Nature Of Mathematics Research Articles

Addressing discontinuous root-finding for subsequent differentiability in machine learning, inverse problems, and control

There are many physical processes that have inherent discontinuities in their mathematical formulations. This paper is motivated by the specific case of collisions between two rigid or deformable bodies and the intrinsic nature of that discontinuity. The impulse response to a collision is discontinuous with the lack of any response when no collision occurs, which causes difficulties for numerical approaches that require differentiability which are typical in machine learning, inverse problems, and control. We theoretically and numerically demonstrate that the derivative of the collision time with respect to the parameters becomes infinite as one approaches the barrier separating colliding from not colliding, and use lifting to complexify the solution space so that solutions on the other side of the barrier are directly attainable as precise values. Subsequently, we mollify the barrier posed by the unbounded derivatives, so that one can tunnel back and forth in a smooth and reliable fashion facilitating the use of standard numerical approaches. Moreover, we illustrate that standard approaches fail in numerous ways mostly due to a lack of understanding of the mathematical nature of the problem (e.g. typical backpropagation utilizes many rules of differentiation, but ignores L'Hopital's rule).

Exploring the multifaceted roles of mathematics learning in predicting students' computational thinking competency

BackgroundThere exist shared competencies between computational thinking (CT) and mathematics, and these two domains also mutually benefit from various teaching approaches. However, the linkages between mathematics and computational thinking lack robust empirical support, particularly from student-centered learning perspectives. Our study aimed to enhance our understanding of the connections between students' mathematics learning and computational thinking. To assess students' mathematics learning, we measured their beliefs about mathematics learning and their level of mathematical literacy (ML). Our hypothesis posited that students' beliefs concerning mathematics learning, encompassing their views on the nature of mathematics and their attitude towards the subject, can both directly and indirectly influence their CT, with ML serving as a mediating factor. Our data were gathered through surveys and tests administered to eighth- and ninth-grade students. Data were analyzed using partial least squares–structural equation modeling (PLS–SEM).ResultsThe evaluation of the measurement model indicated strong internal consistency for each construct. Both convergent and discriminant validity were also established. Upon assessing the structural model, it was found that beliefs about the nature of mathematics positively predicted attitudes towards mathematics, and this belief also indirectly predicted ML through positive attitudes towards mathematics. In addition, ML directly and positively predicted both CT subscales. Notably, a comprehensive mediating effect of ML on beliefs about mathematics learning and CT was identified in the analysis.ConclusionsThis study advances the understanding of the relationships between mathematics learning and CT. We have further confirmed the importance of mathematical literacy in predicting CT and its mediating role between beliefs about mathematics learning and CT. It is suggested that teachers could promote students’ CT competence by enhancing their mathematical literacy or integrating mathematics and CT into the same learning activities. Finally, we propose that upcoming investigations treat CT assessments as formative constructs, diverging from their reflective counterparts.

Generalized viscoelastic flow with thermal radiations and chemical reactions

Background:A generalized model of mathematical nature is considered to address the viscoelastic flow problem using fractional derivatives. Control/freedom of the flow mechanism is achieved with these derivatives. In simulations of industrial interest, more variations are available with fractional derivatives when compared with ordinary derivatives. Relaxation times are incorporated to handle the abrupt changes in the flow domain. Fluid flow is carried out under the influence of thermal radiations and when a heat source or sink is present. Chemical reactions of the first order are observed in the mathematical modeling of the flow. Methods:Flow is induced with the movement of the lower surface while applying force on the x-plane. Simulations of the governing mathematical problem are computed with the combination of finite element and finite difference algorithms. Significant Findings:It is noted that velocity, temperature, and concentration change with the variation of fractional order derivatives which was not possible with the classical derivatives. Moreover, with greater relaxation times, velocity, temperature, and concentration remained at a lower level. The modeled mechanism can be considered to avoid costly trials in chemical and polymer casting industries.

The Effects of GeoGebra-Assisted Transformation Geometry Instruction on Student Achievement, Attitudes, and Beliefs

This study aims to investigate the effects of using GeoGebra, a popular dynamic geometry software, on students’ mathematics achievement, attitudes toward geometry, and beliefs about mathematics and its teaching in transformation geometry teaching. The study was a quasi-experimental design including experimental and control groups. The sample of the study was 34 7th-grade students from a public middle school in a city in Turkey. The study was conducted in the 2015-2016 academic year, lasting ten lesson hours (three weeks). The data were collected through achievement test and surveys regarding attitudes toward geometry, beliefs about the nature of mathematics, and beliefs about the teaching of mathematics. The quantitative data analyses were carried out by using the Mann-Whitney U test. Results revealed that, although the use of GeoGebra in teaching transformation geometry demonstrated student improvement in achievement, attitudes toward geometry, and beliefs about

How Permeable Are the Beliefs of Future Secondary School Mathematics Teachers to Pre-Service Experiences? Looking Across Their Years of Training

Even though it has been recognized that prospective teachers’ conceptions of the nature of mathematics, and of mathematics learning and teaching shape their teaching decisions, and, thereby, students’ engagement and achievement, to date no research has examined these conceptions from a person-centered perspective taking into account year of teacher education program. Cluster analysis revealed three distinct profiles: anti-socioconstructivist, socioconstructivist, and flexible. Although the same driving dimensions were present at the different stages of the training, the characterization of these three profiles fluctuated, in tandem with the process of construction of a professional identity, the increase of the field experience and the discovery of the complexities of the profession. The analysis of the pedagogical practices support the non-linearity of the beliefs-practices relationship.

Beyond content: exploring the neglected dimensions of mathematics literacy

Abstract Becoming mathematically literate means not only knowledge of the content of mathematics but also understanding of the nature of mathematics and the literate practices involved in the creation, communication, and consumption of its content. This case study examined one mathematician’s view of the nature of mathematics and his literate practices. Data collected include semi-structured interviews with the mathematician, observations of his daily work routines, and his think-alouds during the reading of a disciplinary text. These data were analyzed qualitatively through an iterative process involving multiple readings and identification and refinement of codes. The analysis revealed that the mathematician (a) viewed mathematics as rigorous and demanding, both theoretical and practical, relatively stable but highly rewarding, interconnected with other disciplines, and involving discipline-legitimated discursive practices; (b) engaged in extensive reading/viewing and writing, valued learning from repeated trials and errors, and collaborated with an international network of peers in research; (c) used a range of reading strategies (e.g. close reading, summarizing, questioning, storying, evaluating, annotating) to help him make sense of the disciplinary text; and (d) marshaled both verbal and visual resources to create specialized knowledge, engage in rigorous reasoning, develop logical argument, and construct professional identity. These findings provide important insights that can help teachers design activities that are authentic to mathematics practices and effective for promoting mathematics literacy.

Reduced Radial Curves of Diatomic Molecules.

The prospect of using the concept of a universal reduced potential energy curve (RPC) for a broader class of radial molecular functions is explored by performing appropriate model calculations for the electric dipole moment functions of the hydrogen halides HF, HCl, and HBr. The reduced radial functions of the model systems, constructed from their best available theoretical approximants, coincide so closely that they can be used as few-parameter universal representations of functions available in the literature. Given the mathematical nature of the problem addressed here, the results are not limited to the functions studied but can be applied equally well to all radial molecular functions that have similar shapes, such as electric quadrupole moment and dipole polarizability functions.

Teacher Development for Equitable Mathematics Classrooms: Reflecting on Experience in the Context of Performativity

In this article, we chart the development of one of us—Sue Hough—from a teacher who wanted students to understand to one who gained new critical understandings of student thinking, pedagogy, and the very nature of mathematics. We comment on the role of research interventions and learning communities in this development, with a particular focus on Sue’s encounter with Realistic Mathematics Education and the connections it makes between informal and formal mathematics through the pedagogy of guided reinvention. Development towards teaching that enables all learners to make sense of mathematics requires fundamental changes in pedagogic practice and a reconceptualisation of progress. Bringing about such radical change relies on one further aspect of Sue’s story—the freedom to experiment and learn as a teacher. We note the remoteness of this possibility in a climate of performativity and marketised education, and we discuss the implications of Sue’s journey for our pedagogical responsibilities in professional development today.

Mathematics education faculty members' perceptions of barriers to distance education: Q-method analysis

As a result of the great earthquake disaster in Turkey, mathematics education in universities was reorganized as distance education after the pandemic period. This article aims to identify the barriers to distance education practices in higher education. In line with the aim of this study; Q methodology was used to help determine the perceptions and perspectives of faculty members regarding the barriers to distance education in the process of teaching mathematics in higher education. Q methodology aims to reveal the internal frame of reference of individuals through their preferences. According to the findings of the study, participants identified barriers under 3 factors. These 3 factors, in order of weight, represent student, curriculum and technology barriers. In general, it was seen that the 4 factors revealed by the methodology in the study yielded results compatible with the literature. However, it was determined that the barriers arising from the teacher did not carry a factor load in the study. Another unique result was the identification of barriers arising from the nature of mathematics. It was understood that there are unique obstacles specific to the field of mathematics due to pedagogy and curriculum. In this respect, investigating the barriers to field-specific extended education will provide more accurate findings. In this study, which was conducted with Q methodology and supported by qualitative data, the identified barriers to distance education were consistent with the general literature. Q methodology reveals that it is a compatible tool for various research in similar situations.

First Year Engineering Students’ Difficulties with Math Courses- What Is the Starting Point for Academic Teachers?

The discussion about first-year engineering students’ difficulties in mathematics is continuous in the fields of engineering, mathematics and higher education. The present study aimed to examine the initial barriers academic math teachers need to have in mind if they want to improve students’ performance in engineering math courses through appropriate teaching practices in order to face their initial interindividual differences. During the first phase of the study, we examined first year engineering students’ initial beliefs about the nature of mathematics, their self-efficacy beliefs about mathematics and their basic mathematical knowledge. The math school grade was used for their previous mathematical performance. Results indicated the predominant role of the previous mathematical knowledge and the important role of the formalistic disposition toward mathematics. The lack of experience of using mathematics for problem-solving situations within an engineering framework prevented students from recognizing and appreciating the value of mathematical courses during the engineering studies. The second phase of the study examined, through an interview with a group of students, their perceptions of the teaching practices which were introduced after their teacher attended a training program. The discussion concentrates on how academics can use teaching processes for equity and not equality in order to motivate their students.

The Art of Mathematical Modeling in College Physics: Strategies for Fostering Student Engagement

This study investigates the integration of mathematical modeling strategies in college physics education to enhance student engagement, comprehension, and attitudes. Through a mixed-methods approach involving surveys, assessments, interviews, and observations, the study explores the impact of these strategies on student learning. The findings reveal a significant improvement in student engagement levels, increasing from 65% to 88% after implementation. Assessment scores demonstrated a parallel enhancement, rising from 62% to 90% across three rounds. Qualitative interviews highlighted students' initial lack of confidence in applying mathematical tools to physics scenarios, coupled with frustration due to the abstract nature of mathematics. However, the strategies' contextualization effectively addressed these challenges, leading to increased confidence and positive attitudes. The outcomes align with theoretical frameworks and previous research, underscoring the strategies' ability to bridge theoretical concepts with real-world applications. The study concludes that the integration of mathematical modeling in physics education promotes active learning, deepens understanding, and reshapes students' perceptions of mathematics and physics. It advocates for personalized support to overcome initial challenges and emphasizes collaborative coordination in curriculum planning. Ultimately, the study contributes to the evolving landscape of education by promoting interdisciplinary approaches that enhance student learning experiences.

Inquiry-based mathematics education and attitudes towards mathematics: tracking profiles for teaching

Research on the relationships between the main constructs underlying inquiry-based learning is rarely reported in mathematics education research. Considering this as a complex problem which is worth to be investigated, the present study aims to provide some empirical evidences that might serve as an insight to support further investigations on the relationships between attitudes towards mathematics and inquiry-based learning approaches. Thus, this study adopts a descriptive research design where no variables are manipulated but observed and measured in order to identify changes depicted in data collection. An instructional design focusing on the nature of mathematical inquiry is carried out with the participation of 304 secondary and high school students, and a clustering approach is used to look at how participants are grouped around certain attitudinal profiles before and after such mathematical practice. The results show how the heterogeneity of attitudinal profiles present in the classroom evolves positively in terms of perceived usefulness of mathematics and mathematical self-concept as perception of competence in mathematics. This fact provides some basis that might be used for further research on the idea that certain forms of development in inquiry-based mathematics education (IBME) based on greater immersion in the nature and culture of mathematics can help students to improve their attitudes towards mathematics.

Future mathematics teachers’ implementation of research activities based on dynamic mathematics software GeoGebra

Problem statement. The digital transformation of higher education significantly affects the organization of pedagogical universities students’ research activities. Therefore, two important problems can be identified. Firstly, the matter of discovering information technology tools and directions for their use for the implementation of research work of future mathematics teachers studying in 1-2 courses. And secondly, the problem of choosing a mathematical topic, the solution of which with the help of digital technologies would lead to some interesting results that have elements of novelty. The purpose of this study is to identify the line of research of a mathematical nature, the implementation of which relies heavily on the dynamic mathematics software GeoGebra’s features. Methodology. Some features of the GeoGebra dynamic mathematics software and its application in geometric studies are analyzed. To study geometric figures, methods of visualization of mathematical objects and of experimental mathematics are used. Results. Some possibilities of dynamic mathematics software applicability in research and development of pedagogical universities’ mathematical profile students are considered. In these circumstances, the emphasis is on research related to the study of spatial curves and their planar projections, the construction of which naturally relies on the principles of displaying three-dimensional objects on a screen. The role of dynamic mathematics software in such studies mainly lies in the visualization of considered geometric objects, which makes it possible to identify possible properties of the constructed curves, and then, using available mathematical apparatus, to confirm their validity. Conclusion. The direction of designing and studying geometric objects using the GeoGebra dynamic mathematics software, which formed the basis of several research works of undergraduate students, is presented.

Are beliefs believable? An investigation of novice mathematics teachers’ beliefs and teaching practices

Observing 17 teachers for a total of 116 hours, this paper examined whether a relationship between the beliefs of novice mathematics teachers and their classroom practices exists. A Spearman correlation analysis showed a modest relation between beliefs about the nature of mathematics and about learning mathematics, while the relationship between beliefs about mathematics achievement and these other two components was weak and statistically insignificant. Unexpectedly, the relationship between teachers’ beliefs and their teaching practices has been found to be much different than anticipated, and in some cases, the relationships were even negative. The researchers conclude with a discussion of further implications.

Relativistic Corrections in White Dwarf Asteroseismology

Abstract With the precision now afforded by modern space-based photometric observations from the retired K2 and current TESS missions, the effects of general relativity (GR) may be detectable in the light curves of pulsating white dwarfs (WDs). Almost all WD models are calculated using a Newtonian description of gravity and hydrodynamics. To determine if the inclusion of GR leads to observable effects, we used idealized models of compact stars and made side-by-side comparisons of mode periods computed using a: (i) Newtonian and (ii) GR description of the equilibrium structure and nonradial pulsations. For application to WDs, it is only necessary to include the first post-Newtonian (1pn) approximation to GR. The mathematical nature of the linear nonradial pulsation problem is then qualitatively unchanged and the GR corrections can be written as extensions of the classic Dziembowski equations. As such, GR effects might easily be included in existing asteroseismology codes. The idealized stellar models are (i) 1pn relativistic polytropes and (ii) stars with a cold degenerate electron equation of state featuring a near-surface chemical transition from μ e = 2 to μ e = 1, simulating a surface hydrogen layer. A comparison of Newtonian and 1pn normal mode periods reveals fractional differences in the order of the surface gravitational redshift z. For a typical WD, this fractional difference is ∼10−4 and is greater than the period uncertainty σ Π/Π of many WD pulsation modes observed by TESS. Consistent theoretical modeling of periods observed in these stars should, in principle, include GR effects to 1pn order.

Hypothetical Learning Trajectory: Bagaimana Perannya dalam Pembelajaran Matematika di Sekolah Dasar?

In today's modern era, students are required to become learning centers. Thus, students must have prior experience and knowledge in order to be able to connect with the next concept (material). However, because the differences between the nature of elementary schooler and mathematics, teacher needs to structure learning that can be a bridge to neutralize these differences. One way to structure the learning is make a learning design such as Hypothetical Learning Trajectory (HLT). HLT is an alleged learning process as an anticipation of things that might happen, both students' thinking processes and other things in the learning process. The term HLT refers to the teacher's lesson plan which is designed not only to consider the material, but also the level of understanding, knowledge and student characteristics of the material being studied, so that students experience the learning process from the beginning to achieving the expected mathematics learning goals for students. This article is a literature study that aims to discuss HLT in elementary mathematics learning.

Families of hybridized interior penalty discontinuous Galerkin methods for locally degenerate advection-diffusion-reaction problems

We analyze families of primal high-order hybridized discontinuous Galerkin (HDG) methods for solving degenerate second-order elliptic problems. One major problem regarding this class of PDEs concerns its mathematical nature, which may be nonuniform over the whole domain. Due to the local degeneracy of the diffusion term, it can be purely hyperbolic in a subregion and elliptical in the rest. This problem is thus quite delicate to solve since the exact solution can be discontinuous at interfaces separating the elliptic and hyperbolic parts. The proposed hybridized interior penalty DG (H-IP) method is developed in a unified and compact fashion. It can handle pure-diffusive or -advective regimes as well as intermediate regimes that combine these two mechanisms for a wide range of Péclet numbers, including the tricky case of local evanescent diffusivity. To this end, an adaptive stabilization strategy based on the addition of jump-penalty terms is considered. An upwind-based scheme using a Lax–Friedrichs correction is favored for the hyperbolic region, and a Scharfetter–Gummel-based technique is preferred for the elliptic one. The well-posedness of the H-IP method is also briefly discussed by analyzing the strong consistency and the discrete coercivity condition in a self-adaptive energy-norm that is regime dependent. Extensive numerical experiments are performed to verify the model’s robustness for all the abovementioned regimes.

The experiential construction of mathematics teacher identity and the impact of early mathematical failure

It is broadly accepted that teachers’ professional identities influence how they teach and what their pupils learn. In this paper, drawing on semi-structured interviews with 22 experienced primary teachers from the Republic of Cyprus, we explore the construction of informants’ professional identities with a specific focus on mathematics teaching. Analyses, undertaken according to the constant comparison method, yielded three broad themes, implicated in differing ways in the construction and manifestation of informants’ identities: prior experiences of mathematics, mathematical competence, and images of the self-as-teacher. Overall, teachers fell into two groups, which analyses led us to construe as either mathematical victors or mathematical victims. Mathematical victors had experienced success as learners of school mathematics, from which pleasure, pride, and confidence in their mathematical knowledge for teaching emerged. Their teaching, which emphasized pupils’ attainment of similar enjoyment and success, focused on abstraction and mathematical reasoning. Mathematical victims had experienced failure as learners of school mathematics, from which anxiety and a restricted mathematical knowledge for teaching emerged. Their teaching, which emphasized positive pupil experiences, focused on affect rather than cognition and an avoidance of “traditional” teaching. However, both groups, despite their confident assertions, appeared unaware of the potential of their actions for creating new victims. The findings, which are discussed in relation to existing literature, confirm the complex nature of mathematics teachers’ identities and highlight, in particular, the need for further research into the formative role of teachers’ prior experiences of mathematics, whether positive or negative.

Russell and Cassirer as Leibniz’s Interpreters: On the Analytic and Synthetic Nature of Mathematical and Physical Knowledge

Russell and Cassirer as Leibniz’s Interpreters: On the Analytic and Synthetic Nature of Mathematical and Physical Knowledge

Mathematical prerequisites for non-STEM programs

Mathematics is applied in a broad range of academic disciplines and thus takes on a paramount role for college access. However, students often take for granted the necessity of mathematics for STEM subjects but are unaware of or insufficiently informed about the role of mathematics for subjects outside of STEM. In fact, there is no in-depth empirical investigation on which mathematical prerequisites are expected by university instructors from incoming non-STEM undergraduates. The present paper addresses this research gap. We conducted a three-round Delphi study with N = 19/547/337 university instructors in Germany teaching undergraduates in non-STEM programs. Our results show that many more subjects require mathematical knowledge than is commonly assumed. We present a comprehensive catalogue of mathematical prerequisites that students are expected to have at the beginning of their studies. The prerequisites include (1) mathematical content, (2) mathematical processes, (3) views about the nature of mathematics, and (4) personal characteristics. The expected mathematical prerequisites primarily address a basic understanding and do not require formalistic or abstract mathematical knowledge and skills. The Delphi study also revealed five subject groups in which similar mathematical prerequisites are expected. Although the study’s focus is on German instructors, it provides valuable insights for an international audience as well. For example, future research can use our results to adapt instruments to explore predictors of college success. In addition, our results inform stakeholders involved in the transition from high school to college and in college education, such as teachers preparing their students for future study, college instructors making their expectations transparent to incoming students, and policymakers involved in curriculum design.