Advanced Study In Mathematics Research Articles

Emerging perspective of a mathematics teacher educator on connections between advanced and school mathematics

Contemporary research on mathematics teacher education reveals attention by researchers on mathematics connections (MC). Nevertheless, researchers have not investigated experiential views of mathematics teacher educators (MTEs) around MC. This article addresses the scenario by relating findings of an intrinsic case study involving a purposively chosen MTE. MTE responded to interview questions regarding experiential viewpoints on connections between advanced and school mathematics. Narrative data analysis suggested that MTE prefers special mathematics education courses (MECs). Envisaged MEC are supposed to align with school mathematics but comprised of subject matter, which is neither advanced mathematics nor school mathematics. Such courses coupled with school-based mentorship are expected to acquaint mathematics student teachers with MC. MTE could not provide subject matter-based connections except for pedagogic ones. Study of advanced mathematics was encouraged to foster understanding of MC. These findings motivate further research concerning how advanced mathematics ought to be connected to school mathematics concepts.

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.

Online-Offline Teaching for Bio-Pharmaceutical Students During the COVID-19 Pandemic: The Case Study of Advanced Mathematics in Application-Oriented Universities of China.

BackgroundWith the development of the COVID-19 pandemic, the importance of online teaching is becoming more and more prominent, especially for the basic advanced mathematics majoring in bio-pharmaceutical in colleges. However, the only online teaching model loses efficiency when facing the undergraduates in application-oriented universities.PurposeHow to improve the teaching quality of advanced mathematics has always been a concern because the mathematical abilities of students in application-oriented universities are not ideal. In this article, we develop a blending online-offline teaching model that combined online teaching and offline outcome-based education (OBE), as an alternative to traditional offline education.MethodologyThe comparative analysis experiment is carried out to the two classes of undergraduates. The control group and the experimental group are, respectively, the 2020 class students and the 2021 class students majoring in bio-pharmaceutical. The experimental group students receive the combined teaching method, while the control group students receive the traditional offline education.Results(1) From the comparative analysis, we can find that the students under the online-offline teaching model are more differentiated than those under the traditional offline education model. (2) The online-offline teaching model equipped with “case study + knowledge point + applications” process has achieved a good teaching effect in the author's university.ConclusionThe proposed teaching model can well stimulate students' interest in advanced mathematics learning and resonate with students through actual cases, thereby arousing students' autonomous learning drive and allowing them to apply what they have learned to professional fields.

ПРИМЕНЕНИЕ ПОНЯТИЯ «КОРЕНЬ N-ОЙ СТЕПЕНИ» ПРИ РЕШЕНИИ КУБИЧЕСКИХ УРАВНЕНИЙ. ФОРМУЛА КАРДАНО

The main task of teaching mathematics in secondary schools is to ensure that students have a solid and conscious mastery of the system of mathematical knowledge and skills necessary in everyday life and work for each member of modern society, sufficient to study related disciplines and continue their education. Along with solving the main problem, advanced study of mathematics provides for the formation of students ' sustained interest in the subject, the identification and development of their mathematical abilities. In-depth study of the subject should provide preparation for entering the UNIVERSITY and continuing education, as well as for professional activities that require a sufficiently high mathematical culture. The topic "Root of the nth degree" is necessary for the study of many sections of algebra, namely, to find the derivative and primitive. The article deals with the solution of cubic equations using the Cardano formula. The algorithm for applying this feature is described.

Is there a point? Teachers’ perceptions of a policy incentivizing the study of advanced mathematics

ABSTRACT A growing body of international research has highlighted the importance of students studying mathematics at an advanced level. In Ireland, the subject has been assigned a special status with the introduction of an education policy called the Bonus Points Initiative (BPI) in 2012. Students are now awarded an extra 25 points in their final State examination results if they achieve ≥40% in advanced/Higher Level (HL) mathematics. This incentive has had a noticeable impact and the numbers opting for HL mathematics in the final State examination have increased from 15.8% in 2011 to 32.9% in 2019. While this is undoubtedly a positive development, there has been little research about any knock-on effects regarding the teaching and learning of the subject. This article examines HL mathematics teachers’ (n = 266) perceptions of the BPI. The findings reveal many mixed opinions with more teachers (46%) agreeing with the initiative than disagreeing (27%), recognizing advantages such as increasing numbers and rewarding students. However, issues perceived by teachers such as questionable student motivation and the negative impact on the pace of lessons appear to be key reasons for a majority of teachers recommending a review (56%) or discontinuation (23%) of the initiative.

A “new normal”: Teachers’ experiences of the day-to-day impact of incentivising the study of advanced mathematics

ABSTRACT This article examines teachers’ perspectives of the impact in the classroom of a novel approach to boosting participation in upper secondary (Senior Cycle) mathematics in Ireland at the most advanced level (Higher Level). This approach, termed the Bonus Points Initiative (BPI), is an incentive which has coincided with a gradual increase from 16% to 31.5% in the proportion of student cohorts opting to complete Higher Level Senior Cycle mathematics studies in Ireland between 2012 and 2018. Qualitative and quantitative data for this study was gathered through questionnaire responses from 266 teachers of Higher Level Senior Cycle mathematics across Ireland. Data analysis led to the identification of a number of emerging challenges for teachers, mainly due to widening ranges of attainment in their classrooms. These challenges include addition to workloads, worries that higher attaining students are being neglected, pressure to cover the syllabus, and concerns with regards to pace of instruction.

Comparison of Teaching Instruction Efficiency in Physics through the Invested Self-Perceived Mental Effort

The main goal of the research is to determine how certain teaching instruction methods affect the achievement and mental efforts of high school students needed for learning Fluid Mechanics topic in Physics. Determining mental effort or cognitive load as a wider concept helps obtain important data, which can be used to identify teaching instruction menthods, which result in higher performance and motivation. This research is aimed to examine the efficiency of three approaches to teaching physics, which are most common in the Republic of Serbia. These are: an approach based on the use of laboratory inquiry-based experiments (LIBE), an approach based on the use of interactive computer-based simulation (ICBS) and a traditional teaching approach (TA). The article describes an experimental study conducted with two experimental and one control groups. The research was conducted on a sample of six high school classes in a gymnasium with advanced study in Natural Science and Mathematics in Novi Sad, Republic of Serbia. The total sample count was 187 students (mean age 16 years). The main conclusions of the research are that there is a causal link between the teaching instruction method applied and the achievement, or the self-perceived mental effort, of a student. Students, who were learning the teaching content through LIBE or ICS approach, have achieved better results in the knowledge test and estimatd their mental effort to be lower compared to the students, who were learning the same content through traditional teaching approach applied. The reasearch also showed, that LIBE or ICBS teaching approaches achieve higher levels of instructional efficiency and instructional involvement compared to the traditional teaching approach.

Gestures in mathematical function talk

Abstract The mathematical concept of function is an essential underpinning for advanced study in mathematics, such as calculus, and influences success in higher mathematics. Teaching about functions is challenging as many students find it difficult to understand. Effective teaching strategies are reliant on a teacher’s knowledge of the current understandings of their students. The gestures of the twenty-three middle-secondary Australian students (14–15 years old) in this study clarified the meaning of their words as they were interviewed whilst viewing two different computer simulations. Five ways in which students used their hands when providing multiple representations of mathematical functions are presented, which provide insights into students’ thinking about functions not available in their words alone.

ECONOMIC APPLIED TASKS AS A WAY TO ENLARGE MOTIVATION WHEN LEARNING ADVANCED MATHEMATICS IN TECHNICAL UNIVERSITY

The problems of students' motivation in studying advanced mathematics by students of economic specialties are considered. The ways of its increase are shown by the use of applied tasks related to the problems of the region.The authors analyzed the research motives, which encourage students of the first year to study, based on the problem of increasing motivation. One of the ways of its solution suggested by the authors is the application of applied tasks at studying of advanced mathematics. Among several classes of applied tasks, the application of applied tasks that are directly related to the future professional activity of students are considered and based on statistical data taken from specific enterprises of the region. The authors identified the main stages of solving such tasks and designated the main difficulties that may arise. For example, mathematical solution of an economic task can be given in the form of the following steps: the construction of a mathematical model which would reflect those properties of the object that really affect the investigated indicator and are consistent with known standard mathematical schemes; the choice of a solution method or a research of a model that directly depends on the quality of the model; directly the process of solution, which is purely mathematical; economic interpretation of the obtained result. As one of the key stages the authors consider the construction of a mathematical model of this process. In the article the role of mathematical modeling for increasing students' motivation in the study of advanced mathematics is considered. Basing on many years of experience, the authors showed the application of applied tasks in the study of linear algebra by students of economic specialties in the Poltava region. Also there is highlighted a number of specific tasks that the teacher can solve, using the proposed approach in the classes. Namely: the consolidation of theoretical knowledge, the formation of skills for the application of theoretical knowledge in practice, the development of skills for analysis, synthesis, induction, deduction and model construction, increasing motivation to study. Thus, in order to increase the efficiency of the perception of theoretical material and to increase the motivation to study advanced mathematics, it is useful for students to use not only applied tasks related to the specialty on which students study, but also to fill them with specific statistical data taken from enterprises of a certain region.

Universal Algebra I policy, access, and inequality: Findings from a national survey

Many in the US view algebra as a gatekeeper to advanced study of mathematics, and increasing enrollment in algebra courses as a strategy to address unequal access to educational opportunity. As a result, universal enrollment policies, which require all students to complete Algebra I by grade 8 or 9, have garnered attention in school districts or states. Based on a view that school districts are the primary implementers of state and national policy in the US, this study surveyed a nationally representative sample of districts to investigate the prevalence of such policies and their relationship to algebra enrollment. Districts reported substantial increases in Algebra I enrollments in eighth grade, although ninth grade remains the most common year students enroll. Only 26% of districts reported having universal enrollment policies; in these districts, linear regression indicated that an association with higher eighth grade Algebra I enrollment was moderated by poverty level (measured by FRL). As a result, universal policies, while decreasing within-district disparities, may increase disparities between districts. These disparities may be explained by maximally maintained inequality (Raftery & Hout, 1993) and effectively maintained inequality theories (Lucas, 2001), which posit that more affluent groups take deliberate action to perpetuate inequalities.

Support with caveats: advocates’ views of the Theory of Formal Discipline as a reason for the study of advanced mathematics

ABSTRACTThe Theory of Formal Discipline (TFD) suggests that studying mathematics improves general thinking skills. Empirical evidence for the TFD is sparse, yet it is cited in policy reports as a justification for the importance of mathematics in school curricula. The study reported in this article investigated the extent to which influential UK advocates for mathematics agree with the TFD and their views on the arguments and evidence that surround it. Quantitative and qualitative analysis of data from structured interviews revealed four themes: broad endorsement of the TFD; reference to supportive employment data; the possibilities that mathematics education might not always effectively develop reasoning and that study of other subjects might have similar effects; and concerns about causality and the extent of the evidence base. We conclude that advocates broadly support the TFD despite being aware of its limitations.

МЕЖПРЕДМЕТНЫЕ СВЯЗИ В ФОРМИРОВАНИИ МАТЕМАТИЧЕСКОГО МЫШЛЕНИЯ ПОСРЕДСТВОМ ИЗУЧЕНИЯ ГЕОМЕТРИИ (НА ПРИМЕРЕ РАЗДЕЛОВ «ВЕКТОРЫ» И «МЕТОД КООРДИНАТ»)

In recent years, it is said frequently enough about the necessity of applying mathematical knowledge in all fields of science and economy. However, mathematics is difficult for many students to learn. Increasingly frequently, various experts say about the low level of mathematical training of school and university students. Such problems as the lack of background knowledge, the separation of theory and practice, and the lack of continuity in the school and higher mathematics can be observed in the modern school mathematical education. The paper suggests one of the most effective ways of eliminating these issues – the introduction of interdisciplinary relationships to the educational process. The advantages of interdisciplinary relationships application are displayed through the example of studying two sections of school mathematics: vectors and method of coordinates. It is reasonable to show the alternative ways of solving the same task, including the application of methods of the related disciplines, to the students of classes with the advanced study of mathematics the graduates of which are the intending students of technical universities. The paper gives the example of solving the geometrical task by two methods (traditional and using vectors and the system of coordinates) and suggests the comparative analysis of the complexity of the solutions obtained. The analysis of the problems determined in the mathematical education allows speaking about a number of advantages of the interdisciplinary relationships in education. Among them are the creation of the general picture of subject matter that causes its in-depth understanding, the application of the alternative methods of solving tasks that allow escaping stereotypeness, and the elimination of gap between the school and university programs in mathematics.

Reassessing the economic value of advanced level mathematics

In the late 1990s, the economic return to Advanced level (A‐level) mathematics was examined. The analysis was based upon a series of log‐linear models of earnings in the 1958 National Child Development Survey (NCDS) and the National Survey of 1980 Graduates and Diplomates. The core finding was that A‐level mathematics had a unique earnings premium of 7–10% at age 33. In recent years, this finding has contributed to the government's agenda of increasing participation in post‐16 study of advanced mathematics in England. Given that the 1958 NCDS participants are now 57 years old, this paper repeats this important work using the 1970 British Cohort Study (BCS). Updated models are used to investigate whether the A‐level mathematics premium also existed at age 34 for the 1970 BCS participants. There does appear to be a return of approximately 11% when replicating the original model closely but the robustness of this result depends on the model specification, sample size and the handling of missing data. We consider theoretical explanations of these findings and their implications for policy and future research.

Polynomial calculus: rethinking the role of calculus in high schools

Access to advanced study in mathematics, in general, and to calculus, in particular, depends in part on the conceptual architecture of these knowledge domains. In this paper, we outline an alternative conceptual architecture for elementary calculus. Our general strategy is to separate basic concepts from the particular advanced techniques used in their definition and exposition. We develop the beginning concepts of differential and integral calculus using only concepts and skills found in secondary algebra and geometry. It is our underlining objective to strengthen students' knowledge of these topics in an effort to prepare them for advanced mathematics study. The purpose of this reconstruction is not to alter the teaching of limit-based calculus but rather to affect students' learning and understanding of mathematics in general by introducing key concepts during secondary mathematics courses. This approach holds the promise of strengthening more students' understanding of limit-based calculus and enhancing their potential for success in post-secondary mathematics.

Abstract Algebra for Algebra Teaching: Influencing School Mathematics Instruction

ABSTRACTThis article explores the potential for aspects of abstract algebra to be influential for the teaching of school algebra (and early algebra). Using national standards for analysis, four primary areas common in school mathematics—and their progression across elementary, middle, and secondary mathematics—where teaching may be transformed by teachers' knowledge of abstract algebra are developed. In each of the four content areas (arithmetic properties, inverses, structure of sets, and solving equations), descriptions and examples of the transformational influence on teaching these topics are used to depict and support ways that study of more advanced mathematics can influence teachers' practice. Implications for the mathematical preparation and professional development of teachers are considered.

Technologies Used in the Study of Advanced Mathematics by Students who Are Visually Impaired in Classrooms: Teachers’ Perspectives

Introduction This research examined the perspectives of teachers of students who are visually impaired regarding the use and effectiveness of high-tech assistive technology purported to assist visually impaired students in advanced mathematics. Methods The data for this study were collected via a mixed-methods online survey distributed through professional networks to reach teachers with experience supporting students who are braille readers in advanced mathematics. A device matrix was used to ask participants about three interrelated issues. First, which of the 35 assistive technologies presented did they use to aid students? Second, how was the technology implemented? And third, how did they rate the effectiveness of each device used? Open-response items provided space for additional tools and other feedback. Results A total of 82 surveys were completed through the device matrix question. Results conclusively indicated that 20 of the 35 technologies were used; of these, 13 were used regardless of subject. More than half of the participants indicated that the same four technologies were implemented for student information access during class, guided practice, and independent practice. Participants recommended seven technologies not included in the device matrix through the open-response questions. Discussion This survey revealed that teachers of visually impaired students are using assistive technology for multiple functions. A core set of 13 devices emerged, as well as varying subsets for specific tasks across different subjects. Limitations of the study were the small sample size and possible survey fatigue. Implications for practitioners By examining the uses of technology presented in this article, teachers can determine which assistive technology might be worth exploring to use for preparation of materials for students and which to teach others to use independently for reading or preparation of assignments in advanced mathematics courses.

Stein Meets Malliavin in Normal Approximation

Stein’s method is a method of probability approximation which hinges on the solution of a functional equation. For normal approximation, the functional equation is a first-order differential equation. Malliavin calculus is an infinite-dimensional differential calculus whose operators act on functionals of general Gaussian processes. Nourdin and Peccati (Probab. Theory Relat. Fields 145(1–2), 75–118, 2009) established a fundamental connection between Stein’s method for normal approximation and Malliavin calculus through integration by parts. This connection is exploited to obtain error bounds in total variation in central limit theorems for functionals of general Gaussian processes. Of particular interest is the fourth moment theorem which provides error bounds of the order $\sqrt {\mathbb {E}({F_{n}^{4}})-3}$ in the central limit theorem for elements {F n } n≥1 of Wiener chaos of any fixed order such that $\mathbb {E}({F_{n}^{2}}) = 1$ . This paper is an exposition of the work of Nourdin and Peccati with a brief introduction to Stein’s method and Malliavin calculus. It is based on a lecture delivered at the Annual Meeting of the Vietnam Institute for Advanced Study in Mathematics in July 2014.

Independent Study Workbooks for Proofs in Group Theory

This paper describes a small-scale research project based on workbooks designed to support independent study of proofs in a first course on abstract algebra. We discuss the lecturers’ aims in designing the workbooks, and set these against a background of research on students’ learning of group theory and on epistemological beliefs and study habits in higher education. We organise our analysis of student responses around three emerging themes: 1) structured support provided by the workbooks, 2) productive forced study of lecture notes, and 3) engaging with proofs. Discussion of our data in terms of these themes suggests several considerations for the design of tasks for independent study of advanced mathematics.

The Problem of Forming and Developing Students’ Logical Thinking in the Context of Subject Specialization in Secondary School

The article considers the certain features of the development of learners’ logical reasoning in classes with different subject-specialization at secondary school. A short author’s review of literature suggests that three main directions have been formed in the theory of teaching mathematics in the last 50-60 years in . The author investigates them in brief. The analysis shows that the current approach to forming pupils’ logical thinking, significantly varies depending on the subject-specialization and the level of mathematics training. More detailed investigation of particular methods and techniques of teaching the elements of mathematical logic in classes with advanced study of mathematics are given in the article.

Advanced Mathematical Study and the Development of Conditional Reasoning Skills

Since the time of Plato, philosophers and educational policy-makers have assumed that the study of mathematics improves one's general ‘thinking skills’. Today, this argument, known as the ‘Theory of Formal Discipline’ is used in policy debates to prioritize mathematics in school curricula. But there is no strong research evidence which justifies it. We tested the Theory of Formal Discipline by tracking the development of conditional reasoning behavior in students studying post-compulsory mathematics compared to post-compulsory English literature. In line with the Theory of Formal Discipline, the mathematics students did develop their conditional reasoning to a greater extent than the literature students, despite them having received no explicit tuition in conditional logic. However, this development appeared to be towards the so-called defective conditional understanding, rather than the logically normative material conditional understanding. We conclude by arguing that Plato may have been correct to claim that studying advanced mathematics is associated with the development of logical reasoning skills, but that the nature of this development may be more complex than previously thought.