scholarly journals Discrete Mathematics into K-11 and K-12 Grade Education

2021 ◽  
Vol 23 (05) ◽  
pp. 319-324
Author(s):  
Mr. Balaji. N ◽  
◽  
Dr. Karthik Pai B H ◽  
Keyword(s):  
Set Theory ◽  
Mathematical Reasoning ◽  
Mathematical Induction ◽  
Discrete Mathematics ◽  
College Classrooms ◽  
Underlying Principle ◽  
Education Study ◽  
Strongly Connected ◽  
Proof Techniques ◽  
K 12

Discrete mathematics is one of the significant part of K-11 and K-12 grade college classrooms. In this contribution, we discuss the usefulness of basic elementary, some of the intermediate discrete mathematics for K-11 and K-12 grade colleges. Then we formulate the targets and objectives of this education study. We introduced the discrete mathematics topics such as set theory and their representation, relations, functions, mathematical induction and proof techniques, counting and its underlying principle, probability and its theory and mathematical reasoning. Core of this contribution is proof techniques, counting and mathematical reasoning. Since all these three concepts of discrete mathematics is strongly connected and creates greater impact on students. Moreover, it is potentially useful in their life also out of the college study. We explain the importance, applications in computer science and the comments regarding introduction of such topics in discrete mathematics. Last part of this article provides the theoretical knowledge and practical usability will strengthen the made them understand easily.

Soaring with Scope-on-a-Rope at LSU: Development of Innovative Microscopy Technology for K-16 Classroom use

2000 ◽  
Vol 6 (S2) ◽  
pp. 1170-1171
Author(s):  
M. C. Henk ◽  
H. Silverman
Keyword(s):  
Classroom Teachers ◽  
Classroom Setting ◽  
College Classrooms ◽  
Industrial Inspection ◽  
K 12 ◽  
Classroom Use

LSU began introducing a prototype SCOPE-ON-A-ROPE (SOAR) to selected teachers in Louisiana and Tennessee three years ago as part of our K-12 outreach activities. It proved to be an invaluable aid to all K-12 classrooms as well as to college classrooms or laboratories in several disciplines. The SOAR is extremely easy to use in the normal classroom setting, but can also introduce sophisticated concepts usually possible only through complicated microscopy exercises with specialized instrumentation.The professional microscopist who occasionally teaches students how to use microscopes can only begin to appreciate the position of classroom teachers who are routinely faced with inadequate, insufficient microscopes for classes of 20- 30 students at a time. This SOAR, inspired by industrial inspection devices, aids the teacher in introducing valuable concepts in microscopy and scale while easily serving the functions of many different microscopes and accessories. It is a comfortably hand-held device that can be used capably even by a five-year-old to provide excellent,

Type-theoretical checking and philosophy of Mathematics

1998 ◽  
Author(s):  
Nicolaas Govert de Bruijn
Keyword(s):  
Set Theory ◽  
Subject Matter ◽  
Mutual Influence ◽  
Mathematical Reasoning ◽  
Philosophy Of Mathematics ◽  
General Information ◽  
Point Of View ◽  
De Bruijn ◽  
Level Of Understanding ◽  
The Way

After millennia of mathematics we have reached a level of understanding that can be represented physically. Humankind has managed to disentangle the intricate mixture of language, metalanguage and interpretation, isolating a body of formal, abstract mathematics that can be completely verified by machines. Systems for computer-aided verification have philosophical aspects. The design and usage of such systems are influenced by the way we think about mathematics, but it also works the other way. A number of aspects of this mutual influence will be discussed in this paper. In particular, attention will be given to philosophical aspects of type-theoretical systems. These definitely call for new attitudes: throughout the twentieth century most mathematicians had been trained to think in terms of untyped sets. The word “philosophy” will be used lightheartedly. It does not refer to serious professional philosophy, but just to meditation about the way one does one’s job. What used to be called philosophy of mathematics in the past was for a large part subject oriented. Most people characterized mathematics by its subject matter, classifying it as the science of space and number. From the verification system’s point of view, however, subject matter is irrelevant. Verification is involved with the rules of mathematical reasoning, not with the subject. The picture may be a bit confused, however, by the fact that so many people consider set theory, in particular untyped set theory, as part of the language and foundation of mathematics, rather than as a particular subject treated by mathematics. The views expressed in this paper are quite personal, and can mainly be carried back to the author’s design of the Automath system in the late 1960s, where the way to look upon the meaning (philosophy) of mathematics is inspired by the usage of the unification system and vice versa. See de Bruijn 1994b for various philosophical items concerning Automath, and Nederpelt et al. 1994, de Bruin 1980, de Bruijn 1991a for general information about the Automath project. Some of the points of view given in this paper are matters of taste, but most of them were imposed by the task of letting a machine follow what we say, a machine without any knowledge of our mathematical culture and without any knowledge of physical laws.

A formalization of the theory of ordinal numbers

1965 ◽  
Vol 30 (3) ◽  
pp. 295-317 ◽  
Author(s):  
Gaisi Takeuti
Keyword(s):  
Set Theory ◽  
Natural Extension ◽  
Mathematical Induction ◽  
Predicate Calculus ◽  
Transfinite Induction ◽  
First Order ◽  
Recursive Functions ◽  
Ordinal Numbers ◽  
Primitive Recursive ◽  
Order Predicate Calculus

Although Peano's arithmetic can be developed in set theories, it can also be developed independently. This is also true for the theory of ordinal numbers. The author formalized the theory of ordinal numbers in logical systems GLC (in [2]) and FLC (in [3]). These logical systems which contain the concept of ‘arbitrary predicates’ or ‘arbitrary functions’ are of higher order than the first order predicate calculus with equality. In this paper we shall develop the theory of ordinal numbers in the first order predicate calculus with equality as an extension of Peano's arithmetic. This theory will prove to be easy to manage and fairly powerful in the following sense: If A is a sentence of the theory of ordinal numbers, then A is a theorem of our system if and only if the natural translation of A in set theory is a theorem of Zermelo-Fraenkel set theory. It will be treated as a natural extension of Peano's arithmetic. The latter consists of axiom schemata of primitive recursive functions and mathematical induction, while the theory of ordinal numbers consists of axiom schemata of primitive recursive functions of ordinal numbers (cf. [5]), of transfinite induction, of replacement and of cardinals. The latter three axiom schemata can be considered as extensions of mathematical induction.In the theory of ordinal numbers thus developed, we shall construct a model of Zermelo-Fraenkel's set theory by following Gödel's construction in [1]. Our intention is as follows: We shall define a relation α ∈ β as a primitive recursive predicate, which corresponds to F′ α ε F′ β in [1]; Gödel defined the constructible model Δ using the primitive notion ε in the universe or, in other words, using the whole set theory.

scholarly journals How Do Higher-Education Students Use Their Initial Understanding to Deal with Contextual Logic-Based Problems in Discrete Mathematics?

2017 ◽  
Vol 10 (5) ◽  
pp. 72
Author(s):  
Asrin Lubis ◽  
Andrea Arifsyah Nasution
Keyword(s):  
Higher Education ◽  
High School ◽  
Mathematics Curriculum ◽  
Mathematical Reasoning ◽  
Video Recording ◽  
Senior High School ◽  
Discrete Mathematics ◽  
Small Scale ◽  
Education Students ◽  
The Third

Mathematical reasoning in logical context has now received much attention in the mathematics curriculum documents of many countries, including Indonesia. In Indonesia, students start formally learning about logic when they pursue to senior-high school. Before, they previously have many experiences to deal with logic, but the earlier assignments do not label them as logic. Although the students have already experienced much about logic, it does not assure that they have a better understand about it even they purpose to university. Thus, this paper presents several findings of our small-scale study which was conducted to investigate the issues on how higher-education students overcome contextual logic-based problems. Data were collected through pretest, students’ written work, video recording and interview. A fifteen-minute test which consisted of four questions was given to 53 student participants in the third semester who proposed mathematics discrete course. The information towards the main issues was required through the analysis of students’ written work in the pretest and video recording during the students’ interview. The findings indicate that the students’ initial understanding, in general, do not help them much to solve logical problems based on context. In our findings, they apply several strategies, such as random proportions, word descriptions, permutation-combination calculations and deriving conclusion through logical premises.

The Americans With Disabilities Act Implications for Family Counselors: Counseling Students in Higher Education

2020 ◽  
pp. 106648072094886
Author(s):  
Larry K. Phillippe ◽  
Nicole Noble ◽  
Bret Hendricks ◽  
Janna Brendle ◽  
Robin H. Lock
Keyword(s):  
Higher Education ◽  
Students With Disabilities ◽  
Americans With Disabilities Act ◽  
Counseling Students ◽  
College Counselors ◽  
College Classrooms ◽  
Family Counselors ◽  
Transition To Higher Education ◽  
K 12 ◽  
The Impact

Family counselors at times work with families in which a family member with a disability is transitioning into higher education settings. Frequently, these counselors are unaware of the federally protected rights of all students and they may not know how to access this information. This article explains the differences between laws for students with disabilities in K–12 school settings and the components of the Americans with Disabilities Act (ADA) and its subsequent ADA 2008 Amendments to inform family counselors on how to support clients and their families in the transition to higher education. In this article, the authors discuss the ADA and the ADA 2008 Amendments, which dramatically impacted the college experience of all students with disabilities. With record numbers of students with disabilities now attending college, counselors, as they advocate for families, should be aware of federal guidelines that require physical access to educational facilities, the use of universal design, electronic accessibility, and the provision of academic accommodations and modifications in college classrooms. Through family counselors’ awareness of these significant changes in the higher education experience, they can more fully assist families with students with disabilities who are transitioning from high school to higher education. This article describes each of these four facets of the ADA 2008 Amendments as well as the impact each major facet of the amendment has on the higher education landscape for students with disabilities.

scholarly journals Kurt Gödel, 28 April 1906 - 14 January 1978

1980 ◽  
Vol 26 ◽  
pp. 148-224 ◽  
Keyword(s):  
Mathematical Logic ◽  
Set Theory ◽  
Current Practice ◽  
Mathematical Reasoning ◽  
Internal Development ◽  
Kurt Gödel ◽  
The Subject ◽  
Formal Rules

Kurt Gödel did not invent mathematical logic; his famous work in the thirties settled questions which had been clearly formulated in the preceding quarter of this century. Despite sensational presentations by crackpots, philosophers and journalists (or even in poems, for example, by H. M. Enzensberger, set to music by H. W. Henze), Gödel’s results have not revolutionized the silent majority’s conception of mathematics, let alone its practice; much less so than the internal development of the subject since then. Certainly, those results refuted most elegantly each of the grand foundational ‘theories’ current at the time, of which Hilbert’s, on the place of formal rules in mathematical reasoning, and those associated with Frege and Russell, on its reduction to universal systems like set theory, were most popular. (Gödel’s own and related results also deflate the particular ‘anti-formalist’ foundations of the time, Poincaré’s and Brouwer’s constructivist and Zermelo’s infinitistic schemes being extreme examples; they are taken up in the last sections of parts II-IV.) For obvious reasons, in his original publications Gödel made a point of formulating his work in terms acceptable to the theories mentioned, and to stress its bearing on them. But it is fair to say that they were suspect anyway, and—less trivially—that they can be refuted more convincingly by simple constatations rather than by (his) mathematical theorems as explained in more detail in part II. Further, as so often with very grand schemes, the refutations put nothing comparable in the place of the discredited foundational views which are, quite properly, simply ignored in current practice.

scholarly journals User-Centric Design for Mathematical Web Services

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Adlin Sheeba ◽  
Chandrasekar Arumugam
Keyword(s):  
User Interface ◽  
Web Services ◽  
Web Service ◽  
Set Theory ◽  
Rapid Development ◽  
Discrete Mathematics ◽  
Virtual Keyboard ◽  
Application Logic ◽  
User Centric ◽  
Sophisticated User

A web service is a programmatically available application logic exposed over the internet and it has attracted much attention in recent years with the rapid development of e-commerce. Very few web services exist in the field of mathematics. The aim of this paper is to seamlessly provide user-centric mathematical web services to the service requester. In particular, this paper focuses on mathematical web services for prepositional logic and set theory which comes under discrete mathematics. A sophisticated user interface with virtual keyboard is created for accessing web services. Experimental results show that the web services and the created user interface are efficient and practical.

A formal system of logic

1950 ◽  
Vol 15 (1) ◽  
pp. 25-32 ◽  
Author(s):  
Hao Wang
Keyword(s):  
Set Theory ◽  
Existence Theorems ◽  
Formal System ◽  
Mathematical Induction ◽  
Restrictive Condition ◽  
Infinite Class ◽  
Class V ◽  
Inconsistent System ◽  
Theory Generation

The main purpose of this paper is to present a formal systemPin which we enjoy a smooth-running technique and which countenances a universe of classes which is symmetrical as between large and small. More exactly,Pis a system which differs from the inconsistent system of [1] only in the introduction of a rather natural new restrictive condition on the defining formulas of the elements (sets, membership-eligible classes). It will be proved that if the weaker system of [2] is consistent, thenPis also consistent.After the discovery of paradoxes, it may be recalled, Russell and Zermelo in the same year proposed two different ways of safeguarding logic against contradictions (see [3], [4]). Since then various simplifications and refinements of these systems have been made. However, in the resulting systems of Zermelo set theory, generation of classes still tends to be laborious and uncertain; and in the systems of Russell's theory of types, complications in the matter of reduplication of classes and meaningfulness of formulas remain. In [2], Quine introduced a system which seems to be free from all these complications. But later it was found out that in it there appears to be an unavoidable difficulty connected with mathematical induction. Indeed, we encounter the curious situation that although we can prove in it the existence of a class V of all classes, and we can also prove particular existence theorems for each of infinitely many classes, nobody has so far contrived to prove in it that V is an infinite class or that there exists an infinite class at all.

scholarly journals A Note on the Difference of Powers and Falling Powers

2021 ◽  
Vol 2021 ◽  
pp. 1-4
Author(s):  
Taoufik Sabar
Keyword(s):  
Generating Functions ◽  
Binomial Coefficient ◽  
Mathematical Induction ◽  
Discrete Mathematics ◽  
Special Series ◽  
Induction Principle ◽  
Combinatorial Sums ◽  
The Difference ◽  
Binomial Identities

Combinatorial sums and binomial identities have appeared in many branches of mathematics, physics, and engineering. They can be established by many techniques, from generating functions to special series. Here, using a simple mathematical induction principle, we obtain a new combinatorial sum that involves ordinary powers, falling powers, and binomial coefficient at once. This way, and without the use of any complicated analytic technique, we obtain a result that already exists and a generalization of an identity derived from Sterling numbers of the second kind. Our formula is new, genuine, and several identities can be derived from it. The findings of this study can help for better understanding of the relation between ordinary and falling powers, which both play a very important role in discrete mathematics.

Sign in / Sign up

Export Citation Format

Share Document