Mathematical Understanding Research Articles

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.

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.

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.

Enhancing mathematical function understanding in university students: a comparative study of design thinking vs. traditional teaching methods

Mathematical education requires innovative didactic strategies to enhance the understanding and application of mathematical concepts, as traditional teaching methods often lack relevance. This methodology aims to develop a problem-solving scientific approach called design thinking as a strategy for learning mathematics functions. The study was applied to a sample of 138 students of biochemical, biological, and industrial engineering careers attending the first academic cycle at the Faculty of Natural and Exact Sciences of the Particular Technical University of Loja-Ecuador. The methodology uses a quasi-experimental design with a convenience sampling method. All participants were divided into a control group (C, D, K) and an experimental group (P, Q, R). Knowledge, skills, perceptions, and engagement were measured through pretest, posttest, workshop, rubric, project, and survey instruments. The pretest results indicate that both groups had similar knowledge of mathematical functions (pretest mean experimental group: 1.42/2 and mean control group: 1.55/2). Moreover, after applying design thinking strategy to the experimental group, variables questionnaire, project, and workshop show statistical differences (p < 0.001) between groups related to the traditional learning strategy, increasing the experimental group’s score in the project (posttest mean experimental group: 1.62/2 points, and mean control group: 1.65/2). The survey opinion indicates that 53.5% of the experimental group highlighted the project’s development as positively impacting their academic training. In conclusion, problem-solving design thinking using scientific projects as a mathematical function learning strategy contributes to improving the comprehension of polynomial functions and developing mathematical competencies, abilities, and skills to generate tangible solutions for real problems.

Mathematical Foundations of Real Numbers and its Application in Computation

The study of Real Numbers (R) is a fundamental pillar of Mathematical Analysis, serving as the cornerstone for a broad spectrum of Mathematical principles and practical applications. Real Numbers are examined both in theory and in practice, impacting fields like pure Mathematics, physics, engineering, and economics. This paper thoroughly explores Real Numbers, looking at their properties, how they are represented, and their significant role in Mathematical research and various applications. We highlight the importance of Real Numbers in advancing our understanding of Mathematics. This research article provides a comprehensive overview of the Mathematical foundations on Real Numbers and its application in computation. It explores historical developments, theoretical frameworks, Mathematical proofs, comparative analyses, applications, and future directions in the study of Real Numbers. The study aims to synthesize existing knowledge, highlight key contributions, theoretical framework and application in computation.

To promote the cultivation of mathematical thinking skills and strategies in games for the senior class of kindergarten children

With the advancement of quality education and an increase in research into preschool education, early childhood education is becoming more focused on individual physical and mental development, as well as the development of comprehensive quality education. Mathematical thinking ability represents a significant aspect of young children’s cognitive development. Given that young children’s brain development is not yet fixed, so their capacity for “plasticity” is considerable. The period between the ages of three and eight represents a critical phase in developing mathematical thinking ability in young children. Children undergo rapid growth in their mathematical understanding and reasoning abilities during this time. This paper will look at promoting the development of mathematical thinking and strategies in play for the senior class of kindergarten children, based on practice, in order to provide new strategies for preschool education. Through the research on the combination of games and mathematics education, we can provide better ideas and methods for the use of games in mathematics education, help the senior class of kindergarten children to develop their mathematical thinking skills, and help them to recognize, articulate, and generalize the quantitative relationships and spatial forms that exist in their daily lives so that they can be well-prepared for the next step of entering the primary school.

Wordwall Application in Improving Numeracy Ability SDN RENDA

Education plays a crucial role in developing individual potential and enhancing quality of life, with numeracy skills being a key aspect of this process. This study aims to explore the effectiveness of using the Wordwall application in mathematics learning to improve numeracy skills among students at SDN Renda. A qualitative approach with a descriptive case study design was employed. Data were collected through observations, interviews with teachers and students, and documentation. The results indicate that the use of Wordwall significantly increased student engagement and motivation, with an average posttest score improvement of 20.4 points compared to the pretest. The application offers various quiz and game templates that make learning more interactive and engaging. However, challenges related to technology access limited the full implementation of the application. The discussion highlights that Wordwall is effective in enhancing mathematics understanding and numeracy skills, as well as increasing student interest in learning. In conclusion, Wordwall is a highly beneficial tool in mathematics education, and schools should provide technological support and training for teachers to maximize its benefits. The study recommends broader implementation of the Wordwall application to improve the quality of mathematics education in elementary schools.

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 trade-off. 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 e-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 e-greedy policy to randomly switch between these two extremes. Small and large values of e govern exploitation and exploration, respectively. By minimizing two benchmark functions and solving an inverse problem of a steel cantilever beam, we empirically show that e-greedy TS equipped with an appropriate e is more robust than its two extremes, matching or outperforming the better of the generic TS and the sample-average TS.

<span><strong>Enhancing students’ mathematical competence in learning spatial geometry in grade 11<sup>th</sup> through practical and experiential activities</strong></span>

This study embarks on enriching 11th-grade students' mathematical competencies, particularly in spatial geometry, through experiential learning. Specifically, the study focused on the practical concepts of perpendicular and parallel relationships in space. Following the qualitative approach, we collected and analyzed the data of the students' behaviors observed within a teaching scenario, which was specifically designed for them to apply their knowledge to the task "determining the classroom's center". The scenario involved three phases: defining the "classroom center," problem-solving in the classroom, and group presentations analyzing each other's methods. The results showed that all student groups successfully solved the task, which was implemented in a serious way using their mathematical understanding in a real-life context. Thus, students enhanced their two mathematical competencies, including mathematical problem handling competency (posing and solving mathematical problems) and mathematical communication competency (communicating in, with and about mathematics). This study underscores the effectiveness of practical and experiential activities in enhancing mathematical competencies among high school students, specifically within spatial geometry. It highlights the benefits of applying mathematical concepts to practical situations, promoting more profound understanding and student engagement.

<span>Enhancing student's mathematical competency in learning the graph and properties of the sine function by Google Sheets app on smartphone</span>

This study explores the integration of the Google Sheets app on smartphones to enhance students' mathematical competency in understanding the graph and properties of the sine function. The paper was an exploratory study involving 32 11th-grade students. The teaching intervention was structured into four phases: familiarizing with the Google Sheets app, creating the sine function value table, sketching the function graph, and analyzing the graph to deduce the sine function's properties. Data were collected through worksheets and audio -video recordings of group work, focusing on manifestations of mathematical competencies. The findings indicate that students completed the tasks and demonstrated significant competencies in mathematical reasoning and using digital tools. Students were able to identify and solve mathematical problems, utilize Google Sheets effectively, and use logical reasoning to enhance their understanding of the sine function. The study concludes that integrating Google Sheets into mathematics education not only aids in visualizing and comprehending mathematical concepts but also promotes interactive and collaborative learning environments, ultimately fostering a deeper understanding and proficiency in mathematics.

EFFECT OF BLENDED LEARNING EDUCATIONAL MODEL ON SECONDARY SCHOOL STUDENTS’ MATHEMATICS CONCEPTUAL UNDERSTANDING

The study was conducted to assess the effect of a blended learning educational model on grade ten students' mathematics conceptual understanding in Ethiopia. A non-equivalent pre-test and post-test quasi-experimental design was employed. Two intact classes from different schools were randomly selected and assigned to experimental (n=38) and comparison group (n=45). From the previous semester, students’ record book revealed that both groups have almost an equivalent mathematics performance. A one-way ANOVA and paired sample t-test were used to analyze the data. Moreover, to check the equivalence of the groups, the same pre-test was developed and administered. The result indicates that both groups were equivalent. The test covers tenth-grade topics relations and functions, polynomial functions, exponential functions, and logarithmic functions. The study shows that students instructed through the blended learning approach surpassed those taught via the conventional method in their comprehension of mathematical concepts. The one-way ANOVA statistical result revealed there is a notable disparity between groups with effect size of η2 =.269, which is a large effect size. Moreover, a paired sample t-test shows the experimental group showed a substantial mean difference of 14.327 with a very large effect size (d=1.016) when compared with the comparison group. Thus, the study shows that blended learning was more effective in improving mathematical conceptual understanding when compared with a conventional method of teaching. Based on the findings, the researcher recommends blended learning in mathematics education and suggests stakeholders and policymakers incorporate the model into the curriculum. Keywords: blended learning, conceptual understanding, educational model, lab rotation model, mathematical concept

Analyzing Junior High School Students’ Mathematical Communication Ability and Self Esteem using Discovery Learning Model

Mathematical reasoning, mathematical communication, mathematical problem solving, conceptual understanding, mathematical understanding, creative thinking and critical thinking are part of the mathematical abilities that students must have. One of the goals of learning mathematics according to NCTM is learning to communicate (mathematical communication). Mathematical communication skills in mathematics learning really need to be developed. This type of research design uses a quasi-experimental research design, namely pretest and post-test with non-equivalent control group design . The research used research subjects into two groups, namely the experimental group and the control group. The subjects of this research were students of class VII I MTs As-Sa'adah Padahanten Majalengka, numbering 64 people. Subjects were selected based on experimental class VIII A, 32 people and control class VIIIB, 32 people. Walkthroughs, questionnaires, and tests are techniques used in data collection. Data processing analysis was carried out qualitatively and quantitatively. The results of this research are that there is a difference in the effectiveness of the Discovery Learning model learning approach with the ordinary (conventional) learning model in terms of the aspects of mathematical communication skills and self-esteem of MTs in statistics material. The implications of the Discovery Learning learning model in this research are that it makes it easy for students to remember the lessons learned, integrate self concept according to their initial abilities, focus, be creative in exploratory projects, and the effect of knowledge transfer on learning outcomes can last a long time.

Gamification of Al-Khawarizmi Number Basis Module for Gifted and Talented Muslim Students

There are a few bright and intelligent students who have a poor level of achievement in mathematics. They think that mathematics is a boring subject and not important in daily life. A special module named the Al-Khawarizmi Number Basis Module will be used as an intervention step for students who need this guidance and support to achieve excellence in mathematics, increase their level of understanding of professional mathematics, and give encouragement to Muslim students in cultivating a mathematical life. The main objective of this research is to develop a game card for gifted and talented Muslim students at Kolej PERMATA Insan, Universiti Sains Islam Malaysia. This module helps students enhance their understanding and achievement in mathematics. This module contains three main topics, namely rational numbers, factors and multiples, and standard form. The effectiveness of this module needs to be evaluated and analyzed carefully. This way, the module can improve the level of achievement and understanding of gifted and talented Muslim students, which can then be shared with students that are in mainstream schools. In completing this module, students are also tested with a game card called Find and Match: Al-Khawarizmi Robot of Number (Riz and Ron). Then, students will undergo a pre-test and post-test to assess their level of achievement. The implementation of this module can increase understanding and strengthen the professional knowledge of mathematics among gifted and talented Muslim students and encourage students to cultivate mathematical knowledge throughout life.

Gifted Students’ Views on Integrating History of Mathematics in Mathematics Lessons

ABSTRACT This qualitative study aimed to explore the perspectives of gifted students on incorporating the history of mathematics into their lessons. The research involved 51 fifth-grade students, identified as gifted, attending a Science and Art Center in Ankara, Turkey. The study spanned 12 weeks, equivalent to one semester, during which the lesson plans were enhanced and diversified with various instructional activities. Data were gathered through the History of Mathematics Student Questionnaire (HMSQ), worksheets, and students’ reflections on the instructional activities. The HMSQ data were subjected to descriptive analysis techniques such as percentages, frequencies, and arithmetic means. Additionally, responses to short-answer questions were independently reviewed by another mathematics teacher, leading to the formation of distinct categories. Reflections from the students were analyzed through the lenses of “history as a tool” and “history as a goal.” Employing history as a tool increased students’ motivation and curiosity while viewing history as a goal enriched their understanding of mathematics’ evolution and the lives of mathematicians. Moreover, survey findings revealed a significant shift in their perceptions regarding the integration of the History of Mathematics into mathematics lessons.

Dialectics, Mathematics and the Cavaillès-Hyppolite Encounter

This essay utilises Suzanne Bachelard's essay ‘Forme et Contenu’ as an entry point to sketch out the possibility (or impossibility) of a dialectic of mathemata following the work of Jean Cavaillès and Jean Hyppolite. Firstly, I outline what Bachelard sees as a ‘dialectic’ of form and content in the history of abstract algebra, highlighting a parallel between Bachelard's reading of algebra and Hyppolite reading of the relation of the human subject to logos. Secondly, I move towards Jean Hyppolite's discussion of Lautman in ‘Mathematical Thought’ which allows for a revealing of convergences between the work of Hyppolite and Cavaillès. I conclude by noting that the similarities between Cavaillès and Hyppolite – despite possible protestations by Hyppolite – go against Hegel's understanding of mathematics, leading to the possibility of opening up a ‘Hegelian mathematics against Hegel’.

Building the educational future: Exploring the dimensions of STEM education through 3D printing and platonic solids in university mathematics teaching

This essay addresses the convergence of STEM (Science, Technology, Engineering, and Mathematics) education, exploring the opportunities and challenges of this multidisciplinary integration in the context of university mathematics education. Focusing on Platonic solids, the study examines how 3D printing becomes a transformative vehicle for mathematical understanding by providing tangible representations of abstract concepts. The tangible dimensions of teaching and the intersection between science, mathematics, and art are highlighted as key factors that enrich the educational experience. Additionally, the technological, ethical, and pedagogical challenges associated with the implementation of 3D printing in STEM education are addressed. This analysis culminates in the conclusion that this integration not only prepares students for an interconnected world but also raises new questions about ethics and responsibility in the evolution of university mathematics education.

Increasing the Ability to Understand Mathematical Concepts through the Team Games Tournament (TGT) Model Assisted by the PhET Application

This research seeks to assess the enhancement of third-grade elementary students' understanding of mathematical concepts comprehension through the Team Games Tournament (TGT) instructional approach, supported by the PhET application, compared to traditional teaching methods. The study employs a quasi-experimental method using a nonequivalent control group design. A total of 63 third-grade students participated, selected through purposive sampling. Data were collected through multiple techniques, such as tests, observations, discussions, and photo and video records. The assessment tool used was a descriptive test measuring mathematical concept understanding and is stated to have met the requirements for content validity, construct validity, reliability, level of difficulty, and differentiation. The data analysis technique for hypothesis testing uses IBM SPSS Statistics 2.6 software, namely the normalized n-gain hypothesis test. As indicated by the research results, students' ability to understand mathematical concepts at SDN Pancasila Lembang experienced a significant increase after receiving learning using the Team Games Tournament (TGT) instructional approach supported by the PhET application. This is proven by the results of the n-gain data test calculation, which is 0.36 in the medium category. The posttest average score for the experimental group was 67.78 and the posttest average score for the control class was 30.32. This indicates that the improvement in mathematical concept understanding in the experimental group surpasses that of the control group. Consequently, it can be concluded that the Team Games Tournament (TGT) learning model, supported by the PhET application, is a recommended alternative for enhancing elementary students' ability to grasp mathematical concepts.