Advanced Mathematics Research Articles

The Teaching Reform and Practice of Photoelectric Advanced Mathematics Course of Science and Technology Integration

The Teaching Reform and Practice of Photoelectric Advanced Mathematics Course of Science and Technology Integration

Teaching Practice of Advanced Mathematics Based on PBL Teaching Method

Teaching Practice of Advanced Mathematics Based on PBL Teaching Method

Do linguistically diverse migrants dominate advanced mathematics? Comparing Greater Sydney with the rest of New South Wales

This study uses ethnic capital theory to explore access to secondary mathematics education among linguistically diverse (LD) migrants in metropolitan and regional New South Wales, Australia. Administrative data from over 50,000 students who completed their Higher School Certificate in 2017 were analysed using multilevel logit regressions and marginal effects. The results indicate that, in Greater Sydney, all linguistically diverse first-generation youth took mathematics courses at higher rates than their peers. So did second-generation migrants from Asian backgrounds. Furthermore, considerably larger proportions of students who spoke East Asian, Indo-Aryan, or Arabic languages studied advanced mathematics. Even when only parents spoke these languages at home, their Australian-born children took advanced mathematics more often. Yet, these second-generation students were less overrepresented than those fluent in parental languages. The paper discusses the potential consequences of LD migrant concentration in Greater Sydney, stressing the importance of equitable mathematics education in metropolitan and regional areas.

What’s mathematics doing here? The role of mathematics in German Physics Olympiad tasks

Physics competitions target at motivating students for engaging in physics and at promoting students talented in physics. Physics competitions also aim at conveying a picture of physics as a scientific discipline, going beyond regular physics classes in school, and should thus also include a more intense mathematization of physics problems. Given that physics students often struggle with mathematics, mathematics in the competition may pose a challenge beyond the intended physics challenge. The present paper therefore presents an analysis of theoretical tasks employed in the German Physics Olympiad. Mathematical requirements were (1) identified by analyzing sample solutions and (2) compared to mathematics curricula. Then, (3) the solutions were categorized regarding whether the mathematics employed was used in a tool-like manner, a structuring manner, or both. Findings indicate that the first competition round typically includes rather basic mathematical requirements which are addressed in lower secondary mathematics education. In the higher rounds, more advanced mathematics is needed, which is only taught in the last years of high school, or not taught at all. The structural role of mathematics proved to be essential in the analyzed tasks. Beside the (intended) physics challenges, participants thus have to deal with mathematics they have not learned at school and have to apply mathematics in a different way than is often done at school when participating in the competition. As a consequence, unintended negative effects may occur, potentially counteracting the aim to promote students talented in physics. Implications for science education and future research are discussed.

Using conceptual analyses to resolve the tension between advanced and secondary mathematics: the cases of equivalence and inverse

Using conceptual analyses to resolve the tension between advanced and secondary mathematics: the cases of equivalence and inverse

Teacher perspectives on adoption of student-made screencasts as a peer learning approach in secondary school mathematics

A secondary school in Melbourne, Australia piloted an innovative programme in Year 10 Advanced Mathematics, in which the students recorded screencasts as a method of student-led peer learning. The four mathematics teachers who piloted the programme were interviewed to gain their perspectives on their experience and the outcomes of the initiative. Thematic analysis revealed the teachers did not recognize a need to adapt their pedagogy to integrate the technology effectively into their teaching. Three of the teachers were also focused on the quality of the screencast product rather than scaffolding the lessons to enable impactful student-led learning. One teacher who did scaffold his lessons felt unable to lead the other teachers in a consistent approach to the project. The study underscores the need for teacher professional development in pedagogical methods for student-centred, technology-enabled learning if the integration of digital technologies in twenty-first century secondary school classrooms is to be successful.

Evaluation of Limits Involving Trigonometric Functions by L’ Hospital Rule

Limit is the basic idea of calculus and it is the theoretical basis of a series of important concepts such as derivatives, continuity, definite integral, etc., In addition, it is a tool for solving many problems, thus correctly mastering the calculation methods and skills of the limit is of great significance to learn more advanced mathematics well. Through induction and conclusion, this article mainly introduces the application of several methods of solving the limit in light of L'Hopital's rule, as well as gives the analysis of example problems for each method. For this purpose, it is expected that readers can get some inspiration for solving problems from them, so that they can solve problems more clearly. Furthermore, a brief introduction is made to the main content of the L'Hospital's rule in advanced mathematics and an analysis is given of the misunderstandings in the teaching and application of the L'Hospital's rule, and a detailed description of many specific application of the L'Hospital's rule in the analysis of limit problems.

The Effect of Gender on Math Performance -- Research based on College Advanced Mathematics Scores

The influence between gender difference and mathematics achievement has always been a concern of scholars at home and abroad. Many scholars have conducted investigations on this issue, and most of them conclude that gender has no direct influence on mathematics learning. In order to verify this conclusion, this study collected the scores of advanced mathematics and family background of 300 college students from all over the country through an online questionnaire, and conducted research on the possible influencing factors of college students mathematics scores in the college entrance examination, mainly including gender, learning level, parents education background and other regression analysis. The results showed that there was no correlation between gender and advanced mathematics achievement, but school level, the frequency of learning mathematics, the mothers education level and the average monthly income of the family had an influence on advanced mathematics achievement. Therefore, girls should eliminate their fear of higher mathematics and devote more time to it. Teachers and parents should change the stereotype of girls and encourage girls to study more.

The role of expectancy-value theory in upper-secondary level students’ decisions to avoid the study of advanced mathematics

ABSTRACT Widening and increasing participation in advanced mathematics studies at upper-secondary level (age 16–18) is a significant challenge for most education systems. Policy makers in Ireland have attempted to address this challenge over the past decade by introducing an incentive to encourage students to study advanced mathematics. This study examines the reasons why students, who would appear to have sufficient prior achievement to enable them to engage in advanced mathematics studies at upper-secondary level, opted not to do so even with the presence of this incentive. Responses to questionnaires completed by 183 students in 10 secondary schools across Ireland were analysed. This analysis indicated that these students tended to avoid engaging in advanced mathematics study at upper-secondary level for a range of reasons. Most cited the expectation that they would struggle or had struggled too much with advanced mathematics. Other commonly cited reasons included the amount of time and effort required to engage effectively in the study of advanced mathematics and the impact this would have on time available to study other subjects.

Catalysts and Deterrents for STEAM Talent Development of Students from Economically-Disadvantaged Families Through Specialized STEAM Talent Residential High Schools

Talent developmental experiences of economically disadvantaged students of Specialized STEM Talent Residential High Schools (SSTRHS) were investigated. Nine students who were preferentially admitted to SSTRHS due to their families’ economic disadvantage were interviewed twice on their experiences from early childhood to college. The data were verified by checking with survey data, national gifted education data and interviews with master teachers. A thematic analysis revealed that STEM talent was recognized early by parents and acknowledged through competitions. However, there were limited resources and opportunities for them to study advanced mathematics systematically and intensely until they entered SSTRHS. At the SSTRHS, they struggled with an extremely accelerated math curriculum. However, with support from advanced peers and teachers, they could cope with their weaknesses caused by limited dosage of STEM learning. Experience conducting research with peers and mentors were the most critical catalysts for their planning for future careers as research scientists.

Principal components of students’ difficulties in mathematics in the purview of flexible learning

The abrupt migration of educational institutions into a more flexible mode of learning due to the COVID-19 pandemic has undoubtedly resulted in students' difficulties. Such difficulties specific to mathematics flexible learning are generalized in this quantitative study. Using Principal Component Analysis, seven (7) factors were identified as the emerging students’ difficulties. Further analyses reveal a moderate degree of seriousness for most of the components. However, the standard deviation suggests that the students' responses are spread across the measuring scale, indicating that the severity of difficulties experienced by other students is higher than moderate. Further comparison of such ratings shows that for students enrolled in advanced and major mathematics courses, difficulties emanating from inadequate learning materials and support, and difficulty in submitting requirements on time are more pronounced. These intertwined difficulties generally stem from the lack of planning and preparation, at the same time from the nature of mathematics as being complex, abstract, and notational. By considering these difficulties, adjustments may be prioritized by students and teachers in the hope of improving the current state of mathematics flexible learning. These improvements will eventually lead to sustainable and fully stable online academic programs that may be offered even after the pandemic.

Teachers’ perceptions of the effectiveness of a planning framework on content sequencing for the teaching and learning of mathematics

Planning is an instrument for effective teaching and learning of mathematics, which can address the dropping enrolments of year 12 students studying advanced mathematics. This study investigated teachers’ perceptions of how a planning framework on content sequencing from junior mathematical knowledge (years seven to 10) to senior mathematical knowledge (years 11 to 12) informs teaching and learning of mathematics in Queensland, Australia. This mixed methods study collected data through a survey and semi structured interviews with 16 high school mathematics teachers. The data reveals that the elements of the framework can enhance the process of content sequencing, promote an environment that enhances development of new knowledge from prior knowledge, and articulate the hierarchical nature of mathematics. The study found that the framework can enhance collaborative planning among teachers within and across year levels. The study argues that using the planning framework on content sequencing can be a significant tool that can play an important role in guiding teachers to plan and teach new mathematical knowledge building from prior mathematical knowledge.

News in immunology.

The field of immunology has undergone a very significant development in recent decades, which has been reflected especially in the beginning of this millennium in significant advances in the understanding of the immune system and in the application of this knowledge in practice. The progress and acceleration of research and advances in the field of immunology was further prompt by the unexpected onset of the COVID-19 pandemic in 2020. The intense scientific work has not only led to the development of our understanding of the immune response to viruses, but also to the rapid conversion of this knowledge into practical pandemic management on a global scale, as exemplified by the development of vaccines against SARS-Cov-2 virus. The pandemic era has further contributed to the acceleration of the application of not only biological discoveries but also technological approaches into practical applications, such as use of advanced mathematics, computer science and, more recently, artificial intelligence which are all are adding to the advances that are significantly moving the field of immunology forward. In this communication, we present specific advances in particular areas of immunopathology, which are mainly allergy, immunodeficiency, immunity and infection, vaccination, autoimmune diseases and cancer immunology.

Adapting a virtual manipulative-based instructional sequence to target maintenance

The ability to maintain mathematical skills learned is critical for students to successfully engage in advanced mathematics and promote independence in daily living skills. The purpose of this study was to evaluate the effectiveness of a graduated manipulative sequence consisting of virtual, representational, abstract, and an extended abstract phase (VRAEa) to teach various types of mathematical problems to three students with autism and mild intellectual disability using a multiple probe across behavior single case design. This study extended previous research on the virtual, representational, abstract sequence (VRA) to address deficiency in maintenance results through incorporating the extended abstract phase. The results indicate the intervention was effective, however, maintenance was variable. Implications for practice and ideas for future research to support the learning needs of students with developmental disabilities are discussed.

Cubic equation and topology of complex numbers

The study of complex analysis is an intriguing and technical topic. Over the past few centuries, mathematicians have produced ground-breaking work in this field. Mathematicians initially defined complex numbers, and then they gradually investigated their algebraic and geometric properties. They stumbled upon and researched intricate calculations hundreds of years later. The solution of the cubic equation has motivated mathematicians to explore the study of complex numbers, but it has also advanced mathematics in many other ways by exposing previously undiscovered issues. First off, when there are multiple real roots to the equation, it is impossible to obtain all of them using Cardano's formula at that time without understanding the fundamentals of complex numbers. Therefore, when they test various examples, mathematicians occasionally discover that there are other roots that are quite clear but are impossible to pass via Cardano's formula. In this paper, it start from the research of cubic equaiton and then discuss in detail topology of complex numbers.

On Intuition of the Concept of Limit: Real-life Examples Given by Secondary Mathematics Teachers

In Sri Lankan advanced level mathematics curriculum, teachers are required only to provide the intuitive idea of the concept of limit. The purpose of this study is to explore the strategies used by mathematics teachers to achieve this. Twelve in-service secondary mathematics teachers working in government and private schools participated in the study. Data was collected through lesson observations and field notes. Video recorded lessons were transcribed and qualitatively analyzed. Teaching-With-Analogy model was used as a guiding framework to evaluate how teachers integrate real-life examples in classrooms. The analysis revealed few appealing approaches as well as various flaws and misconceptions. In fact, seven teachers took deliberate efforts to link the concept to real-life and among them, four teachers gave an intuition on how x approaches a specific value, through real-life scenarios. However, many were not successful in mapping features of the real-life example to the features of the concept. Few teachers superficially and negligently applied irrelevant metaphors, analogies and colloquial terms. There were gaps in teacher knowledge when incorporating real-life connections to the limit concept. We emphasize the importance of teachers being mindful when using real-life examples in fostering intuition since inappropriate choices of examples could contribute to misconceptions.

Is it a function? Generalizing from single- to multivariable settings

Generalizing is a hallmark of mathematical thinking. The term ‘generalization’ is used to mean both the process of generalizing and the product of that process. This paper reports on five calculus students’ generalizing activity and what they generalized about multivariable functions. The study makes two contributions. The first is a fine-grained, actor-oriented characterization of the ways undergraduates generalized. This adds to knowledge in two areas: the use of the actor-oriented perspective and generalization in advanced mathematics. The second contribution is the products of students’ generalizing: what they generalized about what it means for a multivariable relation to represent a function). This adds to the literature about student reasoning regarding multivariable topics by characterizing the powerful ways of reasoning students possess pre-instruction.

Prospective Secondary Teachers’ Emergent Knowledge and Beliefs: Inquiry-Oriented Differential Equations Contributing to Teacher Preparation

This article reports on the evolution of prospective secondary teachers’ knowledge (meanings for frac{dy}{dt}) and beliefs (about teaching and learning mathematics) in a semester-long inquiry-oriented differential equations class. Students entered the course with limited, primarily procedural, meanings for frac{dy}{dt}. Throughout the semester, they engaged in collaborative mathematical inquiry using a research-based curriculum. As viewed through the emergent perspective, students’ meanings for frac{dy}{dt} and their beliefs about teaching and learning mathematics co-evolved with community norms and practices through the classroom discourse. Students’ end-of-term portfolios and portfolio presentations were analyzed for evidence of meanings for frac{dy}{dt} and beliefs about students’ roles, instructors’ roles, and the general nature of learning mathematics. In these, students expressed rich, multifaceted meanings for frac{dy}{dt} and beliefs about learning as an active process of meaning-making. While many prospective teachers do not see the relevance of advanced mathematics coursework to their career, these students reflected on their experiences in the course and volunteered ways in which their emerging knowledge and beliefs would influence their future practice. We emphasize that the classroom norms supported students in pursuing connections themselves, and conjecture that similar norms would support the development of reflective practitioners in other contexts.

Predict Who Your Friends Are?

Your friends can influence you in ways you may not realize. They can influence the way you think and behave, your likes and dislikes, what you buy, how well you do in school. Information on who you know is useful to scientists because we can use it to better understand how ideas, or in some cases diseases, spread. Scientists and mathematicians can use data and advanced mathematics to make educated guesses about how likely something is to occur. For example, we can predict your likelihood of being friends with another person given just a few pieces of information, like what school you attend, your hobbies, and who you know! In this article, you will learn about some of the tools to become a network scientist and make your own mathematical predictions about friendships!

The Development of SOLMAT (Solution Math) Application on Basic Mathematics Material for Improve Learning Outcomes

Only 28% of students in Indonesia are capable of interpreting problems into mathematical models, due to their lack of understanding of basic mathematical concepts. The purpose of stimulating and guided training is to provide students with a good understanding of problem solving in mathematics. Based on this concept, the purpose of developing the Basic Mathematical Solution Application (SOLMATDAS) is to provide understanding to students with guided training using android as the medium. Media development follows the 4D methodology (Define, Design, Develop, Disseminate), with media and ICT validation by each expert and media effectiveness by learning participants. from the results of the validation test, the percentage of learning expert validation was 90% and ICT expert validation was 90.2%, so the media can be categorized as very valid and can be used as a learning media, whereas after dissemination and testing the effectiveness of the media in obtaining pretest and posttest scores it was carried out by teacher to students in one class. The results of processing the pre-test and post-test statistics obtained a significance value of 0.038, because the significance is less than 0.05 which means there are differences in learning outcomes using SOLMAT and direct learning. From these results it can be concluded that the use of the SOLMAT application can assist in advanced mathematics learning activities which requires an understanding of basic mathematical concepts.