bluuebunny/crossref_metadata_2025 huggingface dataset:
Restrictions on enforcement against real estate serving the debtor’s housing needs in the light of the amendment to the Code of Civil Procedure of May 14, 2020
Authors: Demendecki Tomasz
Published: January 2021

The necessity to protect debtors with regard to meeting their basic life needs, including ensuring a place of residence in epidemic emergencies, led to the introduction by the national legislator of new regulations (also in the field of real estate auctions), rules aimed at satisfying the debtor’s housing needs. The consequence of the aforementioned legislative actions is the addition of a new provision to the Polish Code of Civil Procedure – Art. 9521 with effect from
May 30, 2020.
The purpose of this study is a detailed presentation of the restrictions on enforcement against real estate introduced by the amendment to the Polish Code of Civil Procedure of May 14, 2020, aimed at satisfying the debtor’s housing needs, taking into account the problems caused by the application of new legal norms. The solutions discussed are a response to the situations repeatedly found in practice, in which the submission by the creditor of an application for enforcement against real estate was used to harass the debtor and therefore undoubtedly deserve approval. The possibility of using enforcement measures aimed at real estate with the view to satisfy the debtor’s residential purposes in the enforcement of small claims, is clearly contradictory to the basic
assumptions of the debtor’s protection against enforcement infringing his personal dignity. The new rules on enforcement against real estate serving to satisfy the debtor’s housing needs, strengthen considerably the position of debtors in the course of the execution process. It is made at the cost of significant limitation of the possibility to satisfy the creditors with the use of this type of enforcement process and postponement of debt collection in time. The only valuable component of the debtor’s property is often the residential property, which makes property execution one of the most effective methods of execution itself. Limiting execution possibilities of these properties, may in practice mean for many creditors an inability to satisfy the claimed debt for a long time and the necessity to raise additional costs and lead to an increase in the number of unsuccessful executions. However, the envisaged legal solutions, with respect to debtors remaining in a deteriorating economic situation caused by an epidemic threat, in the sense of an extraordinary and only temporary threat, should be assessed positively as a rule, subject to their episodic nature.
ZASADA POMOCNICZOŚCI JAKO ZASADA USTROJOWA
Authors: Chochowski Krzysztof
Published: January 2021

Niniejszy artykuł prezentuje rozważania dotyczące genezy i istoty zasady pomocniczości, jej źródeł oraz znaczenia dla społeczności lokalnych i regionalnych. Zasada pomocniczości determinuje bowiem życie społeczności lokalnych i leży u źródeł samodzielności samorządu terytorialnego, który jest wyjątkowy, bo stanowi prawną wspólnotę lokalną. Wspólnota ta, wyposażona w kompetencje i zadania publiczne, samodzielnie zarządza sprawami publicznymi, które jej dotyczą. To właśnie w zasadzie pomocniczości, ma swoje korzenie zasada decentralizacji władzy publicznej, która legła u podstaw organizacji samorządu terytorialnego w Polsce. Zasada pomocniczości kreuje pożądane postawy jednostki tak w jej życiu prywatnym, jak i społecznym. Umożliwia wzniesienie się ponad własny egoizm i interes prywatny, realizację celów wspólnych dla danej społeczności, czyli tego, co w uproszczeniu można nazwać interesem publicznym.
Current foreign and security policy issues regarding stationing nuclear tactical weapons in Italy
Authors: Takács Lili
Published: January 2021

Italy has been storing U.S. nuclear tactical weapons since the fifties. During the Cold War Rome considered hosting nuclear weapons a tool to strengthen the international ranking of the country, hoping that it provides opportunity to restore trust in Italy’s international image, which had been shaken by World War II. As a consequence of the effect of bipolar logic on Italian domestic politics, hosting nuclear weapons guaranteed that the Italian Communist Party remained in opposition. Although the original reasons of hosting U.S. nuclear weapons have disappeared at the end of the Cold War, Italy is not actively promoting nuclear disarmament.
El Pragmatismo en la CSS de China: integración del genio histórico y elementos modernos
Authors: Yuan Fang
Published: January 2021

Durante los últimos setenta años, China ha sido una estrella fulgurante en el campo de la cooperación internacional. El factor principal que contribuyó al ascenso exitoso del gigante asiático es el pragmatismo reflejado en sus estrategias de Cooperación Sur- Sur (CSS). El desarrollo de las estrategias tiene tres fases- el período de Mao Zedong (1949-1976), la época y la post-época de Deng Xiaoping (1978-2012) y la nueva etapa de Xi Jinping (desde 2013)- y en cada una observan reformas dirigidas, en vez de por las ideologías, por los intereses del Estado en diferentes contextos internacionales. Consecuentemente, el progreso de la CSS de China cambió la estructura política internacional, y ofreció experiencias y lecciones útiles para otros países en desarrollo. El presente trabajo va a analizar desde el vínculo de continuidad del pragmatismo las estrategias de la CSS de China a lo largo de las tres fases, y durante un período de tiempo que va desde la década de 1950 hasta la actualidad.
Da ação à reação: O caso chinês na consolidação do monopólio de Elementos de Terras Raras à demanda na Organização Mundial do Comércio
Authors: Leite Alexandre Cesar Cunha , Araújo Mércia Cristina gomes de , Cia Alves Elia Elisa
Published: January 2021

Qual a estratégia da China em mercados de recursos naturais em que tem grande participação na oferta? A partir de um estudo de caso eminentemente qualitativo, mas com ferramentais quantitativos, tais como índice de poder de monopólio do mercado de Elementos de Terras Raras (ETR), mostramos como a China consolidou-se como monopolista e como reagiu às acusações no sistema internacional de comércio. Os ETR não são terras e tampouco raras, mas caracterizam um importante insumo da cadeia produtiva comercial global, destacando-se, majoritariamente, nos sistemas de controle de mísseis, de defesa e de comunicação. Afinal, qual o interesse da China em obter o controle sobre a cadeia produtiva desse setor? Como o país alcançou isso e quais instrumentos de política industrial e comercial foram empregados? Qual a reação dos outros atores nesse mercado? Sugere-se que o objetivo de monopolizar a cadeia produtiva dos recursos naturais estratégicos de ETR foi em consolidar o poder econômico chinês. Mostramos como a China ocupou uma posição privilegiada no setor, galgando a posição de maior exportador destes elementos e forte poder de monopólio, calculado através de índices de elasticidade-preço da demanda e como foi seu posicionamento no litígio internacional.
Stroje versus lidé – tematika vztahu člověka a umělé inteligence ve školním kurikulu
Authors: Štěrba Radim
Published: January 2021

Th e paper maps the possibilities of including the topic “artifi cial intelligence versus man” in school teaching. Based on the use of classical fi ction, the work of Karel Čapek, Philip K. Dick and Isaac Asimov presents possible projections of this topic into the school curriculum and suggests possible methodological approaches usable in school classroom.
Toward discovery and creativity
Authors: Levy Norton
Published: January 2021

The true education of American youth and the real function of a teacher extend far beyond teaching the solutions of a quadratic equation. These involve the realm of the spirit and the nature of man.
Homework and achievement in plane geometry
Authors: Hines Vynce A.
Published: January 2021

Some more light is cast on the difficult problems pertaining to homework in the high school.
Trigonometric functions of real numbers
Authors: Tierney John A.
Published: January 2021

The trigonometric functions of real numbers are of greater importance in mathematics than the trigonometric functions of angles. Why not make the change in both high school and college courses?
College mathematical training for students specializing in agriculture
Authors: Layton W. I.
Published: January 2021

A summary of a research study on the mathematical requirements of seventy-five agricultural colleges located in forty states.
A study on the use of science counselors
Authors: Mayor John R.
Published: January 2021

The efforts of various groups to organize for an attack on the various problems of instruction are indicative of intense interest in the status of secondary mathematics and secondary science.
Points and Viewpoints: If it's good mathematics, it's easier to learn
Authors: Engen H. Van
Published: January 2021

A column of unofficial comment
The comparative effectiveness of one-level and three-level assignments in plane geometry
Authors: Griff Ernest R.
Published: January 2021

Many teachers carry out small research projects during the year. This study of the contract plan vs. the one-assignment plan illustrates what can be done by an individual teacher.
Length of chords and their distances from the center
Authors: Pickett Hale
Published: January 2021

Stimulating supplementary material for classroom use is always welcomed by geometry teachers.
Foundations of algebra
Authors: Meserve Bruce E.
Published: January 2021

The structure and logic of modern algebra are a fundamental “piece of knowledge” for high school teachers of modern mathematics.
Tips for Beginners: Prime power decomposition
Authors: Mock Gordon D.
Published: January 2021

The fact that a number can always be expressed as a product of prime numbers can he used to develop new understanding about equivalent fractions and about the square root of a number. Prime power decomposition is merely the language of number theory for saying that any number can be expressed as a product of prime numbers or powers of primes, e.g., 10 = 2-5 and 27 = 3·3·3 = 33. This idea can be used in eighth and ninth grade general mathematics.
Points and Viewpoints: Helping our teachers
Authors: Fehr Howard F.
Published: January 2021

A column of unofficial comment
Mascheroni constructions
Authors: Hlavaty Julius H.
Published: January 2021

A good, understandable approach to Euclidean constructions performed with compasses alone.
The foundations of algebra
Authors: Brumfiel Charles
Published: January 2021

Some excellent ideas anent introducing proofs in high school algebra.
General mathematics for college freshmen
Authors: Mires Kathrine C.
Published: January 2021

A study of students’ functional competency—and a proposal.
Report of the Membership Committee
Authors: Rogers Mary C.
Published: January 2021

The Membership Committee of the National Council of Teachers of Mathematics is happy to announce that an alltime high in membership total has been reached and wishes to thank you for your fine co-operation and support. It has been the individual assistance of each of you that has been largely responsible for this accomplishment. The Committee is indeed grateful to you for your helpfulness in this service to the Council.
The “make 'em run” theory
Authors: Bengtson Ray
Published: January 2021

A teacher of some experience sets forth his ideas on how to do the most for students with superior ability.
Testing Time: A test for senior day
Authors: Manheimer Wallace
Published: January 2021

The following test was given to my advanced algebra class on Senior Day when high spirits and fancy costumes were much in evidence. After a preliminary groan or two, the class awoke to the fact that the test was something of a mathematical joke, and they entered gaily into the spirit of it.
A high school digital computer program
Authors: Buchman Aaron L.
Published: January 2021

The lead article in this issue has emphasized the extensive use of computers. This article tells what one teacher has done with a homemade digital computer for the high school.
Mathematics in the Junior High School: The stock market as a teaching device
Authors: Scott Douglas E.
Published: January 2021

The practical teaching unit in a high school general mathematics course should certainly involve more than one or two skills or processes, if possible. Ideally, too, practice in computation should be attained by use of techniques other than drill, since students, by the time they reach the ninth grade, often fail to profit from drill-type activities.
Let's teach statistics
Authors: Olander Clarence E.
Published: January 2021

Statistics is being recognized as a suitable course for high schools. Here is one high school's answer to the demand for more statistics in the high school program.
Testing Time: How a mathematics test is born
Authors: Epstein Marion , Myers Sheldon
Published: January 2021

Any mathematics test should be conceived, constructed, and used for a specific purpose. Among the different purposes to be served by tests are diagnosis, selection, placement or sectioning, and appraisal of status or growth. A test which is particularly suited for one of these purposes will often not be suitable for another purpose.
On teaching trihedral angle and solid angle
Authors: Fehr Howard
Published: January 2021

This is the third in a series of articles dealing with angle measures and definitions of angles.
Remedial teaching that builds understanding
Authors: Potter Mary A.
Published: January 2021

An experienced teacher gives some helpful hints on how to teach so that reteaching will he minimized.
Mathematics in the Junior High School: Mobile geometric figures
Authors: Ross Alvin E.
Published: January 2021

This is a project which can be used in a seventh- or eighth-grade mathematics class, giving the teacher an opportunity to broaden the horizontal scope of learning situations. Having students construct mobiles facilitates reviewing of mathematical formulas, and intense motivation for the activity can be obtained by hanging a teacher-made mobile in the classroom.
Points and Viewpoints: What algebra for seventh and eighth grades?
Authors: Rosskopf Myron F.
Published: January 2021

During the fall semester, and particularly after the progress made in putting satellites into the sky, many teachers spoke to me about a speeded-up program in mathematics. What concerns me is the pressure on the mathematics program in the junior high school.
Tips for Beginners: An eighth-grade unit on number systems
Authors: William C. Lowry. , Ingham Carolyn J. , Payne Joseph N.
Published: January 2021

It is a continuing aim of mathematics instruction to develop an appreciation for and an understanding of our number system. Students come into junior high school mathematics with a correct but somewhat incomplete conception of place value, one feature of our system. They recognize that in the number 463, the 4 represents the number of hundreds, 6 the number of tens, and 3 the number of ones.
Mathematics, missiles, and legislation
Authors: Cairns Stewart Scott
Published: January 2021

While our schools need to fight anti-intellectualism, we still have faith that, “In matters of intellect as well, the products of a free educational system will prove superior in the long run to those of a system dedicated to the efficient production of servants of the state or of a particular political and economic order.”
Computations with approximate numbers
Authors: Delury D. B.
Published: January 2021

Contrary to present practices in the schools, there are no simple answers and no general rules for computing with approximate numbers.
Coordination and possible curricular changes
Authors: Neelley J. H.
Published: January 2021

How can the colleges work more effectively with the high schools? What is the best way to get continuity in high school courses and the necessary mathematical competence?
Tips for Beginners: Rapid calculation for the teacher
Authors: Alfred Brother U.
Published: January 2021

One of the interesting by-products of the teaching profession is the contrast between the self-image—what we conceive ourselves to be and what we regard as important in our work—and the image we leave in the minds of our students.
Geometric proof in the eighth grade
Authors: Rosskopf Myron F.
Published: January 2021

There have been many recommendations in recent years concerning the secondary school geometry program. That the assertion is true could be documented by reference to the publications of a variety of committees, commissions, and projects. One suggestion has been singled out for a closer inspection in this paper.
The additive inverse in elementary algebra
Authors: Rheins Joel J. , Rheins Gladys B.
Published: January 2021

Some wags a teacher mag wish to use the field properties.
Suggestions to the applicant for a National Science Foundation institute
Authors: Myers W. H.
Published: January 2021

Having been charged with the selection of participants for two National Science Foundation Summer Institutes and having served in a department that has conducted at least eight such institutes, the author has been impressed by the many things which applicants do or fail to do which result in their being denied the right to participate. Perhaps a listing of some of the common errors will aid teachers in making successful applications in the future.
Trammel method construction of the ellipse
Authors: Lubin C. I. , Mazkewitsch D.
Published: January 2021

An extension of the trammel method for construction of an ellipse
Tips for Beginners: Pythagorean converse
Authors: Hirsch Martin
Published: January 2021

In the past few years, various proofs of the Pythagorean theorem have appeared in The Mathematics Teacher. It is undoubtedly known to the reader that there are many different proofs of this well-known theorem.
Setting up an approximate antilog table
Authors: Bayer George L.
Published: January 2021

Approximating antilog x by inverting some key entries in a log table and using interpolation
Enriching mathematics instruction with creative activities
Authors: Johnson Donovan A.
Published: January 2021

Some ways to identify and nourish creativity in the mathematics classroom
Nasc or iff or wow
Authors: Santner Joseph F.
Published: January 2021

Ways of stating conditionals and biconditionals
Tips For Beginners: Lively points
Authors: Ryle George W.
Published: January 2021

Your students do not wait until their first algebra course to become acquainted with rectangular coordinate systems. And you do not have to wait until then, either, to help them gain understanding of basic concepts involved in these systems.
Experimental Programs: Coordinate Geometry with an Affine Approach
Authors: Sitomer Harry
Published: January 2021

In the spring of 1961, the School Mathematics Study Group convened a group of college mathematicians and high school teachers of mathematics to consider plans for writing an alternate high school geometry course, in which coordinates would be introduced and used as early as feasible.
Continued Fractions
Authors: Holmes Joseph E.
Published: January 2021

CONTINUED fractions have been known for centuries and, during the 1800's, were considered to be a standard part of higher algebra taught at the college level. For the first half of this century these fractions have received little attention. During the last ten or fifteen years, however, there has been a growing interest in the use of continued fractions.
Several Ways of Translating a Conditional Sentence: A Dialogue
Authors: Szabo Steven
Published: January 2021

CONDITIONAL sentences are sentences of the form “if p, then q,” and they occur quite frequently in mathematical discourse. In some contexts, it is often convenient to say in other ways what could be said with a conventional conditional sentence.
Finger Reckoning
Authors: Prielipp Robert W.
Published: January 2021

OVER the years many young school children (and some not so young) have discovered how helpful their fingers can be in performing simple addition problems. One can't help but wonder how many people have ever given serious thought to multiplying numbers on their fingers. Can such a thing be done? Well, let's see.
International Mathematical Education: The Czechoslovak National Mathematical Olympiads
Authors: Hlavaty Julius H.
Published: January 2021

SINCE 1950 every young hopeful future mathematician in a secondary school in Czechoslovakia, or even in elementary school, looks forward to the various rounds of the annual mathematical olympiad. He has hopes for valuable prizes (mostly books on mathematics)- but even more to the sheer glory of recognition (in the form of certificates) on a local, regional, or national level. If he is in the last year of a secondary school, he may even reach the International Olympiad conducted annually (since 1959) by the East European countries.
Verifying the Associative Property for Finite Groups
Authors: Hammel Arnold
Published: January 2021

Many of the group axiom for the operation o of a finite group can be checked by referring to the multiplication table for the group. This table is an n-by-n array constructed by listing the elements of the group along the left side and along the top. If r is the element heading theith row and s is the element heading the jth column, then the contents of the box in the ith row and jth column is the group element r o s. Throughout this paper the group operation o will be written as ordinary multiplication. Thus r o s will be written as rs.
Points and Viewpoints: On “Digital Problem 196X,”
Authors: Brune Irvin H.
Published: January 2021

It began in 1965. Observing that digital problems have long captured the mathematical imagination, Jerome Niebaum proposed the following problem:
On Similarity Transformations
Authors: Hardesty James
Published: January 2021

In discussing the parabola with students, one frequently encounters questions such as “Are all parabolas similar?” or “Do all parabolas have the same shape?” Before an attempt can be made to answer such questions, some kind of clarification must be made of the term “similar” so that it applies to plane curves other than simple closed polygonal paths.
Concerning the Rational Zeros of Polynomials
Authors: Schaumberger Norman , Just Erwin
Published: January 2021

Mathematics at all levels of sophistication is a growing body of knowledge. Here is a direct, interesting example.
What Should High School Geometry Be?
Authors: Buck Charles
Published: January 2021

The question “What to do about geometry?” has for decades beset the planners of both high school and college mathematics curricula. Until the nature of the first course in high school geometry is settled, the high school mathematics curriculum cannot stabilize. If the high school geometry question could be answered, this would help the colleges to reset geometry in their curricula.
A Simple General Criterion of Divisibility
Authors: Boley Daniel L.
Published: January 2021

While criteria of divisibility for some simple integers (such as 2, 3, 5, 9, 11, etc.) are well known, it is often necessary to have a general scheme by means of which divisibility by any integer can be determined. Such a scheme is presented here.
Research in Psychomathematics (Research in the Emerging Discipline of Psychomathematics)
Authors: Scandura Joseph M.
Published: January 2021

DOING basic research on mathematics learning is a risky business. In the first place, it is hard to know whether one is asking the right questions, and, in the second place, it has been difficult to formulate significant questions involving mathematics learning in a researchable form. One of the major reasons for this state of affairs has been the lack of any suitable theoretical superstructure from which to work. The purpose of this paper is to outline some of the theoretical work under way at the University of Pennsylvania and to show how this work has helped to improve our understanding of how people learn mathematics.
A Method for Finding the Period and Phase Shift of Trigonometric Functions
Authors: Fraser Donald G.
Published: January 2021

Finding the amplitude of trigonometric functions has caused my students little trouble. The same could not be said for the period and the phase shift until the following analysis was presented.
Historically Speaking— The Lore of Number
Authors: Barit Julian
Published: January 2021

The origins of counting and number are shadows of prehistory. Certainly they must have sprung from the needs of the earliest herdkeeping beings who, finding themselves with possessions, had to keep track of them. It is equally certain that number became far more than a tool of convenience and need—it penetrated inexorably into men's beliefs, mores, and very lives. Evidence of number's pervasive function may be found in every oral and written record left by the human race. Through the years a huge quantity of fact and fancy, science and symbolism, sense and supersition has grown up. It is the purpose of this paper to discuss and collate some of this lore of number.
Classics in Mathematics Education: A Philosophy for the Mathematics Teacher
Authors: Crosswhite F. Joe , Newsom† Carroll V.
Published: January 2021

We who are teachers of mathematics are proud. Our ancestors in the profession include some of the greatest teachers in history. We are direct descendents of Pythagoras, Euclid, and the great teachers of Plato's Academy. The students we and our forebears have trained have made history; they have assisted in the development of methods and devices that have been responsible for the high level to which man has attained.
Have You Read …?
Authors: Peak Philip
Published: January 2021

Everyone knows there are lots of ways to handle money. We also know that many people who can handle mathematics on a high level can handle money only on a low level, and the converse.
Experimental Programs: Let's Play “Wff'n Proof,”
Authors: Jeffryes James
Published: January 2021

Wouldn't it be a great deal of fun to learn logic if the learning activity were primarily one of playing games? The conviction that this is true was the basic belief of Layman Allen, assistant professor of law, Yale Law School, when he constructed WFF'N PROOF', the Game of Modern Logic. 1
Digital Sums of Perfect Numbers and Triangular Numbers
Authors: Prielipp Robert W.
Published: January 2021

BEFORE a theorem can be proved it must first be discovered. One of the areas of mathematics in which some very intriguing results can be uncovered, even at a relatively elementary level, is that of number theory. We proceed to present evidence to substantiate the preceding remark.
Have You Read …?
Authors: Peak Philip
Published: January 2021

One of the basic principles we follow in our teaching is to relate new ideas with old ideas. Dr. Forbes has done just this in his article about extending the concept of rational numbers to real numbers. He points out how this extension cannot follow the same pattern as that of extensions positive to negative integers or from integers to rationals. If we look to a definition for motivating the extension we at best can only say, “Some polynomial equations have no rational number solutions and do have some real number solutions.” We might use least-upperbound idea, or we might try motivating through nonperiodic infinite decimals. However, Dr. Forbes rejects all of these and makes the tie-in through a geometric approach.
Pi-Three Versus Pi-Four
Authors: Greenblatt M. H.
Published: January 2021

PAGES 432a and 434a of the December 1965 issue of the American Scientist describe certain stone “round things” (I cannot call them circles), which have been found at several places in the British Isles. It is said, on page 434a of that magazine, that these misshapen round things may have been the result of a prehistoric British attempt to make the value of pi equal to 3. The existence of that description and the immediately preceding article, “Legal Value of Pi,” bring to mind the fantasy of the “Pi-Three-ers” and the “Pi-Four-ers.”
Experimental Programs: Using Programmed Instruction to Investigate the Effects of Group Interaction on Learning Mathematics
Authors: Nichols Eugene D. , King Robert W.
Published: January 2021

Programmed instruction, now well into its second decade of existence, has made a place for itself in the educational community. There is no longer any doubt that students can and do learn using programmed material. Many programs have been written and are being used at all academic levels from preschool to graduate school. Programmed instruction has also been widely used in educational research to investigate various aspects of learning theories and teaching practices.
The Demise of Analytic Geometry
Authors: Nadler Maurice
Published: January 2021

Analytic geometry has been dying as a separate course. For some years this subject has been losing its status as a mathematics course, and its content has been given only partial or token recognition by inclusion in precalculus or in elementary calculus courses. While the pedagogical concept of integration of subjects is commendable, the actual result has frequently been a substantial eradication of the subject from the mathematics curriculum.
Historically Speaking— a Visit to a Mathematical Shrine
Authors: Eves Howard , Ayton Joseph
Published: January 2021

During the past summer I enjoyed a vacation in Ireland. Recalling the legend of the circumstances surrounding Sir William Hamilton's invention of noncommutative multiplication for quaternions, I decided to try to find the bridge in Dublin. According to Cajori's History of Mathematics, it is the Brougham Bridge (pronounced “Broom”) over the Royal Canal, and after some search I located it and photographed it.
Historically Speaking— Niels Henrik Abel
Authors: Prielipp Robert W.
Published: January 2021

Niels Henrik Abel (1802-1829) is without a doubt the greatest mathematician Norway has ever produced. Though he died before reaching the age of twenty-seven, struck down by tuberculosis, he left a bountiful legacy for mathematics. At the time of his death the world was really just beginning to awaken to his genius. A young man who was well acquainted with poverty, Niels Henrik's family life was not always of the best. Neither his father, who died in 1820, nor his mother was a saint (for a very readable account of Abel's life see [2]), and the family was usually quite heavily in debt. Abel's studies were financed by his professors and by the Norwegian government. He was granted a fellowship to travel and study in Europe, and the year and a half which he spent in Germany, Italy, and France was one of the happiest periods in his life. When Abel returned to Norway, he could only obtain a temporary position. The last years of his life were beset by many difficulties.
Have You Read …?
Authors: Peak Philip
Published: January 2021

Indirect proof has in generai always been the center of controversy. Aristotle considered it inelegant but indispensable; some have said it was inconclusive or even unnecessarv, The au thor of this article explains its indispensability and its logical legitimacy. He does this through the proof of the irrationality of .
Reviews and Evaluations
Authors: Eakin Richard R.
Published: January 2021

In this book the reader will find an excellent presentation of Great Britain's revision of tra ditional Euclidean geometry with the geometry of vectors and transformations. This geometry would use the fundamental transformation ideas of translation, rotation, stretch, and reflection. The suggestions for this replacement are not new; they have been suggested for years but many of these new topics are just now being ac cepted in high schools in the United States.
A Proposal for the Mathematics Methods Course
Authors: Shafer Dale M.
Published: January 2021

The twentieth century has witnessed the establishment of numerous commissions and committees whose purpose has been to study the optimum preparation that colleges and universities should provide future teachers of secondary school mathematics. Generally these study groups recognized the existence of two basic problems. The first problem was to determine the type of mathematics courses the future teachers should study while attending college, and the second was to determine how the individual can best be prepared pedagogically to teach mathematics. In response to the first problem, the groups were quite diligent in outlining their recommendations concerning an appropriate sequence of mathematics courses and the content of those courses. In response to the second problem, however, the groups either omitted making recommendations or made them in very general terms. The structure of the methods course in mathematics has, therefore, remained undefined in comparison with the content courses in mathematics.
Developing a Meaningful Algorithm for Factoring Quadratic Trinomials
Authors: Baker Betty L.
Published: January 2021

IT WILL be the purpose of this paper to develop a meaningful procedure for factoring quadratic trinomials. Children studying algebra are frequently dismayed when faced with the task of factoring such expressions by the so-called guess method. Careless errors are sometimes made, and often the student fails to see the distributive principle operative in this kind of factoring even though he finds the product of two binomials using the distributive principle.1
Modern Mathematics or Traditional Mathematics
Authors: Buker Werner E.
Published: January 2021

In these days of “modern mathematics” versus “traditional mathematics” it occurs to the writer that these terms have been overworked, and should be candidates for honorable retirement. Perhaps it would be more relevant to speak of “good mathematics” and “bad mathematics,” “interesting mathematics” and “dull mathematics”; or, to paraphrase Hardy, “beautiful mathematics” and “ugly mathematics.” Unfortunately, and perhaps unavoidably, most of the mathematics a secondary student of mathematics sees is of the less interesting variety.
Reviews of Films
Authors: Coxford Arthur F.
Published: January 2021

This film is an excellent introduction to the fundamental concepts of probability. A simple probability scale from 0 to 1 is used to illustrate the probabilities of events that are certain to occur and those whose occurrence is impossible. The formula for the probability of an event is defined to be the ratio of “those events which interest us” to “the total events in the sample space.” This formula is developed with several clearly illustrated examples involving coins and dice. Two more examples, only slightly more complicated, further strengthen this develop ment. One of these, on the effectiveness of a vaccine, shows how a sample space may be created by experiments.
If We Just Turn The Crank
Authors: Hutcheson James E.
Published: January 2021

Teachers of mathematics in elementary school, high school, and college seldom have an opportunity to share ideas. All too often these opportunities come only at annual conferences or conventions. Communication is generally restricted to publications and journals. Teachers at all level of instruction should devote a portion of their professional reading to current and older journals that emphasize instruction at a level other than their own. Ideas are presented that have application at many levels—if only one made a slight modification or “turned the crank.”
A Quality Inequality
Authors: Sloyer Clifford W.
Published: January 2021

The inequality involving the arithmetic and geometric means is a powerful tool in dealing with applications of mathematics. Indeed, it is the cornerstone of a relatively new area of applied mathematics known as geometric programming. The purposes of this article are to discuss briefly this inequality and to demonstrate two “real world” applications that secondary teachers in in-service institutes have found interesting and useful. The mathematics used is directly accessible to secondary teachers and students.
Applications of Groups and Isomorphic Groups to Topics in the Standard Curriculum, Grades 9-11: Part I
Authors: Usiskin Zalman
Published: January 2021

The subfield of pure mathematics that has grown most significantly in the past few decades is that of algebra, by which is meant “higher” or “abstract” algebra and linear algebra. Twenty years ago courses in algebra were at the advanced undergraduate and graduate level, and it was easy to become a certified mathematics teacher without having any knowledge of groups, rings, fields, or vector spaces. Today virtually all prospective teachers take a course in which some of these structures are studied.
A General Intersection Formula for Subspaces of N-Dimensions
Authors: Hollist J. Taylor
Published: January 2021

There are many approaches one could take in making leaps from the plane and space into four dimensions and then into n-space. One approach would be to extend the algebraic model with which most high school students are familiar. In this article, I want to look at an interesting formula concerning intersections by starting with the familiar analytic model for Euclidean geometry.
√2 as a Sequence of Rational Numbers
Authors: Scott Douglas E.
Published: January 2021

In the usual secondary school algebra course real numbers are typically defined (if any explicit definition is given at all) either as infinite decimals or simply as the set of “all” numbers, rational and irrational. The rational numbers, however, are usually carefully defined as quotients of integers; their properties are then derived from the properties of the integers.
A Procedure for Generating Data for Which the Correlation is Exactly .50
Authors: Huck Schuyler W.
Published: January 2021

Almost anyone can come up with a set of scores for two variables, say X and Y, such that the Pearsonian correlation is exactly equal to +1.00. By rearranging the order of the scores associated with one of the two variables, one can just as easily “create” a set of data in which there is a perfect negative correlation. While these two tasks are so easy that they can he accomplished blindfolded, most applied researchers and teachers of statisties would probably have trouble generating—within a one-minute time period—a set of data such that the correlation is exactly equal to +.50 (or - .50), This paper has been written to show how this can be done.
Reader Reactions
Authors: Vaidya A. M.
Published: January 2021

In the article “Numeration Systems with Unusual Bases” (May 1974), David Ballew has missed a very interesting fact. He mentions negative bases such as —7. The exciting fact is that if you use a negative number as a base, all negative integers (as well as positive ones) can) be written without use of the minus sign.
The Eye of Horus is Upon You
Authors: Kavett Hyman , Kavett Phyllis F.
Published: January 2021

Here is how the ancient Egyptians wrote numerals for whole numbers and fractions. Their numeration systems were not at all “primitive.”
Factoring Puzzles
Authors: Hollingsworth Caroline , Dean Eleanor
Published: January 2021

Factor puzzles are a clever technique for building readiness to tackle the complex algebraic task of factoring trinomials.
The Design, Proof, and Placement of an Inclined-Gnomon Sundial Accurate for Your Locality
Authors: Wolf Charles T.
Published: January 2021

A description of a sundial model that students can make and use, as well as the mathematics on which the model is based.
Paper Folds and Proofs
Authors: Fehlen Joan E.
Published: January 2021

Paper folds can provide some sophisticated mathematical relations. The mathematical proofs are good supplements to the usual textbook exercises.
Complex Modular Numbers: Complex Numbers Need not be Complex
Authors: Grant Susan J. , Stewart Ward R.
Published: January 2021

Most students are faced with the task of solving the equation x2 + 1 = 0 over the real numbers at some time in their algebra classes. After they substitute values for x unsuccessfully, they usually attempt to solve the equivalent equation x2 = -1. They soon realize that it is impossible to square a real number and obtain a negative number. At this point their teacher may define the imaginary number i to be and then proceed to develop the complex number system.
Cosine Triples
Authors: Moore Charles G.
Published: January 2021

This investigation of cosine triples was instigated by a chance observation. For years I have used triangles with integral sides for the purpose of quizzing students on the law of cosines. The law of cosines states that the square of the length of any side of a triangle is equal to the sum of the squares of the lengths of the other two sides minus twice the product of the lengths of those two sides and the cosine of the angle between them. For example, given the triangle with sides 6 and 7, and included angle 20 degrees, the student is expected to find the remaining side using the law of cosines (fig. 1).
From the Golden Rectangle and Fibonacci to Pedagogy and Problem Posing
Authors: Brown Stephen I.
Published: January 2021

There is a wealth of valuable material on the golden section, and “fallout” from it as well, Indeed, the problem of constructing the golden rectangle with straightedge and a pair of compasses, and its relationship to constructing the regular pentagon with similar tools, is documented in Euclid's Elements (Euclid 1956; vol. 1, bk. 2, prop. 11; vol. 2, bk. 4. prop. 11). In the thirteenth century, the Italian mathematician Leonardo Fibonacci found connections between that aspect of geometry and number theory, and today there is a mathematical journal that is devoted exclusively to such issues. The golden section has appeared not only in mathematics but in architecture and art as well. The Parthenon in Greece, for example, has the proportions of the golden rectangle, and various portions of Michelangelo's David, from the joints of the fingers to the relative placement of the navel with respect to the total height, exemplify the golden ratio.
The Algebra and Geometry of Polyhedra
Authors: Troccolo Joseph A.
Published: January 2021

Given a box of squares and a box of triangles. you are to build convex polyhedrons. With the constraints, how many polyhedra are possible and how are they constructed?
Calculator Charades
Authors: Boyle Patrick J.
Published: January 2021

Calculators can be jim as well as practical.
A Search for Root-Multiples
Authors: Francis Richard L.
Published: January 2021

Learn what “root-multiples” are and then use them to generate interesting progressions.
1001 Properties
Authors: Norris Fletcher R.
Published: January 2021

The properties discussed in this paper make 7.11, and 13 all lucky numbers.
Some Unusual Tests of Divisibility
Authors: Yazbak Najib
Published: January 2021

Use verifications of these divisibility tests to help junior high school students gain additional practice with basic operations.
A Mathematician's Repertoire of Means
Authors: Maor Eli
Published: January 2021

Unified treatment of the family of means—based on an appealing geometric model.
The Cardiologist'S Theorem
Authors: Staib John H.
Published: January 2021

You and your students will enjoy this “Polya” search for an elegant solution to a real problem in diagnosing heart abnormalities.
Models of the U.S. Economy
Authors: Dunn Samuel L. , Wright Lawrence W.
Published: January 2021

The field of economics provides many opportunities for applying mathematics. The quantification of economics that has occurred in the last thirty years has made it necessary that economists be trained in the uses of higher mathematics. Algebra, geometry, calculus, probability theory and statistics, higher analysis, linear algebra, and computer science are some of the tools being used in contemporary approaches to economics.
The Key Duplicator: A Congruence Machine
Authors: Hollingsworth Caroline D.
Published: January 2021

Opening your door with a key requires the use of a “congruence machine. Do you know why?”
Functions of the Square
Authors: Sconyers James M.
Published: January 2021

Do you have difficulty finding the coordinates of various points on a unit circle? Try beginning with functions defined on a unit square.
On Meeting the Needs of the Mathematically Talented: A Call to Action
Authors: House Peggy A. , Gulliver Markita L. , Knoblauch Susan F.
Published: January 2021

What is the status of the gifted in your school? This article should provoke some thinking and suggest a further study of the issue.
Patterns of Repeating Decimals: A Subject Worth Repeating
Authors: Sgroi James T.
Published: January 2021

Mathematics is the Queen of the Sciences. It has numbers that will dazzle your mind and tickle your fancy. It is the soul of men's minds—a power that can he tamed and put to creative work. Yes, my friend, I can show you some secrets of this power: if you only remember the key word to unleashing this power—PATTERNS. Say it loud and say it often, for it is the building block to developing mathematics.
Dual Concepts—Graphing with Lines (Points)
Authors: Stertz Deloyd E. , Teeters Joseph L.
Published: January 2021

Mapping functions is always interesting—with parallel axes the envelopes produced are often surprising.
Cool'S Clams Casino
Authors: Cobey Paul , Williams David E.
Published: January 2021

You can create your own game using clam shells.
Activities: Magic Triangles
Authors: Caldwell Janet
Published: January 2021

This section is designed to provide mathematical activities suitable for reproduction in worksheet and transparency form for classroom use. Appropriate laboratory experiences, discovery activities, and model constructions of mathematical significance drawn from the topics of the seventh, eighth, and nine grades arc most welcome. Send your suggestions, in a four-page format similar to the “Activities” already published, to the managing editor for review.
Basic Competencies—Twenty-Five Years Ago, Ten Years Ago, and Now
Authors: Beckmann Milton W.
Published: January 2021

Our students can perform as well in many areas as students in the late 1940s.
Changes Needed in The Current Direction of Minimal Competency Testing in Mathematics
Authors: Reys Robert E. , Kasten Margaret B.
Published: January 2021

Will minimal competency testing help or hurt mathematics education?
Sharing Teaching Ideas
Authors: Hirsh Ronnie M.
Published: January 2021

Finding ways to motivate a high school mathematics student to work on computational skills is a difficult task. I have developed a game called “Operation: Four” that has met with great success in my remedial classes.
Can We Require Students to Learn?
Authors: Wood Dorothy F.
Published: January 2021

Will students learn because we pass a law that says they must?
Activities: Tiling
Authors: Freilag Richard A.
Published: January 2021

Materials: Student worksheets, blank paper, paste, scissors, and crayons.
Back To Basics: Friend or Foe?
Authors: Kraus William H.
Published: January 2021

A look at important issues related to the “back to basics” movement.
Who Killed the Cook?
Authors: Corbet James J. , Milton J. Susan
Published: January 2021

An example of the use of probability that should capture the attention of students.
Minicalculators and Repeating Decimals
Authors: Lichtenberg Donovan R.
Published: January 2021

Here we go again on a topic that continues to interest teachers and students. This time, with invaluable insights from our calculators … Here we go again on a topic …
Mathematics in A Pumpkin Patch
Authors: Taffe William J.
Published: January 2021

The “Great Pumpkin” visits the mathematics teacher. Are you ready?
An Easy-To-Paste Model of The Rhombic Dodecahedron
Authors: Wahl M. Stoessel
Published: January 2021

“Crystal” clear instructions for a model with many applications.
Adding Fractions Incorrectly?
Authors: Carmony Lowell
Published: January 2021

A presentation of exceptions to an established rule Students may be motivated to look for other exceptions as they practice basic operations.
Math'e Ma Tish'Ans
Authors: Metz James , Canella Joseph
Published: January 2021

The following names and their pronunciations are offered as an aid to classroom teachers who may have only read the names and not heard them pronounced. Only a few names are listed and you may wish to add others.
Right Isosceles Triangles on The Geoboard
Authors: Austin Joe Dan
Published: January 2021

A wide range of problems flow from a simple isosceles triangle.
Mirror Multiplication
Authors: Manheim Jerome H.
Published: January 2021

Searching for products where reversing the digits does not change the answer.