Multiple Representations
and Meaningful Learning in Mathematics Education During the Transition to
Higher Education
Representaciones
múltiples y aprendizaje significativo en la enseñanza de la matemática durante
la transición a la educación superior
Nancy Karina Tapia Yagual*
Silvia Maribel Placencia Ibadango*
Gustavo Adolfo Román García*
Jorge Wilson Flores Rodríguez*
ABSTRACT
The
transition from secondary education to higher education represents one of the
major challenges in students’ mathematical development. During this stage, many
first-year university students experience difficulties in understanding
fundamental mathematical concepts due to limited connections among different
forms of mathematical representation and the predominance of procedural
learning approaches. In this context, the present study aimed to analyze the
contribution of multiple representations to meaningful learning of mathematical
concepts among first-year university students. A mixed-methods approach with a
quasi-experimental design was employed, involving students enrolled in
introductory Mathematics and Precalculus courses. The instructional
intervention was grounded in Duval’s theory of semiotic representation
registers and Ausubel’s theory of meaningful learning, incorporating activities
that promoted the integration of algebraic, graphical, numerical, tabular, and
verbal representations. The findings revealed significant improvements in
conceptual understanding, graphical interpretation, coordination among
representation registers, and mathematical problem-solving skills. In addition,
students reported positive perceptions of the instructional strategies,
emphasizing that the use of multiple representations facilitated content comprehension
and supported the construction of meaningful knowledge. The study concludes
that multiple representations constitute an effective pedagogical strategy for
enhancing meaningful learning in mathematics during the transition to higher
education. Furthermore, they contribute to the development of mathematical
reasoning, conceptual understanding, and the application of knowledge in
diverse contexts. These findings provide relevant evidence for the design of
educational practices aimed at improving mathematics teaching and learning in
the early years of university education.
Keywords:
Multiple representations; meaningful learning; higher education; mathematics
education.
RESUMEN
La
transición de la educación secundaria a la educación superior representa uno de
los principales desafíos en la formación matemática de los estudiantes
universitarios. Durante esta etapa, muchos estudiantes presentan dificultades
para comprender conceptos fundamentales debido a la escasa conexión entre los
diferentes registros de representación matemática y al predominio de
aprendizajes centrados en procedimientos mecánicos. En este contexto, la
presente investigación tuvo como objetivo analizar la contribución de las
representaciones múltiples al aprendizaje significativo de conceptos
matemáticos en estudiantes de primer ingreso universitario. Se desarrolló un
estudio con enfoque mixto y diseño cuasi experimental, en el que participaron
estudiantes matriculados en asignaturas de Matemática Básica y Precálculo. La
intervención didáctica se fundamentó en la teoría de los registros de
representación semiótica de Duval y en la teoría del aprendizaje significativo
de Ausubel, incorporando actividades que promovieron la articulación entre
representaciones algebraicas, gráficas, numéricas, tabulares y verbales. Los
resultados evidenciaron mejoras significativas en la comprensión conceptual, la
interpretación gráfica, la coordinación entre registros de representación y la
resolución de problemas matemáticos. Asimismo, los estudiantes manifestaron una
valoración positiva de las estrategias implementadas, destacando que la
integración de diferentes formas de representación facilitó la comprensión de
los contenidos y favoreció la construcción de significados. Se concluye que las
representaciones múltiples constituyen una estrategia didáctica efectiva para
fortalecer el aprendizaje significativo de la matemática durante la transición
a la educación superior, contribuyendo al desarrollo del razonamiento
matemático, la comprensión conceptual y la aplicación de conocimientos en
contextos diversos. Estos hallazgos aportan evidencia relevante para el diseño
de propuestas pedagógicas orientadas a mejorar la enseñanza de la matemática en
los primeros años universitarios.
Palabras
clave: Representaciones múltiples; aprendizaje
significativo; educación superior; enseñanza de la matemática.
INTRODUCTION
Teaching mathematics in the early years of higher
education is one of the most significant challenges facing universities,
especially in degree programs where this discipline serves as the foundation
for the development of scientific, technological, and professional
competencies. The transition from high school to college involves not only a
change in educational level but also a profound transformation in the ways
students study, reason, argue, and represent mathematical knowledge. During
this process, many students struggle to understand fundamental concepts,
interpret problems, establish connections between procedures and meanings, and
apply what they have learned to new academic contexts. These difficulties often
become evident in introductory courses such as Basic Mathematics, Precalculus,
Differential Calculus, Physics I, and Statistics, in which mastery of
functions, graphs, equations, tables, and models is essential for academic
progress.
Various studies have indicated that the transition
from high school mathematics to college mathematics represents a critical
moment in students’ educational trajectories. Di Martino et al. (2023), in a
systematic review of the secondary-to-tertiary transition in mathematics
education, argue that this transition is marked by cognitive, institutional,
and pedagogical discontinuities that affect how students engage with
mathematical knowledge. In high school, learning is often organized around
algorithmic procedures, repetitive exercises, and expected answers; in
contrast, college requires greater intellectual autonomy, conceptual
understanding, formalization, interpretation of models, and the ability to
reason. This difference creates tensions, especially when students have
developed a predominantly instrumental relationship with mathematics, based on
the mechanical application of formulas rather than a deep understanding of
concepts.
In this context, the concept of a function plays a
central role. Functions allow us to model relationships between variables,
interpret phenomena, analyze changes, and establish connections between
different areas of knowledge. However, this concept is also often one of the
most problematic for students entering college. Trujillo et al. (2023) state
that the function can be considered a threshold concept in mathematics, since
understanding it is essential for accessing more complex knowledge later on. Nevertheless,
many students struggle to recognize a function in different forms of
representation, interpret the meaning of its parameters, relate an algebraic
expression to its graph, or verbally explain the behavior of a mathematical
relationship. These limitations reveal that learning mathematics cannot be
reduced to symbolic manipulation but rather requires the construction of
meanings that are articulated across various forms of representation.
From this perspective, multiple representations
constitute a highly valuable instructional strategy for promoting meaningful
learning in mathematics. Ainsworth (1999) argues that multiple external
representations can serve different functions in learning, such as
supplementing information, limiting misinterpretations, and building deeper
understandings. Subsequently, Ainsworth (2006) developed the DeFT framework, in
which he argues that the effectiveness of multiple representations depends on
three fundamental dimensions: the design of the representations, the functions
they serve, and the cognitive tasks students must perform when interacting with
them. This means that it is not enough to present an equation, a table, or a
graph in isolation; it is necessary to design activities that require students
to compare, translate, interpret, and coordinate these representations.
In mathematics education, multiple representations
include algebraic, graphical, numerical, verbal, tabular, geometric, and
contextual representations. For example, a linear function can be expressed as
an equation, plotted on a Cartesian graph, organized in a table of values,
described verbally as a relationship of constant change, or applied to a
real-world situation involving proportionality. Each representation offers a
particular perspective on the mathematical object, but none on its own
guarantees understanding. Understanding emerges when students are able to
establish connections among them, recognize their equivalences, interpret their
differences, and use them flexibly to solve problems. In this sense, the use of
multiple representations should not be understood as an accumulation of visual
or symbolic resources, but rather as a cognitive mediation for constructing
mathematical meaning.
Duval’s theory of semiotic representation registers
provides a key foundation for understanding this issue. Duval (2006) argues
that access to mathematical objects is possible only through representations,
since such objects are not directly observable. According to this theory,
learning mathematics involves being able to perform operations within a single
register and make conversions between different registers. Processing occurs,
for example, when a student transforms an algebraic expression without leaving
the symbolic language; conversion, on the other hand, occurs when the student
moves from an equation to a graph, from a table to an algebraic expression, or
from a verbal situation to a mathematical model. For Duval, conversion between
registers constitutes one of the most complex—and, at the same time, most
necessary—cognitive activities for mathematical understanding.
This idea is particularly relevant in the early years
of college, where students must interpret concepts within a more formalized
framework. Many errors are not solely due to a lack of procedural mastery but
to an inability to coordinate registers. A student may solve an equation
correctly yet fail to understand what the result represents graphically; may
construct a table of values but fail to identify the trend of the function; or
may observe a graph but be unable to explain the meaning of its slope, its intercepts,
or its intervals of growth. These difficulties reflect a fragmented
understanding, in which procedures are carried out without being integrated
into a meaningful conceptual structure.
Meaningful learning, from Ausubel’s perspective,
implies that new knowledge is related in a substantive and non-arbitrary way to
the student’s prior cognitive structure. Although Ausubel’s theory was
initially formulated within a broader framework of educational psychology, its
application to mathematics education helps us understand the importance of
connecting new content to experiences, prior concepts, familiar
representations, and contextualized problems. Koskinen and Pitkäniemi (2022),
in a synthesis of research on meaningful learning in mathematics, emphasize
that this type of learning is fostered when instruction promotes connections
between ideas, active interaction, conceptual understanding, and student
participation in the construction of meaning.
Consequently, the use of multiple representations can
be considered a way to promote meaningful learning, provided that these
representations are pedagogically linked to students’ prior knowledge and
cognitive needs. It is not merely a matter of showing various ways to represent
a concept, but rather of creating conditions for students to understand why
these representations are equivalent, what information each one provides, and
how they can be strategically used to solve problems. Polman et al. (2021) note
that meaningful learning in mathematics involves students perceiving meaning,
connection, and utility in what they learn. From this perspective, multiple
representations can help shift the perception of mathematics from a set of
isolated procedures to a language for interpreting relationships, phenomena,
and situations.
The issue takes on greater importance when one
considers that first-year college students come from diverse educational
backgrounds. Some have had a school education focused on calculation and
repetition; others have accumulated conceptual gaps; and many have not
developed independent study habits or reasoning strategies appropriate for
higher education. This diversity highlights the need to design instructional
approaches that enable the assessment, support, and strengthening of
mathematical understanding during the first semesters. In this regard, teaching
based on multiple representations can serve as a transition strategy, as it
helps bridge the gap between high school knowledge and university-level
demands.
Furthermore, this strategy aligns with the current
demands of higher education, which are oriented toward the development of
competencies, problem-solving, critical thinking, and the application of
knowledge in real-world contexts. University-level mathematics should not be
limited to the transmission of techniques but should instead foster an
understanding of structures, relationships, and models. To achieve this, it is
necessary to move beyond approaches focused exclusively on lecture-based
instruction and to promote learning experiences in which students interpret,
argue, compare procedures, and justify their answers. Multiple representations
offer a suitable framework for this purpose, as they require students to engage
different modes of thinking and to recognize that a single mathematical object
can be expressed in various ways.
Despite their importance, incorporating multiple
representations into university-level instruction still presents challenges. In
many cases, graphs are used only as visual aids, tables as a preliminary step
to calculation, and equations as the dominant form of validation. This
hierarchy often reinforces the idea that algebraic representation is the most
important, while the others serve secondary functions. However, research in
mathematics education has shown that deep understanding requires precisely the
coordination among representations, not the subordination of one to another.
Therefore, instruction centered on multiple representations must be
intentionally planned, using activities that promote conversions,
interpretations, and discussions about the meaning of mathematical objects.
From a research perspective, addressing this topic
allows for an analysis not only of academic performance but also of the
comprehension processes that students develop. A study on multiple
representations and meaningful learning during the transition to higher
education can provide evidence regarding students’ initial difficulties, the
ways in which they interpret mathematical concepts, and the effects of a
teaching intervention focused on the coordination of representations.
Furthermore, it can contribute to the design of pedagogical strategies for
introductory courses, especially in institutions where failure or dropout rates
in mathematics courses are a cause for academic concern.
Within this framework, this article aims to analyze
the contribution of multiple representations to the meaningful learning of
mathematical concepts among students transitioning to higher education. The
study is based on the premise that mathematical understanding is strengthened
when students are able to establish relationships between algebraic, graphical,
numerical, verbal, and contextual representations. Furthermore, it is assumed
that meaningful learning does not depend solely on exposure to content, but
rather on the ability to relate that content to prior knowledge, academic
experiences, and problem-solving situations that give meaning to learning.
Thus, this article seeks to address the need to
strengthen mathematics instruction in the early years of college through
teaching strategies that promote conceptual understanding, the integration of
different representations, and problem-solving. The study’s relevance lies in
its focus on a recurring issue in higher education: the gap between the
mathematical knowledge acquired in high school and the cognitive demands of
college-level study. In light of this situation, multiple representations are
presented as a pedagogical alternative capable of promoting deeper, more
flexible, and transferable learning, thereby contributing to a more solid and
meaningful academic transition.
The relevance of studying multiple representations in
higher education is also grounded in the changes that contemporary approaches
to the teaching and learning of mathematics have undergone. Over the past few
decades, research in mathematics education has shifted from perspectives
focused on the acquisition of procedures toward approaches that prioritize
conceptual understanding, the construction of meaning, and the development of
mathematical reasoning. In this context, the ability to interpret and coordinate
different representations of the same mathematical object has become a
fundamental indicator of understanding. Various studies have shown that
students who are able to establish connections between algebraic, graphical,
numerical, and verbal representations develop a deeper understanding of
concepts and demonstrate a greater ability to transfer their knowledge to new
situations (Lesh, Post & Behr, 1987; Kaput, 1989). This perspective
recognizes that mathematical learning consists not only of mastering
operational techniques, but also of constructing networks of meaning that
enable students to interpret, model, and solve problems in different contexts.
The importance of multiple representations is
particularly evident in the study of mathematical functions, which numerous
authors consider one of the most relevant and complex concepts in school and
college mathematics. Functions constitute the language through which phenomena
of change, growth, motion, and dependence among variables are described;
therefore, understanding them is essential in disciplines such as physics,
chemistry, economics, and engineering. However, research conducted in various
educational contexts has shown that many students have a fragmented
understanding of the concept of a function, often limited to the symbolic
manipulation of equations without adequate graphical or contextual
interpretation (Carlson, Oehrtman & Engelke, 2010). This situation creates
difficulties that persist throughout the early years of college and affect the
learning of subsequent content related to differential and integral calculus.
From the perspective of the theory of semiotic
representation registers, these difficulties can be explained by the inability
to coordinate different forms of representation of the same mathematical
object. Duval (2006) argues that authentic mathematical activity requires the
simultaneous mobilization of several semiotic registers and that conceptual
understanding emerges precisely from the ability to convert between them.
Consequently, when students limit themselves to operating within a single
register, their learning tends to be mechanical and superficial. Conversely,
when students are able to relate an algebraic expression to a graph, interpret
a table of values, or verbally describe the behavior of a function, they
develop a more flexible and meaningful understanding. This idea has been
supported by numerous studies that highlight the coordination of
representations as an essential component of advanced mathematical thinking
(Arcavi, 2003; Presmeg, 2006).
At the same time, technological advancements have
expanded the possibilities for using multiple representations in the university
classroom. Tools such as GeoGebra, Desmos, MATLAB, and various computer algebra
systems allow for the dynamic visualization of mathematical concepts that were
previously presented exclusively in symbolic form. The incorporation of these
resources has fostered new ways of interacting with mathematical objects,
allowing students to explore relationships, formulate conjectures, and verify
results using different representations. According to Borba, Askar,
Engelbrecht, Gadanidis, Llinares, and Aguilar (2016), digital technologies have
transformed mathematics education by facilitating visualization and modeling
processes that enrich conceptual understanding. However, the authors caution
that the educational potential of these tools depends on instructional designs
that promote reflection and connections between representations, preventing
technology from becoming merely a calculation tool.
Another relevant aspect relates to the characteristics
of students currently entering higher education. Contemporary university
cohorts are characterized by increasing diversity in terms of prior academic
background, educational experiences, learning styles, and mathematical
competencies. This heterogeneity requires pedagogical strategies capable of
addressing different levels of understanding and fostering inclusive learning
processes. In this regard, multiple representations offer opportunities for
students to access mathematical knowledge from different cognitive entry
points. While some students may better understand a concept through a graphical
representation, others may find greater meaning in a table of values or a
contextualized situation. Integrating these representations broadens the
possibilities for understanding and fosters the construction of shared meanings
within the university classroom.
Likewise, the specialized literature has highlighted
the close relationship between multiple representations and the development of
higher-order mathematical competencies. These competencies include modeling,
reasoning, mathematical communication, and complex problem-solving. Niss and
Højgaard (2019) note that mathematical competence involves the ability to
understand, use, and interpret mathematics in different situations, which
necessarily requires the mobilization of various representational systems. From
this perspective, multiple representations are not merely a teaching strategy
but an inherent component of the development of mathematical thinking and the
comprehensive education of college students.
The connection between multiple representations and
meaningful learning takes on particular relevance when analyzed from the
perspective of cognitive learning theory. Ausubel (2002) argues that the
acquisition of new knowledge depends on the ability to relate it in a
meaningful way to concepts already present in the student’s cognitive
structure. Consequently, meaningful learning requires that the content
presented possess logical and psychological coherence, and that the student
adopt a positive attitude toward understanding. Multiple representations
contribute to this process because they facilitate the establishment of
connections between different forms of knowledge, allowing students to relate
abstract concepts to prior experiences, concrete situations, and contexts of
application. In this way, representations act as cognitive bridges that promote
the construction of more stable and lasting meanings.
The available empirical evidence supports this
relationship. Various studies have reported significant improvements in
conceptual understanding, problem-solving, and academic performance when
teaching strategies based on multiple representations are implemented
(Ainsworth, 2006; Rau, Aleven & Rummel, 2015). This research suggests that
the coordination of representations fosters more elaborate reasoning processes,
reduces reliance on rote memorization, and promotes a deeper understanding of
mathematical concepts. Furthermore, it has been observed that students develop
a greater ability to explain their procedures, justify their answers, and
transfer knowledge to new situations—aspects closely linked to the principles
of meaningful learning.
Despite the advances made in international research,
there remains a need to further explore these relationships in Latin American
contexts, particularly in the realm of higher education. Institutional
conditions, curricula, educational trajectories, and the sociocultural
characteristics of students can significantly influence how mathematical
learning is constructed. Therefore, it is important to generate contextualized
evidence that allows us to understand how multiple representations can help
strengthen mathematics instruction during the first years of college and
facilitate the transition from secondary to higher education. This interest is
particularly relevant in science and technology programs, where difficulties in
mathematics often constitute one of the main factors associated with low
academic performance, course repetition, and student dropout.
Consequently, the study of multiple representations
and their relationship to meaningful learning emerges as a relevant line of
research for understanding and improving the processes of teaching and learning
mathematics in higher education. Analyzing how students interpret, coordinate,
and use different representations of mathematical concepts will make it
possible to identify difficulties, design more effective teaching strategies,
and contribute to the development of instructional models aimed at fostering
deep conceptual understanding. From this perspective, this research posits that
the connection between multiple representations and meaningful learning is a
key element in strengthening university students’ mathematical education and
fostering a more successful academic transition to higher levels of study.
MATERIALS
AND METHODS
This research was conducted using a mixed-methods
approach, based on the premise that understanding mathematical learning
processes requires both the objective measurement of academic outcomes and the
analysis of students’ experiences and perceptions during their educational
journey. The combination of quantitative and qualitative methods provided a
comprehensive view of the phenomenon under study, facilitating the
identification of changes in learning and the understanding of the factors
involved in the construction of mathematical meaning. This approach is grounded
in the need to analyze not only students’ performance but also the cognitive
processes associated with the use of multiple representations in understanding
mathematical concepts during the transition to higher education.
The research design was quasi-experimental with
descriptive and explanatory scope. The study involved two groups of students
enrolled in Basic Mathematics and Precalculus courses during their first
semester of college. An experimental group participated in a
instructional intervention based on the systematic use of multiple
representations, while the comparison group engaged in the standard activities
outlined in the course syllabus. Due to institutional conditions and the
existing academic organization, it was not possible to randomly assign
participants; for this reason, a quasi-experimental design was chosen to
evaluate the effects of the intervention in real educational contexts.
The population consisted of first-year students
enrolled in science, engineering, and education programs at a public university
in Ecuador. The sample was selected using a non-probabilistic convenience
sampling procedure, taking into account course availability and the
participants’ voluntary agreement to take part in the study. A total of 120
students participated, divided into two working groups. The selection of
first-semester students stemmed from an interest in analyzing the learning
processes that occur during the transition from secondary to higher education,
a stage characterized by significant changes in cognitive, methodological, and
academic demands.
The instructional intervention took place over eight
weeks and focused on the study of fundamental concepts related to mathematical
functions, variation, graphical interpretation, and the modeling of real-world
situations. These topics were selected due to their relevance in introductory
college mathematics courses and the evidence reported in the literature
regarding the difficulties students face in understanding them. During the
teaching process, activities were designed to promote coordination among different
registers of semiotic representation, including algebraic, graphical,
numerical, tabular, verbal, and contextualized representations.
The activities implemented in the experimental group
were based on Duval’s theoretical principles regarding registers of semiotic
representation and on Ausubel’s postulates of meaningful learning.
Consequently, each topic was addressed through instructional sequences that
required students to interpret, compare, and transform information across
different registers of representation. For example, exercises were proposed in
which students had to construct graphs from equations, derive algebraic
expressions from data tables, verbally interpret the behavior of functions, and
model contextualized situations using appropriate mathematical representations.
These activities were designed to promote conceptual understanding and prevent
learning from being limited to the mechanical application of procedures.
To support the intervention, technological resources
were used to facilitate the dynamic visualization of mathematical concepts.
Among these, the GeoGebra software was used to represent functions, analyze
variations, and explore relationships between different representations. The
incorporation of this tool allowed students to simultaneously observe changes
in algebraic expressions, tables, and graphs, thereby promoting an
understanding of the relationships between the different representations.
Additionally, collaborative work was encouraged through group activities
focused on discussing and justifying mathematical solutions.
Various instruments were used to collect data. First,
a diagnostic test was administered before the intervention began to identify
students’ initial level of understanding of concepts related to functions and
mathematical representations. This test included exercises in graphical
interpretation, table analysis, the construction of algebraic models, and the
solving of contextualized problems. Subsequently, at the end of the
intervention, an equivalent test was administered to assess changes in learning
and compare the results obtained by both groups.
In addition, a perception questionnaire consisting of
Likert-scale items was designed to gather information about the students’
experiences during the learning process. This instrument provided insight into
the participants’ assessments of the use of multiple representations, the
usefulness of the resources employed, the level of understanding achieved, and
the influence of the activities carried out on their mathematical learning. The
internal consistency of the questionnaire was evaluated using Cronbach’s alpha
coefficient, yielding values above 0.80—considered adequate for educational
research purposes.
To gain a deeper understanding of the quantitative
results, semi-structured interviews were conducted with a group of students
selected through purposive sampling. These interviews allowed us to explore the
problem-solving strategies used, the difficulties encountered during the
learning process, and perceptions regarding the relationship among the
different mathematical representations. The information obtained helped
identify patterns of conceptual understanding and provided complementary
qualitative evidence for interpreting the research findings.
The quantitative data were analyzed using descriptive
and inferential statistical techniques. Initially, frequencies, percentages,
means, and standard deviations were calculated to characterize student
performance. Subsequently, tests for comparing means were applied to determine
whether there were statistically significant differences between the results
obtained before and after the intervention, as well as between the experimental
group and the comparison group. Additionally, effect size measures were calculated
to estimate the magnitude of the changes observed in the participants’
mathematical understanding.
Meanwhile, the qualitative data from the interviews
were analyzed using a thematic coding process. The responses were organized
into categories related to conceptual understanding, the use of multiple
representations, learning difficulties, and perceptions regarding the
usefulness of the implemented teaching strategies. Subsequently, an
interpretive analysis was conducted to identify relationships between the
students’ discourses and the results obtained on the academic tests. This
procedure allowed for the integration of quantitative and qualitative
information, strengthening the validity of the conclusions reached.
To ensure the methodological quality of the study,
widely accepted criteria for validity and reliability in educational research
were applied. The instruments were reviewed by experts in mathematics education
and research methodology, who assessed the relevance, clarity, and coherence of
the items. In addition, a pilot study was conducted with students whose
characteristics were similar to those of the final sample, allowing for
adjustments to be made prior to the formal implementation of the study. The triangulation
of data obtained through academic tests, questionnaires, and interviews further
contributed to strengthening the credibility of the results.
From an ethical standpoint, the research adhered to
the principles of voluntary participation, confidentiality, and responsible use
of information. All participants were informed about the study’s objectives and
gave their consent to participate in the various phases of the research
process. The data collected were used exclusively for academic and scientific
purposes, ensuring the students’ anonymity during the analysis and presentation
of the results.
Overall, this methodology allowed for a rigorous
examination of the influence of multiple representations on the meaningful
learning of mathematical concepts among first-year college students. The
integration of different sources of information and the combination of
quantitative and qualitative approaches provided sufficient evidence to
understand both the results obtained and the underlying learning processes,
contributing to the generation of knowledge relevant to improving mathematics
instruction in higher education.
RESULTS
The results show a significant improvement in the
performance of the students who participated in the instructional intervention
based on multiple representations. A comparative analysis of the tests
administered before and after the intervention revealed progress in conceptual
understanding, graphical interpretation, the relationship between different
modes of representation, and mathematical problem-solving.
In the initial diagnostic test, it was observed that
students had greater difficulties with activities requiring them to establish
connections between algebraic, graphical, and tabular representations. Most
participants were able to apply basic algorithmic procedures but showed
limitations in interpreting the meaning of mathematical expressions and
relating them to contextualized situations. These difficulties were
particularly evident in exercises related to functions, variation, and
graphical analysis.
After implementing the instructional strategy, the
results showed a general improvement across all assessed indicators. Graphical
interpretation rose from an initial average of 48% to 82%, while conceptual
understanding increased from 52% to 86%. Similarly, the ability to relate
different forms of representation increased from 44% to 81%, representing one
of the most significant improvements observed during the study. Finally, the
ability to solve contextualized math problems improved from 55% to 88%, demonstrating
students’ greater capacity to apply the knowledge they acquired in diverse
situations.
Figure 1. Comparison of pretest and posttest results
for the evaluated indicators
The statistical analyses conducted revealed
significant differences between the results obtained before and after the
intervention (p < 0.05). These findings suggest that the systematic use of
multiple representations fostered the construction of more solid mathematical
meanings and a better understanding of the concepts addressed. Furthermore, the
effect size was high, indicating that the observed changes were not only
statistically significant but also educationally relevant.
The results of the perception questionnaires showed a
positive evaluation of the implemented strategy. Eighty-nine percent of the
students stated that the simultaneous use of equations, graphs, tables, and
contextualized situations facilitated their understanding of the mathematical
content. Furthermore, 92% felt that the activities carried out allowed them to
visualize more clearly the relationships between the concepts studied and their
practical applications.
The interviews conducted complemented these findings.
The students noted that previously they tended to solve problems mechanically,
focusing solely on the application of formulas. However, during the
intervention, they began to identify relationships between different forms of
representation, which allowed them to better understand the meaning of the
procedures they were performing. Several participants emphasized that graphical
interpretation helped them verify results obtained algebraically and understand
patterns of variation that they had previously perceived as abstract.
Overall, the results suggest that multiple
representations constitute an effective teaching strategy for promoting
meaningful learning of mathematics among first-year college students. The
observed improvement in conceptual understanding, the coordination of
representational registers, and problem-solving supports the idea that
mathematical learning is strengthened when students have opportunities to
establish connections between different ways of representing the same concept.
These findings align with the theoretical approaches of Duval and Ainsworth,
who argue that mathematical understanding emerges from the ability to
coordinate and interpret multiple representational registers.
DISCUSSION
The results obtained in this study lead to the
conclusion that the use of multiple representations is an effective teaching
strategy for promoting meaningful learning of mathematics during the transition
to higher education. The integration of algebraic, graphical, numerical,
tabular, and verbal representations facilitated the understanding of
fundamental mathematical concepts, allowing students to build stronger
relationships between the different modes of representation and develop a
deeper conceptual understanding of the content covered.
The study revealed that one of the main difficulties
faced by first-year college students lies in their limited ability to establish
connections between different ways of representing the same mathematical
object. Although many participants demonstrated the ability to perform
algebraic procedures, they struggled to interpret graphs, analyze data tables,
or verbally explain the meaning of the results obtained. This situation
confirms Duval’s theoretical arguments that mathematical understanding depends
largely on the ability to coordinate and transform different registers of
semiotic representation.
Furthermore, the findings show that the implementation
of activities aimed at promoting conversion between representations contributed
significantly to strengthening conceptual understanding. Students not only
improved their academic performance but also developed a greater ability to
interpret mathematical phenomena, justify procedures, and apply knowledge in
contextualized situations. These results support the idea that mathematical
understanding goes beyond the memorization of formulas and procedures, requiring
processes of active meaning-making.
Another relevant aspect identified during the study
was the relationship between multiple representations and the principles of
meaningful learning. The ability to establish connections between prior
knowledge and new forms of representation allowed students to make greater
sense of the mathematical content studied. In this regard, the research
confirms Ausubel’s postulates by demonstrating that the most enduring and
transferable learning occurs when new information is substantially integrated
into the student’s cognitive structure.
The results also highlight the importance of
incorporating active learning methodologies into introductory college
mathematics courses. Activities based on the interpretation, comparison, and
transformation of representations fostered more dynamic student participation
in their learning process, promoting mathematical reasoning, argumentation, and
problem-solving. This suggests the need to move beyond traditional approaches
focused exclusively on lecture-based instruction and the mechanical completion
of exercises, and to advance toward pedagogical models that prioritize
conceptual understanding and the construction of meaning.
Similarly, the use of technological tools to support
teaching helped strengthen the processes of mathematical visualization and
interpretation. The ability to simultaneously observe different representations
of the same concept facilitated the identification of relationships and
patterns that, in many cases, are difficult to perceive using traditional
methods. This demonstrates the potential of digital resources to enrich
university-level mathematics education when they are integrated into
pedagogically sound instructional approaches.
The observed improvement in problem-solving
constitutes another significant finding of the research. The students who
participated in the intervention demonstrated a greater ability to analyze
situations, select appropriate strategies, and justify their answers using
different representations. This result suggests that multiple representations
not only promote conceptual understanding but also contribute to the
development of higher-order mathematical competencies, which are fundamental to
academic and professional performance in scientific and technological
disciplines.
From an institutional perspective, the findings
highlight the need to strengthen academic support strategies during the first
years of college. The transition from secondary education to higher education
represents a period of adaptation that requires methodologies capable of
bridging the gaps between students’ prior knowledge and the cognitive demands
of the college level. In this context, multiple representations can serve as a
valuable pedagogical tool to facilitate this process and improve retention rates
and academic success.
Finally, it is concluded that mathematics instruction
based on multiple representations fosters the construction of meaningful
learning, strengthens conceptual understanding, and promotes more active
student participation in their educational process. The results obtained
provide empirical evidence supporting the systematic incorporation of this
strategy into introductory higher education courses and suggest the need to
continue conducting research that delves deeper into its impact on different
mathematical content areas, educational contexts, and student populations. In
this way, it will be possible to move toward teaching models that are more
inclusive, comprehensive, and oriented toward the holistic development of the
mathematical competencies required in contemporary university education.
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* Magíster en Educación Mención Enseñanza de la Matemática, Universidad de
Guayaquil
nancy.tapiay@ug.edu.ec
https://orcid.org/0000-0001-7834-0265
*Magíster en Educación Mención Enseñanza de
la Matemática, Universidad de Guayaquil
silvia.placenciai@ug.edu.ec
https://orcid.org/0000-0003-3164-1639
*Magister en Medicina Forense, Universidad de
Guayaquil
gustavo.romang@ug.edu.ec
https://orcid.org/0009-0003-3752-4393
*Magíster en Educación Superior, Universidad
de Guayaquil
wilson.floresr@ug.edu.ec