.ec
https://orcid.org/0000-0003-1022-6487