Mathematical Knowledge Research Articles

Ethnomathematics in Intermediate Phase: Reflections on the Morabaraba Game as Indigenous Mathematical knowledge

Mathematics education remains a worrisome issue in the South African education system. This dilemma presents itself in the underperformance of learners in Intermediate Phase mathematics. In an effort to address this challenge and develop alternative approaches to the teaching of mathematics, a university-based project was initiated. As part of the project, this article reflects on the experiences of five project members who observed the implementation of Morabaraba, which is an Indigenous game, rich in mathematical relevance. This qualitative case study explored how mathematics teachers implemented the Morabaraba game to teach Intermediate Phase mathematics content. The method used to capture the data entailed reflections of the team that were derived from the observation of mathematics lessons in the Intermediate Phase. From these observations the researchers found that teachers invented their own rules when integrating mathematical concepts; they reinvented the game to cater for the complexity of mathematical content and experienced challenges in the playing of the game. We conclude that the Morabaraba game can be implemented, with practical adjustments, to comply with the objectives of the mathematics concept. The recommendation is that ethnomathematical games be employed as a teaching strategy as they are found to encourage active learning. Future research endeavours include exploring how ethnomathematical games can be sustained for future use.

Machine learning of knot topology in non-Hermitian band braids

The deep connection among braids, knots and topological physics has provided valuable insights into studying topological states in various physical systems. However, identifying distinct braid groups and knot topology embedded in non-Hermitian systems is challenging and requires significant efforts. Here, we demonstrate that an unsupervised learning with the representation basis of su(n) Lie algebra on n-fold extended non-Hermitian bands can fully classify braid group and knot topology therein, without requiring any prior mathematical knowledge or any pre-defined topological invariants. We demonstrate that the approach successfully identifies different topological elements, such as unlink, unknot, Hopf link, Solomon ring, trefoil, and so on, by employing generalized Gell-Mann matrices in non-Hermitian models with n=2 and n=3 energy bands. Moreover, since eigenstate information of non-Hermitian bands is incorporated in addition to eigenvalues, the approach distinguishes the different parity-time symmetry and breaking phases, recognizes the opposite chirality of braids and knots, and identifies out distinct topological phases that were overlooked before. Our study shows significant potential of machine learning in classification of knots, braid groups, and non-Hermitian topological phases.

Mathematics teachers and learners as collective agents of change

In this article, we report on findings from the Primary Maths and Social Justice research project, a recent collaboration between the three authors and six teacher researchers from two culturally diverse primary schools in London, UK. The project aimed to develop strategies for addressing issues of social justice in the mathematics classroom, and operationalising the recently developed theoretical concept of socio-mathematical agency, defined as learners’ capacity to apply mathematics in arguing collectively for social change. The teacher researchers’ experiences, as captured through audio recordings of research team meetings and interviews, form the basis of our data analysis. Our findings highlight the processes that enabled teachers to reflect critically on existing practice, prompting them to adopt more collaborative, discursive and inclusive problem-solving approaches that resulted in significantly higher levels of student engagement in mathematical activities. The teacher researchers found that cross-curricular projects provided more opportunities to apply mathematical knowledge in tackling social justice issues than the prescriptive and content-focused mastery-style mathematics schemes of work in place. They reported how students embraced opportunities to use mathematics in more meaningful contexts and in presenting arguments for change. We argue that a focus on developing socio-mathematical agency has the potential to enhance the role of teachers and learners as agents of change, an important element that is often underdeveloped in culturally responsive teaching.

Aksiologi Sains pada Matematika

This research investigates the relationship between the axiology of science and mathematics, focusing on values, ethics, and norms in the context of scientific research. The study underscores the importance of integrating the axiology of science into mathematics, highlighting that values and ethics should be integral considerations in the development of mathematical knowledge. The research methodology employed is a comprehensive literature review, exploring the contributions of ethical values to the development of mathematics and the ethical implications in mathematical practice. The results of the study reveal that mathematics is not a separate entity from values and ethics; rather, it is closely intertwined with them. The research details how personal values, scientific values, and social values influence the mathematical research process and its application in various scientific disciplines. Ethical implications in the development of mathematical theory are also a significant focus, considering how ethics affect axiomatic considerations, definitions, and theorems in mathematics. This research suggests that integrating the axiology of science into mathematics is highly relevant for the advancement of mathematical knowledge. It helps mitigate biases, enhances understanding of virtues, raises ethical awareness, and improves the abilities of mathematicians to adhere to values, ethics, and norms in their research. The research findings hold significance in the contexts of mathematical education and practice and in promoting ethical considerations in the use of mathematics in science and society.

Axial Symmetry in Primary School Through a Milieu Based on Visual Programming

AbstractThis paper outlines the design and application of a didactic sequence aimed at facilitating primary students’ understanding of axial symmetry, utilizing a combination of digital artefacts and paper tasks. We wondered to what extent the designed didactic sequence is able to make primary school students formulate and validate effective programming strategies to construct symmetrical images with respect to an axis and identify the key properties of axial symmetry. Data analysis from a study carried out with fifth-grade students shows a link between the evolution of students’ programming strategies and the construction of mathematical knowledge related to the definition of axial symmetry. The digital artefact and the paper tasks were effective in bringing out programming strategies and some of the key properties of axial symmetry. However, the designed didactic sequence was not enough to allow students to identify all properties related to axial symmetry, and a subsequent intervention by the teacher was necessary. The results of the experimentation led us to expand the paper tasks with additional questions for students.

What Mathematics Teachers’ Perceptions of The Integration of Educational Drama in Mathematics Instruction: A Mixed-Methods Study

This research aimed to explore mathematics teachers' perspectives on the integration of educational drama in mathematics instruction, specifically focusing on mathematical knowledge, classroom discourse and affective aspects. Using a mixed-methods approach and a sequential explanatory design, the study first employed a quantitative phase, applying a scale to gauge teachers’ views on incorporating educational drama into mathematics instruction. The questionnaire covered three key dimensions: the advancement of mathematical knowledge, facilitation of classroom discourse, and emotional elements. The framework was developed based on established theoretical models. Following this, interview protocols were designed, based on the quantitative outcomes, to explore the insights provided in greater depth. The results indicated a predominantly positive reception among mathematics educators for the use of drama in mathematics education, with consistent agreement observed across all aspects of the scale. The mean scores for the development of mathematical knowledge, classroom dialogue and emotional dimensions were notably high, reflecting substantial alignment among the participants. Qualitative analysis of the teacher interviews highlighted the beneficial aspects of employing drama in mathematics education. The teachers noted the role of drama in reinforcing concepts, enhancing specific mathematical skills and positively affecting student engagement. However, concerns were raised regarding the potential for students' focus to be diverted from core mathematical content. In terms of classroom discourse and emotional development, drama was acknowledged for its role in fostering interactive exchanges, refining students’ speaking abilities, and nurturing self-confidence and motivation. Nonetheless, there were caveats concerning the suitability of drama for all student cohorts, pointing to the need for tailored approaches.

German and Taiwanese secondary students’ mathematical modelling task value profiles and their relation to mathematical knowledge and modelling performance

German and Taiwanese secondary students’ mathematical modelling task value profiles and their relation to mathematical knowledge and modelling performance

Communication of Science & Technology in Ancient India and its Relevance in Contemporary Scenario

Communication is an essential part of all social beings and humans especially have been successful in fighting adversities throughout their evolutionary history because they could communicate and invent necessary tools and procedures to help their survival. It goes without saying that a group as a whole can prosper only if it is able to think of ideas and communicate among them as well. Discovering and inventing concepts is something that falls under the domain of science whereas sharing it with others for them to be applied and executed comes under communication. Science and communication have come together to make great strides in our history of development. India’s contribution is also unparalleled when it comes to science and communication. It is quite evident from old texts that our ancestors had exemplary communication skills. The knowledge of mathematics, medicine, and complex philosophy was majorly communicated through chants and verses. The oral tradition transitioned to written texts on copper plates, on barks and on leaves and eventually using paper by the end of 12th and 13th centuries AD. From verses and hymns, they turned to prose explaining Astronomy, Mathematics, Medical Science, Technology and Agriculture. Communication in the ancient past of India was abundant with discussions and debates adhering to rules and regulations. All these rules are quite applicable to the communication of science and technology, even in the modern era. Therefore, more research in ancient Indian communication pattern might actually be quite helpful in uncovering rich tradition of knowledge sharing.

Empowering hearing‐impaired students: A mobile learning intervention in Israeli Arab elementary education

AbstractThis study evaluates an intervention program that uses mobile learning, designed for Israeli Arab hearing‐impaired elementary school students, focused on language and mathematics. Participants included 23 hearing‐impaired students, 39 teachers, and 46 parents. The evaluation combined quantitative and qualitative methods, using questionnaires, knowledge mapping, observations, individual semi‐structured interviews, and focus group discussions. Findings revealed: (1) integrating mobile learning promoted learning enjoyment of learning, and satisfaction with program implementation; (2) content knowledge development contributed to mathematical knowledge and all language components except grammar; (3) inequality among students persisted since only hearing‐impaired students benefited from the program, leading to another kind of inequality; (4) teachers coped with various challenges: lack of technological skills, need for differential teaching for different populations in the classroom, and general lack of attention in the classroom when using apps with audible instructions and (5) parents have vital role in implementing new learning approaches for special needs students.

Foundational mathematical knowledge of prospective teachers: Evidence from a professional development training

Foundational mathematical knowledge of prospective teachers: Evidence from a professional development training

Political Conocimiento in Teaching Mathematics: mathematics teacher candidates enacting their ethical identities

The sanctioned language around “mathematics teacher knowledge” in the US centers pedagogy, content (mathematical knowledge), and students. Yet, this teacher learning approach often promotes (explicitly or implicitly) teaching and learning of mathematics that operationalizes mathematics as rigid, as a gatekeeper, and only useful for global competition in STEM-related fields. These tacit capitalist goals result in teachers being expected to convey “clear and useful” mathematical information to students, not question why mathematics is being taught, who benefits, and/or how to intervene if the learning environment is harmful for students who are Black, Indigenous, students of Color, multilingual, dis/abled, queer, nonbinary, and/or of immigrant backgrounds. In this article, we build upon a form of relational knowing that intertwines mathematics, pedagogies, students, and politics called Political Conocimiento in Teaching Mathematics. Focusing on political knowledge brings the margins to the center by recognizing political issues cannot be separated from other dimensions or added on, as if politics are not already present in mathematics teaching and learning. As three women of Color, we present empirical results from three teacher education programs in the USA with teacher candidates (n = 55) who range in intersectional identities. Using scenario-based activities that support their development of political knowledge, we apply the lens of “The Mirror Test” to highlight how teacher candidates’ intersectional identities influenced the ethical identities they enacted. Some candidates focused more on “accountability” while others focused on “performativity.” We offer implications for future research on ethical identities in mathematics education with teachers and teacher candidates of various intersectional identities.

Peningkatan Hasil Belajar Matematika Menggunakan Model Pembelajaran Problem Based Learning di Kelas XI IPA 1 SMAN 10 Medan

Students' poor performance in mathematics is influenced by the teacher's incompetence in designing an engaging learning environment. This study's overarching goal is to enhance the mathematical knowledge of SMA Negeri 10 Medan's IPA 1 students in the eleventh grade by using the Problem Based Learning paradigm. This study uses Classroom Action Research (PTK), a research technique that has been used in two research cycles. The four phases of research in each cycle are preparation, execution, evaluation, and reflection. In this study, 36 students (16 males and 20 females) from class XI IPA 1 of SMA Negeri 10 Medan participated. In order to gather data, we used observation and a 5-question fill-out exam. Percentage computations were used in the data analysis method. The research results showed that student learning outcomes in cycle I averaged 71.94 with classical completion at 55.55%. Meanwhile, learning outcomes in cycle II averaged 82.22 with classical completeness of 83.33%. Results show that classical learning outcomes have improved, leading us to believe that PBL (Problem Based Learning) Learning Model may help students in Science 1 of class XI at SMA Negeri 10 Medan learn more mathematics.

Mobile Technologies in Teaching Mathematics

Purpose. The article discusses the use of mobile technologies in teaching mathematics to engineering students. The relevance of the paper is due to the need to introduce new methods and means of teaching in connection with the increasing integration of mobile technologies into all areas of life, the interests and characteristics of the new, networked generation of students, the limited possibilities of using innovative methods and means of teaching mathematics due to the conduct of classes outside the computer lab.Materials and methods. The peculiarity of this study is that the integration of mobile technologies into the process of teaching mathematics is considered as a system of three components: technological, informational and social. The analysis of these components made it possible to obtain answers to research questions: where and how mobile devices should be used when studying mathematics; what mobile applications will provide the most effective learning of the discipline; how to organize the educational process using mobile technologies to increase interest and motivation in learning mathematics.Results. The main reasons for the lack of interest in studying mathematics and low motivation include the high level of abstraction of the discipline and the characteristics of the modern, networked generation of students. Analysis of scientific sources and our own observations made it possible to identify such characteristics as problems with understanding complex and long text, the ability to easily read from a small screen and process information in parallel, the choice of sound and images instead of text, the desire for short-term learning and instant results, and attitude towards mobile devices as vital items. Taking into account the interests and wishes of students, and relying on the didactic capabilities of mobile technologies, the paper proposes mobile applications for studying various topics of a mathematical course, assignment options that help increase motivation, develop group work skills and digital literacy. The importance and effectiveness of using social services for visualizing mathematical concepts and calculation schemes, as well as the WhatsApp messenger for instant messaging between students and receiving timely individual and group consultation from a lecturer, is shown. An important characteristic of the messenger is the presence of video communication.Conclusion. The analysis of scientific literature, the study of the requirements of employers and the interests of the modern, networked generation of students, indicated the need to integrate mobile technologies into the process of teaching mathematics. Working with different groups in terms of both the level of mathematical knowledge and the level of proficiency in mobile technologies made it possible to identify and outline ways to solve the main issues of using mobile technologies in teaching mathematics. With limited time to study mathematics, it is difficult for a lecturer to allocate additional time to present the capabilities and effectiveness of mobile applications. A solution to this issue may be to include a section “Mobile Technologies” in a course related to the study of information and communication technologies. It is noted that group work also contributes to a more effective study of the capabilities and techniques of working with mobile applications.

Formation of Schoolchildren Competencies when Teaching Geometry using Geometric Tasks with Life Content

This article discusses the importance of using geometric tasks with life content in the process of teaching geometry in order to develop key competencies of students. The authors note that this approach to teaching not only promotes the acquisition of geometric concepts, but also develops logical thinking, spatial imagination, analytical skills, problem solving skills and informed decision making. Examples of geometric problems with real-life content are provided that demonstrate the application of mathematical knowledge in real-life situations, which makes learning more interesting and practical for students. It also emphasizes the importance of using such tasks in the educational process to successfully prepare students to solve real life problems and adapt to the modern world.

Pedagogical content knowledge in prospective elementary teachers’ descriptions of teaching and learning of fractions

ABSTRACT This study investigates the pedagogical content knowledge identified in six prospective elementary teachers’ written teacher practicum reports. Focusing on the three categories of pedagogical content knowledge included in the Mathematical Knowledge for Teaching framework, the study analyzes the prospective teachers’ descriptions and combines their knowledge of fraction content with knowing about students, teaching, and curriculum. The results indicate that prospective teachers describe teaching and learning of fractions mainly on a general pedagogical level. Moreover, they do not differentiate characteristics for elementary students’ fraction difficulties and misunderstandings as well as fraction core contents in the curriculum. The findings are discussed in relation to the Swedish teacher education context in which the study was conducted. Implications for teacher education are also discussed.

Attitude and Challenges in Learning Mathematics in the Modern Word (MMW): Basis for Intervention Program

Difficulty in learning mathematics is a common and significant problem throughout the school years. It was identified that only 19% of Filipino students met the Low benchmark, meaning they had “some basic mathematical knowledge,” while 81% did not reach this level. Filipino students also lagged behind other countries in the international assessment for mathematics as mentioned in the Trends in International Mathematics and Science Study 2019 (TIMSS).To address this; a researcher conducted a study to explore students’ attitudes and challenges in learning Mathematics in the Modern World (MMW). The data gathered were used in formulating intervention programs. The study utilized the descriptive method of research and employed a survey questionnaire as the main data-gathering instrument. It involved the participation of all college students from the University of Batangas who were taking Mathematics in the Modern World (MMW) during the school year 2022-2023 using a purposive sampling technique. Furthermore, the responses of the respondents were computed using frequency count, percentage, weighted mean, ranking, and chi-square as the main statistical treatments of data. Ethical guidelines were highly considered to maintain the confidentiality of the study. Key findings include that the majority of the students are 19-21 years old, male, and enrolled in Criminology courses. Students strongly agreed and exhibited a positive attitude toward wanting to develop their mathematical skills. It was observed that there is a positive and moderate correlation between the attitude and the challenges encountered by the students in learning Mathematics in the Modern World (MMW). The main challenge identified is the difficulty in remembering math facts, concepts, rules, formulas, sequences, and procedures. The researcher proposed different intervention programs to address the challenges encountered by students in learning Mathematics in the Modern World (MMW).

Mathematics self-concept and response pattern in higher education examinations: differences between genders

The study examined how mathematics self-concept, mathematics anxiety and endorsement of mathematics gender stereotypes may impact female students’ response pattern in multiple-choice examinations in courses with mathematical content. Questionnaires measuring these three variables were completed by 435 undergraduates (327 enrolled in a non-mathematical course and 108 in a course with mathematical content). Achievement was measured via multiple-choice examinations, thus allowing us to analyse students’ response patterns (i.e. not only their mark but also the number of hits, errors and unanswered questions). In females, a poorer mathematics self-concept and higher levels of mathematics anxiety (compared with male classmates) did not impact negatively on their final mark in the course with mathematical content. However, the response pattern in this course differed according to gender and mathematics self-concept: a poorer mathematics self-concept was associated with a higher number of unanswered questions in women and a tendency toward a greater number of errors in men. This suggests that a poorer mathematics self-concept leads women to be more cautious during multiple-choice examinations so as to minimise mistakes, whereas in males it tends to be associated with a risk-taking response strategy. These results raise further concerns about the use of multiple-choice examinations, particularly for assessing mathematical knowledge in students with low mathematics self-concept.

Using Interactive Maps as a Means of Developing Fundamental Mathematical Knowledge

Introduction. Mathematics as a science and academic discipline has rich educational potential. However, due to the lack of motivation among students to master it, difficulties often arise in acquiring fundamental mathematical knowledge. The author believes that questions of the history of mathematics can help to understand how this discipline has developed over the centuries, how it influences modern life, and the tools of the digital educational environment will make it possible to do this in an accessible and understandable interactive form. The aim of the study is to present a way to use digital interactive maps as a means of developing deep mathematical knowledge in students, based on historical aspects of mathematical theory. Methodology: analysis of psychological and pedagogical literature on the research topic, generalization of many years of experience in training future teachers of mathematics and computer science, results of organizing design and research activities of students at Leningrad State University named after A. S. Pushkin (Saint Petersburg). The digital interactive map presented by the author is based on substantive aspects of the development of mathematics from the history of the Saint Petersburg mathematical school. Results. Digital interactive maps are characterized as an option for a teacher to use the functionality of a digital educational environment to fundamentalize students’ mathematical training. Interactive maps increase studentsʼ interest in educational material, integrate various types of data, use various interactivity tools, are the basis for providing detailed reference information, provide productive interaction with the map, and others. The author presents an example of creating an interactive map that reflects various locations in Saint Petersburg associated with historical aspects of the development of mathematical theory. Conclusions. A conclusion is made about the methodologically justified creation and use by the teacher of materials on the history of mathematics for the development of students’ system of fundamental mathematical knowledge, the formation of interest in the subject, the patriotic education of students, which together contributes to the conscious study of each individual topic of the school and university mathematics course, a deep immersion in the essence tasks to be solved.

METHODOLOGY OF STUDENTS PLANNING OF PROJECT WORKS

The methodology of students' project work planning skills formation, using the "Multisim" program installed along with information and communication technologies, systematization, generalization and deepening of the students' knowledge of electrical engineering, electronics, radio engineering, physics, mathematics, and informatics are presented.

Learning through teaching: the development of pedagogical content knowledge among novice mathematics teachers

ABSTRACT Pedagogical content knowledge (PCK) has been widely recognised as an important aspect of the expertise for teaching. However, the extent to which teachers’ own teaching practice can be a learning resource for them to develop PCK has not been systematically explored. This empirical study aimed to explore the unique contribution of the work of teaching in teachers’ PCK growth by concurrently considering other external professional learning opportunities teachers may have on the job. Using longitudinal data from 207 elementary and middle school teachers in the United States, we found that teachers increased their PCK through teaching on their own, albeit at different rates. Our findings were robust when other external learning opportunities teachers had were taken into account. Our findings underscored the importance of teachers’ robust knowledge of school mathematics in the development of their PCK through teaching.