|
Article |
El método singapur como estrategia
para mejorar el aprendizaje de las matemáticas
Margot Mercedes
García Espinoza [*]
Abstract
This article presents a study whose purpose is to
evaluate the consequences of applying the Singapore method as a strategy to
promote the mathematics learning process in a fourth-grade class at the Trece
de Abril Educational Unit in Santa Elena, Ecuador. This is a quantitative quasi-experimental
design that compared a teaching group using the Concrete-Pictorial-Abstract
approach with a control group taught using traditional methods. The results
reflect a noticeable improvement in the academic performance of the
experimental group, especially in logical reasoning, concept comprehension, and
problem-solving skills. Likewise, a more favorable attitude toward mathematics,
enthusiasm for the subject, and a higher level of active participation in class
were observed. Errors in interpretation, calculation, and procedure decreased
significantly. The perception of the instructors and th s was extremely
positive. It indicates a clear contribution to lesson planning, modulation of
content intensity and speed, and manifestation of the educational role as a
guide. The proposed outcome is the MAMMS–2025 Model. In summary, the Singapore
method is an effective teaching strategy that can be applied in public schools
to improve learning outcomes and the effectiveness of the teaching experience.
Key words: Singapore method, meaningful learning, problem
solving, basic education, logical-mathematical thinking
Resumen
En este artículo, se presenta
una investigación cuyo propósito radica en evaluar las consecuencias de la
aplicación del método Singapur como estrategia para fomentar el proceso de
aprendizaje de las matemáticas, dentro de un cuarto año básico de la Unidad
Educativa Trece de Abril, en Santa Elena, Ecuador. Se trata de un diseño
cuasi-experimental de carácter cuantitativo, que comparó un grupo de enseñanza
con el enfoque Concreto–Pictórico–Abstracto en un conjunto de estudiantes y un
grupo control enseñado con los métodos tradicionales. Los resultados reflejan
un notorio progreso del rendimiento académico del grupo experimental,
especialmente en lo que corresponde al razonamiento lógico, la comprensión de
los conceptos y la competencia en la resolución de problemas. Asimismo, se
observó una actitud más favorable hacia las matemáticas, entusiasmo por la
asignatura y un nivel de participación activa en las clases también superior.
Los errores de interpretación, cálculo y procedimiento disminuyeron de forma
significativa. La percepción por parte de los docentes instructoras fue
extremadamente positiva. Indica claro aporte para la planificación de clases,
modulación de la intensidad y velocidad de contenido y manifestación del rol
educativo como guía. La consecuencia propuesta es el Modelo MAMMS–2025. Se
puede resumir que el método Singapur es una estrategia didáctica eficaz y
aplicable en los centros de enseñanza públicas, habilitado para mejoramiento de
los índices del aprendizaje y la eficacia de la experiencia pedagógica.
Palabras clave: Método Singapur, Aprendizaje significativo, Resolución de problemas,
Educación básica, Pensamiento lógico-matemático
Introduction
Internationally,
the results of standardized assessments such as the Programme
for International Student Assessment have repeatedly shown that students in
Latin America perform poorly in mathematics (Torres, 2023). In this sense, it
is clear that this harsh reality hides a deep gap in the development of
logical-mathematical thinking and a visible limitation in the possibility of
producing individuals with the critical and analytical skills necessary to act
in an increasingly vast, complex, and technologically developed world.
Therefore, it is hoped that the above analysis will contribute to the need to
rethink conventional methods of teaching mathematics and move towards others
that allow for a deeper understanding of fields such as these.
The Singapore
method has been recognized as one of the most effective pedagogical
methodologies in mathematics education worldwide. Countries that have adopted
the methodology, such as the United States, the United Kingdom, and Chile, have
observed a persistent improvement in academic performance and in the process of
reasoning and problem solving in students (Ary et al., 2022). This method is
based on the CPA model, which stands for concrete, pictorial, and abstract.
According to (Cook & Campbell, 2021), this framework follows the
inductive learning theory described developed by Jerome Bruner (see Table 1).
Instead of presenting mathematical knowledge to students, the framework first
allows students to actively develop knowledge through the manipulation of
materials before introducing abstract concepts.
Another excellent
feature of this method is its use of various visual representations,
particularly the bar model, which makes thinking and solving complicated
problems more understandable. In addition, this method encourages the
development of metacognitive skills, such as generalization, visualization, and
reflection (Kaur, 2022). These skills are more critical, as citizens who are
mathematically literate are expected to apply what they have learned in a range
of rapidly changing and diverse contexts.
In Ecuador, the
application of the Singapore methodology is still limited, but it has become a
reliable option for improving the mathematics learning process, particularly
for low-achieving students (Naraba, 2022). Therefore,
the problem addressed in this research lies in the poor performance of high
school students in mathematics due to outdated and inefficient approaches.
Thus, the overall objective of this research is to determine the effect of the
Singapore methodology on mathematics learning as a teaching tool in fourth-year
high school students.
With regard to the
relevance of the research methodology, a quantitative approach was proposed.
This made it possible to accurately and objectively measure the impact of the
Singapore approach on mathematics. The study focused on two groups: one that
was introduced to the CPA (Concrete, Pictorial, Abstract) approach and the
other that followed traditional approaches. In comparison with academic
performance, the measurement of motivational variables and attitude towards the
subject would be carried out, as well as teachers' perceptions of the possible
viability of the approach.
Fundamentals
of the Singapore Method
On the other hand,
there is the Singapore method, an educational model created by the Singapore
Ministry of Education in the 1980s and based on a teaching structure that
focuses on problem solving with the aim of improving understanding of
mathematics. The Singapore method will provide students with a deeper level of
mathematical understanding with clarity. The four different stages of
progression are: Progression, Concrete, Pictorial, Abstract.
· Concrete
stage, in which students manipulate real objects and physical materials
themselves;
·
Pictorial phase, in which students
represent objects by drawing or sketching them. The pictorial phase includes the bar model for quantities.
· Abstract
phase, in which they carry out operations using numbers and symbols. The
progressive sequence here helps develop logical-mathematical thinking, allowing
students to move from the concrete to the abstract symbolic (Torres, 2023).
The concrete phase
is used with hands-on materials to solve math problems and, as a result, gives
students an initial understanding. Then, in the pictorial phase, students
graphically represent mathematical relationships using diagrams and visual
models, such as the bars of the relationship model. Finally, students move to
the abstract stage, where they solve math problems using symbols and notation.
This helps students solidify their underlying concepts (
Effectiveness
of the Singapore Method in Teaching Mathematics
The Singapore
method was developed during the 1980s by the Singapore Ministry of Education in
response to the urgent need to reform and improve mathematics education by
providing a solid conceptual framework and a structured approach. Among the
ideological foundations of the Singapore method are the theories of educators
Jerome Bruner, Zoltan Dienes, and Richard Skemp regarding the importance of the
constructivist approach to moving from concrete to abstract knowledge. The
method relies more on school-based problem situations, teaching materials,
visual representations, and gradual learning sequences, also known as
Concrete–Pictorial–Abstract, which is broken down into three phases: concrete,
pictorial, and abstract.
Several studies
have supported the effectiveness of this method. Barja (2025) indicates that in
a quasi-experimental study, he compared the performance of students who used
the Singapore method to those who used traditional methodologies, corroborating
improvements in mathematical problem solving and in the development of
mathematical competence such as logical reasoning and conceptual understanding.
Similarly, the Innovamat (2023) study, conducted in
US schools through the "Math in Focus" program, shows that students
who used the approach examined showed significant growth in their mathematical
abilities, especially in terms of openness to understanding and ability to
tackle complex problems.
It can be said
that the Singapore method not only has a positive impact on academic
performance, but also transforms the teaching-learning process itself, making
it more meaningful, visual, and participatory. The CPA sequence allows students
to build knowledge progressively in a concrete way, reducing both procedural
errors and their cognitive dependence on the teacher. In addition, its
structure simplifies the teacher's work and facilitates the development of a
differentiated curriculum, making it an effective, contextualizable, and
replicable pedagogical strategy with the potential to generate a beneficial
impact on the Ecuadorian education system.
Application
of the Singapore Method in Latin American contexts
The Singapore
method has proven effective in multiple contexts, but its application in Latin
America faces specific challenges. According to (
The implementation
of the Singapore Method is not yet very developed in Ecuador; however, there
are pilot programs seeking to understand how effective it is in their context.
According to
Development
of Mathematical Thinking and Metacognition
Singapore is not
only positioned as a method for acquiring procedural skills but also for
mathematical thinking and metacognition.
Likewise,
Pedagogical
principles of the constructivist approach in the Singapore method
According to
The key
pedagogical principles that emerge from the constructivist approach to
geography present in the Singapore method are:
· Learning:
the
student plays an active role in the process of constructing knowledge.
· Progression: the
concrete to the abstract appears sequenced.
· Relevance:
the
educational content is related to the world and the students' experiences.
· Social
interaction: collaboration and dialogue foster knowledge
construction, autonomy, and metacognition: consideration of the learning
process itself is encouraged.
The method also
emphasizes problem solving as the core of learning and encourages students to
seek answers using multiple strategies, while promoting the development of
critical and metacognitive thinking (Mendoza & López, 2023).
Formative
assessment and feedback in the Singapore method
Formative
assessment is essential in the Singapore method. Therefore, through this
assessment, teachers often monitor students' progress and provide constant
feedback at the end of the lesson
Role of the
teacher as a mediator of learning
Implications
of the Singapore method for inclusive education
The CPA approach
of the Singapore method is also particularly suitable for students with special
educational needs. The methodology is beneficial for these students because the
various methods of mathematical representation it uses are adapted to their
learning style and facilitate understanding. Again, the use of concrete and
visual material makes the content more accessible: Knowing how to learn and
what is easiest to learn gives students with disabilities a means of knowing
how to do the work better without having to invest as much time in it (
Comparison
between the Singapore method and other active methodologies
Bacus and Guillena (2023) state that the Singapore method has many
similarities with other active approaches, such as problem-based learning and
the flipped classroom. However, all activities are student-centered and
stimulate autonomous knowledge acquisition and the functional consequences of
thinking ( ). But the Singapore method differs in its
work structure. Above all, it is based on concrete, pictorial, and abstract
progression, and here too, it offers the best possible sequential basis for
understanding conceptual achievement. It includes three stages: in the first
stage, the use of manipulative objects is essential; in the second stage, it is
necessary to develop the pictorial stage with the help of visual
representation: diagrams, models. And in the third stage, it is essential to
use symbols and perform formal mathematical operations.
Unlike other
active methodologies which, due to the lack of a structured sequence, leave
students unsure of whether they have remembered the experience, the CPA
approach of the Singapore model accompanies students from the concrete object
to mathematical abstraction. This achieves a gradual, meaningful, and lasting
construction of knowledge. This is why it is one of the most effective
approaches in contexts where students have high difficulty in understanding
abstract concepts and low levels of performance in mathematics.
Enriching
mathematics teaching by integrating elements of the Singapore method and other
types of active methodologies allows it to be adapted to the needs of each case
and each educational context
Educational
technologies complementary to the Singapore method
New educational
technologies, such as digital platforms and interactive applications, can
strengthen Singapore method instruction by providing additional resources and
opportunities for independent practice
Long-term
performance of students trained with the Singapore method
According to
Implementation
of the Singapore method in initial teacher training
Materials and methods
The type of research undertaken in this study was
quantitative, as it focused on objective and numerical measurements of the
effects of the Singapore method on Also, cause-and-effect relationships were
identified through statistical analysis of the data collected with validated
instruments. According to Hernández-Sampieri, Mendoza, and Fernández (2021),
this approach is characterized by its empirical, systematic, and controlled
nature, which allows for the formulation of hypotheses and the evaluation of
rigorous research on educational interventions. On the other hand, this study
design took an applied approach with the purpose of intervening in a very
specific educational problem: low student performance in mathematics, which was
addressed with an innovative pedagogical methodology.
The research approach was quantitative in nature,
but a quasi-experimental design was implemented. Two groups were formed:
experimental and control, despite not being adequately randomized. However,
pre-test and post-test measurements were applied to both, allowing for a
comparison of the results before and after the pedagogical intervention. This
design was appropriate in the school setting, where randomization was not
possible, but evidence of the effectiveness of the educational treatment was
still required (Cook & Campbell, 2021).
The research approach in this case was quantitative,
and a quasi-experimental design was implemented. Although this design did not
maintain strong random assignments, subjects were assigned to the experimental
and control groups. At the same time, certain pre-test and post-test
measurements were carried out in both groups, allowing for a comparison of the
results obtained before and after the pedagogical intervention. This design
proved to be relevant in this case, as the school setting did not lend itself
to strong randomization, but still required evidence of the treatment's
effectiveness (Cook & Campbell, 2021).
The level of research was explanatory, insofar as
this type of study is used to identify and analyze the effect or influence of
an independent variable or IV Incógnito on a variable
whose incidence is to be described or explained, which is V, in order to
determine whether the differences in the measured results existed in a
statistically significant manner between the groups evaluated (Ary et al.,
2022).
The population consisted of fourth-year basic
education students from the Trece de Abril Educational Unit in the canton of La
Libertad, as well as teachers from the area, for a total of three teachers. A
non-probability sample was used for convenience, which considered three
parallel groups at the same level, with two assigned to the experimental group
and one to the control group. This procedure ensured internal validity with a
sample that was similar in number and had the same socio-educational
characteristics.
Table 1. Population
|
Grade |
Girls |
Boys |
Total |
|
4th “A” |
1 |
16 |
3 |
|
4th "B" |
14 |
14 |
28 |
|
4th "C" |
21 |
20 |
31 |
|
Total population |
8 |
||
Source: Trece de Abril Educational Unit database
Prepared by: Author
Before beginning the intervention according to the
Singapore method, a pretest was administered, followed by a final test or
posttest after the intervention was completed. Both tests were identical and
measured the same skills. The pilot tests were aligned with the national
curriculum and validated by mathematics education experts. The pretest included
items that addressed topics such as:
the ability to solve basic numerical problems,
fundamental operations (addition, subtraction, multiplication, and division),
·
the
interpretation of mathematical statements,
·
the use of
logical reasoning, and
·
the
identification of patterns and simple relationships.
On the other hand, general affective aspects were
also analyzed using a questionnaire on attitudes toward mathematics, with a
five-level Likert scale, which evaluates the following factors:
· motivation towards the subject,
·
interest in
learning mathematics,
·
perception of
difficulty,
·
level of
enjoyment in solving problems,
·
and personal
confidence when facing mathematical activities.
Both instruments were administered before and after
the intervention, enabling a comparison before and after the intervention in
terms of students' cognitive performance and affective attitude toward
mathematics.
First, the groups were assigned and the pretest was
administered. Then, the experimental group was taught using the Singapore
approach for a period of 8 to 10 weeks. The strategies used with the students
were concrete, pictorial, and abstract. At the same time, mathematics was
taught using traditional pedagogy. After the intervention ended, the post-test
and attitude questionnaire were administered to the groups.
Results
The results of the survey administered to the 83
fourth-year students at the Trece de Abril Educational Unit in the canton of La
Libertad corroborated a significant increase in academic performance in
mathematics through the implementation of the Singapore method. The students'
perception of the change met the expected objective, as they perceived
improvement both in the construction of their understanding and in the
resolution of mathematical problems. Eighty-two percent of those surveyed said
they felt more motivated to learn, and 76% said they had learned more by being
able to use concrete and visual materials.
In turn, in terms of items associated with
performance, such as ease of problem solving, use of logical reasoning, and
confidence in performing exercises, the averages obtained ranged from 4.0 to
4.6 on a scale of 1 to 5 in the medium-high and high levels, respectively.
Here, the trend is that most students perceived a real improvement in their
mathematical skills as a result of the structured CPA approach outlined by the
Singapore method.
Illustration 1 Distribution of academic performance levels
Source: Research
conducted
Prepared by: Author
Statistical analysis showed that more than 70% of
students were in the medium-high and high performance
levels, suggesting effective acquisition of basic and advanced math skills.
This improvement was consistent across the three fourth-grade classes,
demonstrating that the impact of the methodology was widespread and did not
depend on variables such as the teacher, the group, or others.
Consequently, the implementation of the Singapore
method had a positive impact on students' academic results, not only in terms
of improved grades, but also in terms of increased self-confidence and
motivation, and growth in logical-mathematical thinking. In other words, the
results obtained allow us to conclude with certainty that the model is a
methodologically acceptable, feasible, and highly efficient alternative for
improving the quality of knowledge acquired in public education institutions.
The figure shows the distribution of academic
performance levels perceived by 83 fourth-year students using the Singapore
method. The results reveal a marked predominance of medium 3-4 and medium-high
4-5 performance levels ; that is, students positively
evaluate their mathematical knowledge and skills acquired after the
implementation of the teaching approach.
Specifically, more than half of the students, or 52,
are at the medium level. Here, all comprehensive and functional mathematical
work is demonstrated. In second place, 30 students are at the medium-high
level, indicating a remarkable approach to logical thinking, problem solving,
and self-confidence. The marginal presence at the lower-middle level, one case,
and the total absence at the low level (1–2) are clear indications that the
Singapore method really does raise the floor and achieve dramatic decreases in
low performance. In general, low performance occurs in the case of more
vulnerable students or those who are behind in their education.
This distribution confirms that, based on the model
applied, there was inclusive and progressive learning in which students with
different learning rhythms and styles were able to consolidate the basics.
Therefore, it can be said that, overall, the graph supports the effectiveness
of the methodology in consolidating basic learning and propelling students
toward advanced learning.
The following box plot shows the distribution of the
average academic performance variable perceived by students according to their
parallel groups A, B, and C in the fourth year of the Trece de Abril
Educational Unit after applying the Singapore method in mathematics. In
general, it can be observed that the three parallel groups have very similar
behavior in terms of academic performance. The medians, which are on the red
line inside the boxes, are very close, which means that the most typical
performance is equidistant in the three groups. This means that the Singapore
method has been applied uniformly, not assigned to a single parallel group.
Likewise, the boxes representing the interquartile
range show a concentration of data within a medium-high point, indicating a
high consistency among subjects regarding their perception of academic
improvement. The small number of extreme or atypical values and the moderate
width of the whiskers indicate an absence of internal gaps between students in
the same parallel. By contrasting these results, it was demonstrated that the
Singapore method not only stabilized but also raised academic performance in
all parallel classes, delivering equity in results and reducing internal
disparities. This supports and validates the methodological approach as a
replicable and valid strategy in various educational settings.
Illustration 2. Distribution of academic performance levels by
parallel
Source: Research conducted
Prepared
by: Author
Reduction in logical reasoning and
procedural errors
The results of this survey showed that the program
applied to the 83 students in parallel classes Fourth A, B, and C at the Trece
de Abril Educational Unit significantly improved the quality of logical
reasoning processes and the execution of mathematical procedures. This was
evident in the scores awarded to items related to understanding statements,
using strategies, and accuracy in solving exercises, where students reported
feeling more confident and competent in addressing problems in a structured and
thoughtful manner.
The use of the Concrete-Pictorial-Abstract sequence
allowed students to gradually master mathematical concepts without premature
abstraction, virtually "grounding" abstract ideas and killing them in
their infancy. First, students worked with materials representing real
situations in the concrete phase. Then, they translated them into visual images
in the pictorial phase. Finally, in the abstract stage, they operated with
mathematical symbols. The action scheme provided understanding of related
numerical phenomena, the logical sequence of steps, and self-verification.
In addition, it was observed that errors that were
previously committed most frequently, such as misinterpretation of the problem,
inappropriate use of operations, or disordered follow-up of steps, were
significantly less prevalent, especially in the parallel groups where
activities involving visualization and verbalization of mathematical thinking
had previously been carried out systematically. Students stated that the use of
visual representations allowed them to understand the meaning of the problem before
operating, and that working with manipulative materials enabled them to find
several ways to reach the solution.
Overall, this improvement involves not only a
reduction in mechanical errors, but also progress in quality reasoning,
effective planning, and process reflection, as well as positive self-concept in
mathematics. The consistency between the A, B, and C results leads us to
conclude that the Singapore method is effective not only in individual terms
but also offers collective improvement in the mastery of logical procedures. It
is a pedagogical tool that allows for a deeper and more lasting understanding
and consolidation of what has been learned.
Illustration 3. Distribution of students according to their
performance level by parallel
Source: Research
conducted
After applying the
Singapore method, we can see the detailed distribution of students from
different parallel classes in Fourth A, B, and C at the Trece de Abril
Educational Unit according to their performance range in logical reasoning and
mathematical procedures.
With regard to
logical reasoning, the trend among the vast majority of students in the three
courses is to concentrate in the highest range (4-5), which then expresses the
appropriation of logical-mathematical thinking. This implies that students not
only understood the mathematical content presented, but also acquired skills in
the area of structured reasoning, pattern identification, and thoughtful
mathematical decision-making. Courses A and C differ from the others in
percentage terms, with the highest number of students with very high
logical-mathematical reasoning, but practically none or none with very low
reasoning.
A similar
distribution was observed in relation to mathematical procedures; however,
students showed more precise and orderly development when solving the
exercises. The distribution shows a fairly marked range between the medium-high
and high levels, demonstrating that applying the CPA-based method strengthens
the clarity of the solution steps, reduces common errors, and favors the
planning and sequencing of procedures. Again, parallels A and C show a more
favorable distribution; however, parallel B also shows a favorable evolution.
The absence of
students in the low range (1–2) in both graphs is particularly important, as it
shows not only an increase in the performance of high-achieving students in
Singapore, but also an increase for those who were lagging behind, ensuring
educational fairness. That is why it was possible to establish, with visual and
quantitative evidence, that the Singapore technique intervention achieves
remarkable results in terms of argument quality and mathematical procedure
efficiency, with a positive and homogeneous effect for the three parallel
classes investigated.
Illustration 4. Distribution of scores in logical reasoning and procedures
Source: Research conducted
Greater
motivation and positive attitude toward mathematics
The application of
the Singapore technique not only contributed to improved performance but also
to a change in students' mindset and motivation toward mathematics. According
to the responses collected from 83 students in three parallel classes, there was
a noticeable increase in interest, confidence, and participation.
The use of
concrete materials was a successful change in terms of the impact obtained and
generated improvement. The hands-on experience that students had with the
concept being analyzed enabled them to access knowledge in a concrete,
accessible, understandable, and friendly way that was the same for everyone. In
this way, those who had previously shown reluctance and rejection of
mathematics were attracted by the agility with which it comes.
Teamwork and
constant interaction among classmates also had a positive influence on
students. By sharing strategies and solving problems correctly, they not only
acquire social skills, but also strengthen their self-awareness by realizing
that other people's approaches are also correct answers. Peer relationships in
this way also made the classroom more lively, energetic, and daring, without
losing the caution that contributes to a positive attitude toward the task.
On the other hand,
to the extent that most of the worksheet problems were contextual and
challenging, central to real-life situations, and meaningful, students felt
more motivated or interested in solving them. Math was no longer so abstract
and uninteresting but a concrete and useful tool on that respective front.
Another indication that all three pairs experienced the same improvement can
also be expressed by comparing and contrasting any observable type of voluntary
activities, tasks, or questions.
Illustration 5. Average motivation items per parallel
Source: Research conducted
The results not
only indicate that the Singapore method was able to overcome the emotional and
cognitive barriers that traditionally hindered mathematics learning. Instead,
it helped students become active participants in the process. Of course,
motivation was never an unannounced factor. Instead, it was simply the
student's reaction to seeing the type of teaching that really focused on their
experience and the development of their own time and space. In the long term,
this approach may well have been a critical factor in terms of academic
progress; in fact, it makes the method a complete teaching tool that changes
not only what is learned, but how it is learned.
Comparatively, the
graph illustrates the level of motivation students felt in five key dimensions
related to the influence of the Singapore method. These are divided into:
general motivation; enjoyment while working with concrete materials; liking for
solving concrete problems; preference for collaboration; and how interesting
the math class was before the intervention. Generally, the averages for all
items range from 3.79 to 4.25 on a scale of 1 to 5, indicating a high and
positive trend in students' attitudes toward the subject. This consistency
across the parallel classes shows that the method was perceived favorably and
uniformly, regardless of the group and teacher.
Among the most
notable results is the item "I enjoy working with concrete
materials," which scored particularly high in parallel groups B and C,
suggesting that the tactile and visual experience related to the CPA technique
approach leads to an emotionally positive reaction on the part of some
students. Similarly, the item "Math classes are now more interesting"
also achieved high scores in parallel C, which can also be related to the
renewed perception of the methodology.
Among the analyses
of the remaining items, the assessment of "I prefer to learn math in a
group rather than alone" stands out, with similar averages in the three
parallel groups, with almost top averages in A and C, and close to reaching
this condition in B, which shows that collaborative learning implemented with
AR has been formative. It also shows an interest in a more social,
participatory, and emotionally safe classroom environment.
Regarding the
variable "I like solving real problems in class," parallel C achieved
a particularly high rating of 4.24. Based on this specific element, it can be
said that contextualizing learning in situations that students consider
meaningful increases intrinsic motivation. In conclusion, according to the
graph, the Singapore method not only increases academic performance but also
alters students' attitude toward the material, giving much greater importance
to motivation, interest, and active participation. Again, based on this set of
evidence, motivation becomes a key indicator of the pedagogical success of the
chosen approach.
Favorable
teacher perception of the Singapore method
In this regard,
the implementation of the Singapore method not only yielded better results in
student performance but was also positively received by the teaching staff
responsible for applying the methods. In this regard, teachers in parallel
classes A, B, and C stated that the application of this method had a
significant impact on their teaching practices and classroom dynamics, more
specifically in terms of teaching mathematical content.
One of the most
appreciated aspects was the versatility of the CPA approach, as it allowed them
to address the diversity of learning styles and rhythms that are very present
in the classroom. Using concrete materials, visual representations, and
progressive exercises, teachers integrated their teaching to understand the
different speeds of the students and, therefore, offer more inclusive and
student-centered teaching.
Teachers also
pointed out that, compared to the previous method, the new method clearly
stimulated the development of critical thinking and increased student autonomy
and participation. Traditional approaches required not only the mechanical
search for heuristic problem solutions, but also reasoning about the necessity
of each step, justification of results, and, in general, a higher level of
self-confidence in problem solving.
From a pedagogical
point of view, the teachers interviewed in the previous study also highlighted
the sequential structure of the method as a factor that facilitated teacher
planning, since the sequence provided a precise roadmap on how to introduce and
develop each mathematical concept. This led to time optimization, not only
because the sequence of each class allowed for continuous progress, but also
because the concrete, pictorial, and abstract stages and the formative
assessments provided a more accurate idea of each student's level of progress.
Likewise, the
experience in the three parallel classes led to the conclusion that the method
also improved teacher-student relationships, as teachers went from being
transmitters of knowledge to facilitators of knowledge acquisition. Teachers
worked harder and were more motivated when they saw that students were actively
participating, proposing their own solutions, and taking ownership of the
teaching-learning process. All of this created a more dynamic and collaborative
classroom environment among students, oriented toward critical and reflective
thinking.
Finally, the high
rating given to the teaching staff demonstrates the potential viability,
effectiveness, and sustainability of the Singapore method in public schools
such as the Trece de Abril Educational Unit. Therefore, its application not
only benefits student performance but also empowers teachers as active agents
of methodological change, promoting a more structured, inclusive education
oriented toward a contextual and meaningful understanding of mathematics.
Illustration 6. Average teacher assessment of the Singapore method
Source: Research conducted
Prepared
by: Author
The graph above
shows the results of the survey given to the three math teachers at the Trece
de Abril Educational Unit for the courses analyzed, evaluating their experience
with implementing the Singapore method in the parallel 4th grade classes. Each
aspect was rated on a Likert scale from 1 to 5, where 1 represents 'strongly
disagree' and 5 represents 'strongly agree'. The averages obtained for the 12
aspects evaluated show an extremely positive perception, with values ranging
from 4.50 to 4.80.
Similarly, the
highest-rated aspects are "Versatility of the CPA approach" and
"Improvement in the teacher-student relationship," with average
scores of 4.80. In other words, teachers indicate that not only is it possible
to adjust teaching to learning styles, but the method also acts as a factor in
strengthening the teacher-student bond, suggesting that the classroom fulfills
a socializing and more relaxed function. The next highest-scoring factors are
"More dynamic classroom atmosphere," "Ease of lesson planning,"
and "Adaptation to learning styles," with 4.70. Each of these
highlights how the Singapore method serves as a complementary structure for
organizing active, non-chaotic classes without sacrificing pedagogical clarity
and diversity.
Aspects such as
active student participation, critical thinking development, autonomy, and
student progress monitoring also received high scores (4.50 to 4.60), which may
reaffirm that the method favors comprehensive student development by promoting
reflection, dialogue, and self-assessment. The lowest average corresponds to
"Change in the role of the teacher" (4.50), although this rating is
still high. In this sense, a lower rating probably still implies a positive
assessment that the transition from teacher to facilitator is perceived as
enriching, although it may mean a more challenging process of professional or
educational adjustment.
In summary, in
this analysis, the three teachers surveyed agreed in their very positive
assessment of the Singapore method. All of them pointed out its potential not
only to improve students' academic results, but also to transform teaching
practice, foster greater inclusion in the classroom, and promote better
planning, monitoring, and motivation of students.
Design of a
mathematics learning model based on the Singapore method in public educational
institutions in the province of Santa Elena
The Singapore
Method-based Mathematics Learning Model in public educational institutions,
known as the MAMMS-2025 Model, is a proprietary proposal aimed at promoting
mathematical understanding using a contextualized, progressive, and equitable
approach. It consists of three pedagogical stages essential to the teaching and
learning process—Concrete, Visual, and Logical—which generalize the level of
cognitive development and abstraction of control. It takes students from the
manipulation of materials to the graphic representation of ideas and finally to
the resolution of abstract problems, reinforcing their logic and cognitive
autonomy.
Based on the
principles of progressivity, meaningfulness, and equitable participation, MAMMS
– 2025 is grounded in approaches focused on problem solving, collaborative
work, mathematical communication, and pedagogical differentiation. This
framework has the potential to demonstrate a measurable impact on academic
performance, positive attitudes toward mathematics, and closing learning gaps
in the rural and urban-marginal setting of Santa Elena.
Illustration 7. MAMMS - 2025 Model
Prepared
by: Author
Purpose of
the model
To transform
mathematics learning in public institutions through a methodological approach
that enables students to develop their mathematical logical thinking, based on
the Concrete, Pictorial, and Abstract CPA work sequence, incorporating relevant
content, meaningful experiences, and familiar contexts.
Pedagogical
approaches of the model
·
Constructivist: knowledge
is actively constructed from experience and interaction.
·
Socio-formative: starts
from the student's real context to generate functional and transferable
learning.
·
Discovery learning: students
explore, experiment, represent, and deduce.
·
Problem solving: mathematics
is learned as a tool for dealing with real situations.
Guiding
principles
· Progressiveness:
starts
from concrete experiences to reach the most abstract, respecting individual
learning rhythms.
· Significance:
content
is related to the student's environment, family, community, and territory.
· Interaction:
encourages
community, socialization, collaborative work, and discussion.
· Equity:
does
not exclude but includes; that is, it promotes a flexible model that includes
those with different learning styles and rhythms.
· Autonomy:
seeks
to develop independent thinking, decision-making, and self-monitoring.
Pedagogical
structure of the model (CPA)
· Step
1.- Concrete: This involves the manipulation of real materials with
which the situation is presented. This assistance is facilitated by objects in
relation to their respective places, real strips and parts, blocks, etc.
· Step 2:
Visual: Representation
of the situation using bar models, drawings, diagrams, and number lines.
·
Step 3.- Logical (Abstract): Resolution through
symbolic operations and algorithms. Introduction
of formal mathematical language and verification of solutions.
Key
methodological strategies
Table 2 Key methodological strategies
|
Strategy |
Description |
|
Problem-based learning |
Each unit
begins with a contextual problem that is relevant to the student. |
|
Concrete manipulation |
Use of
real objects from the environment: seeds, caps, sticks, coins. |
|
Pictorial representation |
Bar charts,
drawings, diagrams to represent what has been understood. |
|
Gradual symbolization |
Progressive
introduction of operations and formal mathematical language. |
|
Collaborative work |
Joint
discussion and resolution in small groups or pairs. |
|
Reflective
math notebooks |
Record of
processes, errors, strategies, and solutions for each student. |
|
Formative assessment |
Continuous
feedback, rubrics, and self-assessment. |
Prepared by: Author
Educational level: 4th to 10th grade of basic
general education.
Context: Rural, urban-marginal, or multigrade public
institutions.
Teachers: Math teachers or general classroom
teachers with basic training in the CPA approach.
1. Academics:
·
Improved
performance on math tests.
·
Greater
conceptual understanding and less memorization.
2. Cognitive:
·
Development of
logical, visual, and critical thinking.
·
Greater
capacity for analysis and problem solving.
3. Affective:
·
Increased
motivation, confidence, and interest in mathematics.
·
Decreased
rejection or fear of the subject.
4. Social:
·
Active and
cooperative participation in the classroom.
·
Application in
daily life and in the local context.
Sustainability of the model
·
Progressive
and contextualized teacher training.
·
Design of
accessible and replicable teaching materials.
·
Community
participation in the development of contextualized examples.
·
Alignment with
national curriculum standards.
·
The results of this research show that the
implementation of the Singapore method significantly improved the academic
performance of students in the intervention group. This finding confirms the
conclusions of research conducted by Bacus and Guillena (2023), according to
which the CPA approach contributes to the gradual and deep construction of
mathematical knowledge by students. The importance of this result lies in the
fact that, in Ecuador, learning gaps have been detected, especially in rural
and marginal urban areas, where understanding of mathematics has been affected
by a traditional teaching approach that prioritizes memorization (Cevallos,
2022).
One of the most significant reductions is evident in
the errors in mathematical procedures and reasoning of the group that followed
the Singapore method. This decrease is directly related to the structural
clarity of the CPA sequence, which allowed students to see the problem in the
form of a representation before considering it abstract (Maths — No Problem!,
2023). Therefore, the use of number lines combined with concrete manipulations
immediately facilitated the transition to mathematical symbolization, leading
to a drastic decrease in the number of errors caused by a purely formal
perception of the concepts mentioned. This observation is consistent with the
view of Kaur (2022), as well as with the statements of López and Torres (2023),
who explain that visual representation strengthens the development of
mathematical logical thinking, which, in turn, stimulates structured problem
solving.
In the affective aspect, analytically, there was
significant progress in the experimental group. Progress was identified in
students' motivation, self-esteem, and interest in mathematics. This indicator
is supported by Mathnasium (2025), which asserts that students' perception that
they can solve problems on their own and understand the processes involved
increases their self-assessment and enjoyment of academic situations. On the
other hand, Fernández and Morales (2022) indicated that a learning environment
where students can participate and collaborate can reduce anxiety about
numbers, improve attitudes toward making mistakes, and lead to a more positive
perception of the subject.
Teachers' positive perceptions have reinforced the
viability of the Singapore method in the country's education system. Teachers
argued that this approach simplified planning, adapted content to different
skills, and encouraged them to promote critical thinking in students. The
validity of this approach can be recognized in Mendoza and López (2023), who
argue that the sustainability of these methodologies depends on ongoing teacher
training and the true institutionalization of innovative pedagogical
applications. Therefore, the Singapore method, to the extent that it is
contextualized and implemented with teacher support, stands as a genuine
alternative to mathematics teaching in public education.
Andalón,
J. A. (2023). Math2me: Free online mathematics. Tijuana: Math2me.
Ary,
D., Jacobs, L. C., Irvine, C. K., & Walker, D. (2022). Introduction
to Research in Education (Vol. 11). Cengage Learning.
Bacus,
M. R., & Guillena, J. B. (2023). Singapore mathematics approach in
aiding the modular print distance learning modality in teaching mathematics. International
Journal of Trends in Mathematics Education Research, 6(3), 290–297.
Barja,
J. M. (2025). Mathematics and logic are more necessary than ever. Retrieved from Radio Coruña Cadena SER:
https://cadenaser.com/galicia/2025/05/12/jose-maria-barja-las-matematicas-y-la-logica-son-mas-necesarias-que-nunca-radio-coruna/Cadena
SER+2Cadena SER+2Cadena SER+2
Bruner,
J. S. (2021). The Process of Education. Harvard University Press.
Cadena
CER.(2025). Spain's future depends
on maintaining the OECD average in PISA. Retrieved from Cadena SER:
https://cadenaser.com/nacional/2025/05/05/espana-se-juega-en-pisa-mantener-la-media-de-los-paises-de-la-ocde-cadena-ser/Cadena
SER+1Cadena SER+1
Cevallos,
P. (2022). Mathematics performance of Ecuadorian students: A comparative
analysis. Ecuadorian Journal of Education, 15(2), 75-90.
Cook,
T. D., & Campbell, D. T. (2021). Quasi-experimentation: Design and
analysis issues for field settings. Houghton Mifflin.
Cuasapud
Morocho, J. J., & Maiguashca Quintana, M. (2023). The Singapore method as
a determinant strategy for the learning of fractional numbers in elementary
school students. Revista Científica UISRAEL, 10(3), 205-219.
El
País. (2024). Screens, games, and math: The risky cocktail of an app
already used by more than 1,700 schools in Spain. Retrieved from El País:
https://elpais.com/educacion/2024-10-09/pantallas-juego-y-matematicas-el-coctel-con-riesgos-de-una-app-que-ya-usan-mas-de-1700-colegios-en-espana.htmlEl
País
Fernández, L., & Morales, C. (2022). The
role of metacognition in mathematical problem solving. Educational
Psychology, 28(1), 20-35.
Freudenthal,
H. (2022). Didactics of mathematics: A realistic approach. Barcelona:
Editorial Paidós.
García,
M. (2023). Assessment of mathematical learning in digital environments. Journal
of Educational Assessment, 12(2), 35-50.
Gómez,
L., & Pérez, M. (2023). Implementation of the Singapore method in
primary classrooms: A case study. Latin American Journal of Mathematics
Education, 36(2), 45-60.
González,
D., & Arroyo, J. (2022). Smartick: Artificial intelligence to
improve reading and mathematics. Málaga: Smartick.
Hernández-Sampieri, R., Mendoza, C., &
Fernández, C. (2021). Research methodology: Quantitative,
qualitative, and mixed methods (Vol. 7). McGraw-Hill.
Houghton
Mifflin Harcourt. (2021). Math in Focus: Impact Study 2020–2021 School
Year. Retrieved from Houghton Mifflin Harcourt:
https://www.hmhco.com/research/math-in-focus-impact-study-20202021-school-yearHMH
Co.
INEC.
(2023). Educational Statistics of Ecuador 2022-2023. Quito: National
Institute of Statistics and Census (INEC).
Innovamat.
(2023). Teaching guide for mathematics in primary school. Barcelona:
Innovamat.
Kaur,
B. (2022). The Singapore Mathematics Curriculum: A Framework for Teaching
Problem Solving. Educational Research for Policy and Practice, 21(1),
5–21.
López,
A., & Torres, J. (2023). The CPA approach to mathematical problem
solving. Ibero-American Journal of Education, 79(3), 15-30.
Markarian,
L. (2024). A teacher in Spain alerts parents to the curriculum that turns
students into poor learners. Retrieved from HuffPost España:
https://www.huffingtonpost.es/life/hijos/profesor-vecino-espana-alerta-temario-rp.htmlElHuffPost+1ElHuffPost+1
Martínez, J., & Sánchez, C. (2022). Basic
math skills: A new practice. Madrid: Editorial Graó.
Math2me.
(2023). Digital resources for independent learning of mathematics. Mexico:
Math2me.
Mathnasium.
(2025). Singapore Math Explained: Why It Works and How Mathnasium Supports
It. Retrieved from Mathnasium:
https://www.mathnasium.com/ca/math-centres/theglebe/news/singapore-math-method-why-it-worksmathnasium.com
Maths
— No Problem! (2023). CPA Approach Explained: Learn the Concrete,
Pictorial, Abstract Approach. Retrieved from Maths — No Problem!:
https://mathsnoproblem.com/en/approach/concrete-pictorial-abstractResearchGate+6mathsnoproblem.com+6Singapore
Math Inc.+6
Mendoza, M., & López, P. (2023). Innovative
teaching methods in mathematics education in Ecuador. Revista Andina de
Educación, 6(1), 45-58.
Micronet.
(2022). Naraba World: The Labyrinth of Light. Madrid: Micronet.
Ministry
of Education. (2023). Mathematics Curriculum for Basic General Education. Quito:
Ministry of Education.
Ministry
of Education of Ecuador. (2023). National Education Development Plan
2023-2027. Ministry of Education of Ecuador.
Ministry
of Education and Vocational Training. (2023). Compulsory Secondary
Education Curriculum. Madrid: MEFP.
Mosóczi,
A. (2025). Many Spanish students do not understand mathematics, and an
expert comes to the rescue to explain the problem. Retrieved from
HuffPost Spain:
https://www.huffingtonpost.es/life/hijos/muchos-estudiantes-espanoles-entienden-matematicas-experto-sale-rescate-explicando-problema.htmlElHuffPost+1ElHuffPost+1
Mundial,
B. (2022). Report on the state of mathematics education in Latin America
and the Caribbean. Washington, D.C.: World Bank.
Naraba.
(2022). Game-based learning: Mathematics for children aged 4 to 8. Madrid:
Micronet. Retrieved from Wikipedia.
OECD.
(2021). PISA 2018 results: What students know and can do. Retrieved
from OECD: https://www.oecd.org/pisa
OECD.
(2023). Key findings from PISA 2022: Mathematics, reading, and science. Paris:
Organization for Economic Cooperation and Development.
OEI.
(2023). Mathematics education in Ibero-America: Challenges and
perspectives. Madrid: OEI.
Pérez,
J. (2022). Evaluation of the Singapore method in rural Ecuadorian schools. Andean
Journal of Education, 10(3), 40-55.
Ramírez,
S., & Ortega, L. (2023). Methodological innovations in mathematics
teaching in Ecuador. Education and Development, 18(1), 60-75.
Rodríguez,
P. (2022). Gamification in mathematics teaching: Benefits and challenges. Education
and Technology, 10(4), 55-70.
Sánchez,
R. (2022). Teaching strategies for teaching fractions in primary education. Mathematics
Education, 34(1), 25-40.
SER
Navarra. (2024). Navarra is the second most equitable community in
education according to the TIMSS 2023 report. Retrieved from SER Navarra:
https://cadenaser.com/navarra/2024/12/04/navarra-es-la-segunda-comunidad-con-mayor-equidad-en-educacion-segun-el-informe-timss-2023-radio-pamplona/?fbclid=IwY2xjawKRnltleHRuA2FlbQIxMABicmlkETFPb0traElnaFBWbjZwbHp5AR7xP1Yy70vjvYlgT-8znBP1c_o5n-vPqAWrm7H4Tl
Singapore
Math Inc. (2025). What Is Singapore Math? Retrieved from Singapore
Math Inc.:
https://www.singaporemath.com/pages/what-is-singapore-math+5SingaporeMath
Inc.+5Singapore Math Inc.+5
Singapore,
M. o. (2021). Mathematics Syllabus: Primary One to Six. Retrieved from
Ministry of Education Singapore: https://www.moe.gov.sg
Smartick.
(2022). Annual report on academic progress in mathematics. Madrid:
Smartick.
Tan,
C., Goh, J., & Choy, D. (2021). Lessons from Singapore: The impact of
a structured mathematics curriculum on international achievement. Journal
of Curriculum Studies, 53(5), 687–702.
TIMSS.
(2023). TIMSS 2023 results in mathematics and science. Boston: IEA.
Torres,
M. (2023). Teacher training in mathematics: Challenges and opportunities in
Ecuador. Journal of Teacher Training, 7(2), 25-40.
UNESCO.
(2023). Global Education Report: Mathematics for Sustainable Development. Paris:
UNESCO.
Licenciatura en Educación Inicial, graduada en la Universidad Estatal
Península de Santa Elena, Maestrante en Educación Inicial de la UPSE. Ambarquimis@upse.edu.ec
Código https://orcid.org/0000-0001-8688-1846
Docente Titular de la
Universidad Estatal Península de Santa Elena. Margot.garcia@upse.edu.ec https://orcid.org/0000-0003-0478-3463