Delta-Phi: Jurnal Pendidikan Matematika, vol. 3 (3), pp. 188-191, 2025
Received 02 Nov 2024/published 30 Des 2025
https://doi.org/10.61650/dpjpm.v1i1.52

Numerical Ability in Problem Solving Among
MTs NU Miftahul Huda: A Review of Cognitive
Styles that Dependent and Independent of the
Field of Study
Fajriyah Rachmatika1, Hidayatul mukhsinin2, Anisa Abir3
1,2,3Universitas Nadlatul Ulama Pasuruan, Indonesia

E-mail correspondence to: fajriyah@unupasuruan.ac.id

Abstract
Numerical skills and problem-solving are crucial competencies
influenced by students' cognitive styles. This qualitative descriptive
study aims to describe the numerical abilities of eighth-grade students
of MTS NU Miftahul Huda Prigen in solving problems of the System of
Linear Equations in Two Variables reviewed from the cognitive styles
of Field Dependent (FD) and Field Independent (FI). Numerical
abilities are measured through indicators of counting, logical thinking,
organizing information, and identifying patterns, while the problemsolving process is guided by Polya's steps (understanding, planning,
implementing, and re-checking). The research subjects were
determined using the Group Embedded Figures Test (GEFT). Data
were collected through numerical ability test instruments and
interviews, then analyzed descriptively. The results of the study show
significant differences between the two types of cognitive styles.
Students with the FI cognitive style tend to be more independent,
analytical, and organized in processing numerical data and are able to
separate relevant information from the problem context. In contrast,
FD students tend to need additional guidance, have difficulty in
separating important information from complex contexts, and rely
heavily on previously given example problems. This conclusion
highlights the importance of an adaptive teaching approach to
students' thinking characteristics in mathematics learning.

Keyword: Numerical Ability, Problem Solving, Cognitive Style,
Field Dependent, Field

Introduction

Mathematics education plays a central role in shaping students'
logistical and analytical mindsets. Mathematics is not simply a
discipline filled with formulas and numbers (Alam, 2023), but
rather an essential tool for practicing reasoning and systematic
thinking. Therefore, the primary goal of mathematics instruction
in schools is to ensure students not only master concepts but also
apply them in broader contexts, both in academic matters and in
everyday life. One of the core competencies that must be achieved
in mathematics learning is problem-solving ability (Gunawan,
© 2024 author (s)

2020; Hanin & van Nieuwenhoven, 2020). Problem-solving is a
process that challenges students to identify, analyze, and find
appropriate solutions to previously unseen situations (Afifah &
Putri, 2021). This ability is often considered the heart of the
mathematics curriculum, as it encourages students to think
creatively and use the various knowledge they have acquired in an
integrated manner.
In the context of problem-solving, numerical skills are an integral
foundation. Numerical skills encompass an understanding of basic
numbers, arithmetic operations, and the efficient manipulation of
quantitative data (Michael, 2024; Stagnaro, 2023). Specifically, in
ensuring algebraic material such as Systems of Linear Equations in
Two Variables (SLSV), strong numerical skills are essential for
accurately completing procedural steps such as substitution,
elimination, and final calculations. Without adequate numerical
mastery, the mathematical problem-solving process will be
hampered, even at the conceptual analysis stage.
The System of Linear Equations in Two Variables (SLSV) is
essential material at the intermediate level, requiring a synthesis
of algebraic concepts and numerical skills (de Chambrier, 2021).
SLSV presents contextual problems that must be modeled
mathematically before being solved using methods such as
substitution, elimination, or mixed methods (Grace & Sekhon,
2024). The complexity of the modeling and the need for precision
in calculations in SLSV often leads to conceptual and procedural
difficulties for students, requiring a strong mastery of SLSV
concepts and high numerical accuracy (Helm, 2020; Tian, 2023;
Topsakal, 2022). Although the learning objectives are the same,
the effectiveness of the learning and problem-solving processes is
greatly influenced by the unique characteristics of each student.
These individual differences are known as learning styles and
cognitive styles. Cognitive style refers to the consistent way in
which an individual receives, processes, stores, and uses
information. Understanding cognitive styles is key to explaining
why some students excel at certain types of tasks over others
(Qomaria, Afifah, & Manivannan, 2025).

Usmiyantun, et al.: Exploration of Mathematical Concepts… Delta-Phi: Jurnal Pendidikan Matematika, 3
(3),

One dimension of cognitive style relevant to mathematics learning
is Field Dependent (FD) and Field Independent (FI). Students with
a Field Dependent cognitive style tend to view a problem
holistically, influenced by the surrounding context or "field." They
require a clear external structure and tend to rely on the available
direction and context (Gutsche, 2023). In contrast, Field
Independent students tend to analyze problems partially, are able
to separate important elements from distracting contexts (fields),
and are more independent in developing problem-solving
strategies. Theoretically, the differences between FD and FI
cognitive styles are closely correlated with how students solve
problems, fundamentally affecting the use and efficiency of their
numerical abilities (Khajanchi, 2021; Vanbecelaere, 2021). FI
students may be quicker at identifying calculation steps and
sorting relevant data for numerical operations. Meanwhile, FD
students may take longer to process numerical information when
presented in complex or less structured contexts. Therefore, indepth research is needed to empirically test this hypothesis in the
context of SPLDV materials.
Although numerous studies have examined cognitive abilities,
focusing on general problem-solving skills, few have specifically
examined the numerical ability profiles of students with FD and FI,
particularly when faced with SPLDV material that demands
precision and abstraction (Zhang, 2020). This study aims to
address this gap by examining in detail how these two cognitive
style groups utilize and demonstrate their numerical abilities
when solving SPLDV problems (Barbieri, 2020; Botting, 2020).
Based on this background, this study aims to describe and analyze
in depth the numerical ability profiles of junior high school
students in solving SPLDV problems, as viewed from the
perspective of Field Dependent and Field Independent cognitive
styles (Baiduri, 2020; Hu, 2024). The results of this study are
expected to provide practical contributions to mathematics
teachers in developing more adaptive learning strategies, more
structured materials, and assessments that accommodate the
diversity of students' cognitive styles, ultimately improving overall
mathematics learning outcomes.

Method

This study employs a qualitative approach with a descriptive
research design to provide an in-depth and holistic description of
students' numerical ability profiles in solving System of Linear
Equations in Two Variables (SPLDV) problems, analyzed through
Field Dependent (FD) and Field Independent (FI) cognitive styles
(Gea, Hernández-Solís, Batanero, & Álvarez-Arroyo, 2023). The

Name
PKR
BSZA
KNF
A
NNS
AWDL
FZ
AAZZR
NLZ
NLZ
LM
ABS
MMR
MFA
RY
AZF
DR
MFA
DNAT
KSA
NWAK

descriptive design is specifically intended to present the subjects'
cognitive processes in their entirety without involving statistical
hypothesis testing. The research was conducted at MTs NU
Miftahul Huda Prigen during the even semester of the 2024/2025
academic year. The initial subject pool consisted of all eighthgrade students, who were subsequently screened using the Group
Embedded Figures Test (GEFT) to determine their respective
cognitive styles. Based on the test results, two primary subjects
representing extreme characteristics were selected: one subject
with a Field Dependent (FD) style from the lowest score group and
one subject with a Field Independent (FI) style from the highest
score group. Both subjects were also required to demonstrate
adequate conceptual understanding of SPLDV to ensure a
comprehensive problem-solving process.
Data collection was carried out through three primary techniques
(Nguyen, 2022). First, the Group Embedded Figures Test (GEFT)
served as a classification instrument to categorize students into FD
and FI groups. Second, a Numerical Ability Test consisting of
SPLDV-themed essay problems was administered to measure four
key indicators: information grouping, pattern or formula
construction, mathematical resolution, and comprehensive
explanation of the solution. Third, semi-structured interviews
were conducted following the written test to clarify the subjects'
thought processes, strategies, and numerical awareness. The data
were then analyzed using descriptive qualitative techniques
through three systematic stages (Roh, 2021): data reduction,
which involved summarizing and focusing on data relevant to the
subjects' numerical profiles; data display, presented through
descriptive narratives supported by tables and excerpts of student
work to highlight the differences between FD and FI profiles; and
conclusion drawing, achieved by comparing findings from the
written tests and interviews (methodological triangulation) to
obtain credible data regarding the numerical ability profiles of
each cognitive style.

Result and Discussion
Result
This study analyzes the numerical abilities of Islamic Junior High
School (MTs) students in solving problems related to the System
of Linear Equations in Two Variables (SPLDV), with a focus on
comparing the performance between subjects characterized by
Field Dependent (FD) and Field Independent (FI) cognitive
styles.

Table 1. Analyxes thje numerical abilities
Tota
Scor
Scor
l
cognitive style
e1
e2
Skor
2
3
5
Field-Dependent
4
5
9
Field-Dependent
6
2
7
Field-Dependent
4
4
8
Field-Dependent
5
3
8
Field-Dependent
4
3
7
Field-Dependent
3
3
6
Field-Dependent
3
5
8
Field-Dependent
5
3
8
Field-Dependent
6
7
13
Field-Independent
6
6
12
Field-Independent
5
5
10
Field-Independent
7
7
14
Field-Independent
6
4
10
Field-Independent
8
8
16
Field-Independent
8
9
17
Field-Independent
8
8
16
Field-Independent
8
8
16
Field-Independent
6
4
10
Field-Independent
9
7
16
Field-Independent
6
7
13
Field-Independent

189

Usmiyantun, et al.: Exploration of Mathematical Concepts… Delta-Phi: Jurnal Pendidikan Matematika, 3
(3),

conclusions that were inconsistent with the problem context.

Based on the Group Embedded Figures Test (GEFT) conducted
with 21 students, the distribution of cognitive styles revealed
that 9 students (40%) were classified as Field Dependent (FD),
while 12 students (60%) exhibited a Field Independent (FI)
cognitive style. The findings indicate that students with an FI
cognitive style demonstrate integrated and effective numerical
abilities across all stages of the SPLDV problem-solving process
based on Polya's framework. These students tend to work more
systematically and structurally. Analysis of subjects FI-1 and FI2 showed that they were highly capable of identifying essential
information and accurately transforming verbal problems into
precise mathematical models. Furthermore, they formulated
logical and efficient strategies, performed algebraic operations
with high consistency and precision, and successfully verified
their final results to ensure relevance to the initial problem.
In contrast, students with an FD cognitive style exhibited
numerical abilities that require further guidance, particularly in
organizing information and structuring solution steps. Subjects
FD-1 and FD-2 demonstrated significant challenges in
abstracting or decomposing primary information from complex
contexts. These subjects often struggled to construct complete
mathematical models and frequently proceeded to calculations
without clear planning. They tend to focus on visible numerical
values rather than the underlying structure of the SPLDV,
leading to errors in variable manipulation and significant
inaccuracies in algebraic operations, such as substitution or
elimination. Additionally, FD students rarely performed
evaluations or re-checks of their results, often leading to

The assessment of numerical abilities in this study was
specifically measured through four key indicators. First, the
ability to categorize and organize information necessary for
problem-solving. Second, the capacity to identify numerical
patterns and formulate mathematical summaries. Third, the
proficiency in solving problems provided through accurate
mathematical procedures. Finally, the ability to provide a
comprehensive explanation of the entire solution process. The
disparity between FI and FD subjects across these indicators
suggests that cognitive styles significantly influence how
students process quantitative data and execute mathematical
reasoning in the context of linear equations.
The analysis of the first subject (SFI) commenced with an
assessment of the first numerical indicator: the ability to
categorize information required for problem-solving. The
subject was presented with a contextual word problem involving
a fruit transaction made by two individuals, Rahman and Malik.
The problem posited that Rahman purchased 1 kg of mangoes
and 2 kg of oranges for a total of IDR 50,000, while Malik bought
1 kg of mangoes and 1 kg of oranges for IDR 35,000. To evaluate
numerical proficiency, the subject was required to perform two
key tasks based on this scenario. First, the subject has to identify
and categorize the specific information from the narrative that is
relevant for the solution. Second, the subject was tasked with
constructing mathematical patterns or formulas derived from
the categorized data to calculate the specific unit prices for 1 kg
of oranges and 1 kg of mangoes

Figure 1. SFI Answer
Based on Figure 1, SFI explicitly organized the problem
components by noting, Rahman x=2 kg oranges and y=1kg
mangoes = 50,000 and Malik x=1kg mangoes and y=1kg oranges
= 35,000. This written evidence demonstrates the subject's
capacity to effectively categorize the information required to
solve the problem. This finding is further corroborated by the
interview data, where in the subject explained the strategy of
separating the information into two distinct segments: Rahman's
transaction (buying 1kg of oranges and 2kg of mangoes for
50,000) and Malik's transaction (buying 1kg of mangoes and 1kg
of oranges for 35,000). When queried regarding the difficulty of
this task, SFI confirmed that the process was not difficult. The
consistency between the written work and the verbal
explanation validates that SFI fulfills the first indicator.
Consequently, the analysis proceeds to the second indicator,
which assesses the student's ability to construct numerical
patterns and their underlying relationships.
Subsequently, SFI formulated the mathematical model by
writing x+2y=50,000 and x+y=35,000, demonstrating the
student's proficiency in constructing numerical patterns and
identifying their corresponding conditions. This observation is
substantiated by the interview data, where in the subject
confirmed the ability to identify numerical patterns or formulas
within the provided problem. When probed regarding the
specific method employed, SFI explicitly explained that the
solution was derived using the System of Linear Equations in
Two Variables (SPLDV) formula. This verbal confirmation,
combined with the written evidence, indicates that the subject is
capable of formulating numerical patterns along with their
technical relationships, thereby fulfilling the criteria for the
second indicator.
189

Regarding the third indicator, which assesses the student's
proficiency in mathematically solving the assigned problem, the
analysis of SFI's answer sheet reveals a systematic algebraic
approach. SFI correctly documented the solution process by
writing the equations x+2y=50,000 and substituting variables to
derive the solution, noted as x(35,000−y)+2y=50,000, which
simplified to 35,000+y=50,000, resulting in y=15,000.
Subsequently, the student solved for the second variable using
x+y=35,000, rearranging it to x=35,000−y, yielding x=20,000.
This written evidence demonstrates SFI's capability to execute
the mathematical procedures required. This finding is further
corroborated by the semi-structured interview, where the
researcher sought to validate the solution process. When asked
to articulate the strategy employed, SFI confirmed the ability to
solve the problem and explicitly detailed the steps taken: "To
find the value of y, I used the formula x+2y=50,000 (and the
substitution process). so the value of y is 15,000. To find the
value of x, I used x+y=35,000... so the value of x is 20,000." The
consistency between the written calculations and the verbal
explanation confirms that SFI is capable of solving the problem
regularly, thus fully satisfying the third indicator.
Regarding the fourth indicator, which evaluates the student's
aptitude for explaining the overall solution process, the analysis
confirms that SFI is capable of articulating the entire problemsolving sequence comprehensively. Based on the interview data,
the subject demonstrated a clear understanding of the problem's
structure by explicitly noting that the task consisted of multiple
components, recognizing that the initial requirement was to
categorize the relevant information presented in the prompt.
The analysis of the Field Dependent (SFD) subject begins with
the first indicator: the ability to categorize information. The

Usmiyantun, et al.: Exploration of Mathematical Concepts… Delta-Phi: Jurnal Pendidikan Matematika, 3
(3),

subject explicitly detailed Rahman's transaction as x=2 kg
oranges and y=1 kg mangoes totaling IDR 50,000, and Malik's
transaction as x=1 kg mangoes and y=1 kg oranges totaling IDR
35,000. These data points demonstrate that SFD is capable of
identifying and grouping information relevant to problem
resolution. Regarding the second indicator, which maintains the
ability to construct numerical patterns and their relationships,
SFD demonstrated adequate competence. Based on interview
data, the subject confirmed the ability to formulate mathematical
models using the principles of the System of Linear Equations in
Two Variables (SPLDV), thus fulfilling the second indicator.

For the third indicator, SFD demonstrated the capacity to solve
the problem mathematically and systematically. The subject
articulated the solution procedure through substitution, where
the step 35,000−y+2y=50,000 is simplified to 35,000+y=50,000,
resulting in a value of y=15,000. Subsequently, the value of x was
determined using the equation x+15,000=35,000, yielding
x=20,000. Based on this procedure, SFD satisfied the third
indicator. However, regarding the fourth indicator, the subject
was unable to articulate the solution process in its entirety.
During the interview, the subject acknowledged this limitation,
stating that while they could explain the categorization of
information and the formation of mathematical models
(x+2y=50,000 and x+y=35,000), they struggled to provide a
comprehensive interpretation of the final results. Consequently,
SFD is categorized as not having fully met the fourth indicator.
These findings fundamentally reinforce the theory that
differences in cognitive styles (Field Dependent/Field
Independent) significantly influence how students organize,
analyze, and abstract data within the context of mathematical
problem-solving. Field Independent (FI) students, possessing the
analytical ability to separate elements from a broader context,
are inherently superior in stages requiring systematic
formulation, planning, and algebraic calculations. Their
numerical abilities are well-integrated within Polya's problemsolving framework. Conversely, Field Dependent (FD) students,
who tend to view contexts holistically, face substantial
challenges in the abstraction stage such as transforming word
problems into mathematical models and in problem
decomposition. Their numerical calculations are often impeded
by difficulties in distinguishing relevant data from distracting
contextual information. These implications underscore the
necessity for educators to provide more detailed and structured
scaffolding, particularly for FD students, to assist them in
visualizing variables, constructing mathematical models, and
executing algebraic procedures sequentially in SPLDV subject
matter.

Discussion
This study fundamentally corroborates Witkin's theory that
differences in cognitive styles (Field Dependent/Field
Independent) directly influence the mechanisms by which
students organize, analyze, and abstract data during
mathematical problem-solving. Based on the findings, students
with a Field Independent (FI) cognitive style demonstrate
numerical abilities that are effectively integrated and systematic
across all stages of Polya's problem-solving framework. In terms
of data analysis (Fauza, Inganah, Darmayanti, Prasetyo, 2022;
Yapatang & Polyiem, 2022), FI subjects are capable of
distinguishing essential elements from narrative contexts and
transforming them into precise mathematical models (variables
x and y).
Procedurally, they exhibit high accuracy in algebraic operations
both substitution and elimination with minimal calculation
errors (Olsson & Granberg, 2024). Furthermore, FI subjects
fulfill all ability indicators, including the capacity to logically and
autonomously articulate the solution process without external
assistance. This aligns with the characteristics of the FI style,
which tends to be analytical and efficient in separating relevant
elements from distracting contexts.
In contrast, students with a Field Dependent (FD) cognitive style
encounter more significant challenges, particularly in the stages
of abstraction and procedural precision. FD subjects frequently
190

struggle to decompose primary information when problems are
presented in complex word formats; they tend to perceive the
problem holistically but fail to attend to technical details.
Numerical weaknesses in this group often manifest as
inaccuracies in basic arithmetic operations such as calculations
involving negative numbers and sign errors during coefficient
elimination. Although they may solve problems in written form,
they often face verbal barriers when asked to rationalize their
steps, indicating that their understanding remains heavily
reliant on scaffolding or external cues.
These findings underscore that numerical ability is not merely a
matter of computational proficiency but is profoundly influenced
by the student's information processing mechanisms. As a result,
FI students require more complex challenges to hone their
independent analytical skills, whereas FD students need
structured scaffolding support such as step by step guides or
variable visualization to navigate the complexities of SPLDV
word problems. Overall, this research emphasizes the urgency
for educators to adopt teaching strategies that are adaptive to
students' cognitive characteristics to optimize learning
outcomes.

Conclusion and Recomendation

Based on in-depth qualitative analysis, this study identifies
significant disparities in numerical ability profiles between
students with Field Independent (FI) and Field Dependent (FD)
cognitive styles in solving problems related to the System of
Linear Equations in Two Variables (SPLDV). Students with an FI
cognitive style demonstrate superior and precise numerical
proficiency, exhibiting the capacity to regularly implement
procedural steps ranging from modeling and algebraic
manipulation to complex arithmetic operations. This advantage is
driven by their analytical characteristics, which enable the
independent verification of critical numerical elements within the
problem context, thereby minimizing procedural errors.
Conversely, students with an FD cognitive style, while capable of
constructing global models, exhibit significant weaknesses in
computational precision. Frequently identified errors include
inaccuracies in operational signs, basic arithmetic errors, and
coefficient inconsistencies. This phenomenon is closely attributed
to their tendency towards holistic information processing, where
microscopic numerical details are often overlooked in favor of the
overall problem structure. Theoretically, cognitive style is proven
to influence numerical accuracy; the analytical approach of FI
students supports the deconstruction of problems into distinct
sub-steps, whereas the holistic approach of FD students acts as a
structural impediment to maintain rigor in subject matter
requiring high algebraic precision.
Based on these findings, several strategic recommendations are
proposed. For mathematics educators, it is recommended to adopt
teaching strategies that are adaptive to cognitive styles,
particularly for FD students. Interventions may include the
presentation of material with explicit step-by-step guidance, the
utilization of visualization techniques (such as color markers or
focus boxes) to highlight operational variables, and the
habituation of re-checking procedures at each calculation stage to
reduce negligence. For future researchers, it is suggested to
conduct experimental studies to test the effectiveness of adaptive
learning models such as the use of structured visual media in
enhancing numerical abilities and reducing procedural errors
among Field Dependent students in SPLDV material or other
procedural mathematics topics.

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