Mathematical Concepts Research Articles

Comparative Analysis of Content, Organization, and Presentation of Mathematical Concepts in Canadian and Japanese Grade Fifth Textbooks

The purpose of this study was to examine how Canadian and Japanese fifth-grade mathematics textbooks addressed the topics, rote learning by memorization and conceptual understanding facets of teaching. Based on the comparison, it appears that both textbooks cover the required content for a grade 5 student based on the curriculum in the intended subject area. Regarding topics covered and content presentation, the Japanese textbook provides a more engaging and, overall, more successful approach to teaching student’s mathematical content. With student-driven differentiated instruction and open-ended questions, students can better engage with learning through an easy-to-follow textbook filled. With extras, students have the information they require to solve questions and understand examples. Elements of visual design once again predominate within the Japanese math textbook through the various characters, visual aids, and overall colour choice for the visual aspect of the textbook promote student engagement and help create positive emotions in mathematical content, which helps students create better learning relationships with the content being taught. Overall, the research finds that regarding the impact of students learning through the textbooks, the Japanese textbook has the advantage for student learning. However, from a teaching perspective, the Japanese open-ended student lead approach is more complex, which means if not properly taught, it could limit the success of the textbook and student learning. The Ontario textbook is still a comparable book, but making the textbook more engaging visually could improve students' overall performance when being taught with this book in the classroom.

Radiation adaptive response: the biophysical phenomenon and its theoretical description.

The radiation adaptive response (or radioadaptation) effect is a biophysical and radiobiological phenomenon responsible for, e.g. the enhancement of repair processes, cell cycle and apoptosis regulation or enhancement of antioxidant production in cells/organisms irradiated by low doses and low dose-rates of ionising radiation. This phenomenon, however, is not always present, which creates many problems both for experimenters and theoreticians. Here we propose a comprehensive and complete theoretical model of radioadaptation grounded in mathematical concept of dose- and time-related probability function of the adaptive response appearance. This can be used in the context of two special cases of the adaptive response: the Raper-Yonezawa (priming dose) effect or constant low-dose-rate irradiation (e.g. for high natural background). This complete theoretical approach is supported by Monte Carlo simulations and real-experimental data used for model calibration and validation.

Teaching mathematical concepts based on the principle of "Consistency in Explanation"

This paper explores the importance and methods of achieving "self-consistency" in the teaching of mathematical concepts. It first emphasizes the central role of mathematical concepts in learning mathematics and, through the example of teaching the eccentricity of an ellipse, points out the complexity of understanding mathematical concepts. Then, by exploring the history of "eccentricity," the paper demonstrates the human conventions and historical development behind mathematical concepts, highlighting the need for teachers to delve into the origins and meanings of concepts in their teaching. The paper proposes four strategies to achieve "self-consistency": sorting out the history of mathematical development, finding applications of mathematics in daily life, rebuilding connections between mathematical knowledge, and utilizing perspectives from other disciplines. Finally, the paper argues that "self-consistency" is not only a teaching method but also a teaching philosophy, aimed at helping students gain a deep understanding of mathematical concepts.

Ethnomathematics: Learning Geometric Transformation Through the Formation of Lagosi Motif

The Lagosi motif, a prominent design in the traditional fabric of the Wajo district, is well-known but increasingly rare among local weavers. This motif, drawing inspiration from natural elements such as flowers and stems, holds significant philosophical and cultural meaning for the Wajo people, serving as a key element of their endangered cultural heritage. Despite its cultural importance, the motif has been minimally explored in the context of ethnomathematics. Consequently, this study investigates the mathematical concepts inherent in the design of Lagosi motifs, aiming to utilize them as educational tools for students and to support cultural preservation efforts. Employing a qualitative research methodology with an ethnographic approach, data were collected through observations, interviews, and documentation. The validity and reliability of the data were ensured through triangulation and confirmability. The study reveals that geometric transformations, specifically translation and reflection, play a central role in the creation of Lagosi motifs. These findings highlight the natural emergence of mathematical concepts within local cultural practices, offering potential for their integration into contextualized mathematics education. Furthermore, the results aim to instill a sense of cultural pride in students, thereby contributing to the preservation of Wajo's cultural heritage. The study also underscores the opportunity to leverage Lagosi motifs as culturally relevant educational tools and to foster interdisciplinary collaboration between mathematics and cultural studies.

Unveiling the Mathematical Tapestry: A Comparative Journey of Bharatvarsh and the World

The study aims to investigate the fundamental knowledge of the number system and its evolution, particularly in the context of Bharatvarsh (the Indian subcontinent) and its historical regions such as Awadh, Mithila, Pandecherri, Nepal, Gandaki desh, Bajhang, Tamsaling, Goa, Sindh, Delhi, and Gwalier.The research compares the mathematics developed in Bharatvarsh or South Asia with that of the European Union, drawing insights from the field of Jyotish Shastra (Astrology), where mathematics plays a crucial role. The findings highlight the unique aspects of mathematics in Bharatvarsh, revealing its distinct applications and development within the framework of Jyotish Shastra. The study emphasizes the importance of studying mathematics for undergraduate students and provides suggestions to facilitate their understanding of this subject. The research offers a fresh perspective on the evolution of the number system and of mathematical concepts, contrasting the approaches taken in Bharatvarsh and the European Union while underscoring the significance of Jyotish Shastra in shaping the mathematical traditions of South Asia.

درجة امتلاك طالبات المرحلة المتوسطة في مدينة أبها لمهارات الحساب الذهني من وجهة نظر المعلمات

The research aimed to determine the degree to which middle school students in the city of Abha possessed mental arithmetic skills from the teachers’ point of view. The research used a social survey approach, and the sample consisted of (94) female teachers. The research concluded that the degree to which middle school students possessed mental arithmetic skills was low with an average arithmetic ( 1.50) from the point of view of teachers in the city of Abha, where the arithmetic averages of the research sample’s responses ranged between (1.91 – 1.11), The research recommended the necessity of holding training courses to develop the skills of mathematics teachers to train female students to use mental arithmetic skills, the necessity of diversity among mathematics teachers in using modern teaching methods to teach mathematical concepts, and the necessity of preparing programs and applications to enhance mental arithmetic skills among middle school female students.

Trends of abstraction research in mathematics education: A bibliometric analysis

Abstraction is fundamental in mathematics learning because students can discover the studied concepts through abstraction. Bibliometric analyses of abstraction research in mathematics education have yet to be published. A bibliometric analysis is conducted to explore trends in abstraction research. The mathematics education researchers will gain insights from studying the development of abstraction research over the last fifteen years. The primary objective of this study is to evaluate the primary journals published, the most productive authors, universities, and countries and to identify current trends in abstraction research. Data were collected from the Scopus database and analysed using VOSviewer and R software. A thorough review was conducted on 271 articles published between 2008 and 2022. The collected data was analysed and presented using R studio and VOSviewer software. The publication of abstraction research has increased every year. Abstraction studies related to geometry, computational thinking, and preschool are trend and abstraction studies related to gesture, preschool child, arithmetic, physiology, mathematical concepts, geometry, language, and cognition. Abstraction research is exciting because it will still trend until 2022. This study offers valuable insights to researchers interested in mathematics education for exploring alternative research directions to the primary research trends. Based on these results, recommendations for further research are given so that they can explore various options for research trends.

The Effect of Interactive Learning Media on Students' Understanding of Basic Mathematical Concepts in SDN 1 Tebaban

This study aims to determine the effect of the use of interactive learning media (wordwall) on the understanding of basic mathematical concepts of grade IV students at SDN 1 Tebaban. This type of research is a quantitative research of experimental methods with a quasi-experimental design of pretest-posttest control group design. The population of this study is class IV A and IV B which totals 35 students. The sample of this study uses a saturated sampling technique. The data collection technique in this study uses observation sheets, questionnaire sheets and pretest and posttest test sheets. The data analysis techniques in this study include observation and questionnaire data analysis, descriptive analysis, normality test, variance homogeneity test and independent samples test. The results of the observation sheet showed that the students' understanding of basic mathematical concepts in the experimental class reached 75% ('Very Good'), higher than the control class which only reached 62.99% ('Good'). Based on the results of the questionnaire, the use of wordwall interactive media was proven to be more effective in improving students' understanding than mind mapping learning media and teaching aids in the control class, with 91% of students in the experimental class supporting the use of wordwalls, compared to 80% in the control class. The t-test showed a value of Sig. (2-tailed) 0.02, so that the alternative hypothesis was accepted, indicating a significant influence of interactive learning media on the understanding of mathematical concepts of grade IV students at SDN 1 Tebaban.

TEACHING ANGLES AND POLYGONS USING GEOGEBRA SOFTWARE

The objective of this paper is to describe a didactic-pedagogical sequence developed to teach angles and polygons to an 8th grade class of a public state school in Palmeira dos Índios, Alagoas, Brazil. The sequence is part of the Extension Project "Mathematics and Active Methods" and addresses some common difficulties in teaching geometry, such as the abstraction of formal concepts and the students' lack of understanding of the connection of these concepts to real situations. Studies indicate that the use of digital technologies facilitates the visualization and interaction with mathematical concepts, promoting more dynamic and meaningful learning. In view of this, the sequence, applied in four classes, involved activities contextualized to sports, using the GeoGebra software. The results show that, although some students still face difficulties, the interactive and contextualized approach improved engagement and understanding of formal mathematical concepts, especially geometric ones.

Problem Posing as a Way of Promoting Individual Mathematical Thinking in STEM Contexts – The Case of Climate Change

AbstractProblems encountered in Science, Technology, Engineering and Mathematics (STEM) contexts cannot be adequately described or solved with the knowledge of a single discipline. Instead, a high level of inter- and transdisciplinary knowledge and methods is required to overcome them. These help to pose problems about the complex challenges and solve them in a creative way. The better the knowledge within one and of different disciplines is interlinked, the more targeted questions can be formulated and answered. Mathematical concepts serve as the foundation for many of the pressing problems of our time. At the same time, these problems offer a wide range of opportunities for individualized exploration. They are equally suitable for students with different interests and levels of ability, as everyone can identify and work on an individual problem within the given context. However, numerous studies have shown that posing adequate mathematical problems must be learnt as well as knowledge from different disciplines is not automatically linked or transferred to other situations. The ability to grasp the formal structure of a problem, recognize problems, and find connecting problems are characteristics of mathematically gifted children and young people that need to be promoted. In our theory-based contribution, we use a concrete context to illustrate which possible mathematically rich problems can be posed by students of different ages and abilities. This approach facilitates the development of their individual abilities according to their interests and potential.

The Impossible Biangle and the Possibility of Geometry

Abstract Kant repeatedly uses the biangle as an example of an impossible figure. In this paper, I offer an account of these passages and their significance for the possibility of geometry as a science. According to Kant, the constructibility of the biangle would signal the failure of geometry. Whereas Wolff derives the no-biangle proposition from the axiom that between two points there can be only one straight line, Kant gives it axiomatic status as a synthetic a priori principle possessing immediate certainty. Because we are unable to generate a schema for the biangle, the failure of the attempt to construct it is intuitively clear. The parallel between mathematical and empirical concepts is instructive because both involve the synthesis of disparate intuitions into a unity. We do not, strictly speaking, even possess a well-formed concept of the biangle, because its representation cannot fulfill certain basic requirements of concept formation.

A Review of Implementations of Math Competency Oriented Teaching-Learning Systems

Mathematical skills are fundamental to students' cognitive and practical development and are essential to their academic and professional success. The implementation of effective teaching-learning systems is crucial to facilitate the acquisition of these competencies, providing students with tools and methodologies that promote a deep and lasting understanding of mathematical concepts, these systems must be interactive, adaptive and designed to address the individual needs of students, ensuring that each student can reach his or her maximum potential. This paper seeks to identify the methods used in the implementation of these systems, the technology and the relevant factors for their construction. The results show that gamification and game-based learning are popular and effective methods, while block-based programming is considered to be efficient. Factors include curriculum design, interactivity and content. Among the most used technologies, scratch and virtual reality (VR) are mentioned.

Collatz Conjecture

This paper presents an analysis of the number of zeros in the binary representation of natural numbers. The primary method of analysis involves the use of the concept of the fractional part of a number, which naturally emerges in the determination of binary representation. This idea is grounded in the fundamental property of the Riemann zeta function, constructed using the fractional part of a number. Understanding that the ratio between the fractional and integer parts of a number, analogous to the Riemann zeta function, reflects the profound laws of numbers becomes the key insight of this work. The findings suggest a new perspective on the interrelation between elementary properties of numbers and more complex mathematical concepts, potentially opening new directions in number theory and analysis.This analysis has allowed to understand that the Collatz sequence initially tends towards a balanced symmetric arrangement of zeros and ones, and then it collapses, realizing the scenario of the Collatz conjecture.

Arguing for Access: Teachers’ Perspectives on the Use of Argumentation in Elementary Mathematics

This study investigates teachers’ perspectives on the use of a mathematical argumentation teaching strategy in elementary mathematics in which students disprove mathematical statements they already know to be false. Mathematical argumentation is a process through which students develop an argument about a mathematical concept and rationalize its truth or untruth through mathematical reasoning but is if often underused in mathematics. In this study we focused on a argumentation tasks which involved providing students with number statements which they already knew to be false, and inviting them to argue, using visuals, numeric notation, and/or written explanations, why it was false. Through practical action research, seven teachers from two different schools implemented this approach in their mathematics programs to one hundred and thirty-one students over the course of five months. Findings indicate this approach was easy to implement, improved student engagement, supported learners who struggled and deepened students’ mathematical knowledge. We believe that this approach can be used as a precursor to more formal proofs and provide more access for teachers and students in exploring mathematical proofs in elementary classrooms.

Challenges in sensemaking and reasoning in the context of degenerate perturbation theory in quantum mechanics

We discuss an investigation of student sensemaking and reasoning in the context of degenerate perturbation theory (DPT) in quantum mechanics. We find that advanced undergraduate and graduate students in quantum physics courses often struggled with expertlike sensemaking and reasoning to solve DPT problems. The sensemaking and reasoning were particularly challenging for students as they tried to integrate physical and mathematical concepts to solve DPT problems. Their sensemaking showed local coherence but lacked global consistency with different knowledge resources getting activated in different problem-solving tasks even if the same concepts were applicable. Depending upon the issues involved in the DPT problems, students were sometimes stuck in the “physics mode” or “math mode” and found it challenging to coordinate and integrate the physics and mathematics appropriately to solve quantum mechanics problems involving DPT. Their sensemaking shows the use of various reasoning primitives. It also shows that some advanced students struggled with self-monitoring and checking their answers to make sure they were consistent across different problems. Some also relied on memorized information, invoked authority, and did not make appropriate connections between their DPT problem solutions and the outcomes of experiments. Advanced students in quantum mechanics often displayed analogous patterns of challenges in sensemaking and reasoning as those that have been found in introductory physics. Student sensemaking and reasoning show that these advanced students are still developing expertise in this novel quantum physics domain as they learn to integrate physical and mathematical concepts. Published by the American Physical Society 2024

Choose your own T-cell fate: creation of a narrative-based, decision-making activity to engage students in immunology.

Undergraduate courses in immunology are content-heavy and combined with a new, complex vocabulary, can be an overwhelming subject for students. In-class active learning approaches have been found to improve understanding of difficult concepts in science, technology, engineering and mathematics (STEM) disciplines; however, many undergraduate courses maintain a high dependence on lecture-style teaching because of time constraints, content demands and student resistance. We designed an online, out-of-class activity, the "Life and Death of a Tcell", to complement a lecture on a complex immunological concept, T-cell development. Inspired by the "Choose Your Own Adventure" children's books, a fictional narrative was created in which students assume the role of a cell with a dream of becoming a helper Tcell. Decision-making scenarios then prompt students to draw on their knowledge from the lecture to successfully navigate the steps of T-cell development. The activity was built on two platforms, Google Forms and H5P (HTML 5 Package), both of which are readily accessible and allow the inclusion of branching logic and the creation of a decision tree-based activity. An anonymous survey revealed that students found this interactive approach enjoyable, and their perceived understanding of the content significantly increased. Students appreciated the inclusion of a novel learning resource, with requests for similar activities to be developed for other immunological concepts. In conclusion, we developed a narrative-based, decision-making activity to complement a lecture on T-cell development. As an out-of-class activity, this style of learning approach can potentially capitalize on the benefits of active learning, while also overcoming barriers of student resistance.

Leveraging mathematical modeling framework to guide regimen strategy for phage therapy

Bacteriophage (phage) cocktail therapy has been relied upon more and more to treat antibiotic-resistant infections. Understanding of the complex kinetics between phages, target bacteria, and the emergence of phage resistance remain hurdles to successful clinical outcomes. Building upon previous mathematical concepts, we develop biologically-motivated nonlinear ordinary differential equation models to explore single, cocktail, and sequential phage treatment modalities. While the optimal pairwise phage treatment strategy was the double simultaneous administration of two highly potent and asymmetrically binding phage strains, it appears unable to prevent the evolution of resistance. This treatment regimen did have a greater lysis efficiency, promoted higher phage population sizes, reduced bacterial density the most, and suppressed the evolution of resistance the longest compared to all other treatments strategies tested. Conversely, the combination of phages with polar potencies allows the more efficiently replicating phages to monopolize susceptible host cells, thereby quickly negating the intended compounding effect of cocktails. Together, we demonstrate that a biologically-motivated modeling-based framework can be leveraged to quantify the effects of each phage’s properties to more precisely predict treatment responses.

Learning Opportunities for Solving Context-Based Calculus Tasks in Nepal’s High School Mathematics Textbooks

ABSTRACT A vital skill for students to acquire in school mathematics involves solving problems embedded within real-world contexts. The extent to which students achieve intended learning outcomes largely depends on whether textbooks offer opportunity-to-learn (OTL) the assessed concepts and skills. The National Assessment of Students’ Achievements (NASA) reveals a significant deficiency in the performance of Nepali students in solving context-related problems. We conducted a content analysis of the most commonly adopted Nepali mathematics textbooks for grades XI and XII to assess the OTL they afford for tackling context-related problems in calculus. Results show that almost all mathematical concepts were introduced without providing related contexts, and around 95% of worked examples and exercise problems did not have any context and offered little or no opportunities for modeling and mathematization. Although most of the contexts presented in the problems were categorized as real and essential, the cognitive demands of many tasks were lower. The investigated textbooks were not necessarily aligned with Nepal’s updated National Mathematics Curriculum (2018), which underscores the integration of mathematics with real-world applications. Results of this study indicate that the limited access to quality mathematics textbooks might be a significant barrier hindering students’ ability to solve context-related problems.

Decolonising mathematics education: Teachers’ initial experiences of using ethnomathematical games in the intermediate phase

The African Agenda 2063 strives to attain the goals of peace and prosperity, and to do so, there must be a process of political liberation and decolonisation. In the South African context, decolonisation involves transforming the mathematics curriculum to one that is African by employing cultural approaches such as ethnomathematics. However, South African mathematics teachers still rely on Eurocentric methods for teaching mathematics. To decolonise mathematics in the Intermediate Phase, a project was initiated that utilised indigenous games for teaching mathematics. This paper aims to identify whether such efforts to decolonise mathematics education will be accepted or rejected by the mathematics teachers involved in this study. The paper is framed by the Critical Mathematics Pedagogy Theory. A qualitative case study approach was employed to collect data through a focus group interview with ten mathematics teachers who teach in rural and semi-rural areas of KwaZulu-Natal, South Africa. The findings demonstrate that ethnomathematical games encourage mathematics learning, that teachers’ initial apprehensions gave way to enthusiasm, that learners were excited to play and learn, and that teachers found it effortless to integrate mathematical concepts such as addition, subtraction, multiplication, factorisation, and quadrilaterals with the indigenous games of Morabaraba, Tshetershere, Diketo, and Kgati. Recommendations include that mathematics educational policies incorporate ethnomathematics as a policy directive and that teachers receive training in alternative cultural pedagogical practices for teaching mathematics, such as the use of ethnomathematics. Future research endeavours should focus on finding avenues to sustain ethnomathematics in the classroom.

Pedagogical responsiveness of mathematics instruction at inclusive secondary schools: A particularistic case study

This qualitative particularistic case study explores the perspectives and experiences of learners with visual and hearing impairments regarding the teaching of mathematics. Extreme-case sampling was employed to select four learners (three hearing impaired and one visually impaired) from the only two inclusive secondary schools in the northern part of Namibia. The study qualitative study utilised open-ended questionnaires and interviews. Verbatim quotes from the research participants were used as inferences to identify descriptors within the first-generation Cultural Historical Activity Theory, which served as a theoretical and analytical tool. The findings revealed that the activity system of learners with visual and hearing impairments exhibited tensions that negatively affected their performance in mathematics. Learners experienced marginal responsive pedagogy in mathematics classrooms due to teachers' lack of knowledge and pedagogical skills for delivering meaningful mathematical instruction. While learners without disabilities struggled with comprehension and effective communication skills, learners with visual and hearing impairments faced a lack of equipment to support the conceptualisation and visualisation of mathematical subject content. The authors emphasise the critical importance of addressing communication barriers and adapting teaching practices to support the learning and inclusion of visually and hearing-impaired learners in mathematics, with an emphasis on mathematical language that promotes a better understanding of mathematical concepts. Teacher training should also incorporate responsive pedagogy to enhance outcomes and foster a more inclusive learning environment that promotes academic success and positive social experiences for all learners. Recommendations at a policy level, including advocating for a decolonised curriculum were made.