Fundamental Mathematical Ideas Research Articles

The concept of infinity and the development of set theory solutions

The concept of infinity is intricately connected to and comprehended via the framework of cardinal and ordinal numbers. Cardinal and ordinal numbers are fundamental mathematical ideas within the field of set theory. The cardinal number is used to denote the quantity of items inside a given set while ordinal defines basic algorithms. This article demonstrates how the discipline of set theory may be used as a tool to investigate the nature of infinity, or at the very least give some insights into the subject. The idea of infinity may be better understood by looking at it through the lens of set theory and the commutative property. In addition to that, this research presents the connection and compatibility examination of Cohens operation upon the Zermelo Freankel axiomatic framework. These accomplishments are discussed in this article as illuminating insights for readers to consider while imagining the ultimate solution to the problems posed by the Continuum Hypothesis and the nature of infinity. Through the use of examples from contemporary researchers, the multiverse and indeterminism are introduced as potential approaches to the problem of how to solve Cantors legacy in the future.

A new approach to the study of fixed points based on soft rough covering graphs

<abstract><p>Mathematical approaches to structure model problems have a significant role in expanding our knowledge in our routine life circumstances. To put them into practice, the right formulation, method, systematic representation, and formulation are needed. The purpose of introducing soft graphs is to discretize these fundamental mathematical ideas, which are inherently continuous, and to provide new tools for applying mathematical analysis technology to real-world applications including imperfect and inexact data or uncertainty. Soft rough covering models $ \left(\text{briefly}, \text{ }\mathcal{SRC}\text{-Models}\right) $, a novel theory that addresses uncertainty. In this present paper, we have introduced two new concepts $ \mathcal{L}\mathfrak{i} $-soft rough covering graphs ($ \mathcal{L}\mathfrak{i} $-$ \mathcal{SRCG} $s) and the concept of fixed point of such graphs. Furthermore, we looked into a some algebras that dealt with the fixed points of $ \mathcal{L}\mathfrak{i} $-$ \mathcal{SRCG} $s. Applications of the algebraic structures available in covering soft sets to soft graphs may reveal new facets of graph theory.</p></abstract>

Glimpses of contributions of some top Indian mathematicians: A review article

Indian mathematics has its deep roots in the Vedas, different from what is known as Vedic Mathematics. Vedic age gave rise to a new era of progress in the field of Science, Technology and Development. The Hindu Scripture Vedas is synonymous with all kinds of original source of knowledge and intellectual wisdom in the universe leading to modern knowlege in modern mathematics. Indian mathematicians made tremendous contributions to the entire world of Mathematics and Science. Decimal number system as well as the invention of zero (0) are among the greatest contributions of Indian mathematicians. The theory of trigonometry, Mathematical Modelling, Algebra, algorithm, modern arithmetic, sine and cosine functions leading to modern trigonometry, Diophantine equation, square root, cube root, negative numbers are also developed by Indian mathematicians. In this review article, the work of some of the renowned Indian mathematicians from Indus Valley civilization and the Vedas to modern times are covered in short with the hope that it may reveal hidden fundamental mathematical ideas as basic ideal tools which may usefully motivate for further research work in every domain of mathematical sciences, natural and applied sciences, engineering and social sciences. Moreover, there are many more remarkable Indian mathematicians who contributed to the origin of mathematical sciences. They have made several general contributions to mathematics that have significantly influenced scientists and mathematicians in the modern times.

Research as a resource in a high-school calculus curriculum.

Documents specifying a national mathematics curriculum for grades K-12 have recently been written in Israel. We focus on the calculus component for the highest of three matriculation bound levels, and specifically on the influence of research on this component. In addition to issues of content, we identify three principles that have led the writing team, namely, the manner in which sample tasks in the curriculum document incorporate fundamental mathematical ideas and mathematical Reasoning, the Impact expected from connecting mathematics to its role in everyday life and science, and the Cultivation of fertile intellectual ground from which new concepts may emerge naturally. We demonstrate how these principles are implemented in the unit on integration. We show that mathematics education research, though not mentioned explicitly, has had a profound and pervasive influence on the content and principles of the curriculum document, from the design of entire units down to the formulation of sample tasks.

The impact of a conceptual model-based mathematics computer tutor on multiplicative reasoning and problem-solving of students with learning disabilities

The study explored the impact of Please Go Bring Me-COnceptual Model-based Problem Solving (PGBM-COMPS) computer tutoring system on multiplicative reasoning and problem solving of students with learning disabilities. The PGBM-COMPS program focused on enhancing the multiplicative reasoning and problem solving through nurturing fundamental mathematical ideas and moving students above and beyond the concrete level of operation. This is achieved by taking advantages of the constructivist approach from mathematics education and explicit conceptual model-based problem solving approach from special education. Participants were three elementary students with learning disabilities (LD). A mixed method design was employed to investigate the effect of the PGBM-COMPS program on enhancing students’ multiplicative reasoning and problem solving. It was found that the PGBM-COMPS program significantly improved participating students’ problem solving performance not only on researcher developed criterion tests but also on a norm-referenced standardized test. Qualitative and quantities data from this study indicate that, in addition to nurturing fundamental concept of composite units, it is necessary to help students to understand underlying problem structures and move toward mathematical model-based problem representation and solving for generalized problem solving skills.

Developing preservice elementary teachers’ specialized content knowledge: the case of associative property

Specialized content knowledge (SCK) is a type of mathematical content knowledge specifically needed for teaching. This type of knowledge, although serving as a critical component for preservice teacher education, is often challenging to develop with preservice elementary teachers (PTs). The purpose of this study is to investigate PTs’ development of SCK for teaching fundamental mathematical ideas in a university methods course. Focusing on the case of the associative property (AP) of multiplication, the author as the course instructor identified three instructional opportunities (a formal introduction and two delayed revisits) to stress two SCK components, representations, and explanations. PTs’ learning progresses were assessed through three diagnostic tests (a pretest, a mid-term exam, and a final exam) and two prompts, which informed the upcoming lesson design. Meanwhile, the course instructor conducted ongoing reflections on PTs’ learning, which also informed the corresponding lessons. It was found that PTs initially generated abstract number sentences without reasoning about the contexts of word problems. This representational sequence indicates a symbol precedence view. When prompted for explanations, PTs focused on individual numbers rather than quantitative relationships, and they could not consistently apply the basic meaning of multiplication for reasoning. The methods course, when designed to address these issues, promoted PTs’ SCK development. At the end of the course, the majority of PTs were able to generate number sentences based on the word problem structures and provided reasonable explanations; however, the methods course also faced dilemmas due to PTs’ robust symbol precedence view and the tension between PTs’ learning and children’s learning. Very few studies have explored ways to support PTs’ knowledge growth in SCK, especially for teaching fundamental mathematical ideas. This study, by carefully documenting the successes and challenges in developing PTs’ SCK, contributes to the existing literature. Based on our findings, this study highlights the importance of stressing basic meanings so as to develop PTs’ explanation skills. Meanwhile, to develop PTs’ representation skills, university instructors should be aware of the tension between PTs’ and children’s learning as manifested by PTs’ symbol precedence view. Finally, to support PTs’ SCK growth, it is also important to emphasize the role of elementary textbooks.

Opportunities to Learn: Inverse Relations in U.S. and Chinese Textbooks

ABSTRACTThis study, focusing on inverse relations, examines how representative U.S. and Chinese elementary textbooks may provide opportunities to learn fundamental mathematical ideas. Findings from this study indicate that both of the U.S. textbook series (grades K-6) in comparison to the Chinese textbook samples (grades 1–6), presented more instances of inverse relations, while also containing more unique types of problems; yet, the Chinese textbooks provided more opportunities supporting meaningful and explicit learning. In particular, before presenting corresponding practice problems, Chinese textbooks contextualized worked examples of inverse relations in real-world situations to aid in sense making of computational or checking procedures. The Chinese worked examples also differed in representation uses especially through concreteness fading. Finally, the Chinese textbooks spaced learning over time, systematically stressing structural relations including the inverse quantities relationships. These findings shed light on ways to support students’ meaningful and explicit learning of fundamental mathematical ideas in elementary school.

Элементы математизации теоретических основ начертательной геометрии

Active development of engineering and technology
 rightly necessitates reform of higher education. If before teacher
 was enough to make a course of lectures and read it throughout his
 labor activity, but now, in a time of rapid change reality necessary
 periodically rebuild the curriculum, focusing on the achievements
 of modern science and emerging technologies. Impossible further
 to teach tomorrow´s to engineers yesterday´s the provisions and
 former methods. Today for engineer it is important the acquisition
 of modern professional competencies with the ability to self-develop
 the skills required and if necessary, switch to popular types of
 activities. Therefore, modern engineering and technical education
 must firstly be adequate to modern scientific and technological
 reality, and secondly is based on modern scientific basis and regulations
 and not "braked" with classic content and with traditional
 techniques. Classical descriptive geometry, which in its time had
 an active influence on the development of production and industry
 initially in France and then throughout the world, today in the
 digital age of information must be updated, to enrich scientific
 achievements of modern scientists geometry and thereby become
 a popular academic discipline.
 Article is devoted to development of descriptive geometry in
 the direction of the fundamental mathematical ideas – sets, relations,
 algebraic operations; concepts – figures, conditions; analysis
 of the logical foundations. The authors suggested the development
 of elementary and higher descriptive geometry, uniting the known
 facts, methods and algorithms and leading them into a system based
 on common mathematical and logical ideas. From this perspective,
 we consider three section of descriptive geometry: search for linear
 objects based on the analysis of conditions, designing of surfaces,
 solution of position tasks.

The mathematics education of pre-school teachers

The foundation for children's mathematical development is established in the early childhood through experiences which challenge infants and preschoolers to explore ideas related to patterns, shapes, numbers and space. These early experiences influence significantly children's disposition for mathematics, hence they need to be positive and supportive. The ability of pre-school teachers to provide young learners with such experiences requires both profound understanding of fundamental mathematical ideas and positive attitude toward mathematics. Analysis of the results on the entry test administered to first year students, prospective pre-school teachers, at the Faculty of Pedagogy 'St. Kliment Ohridski ' in Skopje, at the beginning of the semester during five consecutive academic years (2006/07-2010/11) show extremely low levels of initial mathematics competencies of the majority of the students. These results together with research studies in this field establish the need for a high-quality mathematics education to be provided for prospective pre-school teachers within their university studies. The question of what kind of mathematics education is required to equip future pre-school teachers with the competencies to promote young learners' development of mathematical abilities is addressed through theoretical considerations of research studies of mathematics education in early childhood.

The role of framing in productive classroom discussions: A case for teacher learning

In this paper, we contrast two mathematical arguments that occurred during an algebra lesson to illustrate the importance of relevant framings in the ensuing arguments. The lesson is taken from a graduate course for elementary teachers who are enrolled in a mathematics specialist program. We use constructs associated with enthnography of argumentation to characterize the framings for warrants and backings that supported the ensuing arguments. Our findings suggest that teachers fully participated in argumentations that were framed by problem situations that were familiar to them, ones that were couched in elementary, fundamental mathematical ideas, and that these types of argumentations were arguably more productive in terms of opportunities for learning.

Elementary Teachers' Learning to Construct High-Quality Mathematics Lesson Plans

AbstractThis study explored a group of elementary teachers' (n = 35) learning to construct high-quality lesson plans that foster student understanding of fundamental mathematical ideas. The conceptual framework for this study was gleaned from the recently released Institute of Education Sciences (IES) recommendations, including (a) interweaving worked examples and practice problems, (b) connecting concrete and abstract representations, and (c) asking deep questions to elicit student self-explanations. Comparisons between teachers' pre- and postsurveys, and among teachers' initial, revised, and end-of-course lesson plans, indicated teachers' growth in using worked examples, representations, and deep questions during their lesson planning. Issues related to teachers' learning as they constructed lesson plans that aligned with the IES recommendations were also revealed.

Common Mathematical Foundations of Expected Utility and Dual Utility Theories

We show that the main results of the expected utility and dual utility theories can be derived in a unified way from two fundamental mathematical ideas: the separation principle of convex analysis, and integral representations of continuous linear functionals from functional analysis. Our analysis reveals the dual character of utility functions. We also derive new integral representations of dual utility models.

Assessing K‐5 Teacher Leaders' Mathematical Understanding: What Have the Test Makers and the Test Takers Learned?

This article describes the design and implementation of an assessment instrument for Numbers and Operations, the first course in a program to train elementary mathematics specialists. We briefly describe the course and its content, and then we elaborate on the process we used to develop the assessment instrument and the corresponding rubric for evaluating particular items. We discuss the assessment process and its significant role in the course. We highlight particular items from the instrument to describe the gains teachers made in their understanding of fundamental mathematical ideas. We also use these items to argue for the importance of this course in helping mathematics specialists explore the content that underpins the elementary curriculum. The results are presented which indicate that course participants made significant gains as a consequence of their course experiences.

Common Mathematical Foundations of Expected Utility and Dual Utility Theories

We show that the main results of the expected utility and dual utility theories can be derived in a unified way from two fundamental mathematical ideas: the separation principle of convex analysis, and integral representations of continuous linear functionals from functional analysis. Our analysis reveals the dual character of utility functions. We also derive new integral representations of dual utility models.

Closing the Gap between current knowledge level of UAE mathematics teachers and the ideal level

This research project aims to characterize the gap between the mathematics teachers’ current knowledge of mathematics and the ideal mathematical understanding for teaching in the United Arab Emirates (UAE) context. In particular, as part of a research project proposal that was submitted to UAE National Research Foundation and going through the final stage of review and approval, this study aims to investigate “how in-service mathematics teachers understand and think about fundamental mathematical ideas in major strands (geometry, algebra, measurement, probability and statistics, and numbers) as well as “the nature of the gap between ‘what teachers know’ and ‘what they should know’ to teach mathematics effectively in UAE schools. To answer these questions, a profile of UAE mathematics teachers will be determined, and investigated through qualitative and quantitative research techniques. Such work is expected to inform researchers, mathematics teacher education programs and policy makers.

Cultural Differences in Mathematical Concept Learning

THE purposes of this paper are two. The first is to examine the acquisition of certain mathematical concepts by young children in two different cultures. The data from a concept formation experiment performed with children in California will be compared with data from the same type of experiment performed with children of the same age level in Accra, Ghana.' The concepts to be acquired in these experiments are two fundamental concepts in mathematics, identity of sets and identity of ordered sets. The second purpose is to apply mathematical learning theory, and in particular a learning model, to the data from both experiments. The model has proved to be a reasonably good first approximation to actual response behavior in a variety of experimental situations. One objective here is a further application of the theoretical model in a widely different cultural context. Specifically,. my interest is in a comparison of the fit of the model to data from two experimental situations which are, with one major exception, almost identical. The cultural backgrounds of the subjects are widely disparate. The two purposes are in a sense complementary: On the one hand, we can identify differences or similarities across cultures in children's formation of mathematical concepts. From the standpoint of the teaching of mathematics these culturally determined differences are of utmost significance. On the other hand, we assume that the basic behavioral processes involved in learning are the same, and a general learning theory should deal with, represent, or analyze these fundamental processes. We should expect a theoretical model to go beyond what are perhaps adventitious differences in its prediction of actual response behavior. Therefore, we should expect the model to fit the data from two dissimilar cultures equally well when the experimental situations are in other respects comparable. Before going on to a description of various aspects of the experiments, I should like to comment on the immediate relevance of comparisons of children's concept learning in different cultures. The revolution in the teaching of mathematics or in curriculum in mathematics is spreading throughout the world. In particular, many new programs and methods developed in one country are being adopted or adapted in other countries. With this exchange of ideas, materials, curricula, methods, etc., in mathematics teaching, any evidence concerning cultural differences in the concepts, children already possess, or their manner of acquiring new concepts, or the background of experience and language necessary to the formation of fundamental mathematical ideas is essential. The experiments to be described were performed primarily to provide evidence for just such a purpose. They were carried out in connec-

On solving the problem of an ideal incompressible fluid flow around large-aspect-ratio axisymmetric bodies

Based on Babenko’s fundamental mathematical ideas, principally new (unsaturated) algorithms are developed for the numerical solution of problems of a potential axisymmetric ideal fluid flow around bodies of revolution, in particular, an ellipsoid of revolution with an aspect ratio equal to 1000.

Tracing fourth graders’ learning of fractions: early episodes from a year-long teaching experiment

Considerable and significant research 1 The data reported here was first presented at the American Education Research Association Annual Meeting, New Orleans, April 24–28, 2000. 1 has been conducted focusing on what students know about fraction ideas. 2 See, for example, Behr, Harel, Post, and Lesh (1993); Behr, Lesh, Post, and Silver (1983); Kamii and Dominick (1998); Koyama (1997); Mack (1993); Maher and Alston (1989); Maher, Davis, and Alston (1991); Streefland (1991, 1993); Watanabe (1995). 2 However, much of the research looks at student understanding following some instruction. The research 3 This work was supported by grant MDR 9053597 from the National Science Foundation and by grant 93-992022-8001 from the NJ Department of Higher Education. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the National Science Foundation or the NJ Department of Higher Education. 3 reported in this paper differs from other studies in three significant ways: (1) the students were 9- and 10-year-old fourth graders; (2) they had not yet had an introduction to operational rules and definitions about fractions; and (3) the research design emphasized a student-centered approach. The research is based on the view that given particular conditions in which students are invited to work together and conduct thoughtful investigations with appropriate materials, they can build fundamental mathematical ideas.

Proofs and fundamentals: a first course in abstract mathematics

Proofs and Fundamentals: A First Course in Abstract Mathematics 2nd edition is designed as a transition course to introduce undergraduates to the writing of rigorous mathematical proofs, and to such fundamental mathematical ideas as sets, functions, relations, and cardinality. The text serves as a bridge between computational courses such as calculus, and more theoretical, proofs-oriented courses such as linear algebra, abstract algebra and real analysis. This 3-part work carefully balances Proofs, Fundamentals, and Extras. Part 1 presents logic and basic proof techniques; Part 2 thoroughly covers fundamental material such as sets, functions and relations; and Part 3 introduces a variety of extra topics such as groups, combinatorics and sequences. A gentle, friendly style is used, in which motivation and informal discussion play a key role, and yet high standards in rigor and in writing are never compromised. New to the second edition: 1) A new section about the foundations of set theory has been added at the end of the chapter about sets. This section includes a very informal discussion of the Zermelo Fraenkel Axioms for set theory. We do not make use of these axioms subsequently in the text, but it is valuable for any mathematician to be aware that an axiomatic basis for set theory exists. Also included in this new section is a slightly expanded discussion of the Axiom of Choice, and new discussion of Zorn's Lemma, which is used later in the text. 2) The chapter about the cardinality of sets has been rearranged and expanded. There is a new section at the start of the chapter that summarizes various properties of the set of natural numbers; these properties play important roles subsequently in the chapter. The sections on induction and recursion have been slightly expanded, and have been relocated to an earlier place in the chapter (following the new section), both because they are more concrete than the material found in the other sections of the chapter, and because ideas from the sections on induction and recursion are used in the other sections. Next comes the section on the cardinality of sets (which was originally the first section of the chapter); this section gained proofs of the SchroederBernstein theorem and the Trichotomy Law for Sets, and lost most of the material about finite and countable sets, which has now been moved to a new section devoted to those two types of sets. The chapter concludes with the section on the cardinality of the number systems. 3) The chapter on the construction of the natural numbers, integers and rational numbers from the Peano Postulates was removed entirely. That material was originally included to provide the needed background about the number systems, particularly for the discussion of the cardinality of sets, but it was always somewhat out of place given the level and scope of this text. The background material about the natural numbers needed for the cardinality of sets has now been summarized in a new section at the start of that chapter, making the chapter both self-contained and more accessible than it previously was. 4) The section on families of sets has been thoroughly revised, with the focus being on families of sets in general, not necessarily thought of as indexed. 5) A new section about the convergence of sequences has been added to the chapter on selected topics. This new section, which treats a topic from real analysis, adds some diversity to the chapter, which had hitherto contained selected topics of only an algebraic or combinatorial nature. 6) A new section called ``You Are the Professor'' has been added to the end of the last chapter. This new section, which includes a number of attempted proofs taken from actual homework exercises submitted by students, offers the reader the opportunity to solidify her facility for writing proofs by critiquing these submissions as if she were the instructor for the course. 7) All known errors have been corrected. 8) Many minor adjustments of wording have been made throughout the text, with the hope of improving the exposition.

Stability and coefficients properites of polynomials of linear discrete systems

In this paper, by examining the coefficients of a given polynomial, we derive sufficient conditions for the zeros of the polynomial to be either inside the unit disk in the complex plane or at least one zero not inside the unit disk. The results are easily verifiable and provide handy ways of checking. Most results are proved using either Rouche's Theorem or fundamental mathematical ideas. Some of the results are extensions of known coefficient properties found in the literature.