JURNAL PENDIDIKAN MATEMATIKA DAN IPA
Vol.
No.
http://jurnal.
id/index.
php/PMP DEEP LEARNINGAeBASED PROBLEM-BASED LEARNING:
AN EFFORT TO ENHANCE THE MATHEMATICAL
REFLECTIVE ABILITIES OF JUNIOR HIGH SCHOOL
STUDENTS
Indah Frantia Sari 1.
Hepsi Nindiasari 2* Mathematics Education.
Faculty of Teacher Training and Education.
Sultan Ageng Tirtayasa University.
Jl.
Ciwaru Raya.
Cipare.
Serang District.
Serang City.
Banten Email Corresponding: Hepsinindiasari@untirta.
DOI: http://dx.
org/10.
26418/jpmipa.
Abstract This study aims to investigate the implementation of learning using a Problem-Based Learning (PBL) approach based on deep learning to improve studentsAo mathematical reflective thinking skills.
The population of the study consisted of all eighth-grade students of a junior high school in the odd semester of the 2024/2025 academic year, with one class selected as the research subject.
The sample was taken using a purposive sampling technique.
This study employed a quantitative approach with a quasi-experimental design of the One-Group PretestAePosttest type.
The data were collected through tests of studentsAo mathematical reflective thinking skills and classroom observations, and analyzed using descriptive and inferential statistics.
The results showed that the average pretest score was 58.
96, which increased to 74.
96 in the posttest.
Classroom observations indicated that students actively engaged in problem identification, group discussions, evaluation of solution strategies, and reflective reasoning during the learning process.
The KolmogorovAeSmirnov normality test indicated that the data were normally distributed .
= 0.
, and LeveneAos test showed homogeneous variances .
= 0.
The paired-sample t-test revealed a significant improvement in studentsAo reflective thinking skills .
= -59.
p < 0.
The Normalized Gain (N-Gai.
calculation showed an average of 0.
40, categorized as medium.
Thus, the implementation of PBL based on deep learning can significantly enhance studentsAo mathematical reflective thinking Keyword: Deep Learning.
Mathematical Reflective Thinking.
Problem-Based Learning Received Revised Accepted : 2025-12-10
: 2026-02-10
: 2026-02-23
This work is licensed under a Creative Commons Attribution 4.
0 International License Jurnal Pendidikan Matematika dan IPA Vol.
No.
INTRODUCTION
Mathematical reflective thinking is the ability of students to review the manner in which they solve mathematical problems .
This ability enables students to understand the reasoning underlying their answers and to consciously correct errors .
mathematics education, reflective thinking plays a crucial role, as it assists students in connecting procedures, concepts, and logical reasoning, thereby rendering learning more meaningful and continuous .
Furthermore, within mathematics education, reflective thinking serves as a foundation for the development of higher-order thinking skills, as it encourages students to analyze, evaluate, and synthesize information more deeply .
However, in practice, many students still experience difficulties in reflecting upon the steps involved in problem-solving.
Instruction that is overly focused on procedures and final answers tends to result in superficial and less meaningful understanding.
Students are rarely encouraged to reexamine the decisions they have made, evaluate errors, or consider alternative and more efficient strategies, even though such activities constitute the core of higher-order and reflective thinking.
This is consistent with research .
, which states that reflective thinking is often neglected due to excessive emphasis on final answers, thereby indicating that students require greater ability to evaluate and reconstruct their problemsolving processes.
Similar findings were reported by .
, showing that most students are not yet capable of effectively evaluating problem-solving steps, leading to the recurrence of similar errors.
Furthermore, research .
emphasizes that low levels of mathematical reflection are closely with teacher-centered instructional patterns, which limit studentsAo opportunities to construct understanding independently .
addition, studentsAo low academic achievement indicates that classroom instruction remains dominated by an expository approach, in which the teacher serves as the primary source of information, resulting in students being passive and less engaged in deep thinking processes .
These findings reinforce the view that mathematical reflective ability remains an aspect requiring further development.
therefore, an instructional model that facilitates evaluation, and reconstruction of problem-solving strategies is needed.
As a response to this issue, the Problem-Based Learning (PBL) model is considered capable of enhancing the quality of mathematics instruction.
Problem-Based Learning (PBL) is a contextual learning model that employs problems as the central focus of instruction and can be implemented to increase studentsAo learning activities .
In learning through the PBL model, students are presented with a problem to be examined from the outset .
This approach encourages students to design learning strategies, assess their thinking processes, and reflect on the extent to which the decisions they make are effective throughout the learning process .
Hayun and Syawaly .
, as cited in .
, explain that the PBL model consists of five principal stages:
introducing students to the problem to be examined, organizing studentsAo Indah Frantia Sari.
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learning activities, providing guidance both individually and in groups, facilitating the development of presentation of group reports, and conducting analysis and evaluation of the problem-solving process that has been undertaken.
The PBL model also possesses implementation in the topic of Systems of Linear Equations in Two Variables (SPLDV), students are required to connect their prior understanding of single-variable linear equations with the concept of two variables within real-world problem contexts.
When students have not yet mastered the fundamental concepts of linear equations, they tend to encounter difficulties in identifying variables, constructing mathematical models, and determining appropriate solution Moreover, some students are not accustomed to conducting information searches or preliminary studies from various sources to assist in problem-solving.
Consequently, differences in studentsAo prior abilities influence the accuracy with which they relate previously learned concepts to new situations or problems .
address this issue, the implementation of a deep learning approach is required, as it encourages students to develop a understanding, establish connections among concepts, and apply knowledge in authentic contexts.
This approach also facilitates students in evaluating the solution processes they undertake, identifying errors, and reflecting on the appropriateness of the strategies Thus, the limitations of PBL related to minimal information exploration and insufficient integration of knowledge can be mitigated .
Deep learning in the field of education is an instructional approach engagement, reflection, and the development of higher-order thinking skills .
The deep learning approach is grounded in three principal tenets:
mindful, meaningful, and joyful learning .
Mindful learning requires students to maintain focus and full awareness of the learning process, enabling them to reflect upon and understand their own patterns of thinking .
The principle of meaningful learning lies in efforts to connect instructional material with real-life situations that are closely related to studentsAo experiences, so that the information learned is not only understood conceptually but can also be applied, retained longer, and perceived as beneficial in everyday contexts .
Joyful learning is a principle that emphasizes the creation of an enjoyable learning environment.
its implementation in the classroom involves presenting interactive and engaging learning activities to enhance studentsAo involvement .
In the context of this study, the deep learning approach is relevant to the development of mathematical reflective thinking skills, which encompass the ability to interpret cases based on mathematical concepts, identify concepts or formulas in nonroutine problems, evaluate the validity of arguments, analyze and generalize mathematical ideas, and solve The mindful principle encourages students to be aware of and examine their thinking processes when determining the concepts employed Indah Frantia Sari.
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and evaluating the correctness of solution steps.
The meaningful principle assists students in connecting mathematical concepts with real-life situations, enabling them to interpret cases and generalize ideas more Meanwhile, the joyful environment that supports discussion and reflection, allowing students to be more open in identifying errors and refining problem-solving strategies.
Thus, these three principles directly support the indicators of mathematical reflective thinking ability examined in this study.
The effectiveness of PBL can be further optimized when integrated with an approach that emphasizes deep reflective processes, such as deep learning .
The PBL and deep models share similar paradigmatic foundations, both being oriented toward active, meaningful, and student-centered learning .
, .
In PBL, students engage in the exploration of real-world problems to discover solutions through dialogue and collaborative work .
Deep meaningful, and joyful .
The deep learning approach enhances the success of PBL by helping students grasp the conceptual meaning underlying the problems they solve, rather than merely obtaining answers .
When students encounter contextual problems in PBL, deep learning encourages them to reflect on the strategies employed, connect learning experiences with real-life situations, and construct personal meaning of the mathematical concepts studied .
Nevertheless, approaches remains infrequently optimized in mathematics instruction, mathematical reflective thinking, which has been shown to remain low in various previous studies .
In fact, requires deep thinking, reanalysis of solution steps, and awareness in evaluating the strategies employedAi elements directly facilitated by the combination of PBL and deep learning .
Therefore, the application of a Deep LearningAebased ProblemBased Learning model represents a relevant and urgent solution for Through integration, students are expected to critically review their thinking processes, understand the meaning behind each problem-solving step, and connect mathematical concepts in a deeper and more meaningful way.
Based on this rationale, the present
study aims to investigate whether the implementation of Deep LearningAe based Problem-Based Learning can enhance the mathematical reflective thinking abilities of junior high school
METHOD
This quantitative approach with a quasiexperimental design of the One-Group PretestAePosttest type.
This design was chosen because the researchers used only a single group without a control group, yet administered both a pretest and posttest to examine the effect of studentsAo mathematical reflective abilities .
Moreover, according to .
, this design is effective for Indah Frantia Sari.
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observing changes in studentsAo abilities before and after receiving the treatment, thereby facilitating the researchersAo assessment of the effectiveness of the implemented instructional model.
Research Design Scheme:
OCA Ai X Ai OCC Description:
OCA : Pretest of Mathematical Reflective Thinking Ability X : The treatment consisted of Deep LearningAebased Problem-Based Learning instruction.
OCC : Posttest of Mathematical Reflective Thinking Ability The subjects of this study were eighth-grade students at a junior high school in Cilegon City.
The sample was selected using purposive sampling .
, choosing a class that met the criteria of learning readiness and schedule suitability.
The number of students in the selected class was 25.
Students were administered a pretest to determine their initial abilities, then received treatment in the form of Deep LearningAebased Problem-Based Learning instruction, and subsequently were given a posttest to assess the improvement in their mathematical reflective thinking abilities following the intervention.
This involved two main variables: the independent variable, namely the Deep LearningAebased Problem-Based Learning model (PBL-DL), which refers to the implementation of problem-based instruction integrated with deep learning strategies to promote studentsAo deep thinking processes .
, .
, .
and the dependent variable, namely studentsAo mathematical reflective ability, defined as their capacity to review, evaluate, and improve the problem-solving processes they undertake .
The implementation of the Deep LearningAebased Problem-Based Learning model (PBL-DL) in this study was carried out through several stages: .
orientation to contextual problems related to studentsAo real-life .
, .
organization of students into groups to discuss and identify relevant concepts, .
investigation and collection of information from various sources, .
development and presentation of discussion results, and .
analysis and evaluation of the problem-solving The consisted of a test of mathematical reflective thinking ability, developed based on the reflective thinking indicators according to .
, which include the ability to interpret cases based on the involved mathematical concepts, identify concepts or formulas in non-routine problems, evaluate the validity of arguments based on the properties or concepts used, analyze and generalize mathematical ideas, and solve mathematical problems.
This instrument was structured in the form of five essay items representing each of these indicators.
In addition to the test of mathematical reflective thinking ability, the study also employed an observation sheet to monitor the implementation of the Deep LearningAe based PBL model during the learning The observation sheet was used to ensure that the principles of mindful, meaningful, and joyful learning were carried out in accordance with the designed instructional syntax.
Indah Frantia Sari.
Hepsi Nindiasari Deep LearningAeBased Problem-Based Learning: An Effort to Enhance the Mathematical Reflective Abilities of Junior High School Students Jurnal Pendidikan Matematika dan IPA Vol.
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The pretest and posttest data were analyzed using descriptive and Descriptive analysis was employed to observe the increase in mean scores.
This increase was calculated using the N-gain formula, as follows:
ycA Oe yciycaycnycu = ycIycoycuyc ycEycuycycycyceycyc Oe ycIycoycuyc ycyycyceycyceycyc ycIycoycuyc ycoycaycoycycnycoycayco Oe ycIycoycuyc ycyycyceycyceycyc Table 1.
N-Gain Categories .
N-gain Score .
g > 0,7 0,3 O g O 0,7 g < 0,3 Category High Medium Low Meanwhile, inferential analysis was conducted using a paired sample ttest with the assistance of IBM SPSS to determine the significant difference between the pretest and posttest scores.
The testing was carried out at a significance level of = 0.
05, so a difference was considered significant if the p-value < 0.
Prior to the t-test, the KolmogorovAeSmirnov test for normality and the Levene Test for homogeneity were conducted.
If the test results indicated a significance value less than 0.
05, the null hypothesis (HCA) was rejected, meaning the data did not meet the tested Conversely, if the significance value was greater than 05.
HCA was accepted, indicating that the data met the assumptions of normality and homogeneity and were therefore suitable for analysis using the paired sample t-test .
RESULTS AND DISCUSSION
This study aimed to determine whether the implementation of Deep LearningAebased Problem-Based Learning (PBLAcDL) can improve studentsAo mathematical reflective To assess the attainment of studentsAo reflective thinking ability, a descriptive analysis was conducted on the pretest and posttest scores.
Table 2 Descriptive Analysis Results Variable Pretest Posttest Mean Std.
Deviation The average pretest score of studentsAo reflective thinking ability 96, while the average posttest score increased to 74.
This improvement was not only evident quantitatively but was also supported by observational results during the implementation of Deep LearningAe Std.
Error Mean based Problem-Based Learning.
Based on the observations, the learning process was active and studentcentered.
At the problem orientation stage, students appeared able to identify key information from the given contextual problems and began to formulate the variables involved in Indah Frantia Sari.
Hepsi Nindiasari Deep LearningAeBased Problem-Based Learning: An Effort to Enhance the Mathematical Reflective Abilities of Junior High School Students Jurnal Pendidikan Matematika dan IPA Vol.
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the two-variable linear equation model (SPLDV).
During the group discussion and investigation stage, students engaged in the exchange of ideas and arguments regarding the appropriate solution Some students demonstrated the ability to explain the rationale for using the elimination method and to verify This indicates the presence of reflection and evaluation processes on the solution steps.
Furthermore, presentation and reflection stage, students began to explain solution patterns and compare strategies used This demonstrates that the learning process focused not only on the final results but also on thinking processes and conceptual understanding.
Overall.
Deep LearningAebased PBL was implemented effectively and fostered the development of studentsAo reflective After obtaining a descriptive overview of the data and observations, the next step was to conduct normality and homogeneity tests to ensure that the data met the assumption of a normal distribution before performing the t-test.
Table 3.
Results of the KolmogorovAeSmirnov Normality Test Variable Pretest Posttest K-S Statistic 0,084 0,083 Sig.
0,20
0,20
Table 4.
Results of the LeveneAos Test for Homogeneity Score Equal variances assumed LeveneAos Test for Equality of Variances Sig.
The normality test using the KolmogorovAeSmirnov method showed that the significance value for the pretest data was 0.
20 and for the posttest data was 0.
20, indicating that the posttest data were normally distributed (Sig.
> 0.
The homogeneity test using LeveneAos test showed a significance value of 0.
which is greater than 0.
05, indicating that the variance between the pretest and posttest scores was homogeneous.
Since the data met the assumptions of normality and homogeneity, it was appropriate to perform a t-test.
Subsequently, a paired-sample t-test Std.
Error Difference was conducted to compare the pretest and posttest scores.
The test results showed a t-value 084 with Sig.
-taile.
= 0.
< 0.
05, indicating that there was a significant increase in studentsAo reflective thinking ability after participating in Deep LearningAebased Problem-Based Learning.
This improvement was further supported by observation results during the learning process, which showed that students actively identified problems, discussed solution strategies, evaluated the steps taken, and reflected on the outcomes Indah Frantia Sari.
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These activities align with the indicators of reflective thinking.
Table 5.
Results of the Paired-Sample t-Test Paired Variables Mean Difference Std.
Dev Std.
Error t-Value Sig.
Pretest Ae Posttest Ae16,08 1,41 0,28 Ae59,084 0,000 Pedagogically.
Deep LearningAebased PBL model is effective because it encourages exploration of contextual problems and collaborative discussion.
The learning process focuses not only on the final answers but also on the analysis, evaluation, and reflection conducted by the students.
Thus, the effectiveness of Deep LearningAe PBL mathematical reflective thinking ability is demonstrated not only statistically but also through the dynamics of classroom learning To determine the magnitude of the increase in studentsAo reflective thinking ability, an analysis was Normalized Gain (N-Gai.
Table 6.
N-Gain Test Results Statistic Mean
Maximum Score
N-Gain
Pretest
58,96
0,40
The increase in studentsAo reflective thinking ability was calculated using Normalized Gain (NGai.
The average N-Gain of the students was 0.
40, which falls into the medium category.
This indicates that the implementation of the Deep LearningAebased Problem-Based Learning model was able to studentsAo reflective thinking ability compared to their pre-instructional condition.
As an illustration of the improvement in reflective thinking indicated by the statistical data, the example of a studentAos answer sheet in Figure 1
Postest
74,96
Medium Category shows how students not only wrote the final answers but also explained the steps and considerations they followed during the problem-solving process.
Figure 1 displays a studentAos answer in solving a two-variable linear equation problem (SPLDV).
From this worksheet, it is evident that the student was able to interpret the case by understanding the problem context and formulating mathematical equations according to the given conditions.
The student also successfully identified the relevant mathematical concepts or formulas, namely algebraic concepts and the elimination and substitution Indah Frantia Sari.
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techniques for solving the equations.
Additionally, the student was able to evaluate and verify the validity of the arguments by substituting the value of x into the original equation to ensure that the value of y was correct.
The worksheet also demonstrates the studentAos ability to generalize and analyze generalizations, as the steps taken were systematic and could be applied to other SPLDV problems with similar patterns.
Ultimately, the student successfully solved the mathematical problem by finding the final solution, x = 4 and y = 2, through a complete and logical procedure, thereby fulfilling all indicators of mathematical reflective thinking.
This aligns with the significant increase in posttest scores, confirming that the implementation of Deep LearningAe based PBL effectively enhances studentsAo reflective thinking ability.
Figure 1.
Student Worksheet Solving a Two-Variable Linear Equation Problem
(SPLDV)
The findings from the student worksheets align with the results of the quantitative analysis, which showed an overall increase in reflective thinking The average pretest score of 96 indicates that studentsAo reflective thinking was relatively low at the initial stage.
After the intervention using Deep LearningAebased ProblemBased Learning, the average posttest score increased to 74.
This improvement indicates that the learning not only enhanced the accuracy of studentsAo answers but also strengthened their reflective processes in checking, reviewing, and justifying the steps taken in problem solving.
The improvement in studentsAo mathematical reflective ability following the implementation of Deep LearningAebased ProblemBased Learning is clearly evident from the statistical results, where the posttest scores showed a significant increase compared to the pretest.
This aligns with findings that PBL is effective in enhancing studentsAo higher-order thinking skills .
Analysis of studentsAo responses also demonstrated their ability to interpret problems, construct mathematical models, and independently evaluate the correctness of their solution steps, consistent with findings .
indicating Indah Frantia Sari.
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that PBL facilitates deep reflection and evaluation of strategies, and supported by studies .
showing that the PBL approach can develop conceptual understanding through analytical and reflective activities.
The implementation of Deep LearningAebased Problem-Based Learning contributes significantly to the enhancement of reflective thinking ability because both approaches inherently require students to engage in higher-order thinking processes .
The PBL model encourages students to relationships between concepts, and make strategic decisions through investigation .
Meanwhile, the deep learning approach emphasizes indepth analysis, re-examination of the steps taken, and justification of the chosen solutions .
Therefore, the combination of both approaches creates a learning environment that not only guides students on how to solve problems but also enables them to understand the reasoning behind each step taken.
Pretest Postest Interpretation Method Identification Result Generalization Evaluation Problem Solving Figure 2.
Comparison of Average Pretest and Posttest Scores for Each Indicator of Mathematical Reflective Thinking Ability Based on Figure 2, all indicators of mathematical reflective thinking ability showed improvement, as evidenced by higher average posttest scores compared to the pretest.
This increase reflects studentsAo progress in understanding problem contexts, two-variable equation models (SPLDV) using appropriate methods, logically reevaluating the correctness of solutions, and generalizing solution patterns to similar situations.
Thus, the score improvements are not numerical but indicate a tangible development in the quality of studentsAo reflective thinking processes.
Overall, this study demonstrates that Deep LearningAebased PBL is capable of enhancing reflective thinking ability while also providing a foundation for further efforts to studentsAo Nevertheless, limitations of the study should be These results should be interpreted considering the limited sample size and Indah Frantia Sari.
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the context of a single school, meaning that their generalizability needs to be tested in a broader population.
These findings have practical mathematics teachers need to design PBL scenarios that require in-depth analysis, reflection on solution steps, and justification of answers, as these have been shown to enhance studentsAo reflective thinking ability.
Overall, this indicates that Deep LearningAebased Problem-Based Learning is effective in improving the mathematical reflective thinking skills of junior high school
CONCLUSION
The implementation of Deep LearningAebased Problem-Based Learning (PBL) can enhance the mathematical reflective thinking ability of junior high school students.
Based on pretest and posttest results as well as classroom observations, studentsAo reflective thinking ability improved after participating in PBL with a deep learning approach.
Nevertheless, this study was limited to a single class and specific material, so the results cannot yet be generalized Further recommended to examine the application of Deep LearningAebased PBL on other topics or at different educational levels.
REFERENCE