Mathematical Understanding Research Articles

Introduction To Basic Mathematical Concepts Through Learning Media

Understanding fundamental mathematical concepts is essential for developing students' logical reasoning, critical thinking, and problem-solving abilities. This study employs a literature review methodology to analyze foundational mathematical concepts—such as numbers, arithmetic operations, geometry, algebra, and statistics—and their impact on students' learning and cognitive development. The findings indicate that interactive tools, cultural contexts, and hands-on learning media significantly enhance students' comprehension and retention of these basic concepts. Studies reviewed include applications of traditional games, visual aids, and realistic mathematics approaches, which help bridge abstract concepts with everyday life, thus fostering a deeper, more applicable understanding of mathematics. This research provides insights for educators to utilize varied, engaging approaches that support students' foundational knowledge, encouraging better academic outcomes and practical skills. The findings suggest that contextual and interactive methods in early mathematics education are crucial for cultivating a robust foundation for further learning.

PENGEMBANGAN KOMPETENSI MATEMATIKA MELALUI KEGIATAN KEBHINEKAAN

This research discusses the development of mathematical competence through diversity activities, focusing on student learning through the Nusantara Module in the Independent Student Exchange Program in the Tumpang Valley. The background of this research is rooted in the importance of improving the quality of mathematics education and strengthening the values of diversity in the university environment. The purpose of this study was to investigate the effectiveness of Nusantara Module in improving the understanding and mathematical skills of students from various cultural and ethnic backgrounds. The research method used is a qualitative approach with interview, observation, and documentation analysis techniques. The results showed that learning through the Nusantara Module succeeded in building a deep understanding of mathematics and problem-solving skills among students, while appreciating cultural diversity. In addition, there is an increase in active participation, creativity, and enthusiasm for student learning in the context of diversity. In conclusion, the Nusantara Module in the context of the Independent Student Exchange Program not only improves mathematical competence, but also strengthens understanding of the values of diversity and cross-cultural cooperation. These findings provide the basis for the development of diversity-based learning strategies in higher education, supporting students' academic and social growth in a multicultural atmosphere.

Defragmentation of Students' Conceptual Understanding in Solving Non-Routine Mathematics Problems

This study investigated the impact of concept fragmentation on students' understanding of mathematics and examines efforts to reduce it. Concept fragmentation hinders problem-solving abilities, often occurring when students struggle to create meaningful connections or new representations from existing ones. Limited interventions, such as cognitive conflict and scaffolding, are suggested to address this issue. This study employed a qualitative descriptive approach, focusing on two seventh-grade students in Sukoharjo, Indonesia, who exhibited concept fragmentation. Data collection involved tests, interviews, and observations, with analysis following qualitative methods. The findings indicate two main types of fragmentation: translational thinking fragmentation and meaningless connection fragmentation. These arise when students attempt to build new representations but make errors due to disconnected prior knowledge. Interventions revealed a pattern of developing schemas, where students knit together concepts to minimize problem-solving errors. Techniques such as rereading problems and substituting information into formulas improved concept comprehension. The study concludes that defragmentation aids students in connecting existing knowledge with new information, enhancing their problem-solving strategies. Future research should investigate other fragmentation types and effective interventions for reducing fragmentation in mathematics learning.

On the Activity of Students in Mathematics Lessons the Effect of the Pygmalion Effect

The application of mathematical sciences has a great role in the development of society. Mathematical education is a necessary and important part of general education. Therefore, strengthening the mathematical preparation of students is considered the most important task facing secondary schools. Mathematics is taught in a wide range of programs at every level of education in secondary schools. However, it is not perceived in the same way by all students. While some students find mathematics easy to understand, others find it very difficult. There are several reasons for this: poor mastery of mathematical concepts from primary classes, genetic influences, student's lack of independent work, concepts becoming more complex from class to class, age level, environment. We must keep in mind that it is impossible to succeed in life without understanding mathematics. Therefore, it is necessary to look for the cause of difficulties and try to eliminate them. The purpose of the research is to study the impact of the Pygmalion effect on students' performance in mathematics lessons in secondary schools. Using the Pygmalion effect directs students with a poor understanding of mathematics to positive thinking and creates interest in learning mathematics. Many empirical studies also support the predictions of the Pygmalion effect, but it has been found that its effect sizes can vary from low to moderate. As a result of the research, we concluded that applying the Pygmalion effect in teaching mathematics is one of the best ways to achieve success. The research will be useful for mathematics teachers and parents as well as a contribution to the relevant literature.

ANALYSIS OF MATHEMATICAL COMMUNICATION SKILLS IN SOLVING MODELING PROBLEMS WITH MARITIME CONTEXT IN ALGEBRA MATERIAL

This research was motivated the importance of mathematical communication skills in improving students' mathematical understanding. The aim of this study is to describe students' mathematical communication skills on algebraic material when solving modeling tasks in the marine context. The research methodology used in this research is descriptive qualitative research. The research was conducted at SMP Negeri 2 Bintan in June 2024. The research subjects were VIII C students. Data collection techniques were test instruments and interview guidelines. Data analysis techniques are data reduction, data presentation, and inference. The result show that there were 4 students in the high category with a percentage of 19%, 12 students in the medium category with a percentage of 57%, and 5 students in the low category with a percentage of 24%. The researcher suggest that students' mathematical communication skills can be further improved by practicing mathematical problem-solving related to mathematical communication. Keywords: mathematical communication, modeling problems, maritime context, algebra

Students' Learning Obstacles on Sequences and Series Viewed by Pirie Kieren's Theory

Mathematical understanding develops through eight layers, as described by Pirie and Kieren: primitive knowing, image making, image having, property noticing, formalizing, observing, structuring, and inventizing. However, evidence indicates that many students encounter obstacles that hinder their progression through these layers. This study aims to identify and describe the learning obstacles students face in understanding sequences and series, utilizing Pirie and Kieren's theoretical framework. A descriptive qualitative research design was employed, purposive sampling to select participants from 30 Year 11 students in Malang, Indonesia. Data were gathered through tests and interviews and analyzed based on indicators of learning obstacles and the corresponding layers of mathematical understanding outlined by Pirie and Kieren. The findings reveal that many students experience significant difficulties in noticing and formalizing layers within the property. These challenges are attributed to inadequate foundational knowledge (ontogenic conceptual obstacles) and a lack of structured opportunities for developing deeper mathematical understanding (ontogenic instrumental, didactical, and epistemological obstacles). The results underscore the need for further research to address these learning barriers by focusing on enhancing students' foundational knowledge and designing educational experiences that foster the growth of mathematical understanding.

Ethnomathematics Exploration in The Joglo House of Banyumas

Indonesia is known as a country with cultural diversity, one of which is the traditional house, a symbol of each tribe and region. In its implementation, the Joglo house is only used as a residence and a place to protect against exposure to sunlight. Still, after exploring it in more depth, the structure of the traditional house can be used as learning material in the field of education. One field of education that can study culture through the structure of conventional house buildings is mathematics. Learning mathematics that includes culture can help students improve their understanding of abstract mathematics. However, not many mathematics lessons incorporate culture into the material. So, this research aims to explore Banyumas Joglo houses regarding geometric concepts. The type of research used is qualitative research with an ethnographic approach. Data collection methods include observation, documentation and interviews. The data analysis consists of data reduction, data presentation, and conclusion. This research shows that the Banyumas Joglo traditional house is an example of a conventional house rich in ethnomathematics, where elements of traditional culture contain mathematical knowledge in the form of geometric concepts.

Development of Educational Game Learning Media with An Ethnomathematics Approach Based on Banyumas Culture to Improve Students’ Mathematical Understanding Skills

The background of this research is the low mathematical understanding skills of seventh-grade students at SMP Negeri 1 Ajibarang. This study aims to develop educational game learning media with an ethnomathematics approach based on Banyumas culture in Geometry material that is valid and effective in improving mathematical understanding skills. The research method used Research and Development (R&D) with the ADDIE model. The population of this study includes all seventh-grade students, and the sample used in this study consists of class VII-B as the experimental class and class VII-A as the control class, with each class comprising 34 students. The results of this study indicate that the educational game learning media with an ethnomathematics approach based on Banyumas culture in Geometry material is valid, with an average percentage of 90.48% from material experts, 92.50% from media experts, 83.08% from mathematics teachers, 93.67% from small group trials, and 88.13% from field trial. The educational game learning media with an ethnomathematics approach based on Banyumas culture is also declared effective in improving students' mathematical understanding skills in Geometry material, as evidenced by the t-test result Sig. (2-tailed) = 0.000 < 0.05. Since the average N-Gain score of the experimental class is 0.73, which is higher than the average N-Gain score of the control class, which is only 0.43, it can be concluded that the educational game learning media with an ethnomathematics approach based on Banyumas culture is effective in improving the mathematical understanding skills of seventh-grade students in Geometry.

The right answer for the wrong reason: how prospective teachers leverage pedagogical opportunities

ABSTRACT Responding to the pedagogical opportunities afforded by student thinking allows teachers to develop mathematical understanding that is directly related to students’ intellectual needs. Knowing which instances of student thinking provide such opportunities, however, is not always clear – and neither is the most effective way to leverage this thinking. Using a novel type of scripting task, our study explores how prospective teachers ( n = 35 ) would respond to a student’s strategy that produces a correct answer via incorrect reasoning. We first categorise participants’ responses according to whether they recognise this opportunity as a Mathematically Significant Pedagogical Opportunities to Build on Student Thinking (Leatham et al., 2015); for those that did, we next examine the means by which they seek to engender a sense of disequilibrium in the student-characters of their scripts.

STUDI LITERATUR: PERANAN PENGGUNAAN MODEL-MODEL PEMBELAJARAN PADA MATERI INTEGRAL

This article presents a literature review on various learning models applied to improve mathematics learning outcomes in integral calculus. The 4 studies that form the basis of the literature study have discussions in understanding integral concepts, critical thinking skills, motivation and learning outcomes and the dimensions of character education. These four studies provide a comprehensive picture of the various approaches that can be used to improve students' understanding and learning outcomes in integral calculus. The first study highlights the effectiveness of the Problem-Based Learning (PBL) model in enhancing students' understanding of integral concepts. The PBL model has been shown to encourage active student engagement and sharpen critical thinking skills. The second study compares two learning models, Discovery Learning and Accelerated Learning, focusing on students' learning motivation and mathematics learning outcomes in integral calculus. The results indicate that Discovery Learning is more effective in improving students' motivation and their deeper understanding of integral concepts. The third study implements Lesson study with a PBL approach that incorporates character education in the class. This approach not only improves mathematical understanding but also develops students' character, such as collaboration and curiosity. Meanwhile, the fourth study develops a learning device for integral calculus based on a problem-solving model to enhance students' creative thinking abilities. These four studies demonstrate that problem-based learning models, whether PBL, Discovery Learning, or problem-solving-based instructional devices, are effective in improving students' understanding and critical thinking skills in integral calculus.

Mature intuition and mathematical understanding

Mathematicians often describe the importance of well-developed intuition to productive research and successful learning. But neither education researchers nor philosophers interested in epistemic dimensions of mathematical practice have yet given the topic the sustained attention it deserves. The trouble is partly that intuition in the relevant sense lacks a usefully clear characterization, so we begin by offering one: mature intuition, we say, is the capacity for fast, fluent, reliable and insightful inference with respect to some subject matter. We illustrate the role of mature intuition in mathematical practice with an assortment of examples, including data from a sequence of clinical interviews in which a student improves upon initially misleading covariational intuitions. Finally, we show how the study of intuition can yield insights for philosophers and education theorists. First, it contributes to a longstanding debate in epistemology by undermining epistemicism, the view that an agent’s degree of objectual understanding is determined exclusively by their knowledge, beliefs and credences. We argue on the contrary that intuition can contribute directly and independently to understanding. Second, we identify potential pedagogical avenues towards the development of mature intuition, highlighting strategies including adding imagery, developing associations, establishing confidence and generalizing concepts.

Implementasi Model Pembelajaran Matematika Realistik (PMR) dalam Meningkatkan Pemahaman Matematis Siswa Kelas VI MIN 2 Kota Mataram

This research is based on the importance of learning realistic mathematics in mathematics. The purpose of this study is to determine the implementation of learning by using realistic mathematics learning models in improving the mathematical understanding of students of class VI MIN 2 Mataram City. PMR is a learning approach to mathematics that involves the reality and experience of students. In mathematics learning students are required to be more active in finding and finding solutions to a problem, so that learning that occurs will be centered on students. The research method used is a qualiative method for collecting data through observation, interviews and documents. The results showed that the application of realistic mathematics learning was effective in increasing students' mathematical understanding in the discovery of mathematical concepts through contextual learning and relevant to daily life. Thus, this research provides a significant involvement in understanding the role of PMR in increasing the mathematical understanding of students as well as part of an effective mathematical learning approach.

Mathematical understanding – Common themes in philosophy and mathematics education

We present different characterizations of mathematical understanding given by mathematicians, philosophers of mathematics, and mathematics educators. One purpose is to illustrate the diversity of these characterizations. Although the descriptions of understanding may seem incompatible, the paper ends by pointing to some shared themes. They include an emphasis on qualities such as relations and unification. Additionally, we note that re-presentation, including visual representations, is thought to play a role in understanding.

Epsilon-Greedy Thompson Sampling to Bayesian Optimization

Abstract Bayesian optimization (BO) has become a powerful tool for solving simulation-based engineering optimization problems thanks to its ability to integrate physical and mathematical understandings, consider uncertainty, and address the exploitation–exploration dilemma. Thompson sampling (TS) is a preferred solution for BO to handle the exploitation–exploration tradeoff. While it prioritizes exploration by generating and minimizing random sample paths from probabilistic models—a fundamental ingredient of BO–TS weakly manages exploitation by gathering information about the true objective function after it obtains new observations. In this work, we improve the exploitation of TS by incorporating the ε-greedy policy, a well-established selection strategy in reinforcement learning. We first delineate two extremes of TS, namely the generic TS and the sample-average TS. The former promotes exploration, while the latter favors exploitation. We then adopt the ε-greedy policy to randomly switch between these two extremes. Small and large values of ε govern exploitation and exploration, respectively. By minimizing two benchmark functions and solving an inverse problem of a steel cantilever beam, we empirically show that ε-greedy TS equipped with an appropriate ε is more robust than its two extremes, matching or outperforming the better of the generic TS and the sample-average TS.

Pedagogical responsiveness: Focus on the Ukuqonda institute’s mode of engagement in mathematics

This paper was prompted by mathematics teachers’ challenges in implementing the proposed framework that centres on teaching for understanding, as opposed to teaching aimed at producing marks as evidence of achievement. Teaching for understanding in mathematics, among other factors, requires the creation of engaging and inclusive learning environments underpinned by teachers’ pedagogical responsiveness to the diverse needs of learners. We investigated pedagogical responsiveness, focusing on the Ukuqonda Institute’s mode of engagement in mathematics. We specifically pursued two research questions: 1) What are the key elements of the Ukuqonda Institute’s mode of engagement in mathematics? and 2) What characterised the pedagogical responsiveness of mathematics educators at the Ukuqonda Institute? We adopted collaborative autoethnography and used two data sets generated from audio recordings captured by the authors. We employed narrative analysis and organised the findings using the characteristics of pedagogical responsiveness as heuristic devices. The findings show that the participants’ pedagogical responsiveness was stimulated by the iterative process of team planning, deliberate implementation, and reflection. The prevalent characteristics were learner interactions, learner focus, inclusivity, dialogue and relationality, knowledge work, social justice, and equity. We recommend that similar studies be conducted targeting pedagogical responsiveness with a focus on other mathematical themes. Furthermore, this study suggests that, to remain pedagogically responsive, there should be constructive alignment among the types of tasks, questions, classroom interactions, and targeted content that underpin mathematics teaching and learning across different educational levels.

Mathematics and Mathematics Education Values: An Analysis of Implementability in Mathematics Learning at Madrasah

Integrating values into all aspects of the learning process for a particular subject is an important part of education. Students must not only develop cognitively, but the formation of attitudes and characters must also be in line with the overall educational goals. Mathematics, as one of the core subjects that students must learn, must also reflect the values embedded in it. The success of mathematics education can be assessed through the values conveyed during the learning process. This study aims to explore and analyze teachers' understanding of Mathematical Values and Values in Mathematics Education, as well as how these values are conveyed in the classroom. This study aims to explore and analyze teachers' understanding of Mathematical Values and Values in Mathematics Education, as well as how these values are conveyed in the classroom. This study uses a qualitative approach with a case study design. Data were collected through interview techniques from 4 mathematics teacher informants determined by percussive sampling techniques. Data analysis uses an intractic model in line with the data collection process through three activities simultaneously; (1) data condensation; (2) data display, and (3) drawing conclusions/verification. The results of the study indicate that teachers' understanding of Mathematics values [rational values] and Mathematics education values are very important in improving students' logical thinking skills, and providing a comprehensive understanding of mathematics in developing their problem-solving skills. In mathematics teaching practices, teachers convey mathematical values during problem-solving activities, where students must understand various mathematical equations, concepts, and methods. While Mathematics education values (MEV) are conveyed during the problem-solving phase and reinforced at the end of the teaching and learning activities.

Perceptions of Saudi elementary school special education teachers regarding mathematics content and instructional practices for students with intellectual disabilities

This study explored the perceptions of special education teachers of students with intellectual disabilities at three elementary schools in Saudi Arabia regarding the instructional practices and content used for teaching such students mathematics. The research employed a qualitative multiple case study design and used an open-ended questionnaire, artifacts, and teacher interviews with 14 special education teachers. By employing thematic analysis of the data collected, the following major issues were identified: (a) there is a strong focus on teaching only numbers; (b) there is an emphasis on teaching functional life skills rather than academic mathematics skills; and (c) mathematics instruction emphasizes procedural mathematics rather than mathematical understanding, also known as conceptual mathematics. These findings illustrated that elementary school students with ID in the sample did not receive comprehensive mathematics education in terms of either content or instructional practices. It might be beneficial for future studies to replicate and extend this research to incorporate observations of the classroom to obtain objective data on teachers’ actual practices while instructing students with ID in mathematics.

Developing Distractors for Mathematics Multiple Choice Items: A Literature Review

Abstract Introduction: This article aimed at digging deep into distractors used for mathematics multiple-choice items. The quality of distractors may be more important than their number and the stem in a multiple-choice question. Little attention is given to this aspect of item writing especially, mathematics multiple-choice questions. This article provides an engaging but succinct literature review on the development of plausible distractors for mathematics multiple choice items. Methods: A literature review was conducted on developing effective distractors for mathematics tests. The review explored potential strategies for generating distractors that effectively assess students’ understanding and problem-solving skills in mathematics. Results: Alternative sources of distractors other than students’ misconceptions are provided to aid the development of plausible distractors for mathematics multiple-choice items. Discussion: Practical guidelines for judging distractors fit for mathematics questions are provided to help teachers improve their item writing skills based on literature and experience as a mathematics teacher. Common pitfalls in distractor development were identified to enable mathematics teachers to have a clear path for their work. Limitations: This article focused on the development of plausible distractors specifically for mathematics multiple choice questions. Conclusions: Test constructors must ensure aligning the distractors with the objectives of each lesson in order to make the distractors relevant to the demands of the item.

Integration of Calculator.oi Application in Improving Mathematics Learning Outcomes of Students in Elementary Schools

This study aims to evaluate the effectiveness of the Calculator.oi application integration in improving students' mathematics learning outcomes in Elementary Schools. This study focused on fourth-grade students at SDN 22 Ketapang and used a quantitative approach with an experimental method. In this study, the experimental method was applied to compare the differences in students' learning outcomes before and after using the Calculator.oi application. Learning outcome data were collected and analyzed using the Wilcoxon test, after the normality test showed that the data were not normally distributed. The results showed a significant increase in students' mathematics learning outcomes after using the Calculator.oi application. Analysis of the differences between pre-test and post-test scores revealed that all students experienced significant progress in their understanding and achievement in mathematics. The significance value obtained from the statistical test was below 0.05, indicating that this application had a significant positive impact and was effective in improving students' learning outcomes. Based on these findings, it is recommended that the Calculator.oi application be integrated more widely into the mathematics learning process in Elementary Schools. The use of this application is considered an innovative and effective learning medium in supporting the understanding of mathematical concepts. Further research is recommended to involve a larger sample size and different class variations in order to obtain more comprehensive and representative results. In addition, exploration of students' perceptions regarding the use of this application as a learning tool is also important. This will provide a deeper understanding of how the Calculator.oi application can be accepted and utilized in a broader learning context.

THE RELATIONSHIP BETWEEN LEARNING STYLES AND UNDERSTANDING MATHEMATICAL CONCEPTS IN QUADRATIC FUNCTIONS

The research teka-teki is that students are experiencing difficulties in understanding the concept of quadratic functions. The aim of the study is to determine the relationship between learning styles and mathematical concept understanding in solving quadratic function problems among 9th grade students at SMP Negeri 3 Ulunoyo. This research adopts indah ambang quantitative approach with indah ambang correlational study type. The population in this study consists of 20 9th grade students from SMP Negeri 3 Ulunoyo, and the sampling technique used is besaran sampling. The instruments used are questionnaires and tests of mathematical concept understanding. Based on the research results, indah ambang correlation coefficient of 0.64 was obtained, indicating indah ambang positive and strong correlation. The coefficient of determination is 40.96%, showing that the contribution of learning styles to concept understanding is 40.96%, with the remaining contribution attributed to other factors. The calculated t-value is 3.50324, and to determine the critical t-value at indah ambang 0.05% significance taraf, the t-table value is 2.100922. Since the calculated t-value (3.50324) is greater than the t-table value (2.100922), it can be concluded that there is indah ambang relationship between learning styles and mathematical concept understanding among 9th grade students at SMP Negeri 3 Ulunoyo, with the correlation taraf falling into the moderate category. Therefore, the researcher suggests (1) students should recognize their own learning styles and optimize them to facilitate better learning and achieve optimal performance, and (2) teachers are encouraged to adapt their teaching styles to meet the needs of all students, not just those with specific learning styles.