bluuebunny/crossref_metadata_2025 huggingface dataset:
Classics in Mathematics Education: Gestalt Psychology and Mathematical Insight
Authors: Hartmann George W. , Crosswhite F. Joe
Published: November 2020

This paper is presented as an expression of belief and hope that some of the newer formulations in psychological theory will bring about a democratization of mathematical competence somewhat akin to the astonishing elevations of performance often obtained in the field of reading. The symbolism and thought processes involved in using conventional language are not essentially different from those employed in the more universal terminology and operations of mathematics. If reading ability is now better and more widespread among Americans than at any previous time in our history, there is no reason why a better comprehension of numbers, figures, and conceptual relations should not make quantitative inferences as common as simple words and sentences.
The Purpose and The Content of a Course of Mathematics for Technical and Manual Training High Schools
Authors: Gentleman Frederick W.
Published: November 2020

In the treatment of this subject I shall confine myself to those schools only that require a grammar school diploma, or its equivalent, for admission, that have a four year’s course, and that are manual training or technical high schools as distinct from the “trade” school. I have confined my investigation, mainly, to public high schools, for with the institutions supported by the public the problem is more vital, and demands the attention of the public school teacher to a far greater extent than that of private institutions.
Efficiency vs. The Individual
Authors: Passano Leonard M.
Published: November 2020

One of my colleagues, in a recent letter to The Nation, caustically deplores the prevailing practice of “The Czar,” of college presidents, and “even of the deans” — I am quoting his words — their practice of discussing, in public and in print, the sins and shortcomings of the teacher. He says that, just as the inefficiency of the saleswoman in a department store is discussed by the management in its office and in private, so the faults and failures of the teacher should be talked about, sub rosa, in the presidential office or in the sanctum of the Carnegie Foundation.
Freshman Algebra and the Average Freshman
Authors: Gray Ella Durgin
Published: November 2020

Let us first consider our pupils in two classes, those for whom algebra has a practical value and those for whom it is merely a culture study. Now, as probably most of us agree, the practical value of algebra. in lhe sense of the man in the street, is almost negligible. Personally, apart from my professional connection with it, I can recall but one instance of its having been of service to me. If by practical value we mean its value to those who practise it, our first class is limited to those pupils who will use algebra in other courses, at high school or at college. Those of this class who elect review algebra in high school, usually find that their grade in the review has been largely determined during their freshman year by the degree of facility they acquired in interpreting and manipulating notations and by the clearness with which they distinguished certain principles and their applications. For the time allowed review algebra is generally adequate only for those who have already some insight and skill to test these in more complicated and lengthy exercises and to extend somewhat the ground previously covered. The advanced pupil who is hazy and uncertain about fundamental notions and procedures is himself at a great disadvantage and he retards greatly the progress of the others. The advanced pupil who has acquired a merely mechanical proficiency f requently has too large an estimate of his knowledge and therefore does not present so open a mind towards efforts to clarify, deepen and broaden his vision as he would in the discussion of new matter. To the college preparatory student, then, his freshman algebra is highly important.
The Return of the Old Mathematics
Authors: Raphael Brother L.
Published: November 2020

A Great deal of the initial unrest and uproar attending the “new math” revolution has died down, and now most of the teachers and administrators are finding themselves in a period of classroom application, trying to find the text, usually by trial-and-error; in short, the job of working the new programs into the curriculum. The institutes have for the most part done their work of reeducating the older teachers in the terminology, and it seems that just about everyone is quite aware of what the material is all about.
What are Perpendicular Lines?
Authors: Wylie C. R.
Published: November 2020

In a recent issue of The Mathematics Teacher,1 Harry Sitomer described an experimental program in high school geometry originally recommended to the School Mathematics Study Group by some of its advisers but finally undertaken by the Wesleyan Coordinate Geometry Group.
Intuitive Approach To X0 ≡ 1
Authors: Mallory Curtiss
Published: November 2020

One of the problems encountered in teaching mathematics is making the definition x0 ≡ 1, when x ≠ 0, meaningful to the mathematics student who faces it for the first time. Two methods are used to show x0 ≡ 1, where x is a natural number. Similar procedures may be used for number systems other than the natural numbers.
Experimental Programs: How Modern is Today's Secondary Mathematics Curriculum?
Authors: Alspaugh John W. , Delon Floyd G.
Published: November 2020

Revision in the secondary mathematics curriculum is an acknowledged fact among mathematics educators on both the secondary and the college levels. However. only estimates exist as to the extent of this revision in the nation's schools or even within the schools of a given state. No effective adjustment in college mathematics and related courses in the teacher- preparation program can take place until this information is available. A recognition of the need for a survey of the status of the secondary mathematics curriculum led to the study upon which this report is based. While schools in only one state are considered, the information may be indicative of the status of the secondary mathematics curriculum elsewhere.
The Modern Approach to Elementary Geometry
Authors: Veblen Oswald
Published: November 2020

Let us try to see what effect, if any, modern research has had on our views about elementary geometry, and how the latter ought to be presented to students. According to old tradition the Euclidean geometry was the type example of a logically perfect science. Starting from the axioms it proceeded by inexorable logic from one theorem to another. On the other hand, what we now call analysis was admitted to be on rather a shaky foundation, but justified by the importance of its results.
The Direction of Sunset
Authors: Graesser R. F.
Published: November 2020

Everyone has noticed how the direction of sunset varies with the seasons. A formula for this direction is usually derived from a background of spherical trigonometry and some astronomy. It is the purpose of the following discussion to derive such a formula using only plane trigonometry, and without assuming any knowledge of astronomy.
The Square Function: An Abstract System for Trigonometry
Authors: Biddle John C.
Published: November 2020

An interesting diversion for students working with identities in trigonometry is to talk about a “square function.” Any or all of the following three systems can be used in the usual trigonometry class for a breather to break the monotony, to provide enrichment for the better students, or to provide a topic for discussion in the mathematics club.
Experimental Programs: A Study of Two Methods of Checking Homework in a High School Geometry Class
Authors: Small Dwain E. , Holtan Boyd D. , Davis Edward J.
Published: November 2020

“What effect does homework have on mathematics achievement?” is a question that has been of interest to educator's for some time. Several reports on research in this area indicate that results of studies have been inconclusive [2, 6].1 Goldstein [3], however, suggests that the data of many of these studies have been misinterpreted and that regularly assigned homework does favor higher academic achievement.
Historically Speaking—: Credit Where Credit is Due!
Authors: Greitzer Samuel L.
Published: November 2020

To the unbiased observer, it must appear as if mathematical discoveries and the names of their discoverers are deliberately being excluded from what is considered “culture.” Cultured people who can discourse learnedly about Aristophanes, Euripides, Parmenides, Velocipedes, and the Eumenides appear to know nothing about Fermat, Euler, Gauss, or Riemann. To a great extent, the blame for this must be laid on the teacher of mathematics who contents himself with presenting the work of mathematicians without in any way mentioning the story of the discoverer or of the discovery.
Number Theory—a Short Course for High School Seniors
Authors: Hollingshead Irving
Published: November 2020

For several years the traditional mathematics curriculum has been under fire. Though suggest ions for improvement have been diverse, some degree of agreement has been evolved on the content of the mathematics program. At least there is an outline of agreement on the curriculum through Grade 11.
Classics in Mathematics Education: On the Foundations of Mathematics
Authors: Moore Eliakim Hastings
Published: November 2020

The notion within a given domain of defining the objects of consideration rather by a body of properties than by particular expressions or intuitions is as old as mathematics itself. And yet the central importance of the notion appeared only during the last century—in a host of researches on special theories and on the foundations of geometry and analysis.
New Learning and Subverbal Knowledge
Authors: Hajek Roy D.
Published: November 2020

Every student has a hidden asset. Each has a large store of subverbal knowledge and ability that can be tapped to contribute to his progress in mathematics. This knowledge, accumulated through the sum of past experience in the surrounding world, is seldom displayed, but certainly does contribute to attitudes, conceptions, and reasoning processes. Though the knowledge is not neatly catalogued nor readily available for specific recall, it can form an important basis for progress in new mathematical situations. It guides, outlines, and either limits or extends the amount of insight each person is able to exhibit in new situations.
Using a Programmed Text to Provide an Efficient and Thorough Treatment of Solid Geometry Under Flexible Classroom Procedures
Authors: Brown O. Robert
Published: November 2020

The UICSM Mathematics Project has long been concerned with the mathematics curriculum in today's secondary schools. Two areas of curriculum work were joined in producing the programmed solid geometry text under discussion. The content was developed from an Appendix of High School Mathematics, Unit 9 (UICSM, 1962) and an earlier UICSM unit on topics from solid geometry. The instructional procedure reflected findings of the UICSM Programmed Instruction Project (Brown, 1962 and 1964).
A Model for Teaching Mathematical Concepts
Authors: Henderson Kenneth B.
Published: November 2020

An activity on which mathematics teachers (and teachers of other subjects, too) spend much time is the teaching of concepts. One may argue that the teaching of concepts is the central objective of academic instruction. It is the purpose of this article to offer a model for teaching mathematical concepts and then to point out how this model may be useful in the education of mathematics teachers and in the direction of research.
Mathematical Models for Physical Situations
Authors: Van Engen H. , Cleveland Ray W.
Published: November 2020

Recent attempts to improve the teaching of problem solving in mathematics have made it necessary to look carefully at physical situations and the way in which they are represented by conventional mathematics. Relatively recent interest in the newer studies in developmental psychology has focused attention also on the need for a closer examination of experiences' being provided elementary and secondary pupils in connection with certain mathematical ideas as they relate to problem solving. There are instances in the physical world where it is possible, and at times desirable, to represent a physical situation by using mathematical ideas and their properties which are quite different from those commonly used.
International Mathematical Education: Equivalent Forms of the Parallel Axiom
Authors: Bunt Lucas N.H.
Published: November 2020

Editor's Note.—Euclides is a magazine for the didactics of mathematics. It was started in 1924, and it is now the official journal of the associations of mathematics teachers in the Netherlands. There are ten issues a year, of 32 pages each. It contains articles on problems connected with the teaching of mathematics in the Netherlands and in other countries, especially in relation to modernization tendencies, and articles which are aimed at showing the mathematical background of the subject matter which is taught in secondary schools. It also gives reviews of books appearing in the field.
New Mathematics and Old Dilemmas
Authors: Fremont Herbert
Published: November 2020

It is interesting to note how little difference there seems to be these days between articles in professional journals and the advertisements of publishers. In a release of Science Research Associates entitled “Your Guide to SRA Mathematics Materials” the following statement is made.
Intuitive Approach to a−m≡1am
Authors: Murray Pete
Published: November 2020

In the article “Intuitive Approach to X0 ≡ 1” by Curtiss Mallory (The Mathematics Teacher, January 1967, page 41), two approaches to the definition X0≡ 1 were suggested, one depending on the relationship am · an = am + n, the other depending on the pattern that occurs as one writes a sequence of descending powers of some (nonzero) base.
The Floating Teacher— Help for the Mathematically Disadvantaged
Authors: Newman Thomas B. , Seiser William
Published: November 2020

How can a youngster learn about primes and factors when he is not able to multiply? Such deficiencies are common among many pupils entering junior high school in areas of high population density and low income. They have not achieved in mathematics to the extent necessary to be able to cope with the mathematics concepts of the seventh-grade level. Teacher preparation, textbooks, and curriculum materials that are oriented to the “average” pupil have little specific provision for the mathematically disadvantaged pupil. Thus, provision must be made for these pupils to successfully learn mathematics.
Characteristic Functions and Sets
Authors: Glaymann Maurice
Published: November 2020

The simple concept of characteristic function is a powerful tool for the study of the algebra of sets; it has not as yet been used in our teaching. The present paper is written to show the interest of this concept. However, this tool must be used only with pupils who have had much experience with sets. During the long period of learning our task is to form and build up concepts; the child must learn in various situations and at the same time “he accumulates,” says Dienes, “a capital to which he will later on refer.” It is only after this stage that we can provide powerful tools and “elegant” methods.
A Device for Teaching Addition and Subtraction of Integers
Authors: Hannon Herbert
Published: November 2020

Many different approaches have been suggested for teaching the algorisms that are used to add and subtract the integers.
An Elective in Mathematics
Authors: Martin Donna Lou
Published: November 2020

The use of the abacus, slide rule, Napier's “bones,” and other devices as demonstration aids in the teaching of junior high school mathematics has one drawback: the kids love it! And so they encourage the teacher (would pester be a better word?) to “teach me how to use it.” With the crammed-full curriculum that must be conquered in nine short months, there are not enough spare hours to devote to the time-consuming task of teaching computation on the abacus or slide rule or to fully satisfy students in some of the other interests they express.
Points and Viewpoints: Excellent Help
Authors: Brune Irvin H.
Published: November 2020

A column of unofficial comment
Computer-Oriented Mathematics: Patterns in Algorithms For Determining Whether Large Numbers Are Prime
Authors: Buchman Aaron L.
Published: November 2020

Primes and modular arithmetic are two topics in mat hematies which have found use at various grade levels of the high school and even the elcmcntary school. Content from these areas has been used in units in the regular mathematies classwork as vvell as in enrichment units for individual pupil work. The depth to which these topics are pursued depends upon the grade level and the interest of the pupils and the teacher. The materia! that will be developed in this unit carries the study of primes somewhat further than most other units in this area, and it uses some of the concepts of modular arithmetic to explore the organization of the structures of the algorithms that are devised. This presentation can be followed by an able, motivated high school student seeking enrichment experiences in mathematics. Although proofs are indicated for the various theorems, pupils at some levels may accept a number of theorems as “self evident.” Teachers may also find the de velopment useful as a source for a limited or more extended classroom unit, or merely to broaden their own backgrounds in these areas.
Historically Speaking—: Identification of Napier's Inequalities
Authors: Hacker Sidney G.
Published: November 2020

The discovery of logarithms by John Napier (1550-1617) is a well known facet in the history of mathematics. His singular accomplishment in defining the logarith mic function of a real variable by providing a numerical description of it, over a wide range of its argument, at small intervals and to several (decimal) places, antedated by many yeara the development of funda mental concepts which the modern stu dent regards as necessary to achieve even the same limited goals. Napier success fully bridged, solely in regard to this function, these lacunae in the mathematical knowledge of bis day. It has long been of interest to identify the concepts which he intuitively invoked. This is not done, it should be clearly said, with any idea of assigning to him some kind of priority for them, but merely in the interests of a elearer appreciation of the ingenuity he displayed and the power of his methods. Two inequalities that he obtained are the key to his numerical resolution of the problem and his consequent table of logarithms. The analytical identification of these inequalities appears to have been overlooked. Before exhibiting this identi-fication we shall speak of the fundamental role that these inequalities played. In the interests of intelligibility we first recollect a few familiar facts regarding Napier's formulation of the problem.
An Intuitive Approach to Square Numbers
Authors: Edmonds George F.
Published: November 2020

My Experience with students who have had one or two years of algebra and who are unable to use the algebra they have “learned” convinces me that there is something wrong with the way we teach the subject. In trying to teach algebra to students in Kenya who have had practically no experience with algebra in any form and at the same time teach them how to use more up-to-date methods in their own teaching in the primary schools, I have been forced to teach algebra from an intuitive point of view. It occurs to me that a more intuitive approach to the teaching of algebra might make algebra more understandable and usable for all students. I shall describe a series of les sons on squares which I used in teaching algebra to teachers who are trying to update their teaching methods. The object of the lessons was to show the teachers how to use the discovery approach and at the same time to give them some useful algebraic ideas.
On Skewed Regular Polygons
Authors: Ranucci Ernest R.
Published: November 2020

The regular polygone considered in the definition above are those encountered in plane geometry. These are configurations whose elements are coplanar. Polygons whose elements are not coplanar are termed skewed. The triangle, by its very nature, must be a plane configuration (fig. 1). (Three noncollinear points always determine a plane.)
Definitions—Consensus Or Confusion?
Authors: Rappaport David
Published: November 2020

In Recent years, there bas been considerable discussion and argument about the “new” or “modern” mathemat ics, A point on which everyone seems to agree, however, is that one of the charac teristics of the new math is greater precision in language, especially with definitions. But, upon closer examination of the textbooks and manuals in current use, it becomes evident that there is indeed a great lack of precision and uniformity of definition.
Complex Roots of a Quadratic Equation Graphically
Authors: Brady W. G.
Published: November 2020

The graphical method for solving a quadratic equation as it is usually presented indicates real roots and coincident roots as intersections of a certain parabola with the x-axis and signals, by failure of intersection, that the roots are either complex or imaginary. In this note, we shall show a geometric method for finding the complex roots of the quadratic.
Two Incorrect Solutions Explored Correctly
Authors: Allen Merle C.
Published: November 2020

In Teaching plane geometry, one must spend a great amount of time cor recting each statement and reason given in each proof submitted on quizzes and tests throughout the year. The author has found this to be a rather tedious endeavor at times. Usually the solutions given are very routine and unimaginative-mimicking the textbook's and the teacher's pre-sentations in nearly every detail. However, two solutions given recently to fairly easy test problems provided interesting rela tionships that the author had never be fore realized.
Use Of Summation Operators For the Derivation of Common Formulae
Authors: Iman Ronald L.
Published: November 2020

A Recent article1 led me to search for a simpler solution to the problem of the derivation of formulae commonly used to find the sum of the first n integers, the sum of the squares of the first n integers, the sum of the cubes, etc. I was fortunate enough to find just such a method in the manuscript of a forthcoming book, Non parametric Statistics, written by Dr. W. J. Conover of the Department of Statistics, Kansas State University. The basic procedure appears here with the permission of Dr. Conover.
Computer-Oriented Mathematics: Topics in Numerical Analysis For High School Mathematics
Authors: Schery Stephen D.
Published: November 2020

At Cassville High School during the 1968-69 school year, the mathe matics department initiated an explora tory computer mathematics program and received a gratifying response from the students. Whether this response was based upon the popular fascination that com-puters seem to hold in our society or upon the intrinsic interest that working with a computer affords, we are not cer tain. However, we are certain that this high degree of motivation has held the attention of students to the extent that it has permitted a treatment of more sophisticated topics from numerical anal ysis than had been initially thought possible. Topics ranging from numerical integration to calculation of five-by-five determinants and Lagrangian interpolation have successfully been taught.
The First International Congress on Mathematics Education
Authors: Hlavaty Julius H.
Published: November 2020

The first International Congress on Mathematics Education took place Au gust 24-30, 1969, in Lyon, France. It had been organized under the chairmanship of Professor Hans Freudenthal of Utrecht, secretary M. Glaymann of Lyon, and an international committee that included two representatives from the United States, L. Gillman and I. Wirszup.
Mathematics Education in the Arab States
Authors: Shouk Mahmoud A.
Published: November 2020

Every nation depends on its educa tional institutions to provide the man power needed for its development. In order to enable their institutions to prepare the young generation to shoulder the responsi bility of developing their nations, the Arabs have had to face a host of problems. Among the problems are: (1) how to direct the educational systems to those economical and service phases that con tribute to growth and development in science and technology; (2) considering the great increase in population, how to develop those potential skills that will be needed by all children who will be adults of the next generations; (3) giving due consideration to particular local circum stances of the Arabs, how to borrow from other advanced nations for adaptation without mere imitation, taking into ac count the existing lack of special resources; (4) how to develop universal education as a preliminary step for the success of nationwide developmental plans; (5) how to adjust vocational and technical educations to industrialization; and (6) how to make science and mathematics furnish a base for citizenship in the technological societies of the future.
Have You Read …?
Authors: Peak Philip
Published: November 2020

All of you have read D'Arcy Thompson's delightful work “On Growth and Form.” His geometric study of living things opened new vistas of interest for many people. Dr. Oxnard explored the same field through an algebraie analysis and the use of the computer. He has provided some answers to questions liker quantitative relationships within sets of biological shapes, the relationship of shape to function, can functions be predicted from shapes, and is the process of evolution a function of shape.
Historically Speaking—: Black African Traditional Mathematics
Authors: Zaslavsky Claudia
Published: November 2020

The long-overdue upsurge in interest in Africa has led me to investigate the development of mathematics on that con-tinent. For many reasons this is a very difficult task. One must look into history, ethnology, anthropology, archeology, lin guistics, and economics, and still be dis satisfied with the results.
A Card Trick
Authors: Trigg Charles W.
Published: November 2020

A Perennially popular card trick, now resurgent, has a simple algebraic explanation. Understandable and stimulating to second-semester algebra students, it is comprehended with difficulty by those laymen so long removed from school days that they intellectually reject the (-)(-) = + concept and for whom x is externally unknown. They can learn to do the trick mechanically, but still wonder why it works.
Experimental Programs: An Assessment of a Mathematics League as Judged by its Participants
Authors: Nichols Eugene D. , Cromack Norman E.
Published: November 2020

There have been numerous descriptive and factual articles about annual mathematics contests or tournaments; however, very little has been written about mathematics leagues per se. Articles have appeared in The Mathematics Teacher describing the Nassau County [Long Island] Interscholastic Mathematics League and the Bergen County [New Jersey] Interscholastic Mathematics League, as well as Intramural Mathematics League [West St. Paul, Minn.]. More recently, an article appeared in the January ′67 issue of this magazine describing in some detail the operations of the Tri-State Mathematics League of Maine, New Hampshire, and Massachusetts. It was noted in the article that the Massachusetts Association of Mathematics Leagues has fifty schools participating. Other educational journals, and newspapers, have included brief resumés of the activities of other leagues and comments by well known educators on the values (and sometimes detriments) of mathematics competition.
International Mathematical Education: Some Aspects of a Modern Pedagogy of Mathematics
Authors: Steiner Hans-Georg
Published: November 2020

Today's life and modern society are increasingly dependent on, and influenced by, science, industry, and technology.
Rational Points on the Number Line
Authors: Mielke Paul T.
Published: November 2020

It Is now fashionable for eighth-grade mathematics texts to include a chapter on intuitive geometry before the chapter in which rational numbers are assigned points on a number line, but the geometry is seldom brought to bear upon the latter problem. The reason is probably that the usual construction is difficult for eighth graders. Recall that it involves dividing a segment into some integral number of equal parts.
GJY insert, Discovery Teaching-from Socrates to Modernity
Authors: Jones Phillip S.
Published: November 2020

Students are rarely aware when the mathematics they are being taught is new.
Teaching General Mathematics: A Semi-laboratory Approach
Authors: Wilkinson Jack
Published: November 2020

There is no shortage of advice and counsel to the mathematics teacher. Teach mathematics classes with proper motivation. Use a problem-solving approach. Provide variety. Emphasize learning by doing. Provide laboratory experiences. Keep the mathematics honest.
Strategies of Proof in Secondary Mathematics
Authors: Engen Henry Van
Published: November 2020

It has been the experience of the author that graduate and undergraduate students are not very knowledgeable about direct and indirect strategies of proof used in mathematics. In methods courses for secondary teachers, the author has observed that even those students who have an undergraduate major in modern mathematics are not at all conscious as to why books usually check the elements of a solution set to an equation before asserting that it is indeed the solution set. Neither are they aware, even at an intuitive level, of the “internal” logic of indirect proof. This lack of awareness leads one to question whether it is possible for them to do a passable job of teaching high school students how to prove theorems.
Two Unusual Representations for the Set of Real Numbers
Authors: Smith Sanderson M.
Published: November 2020

The rational numbers and the real numbers are both fields under the binary operations of addition and multiplication.
Fibonacci-Type Sequences
Authors: Nygaard P. H.
Published: November 2020

The Fibonacci sequence, 1, 1, 2, 3, 5, 8, 13, …, is generated by finding the sum of two consecutive terms to obtain the next term.
Experimental Programs: Individualized Mathematics the Plan Way
Authors: Matthews Josephine J. , Rahmlow Harold F.
Published: November 2020

In The past, numerous attempts have been made to develop mathematics programs that would meet the needs of individual students. Some of these efforts have stressed varying the content that a student studies. Other attempts at individualization have varied the pace at which the student studies. Others have taken the project approach to provide individual activities that will benefit students. Although each of these approaches has merit in itself, none of the previously attempted approaches deals with more than one or two problems of individualizing the student's mathematical studies.
Introducing Matrix Algebra with Computer Programming
Authors: Montague Margariete Ann
Published: November 2020

The structured, sequential nature of the secondary mathematics program presents an obstacle to integrated studies. Within the curriculum one flexible area does remain— the senior year. The onesemester course in matrix algebra and computer programming to be discussed is intended(1)to bridge the gap between the breadth approach in secondary mathematics and the abstract conciseness of undergraduate mathematics;(2)to anticipate the demand for a reasonable fluency in computer languages by the universities.1 Taken separately the components of the course are not unusual. It is, however, unusual at the secondary level to integrate the two topics: to teach the student to program a computer in order to develop and test the structure of matrix algebra.
Why Not Mathematics?
Authors: Lick Dale W.
Published: November 2020

What do you tell students, parents, or friends when they ask: “Why study mathematics? … What opportunities are there for one with a mathematics background? … What about a career in mathematics? … ”? Do you tell them that the study of mathematics is exciting, stimulating, adventurous, and so on? Do you tell them that a background in mathematics provides them with the most versatile preparation for tackling an almost unlimited number of opportunities? Do you tell them that a career in mathematics can be one of the most challenging and satisfying lifetime endeavors?
Three-Dimensional Tic-Tac-Toe
Authors: Mercer Gene B. , Kolb John R.
Published: November 2020

As Teachers of mathematics we are aware of the fact that students should acquire more knowledge than just that of mathematical concepts and principles. Knowledge of terms and the ability to apply formulas to problems are requisites in mathematical learning, but their importance is dwarfed by higher goals, such as learning the methods of mathematics. Knowledge of theorems, rules, formulas, and concepts may well be forgotten when the student leaves the mathematics classroom. Even if he does remember it, this mathematical knowledge may be of limited usefulness to him. By contrast, if the student is given ample opportunity to acquire some ability in attacking and solving problems, he will have acquired an ability and an attitude that will continue to be useful to him. The ability to solve problems, we hope, will remain with him long after knowledge of specifics has dimmed or been forgotten completely.
Divisibility by Integers Ending in 1, 3, 7, or 9
Authors: Kennedy Robert E.
Published: November 2020

Gauss is said to have remarked that mathematics is the queen of the sciences and that number theory is the queen of mathematics. Without a doubt, number theory is of unparalleled interest to many; it is a playground for both the amateur and the professional mathematician.
New Products
Authors: Hansen Viggo P. , Johnson Randall E. , Koetke Walter J.
Published: November 2020

Editors' Note. In view of the great quantity and wide variety of audiovisual materials currently being marketed, it seems appropriate to devote space this month to reporting some of the bibliographies and publications that report and evaluate audiovisual materials. Edward F. Newren, University of Minnesota, who has had considerable experience in the area of cataloging and evaluating audiovisual materials, provides the following report.
A Discovery lesson on Matrix Inverses
Authors: Mann Nathaniel
Published: November 2020

In the Madison Project text Explorations in Mathematics there are several chapters dealing with the system of 2 × 2 matrices. 1 When investigating some of these chapters with fifth or sixth graders, I have always been amazed to see how quickly they can figure out the identity matrix for multiplication; on the other hand, it seems very difficult for these students to figure out the inverse of a 2 × 2 matrix. The technique proposed in the text of just presenting two matrices whose product is the identity seems too arbitrary for my pedagogical taste, so I have preferred to go through another procedure of looking for the inverse to particular matrices and then trying to find relationships between the elements of the two matrices.
More about an Old Number Trick
Authors: Nelson Allyn H.
Published: November 2020

There is an interesting article by William F. Berry, entitled “Algebraic Proof of an Old Number Trick,” in the MATHEMATICS TEACHER for February 1966.
Analytic Geometry Is Not Dead
Authors: Eggan Lawrence C.
Published: November 2020

This article is intended in part as a rebuttal of the recent article “The Demise of Analytic Geometry” by Maurice Nadler.1 Although certain aspects of analytic geometry are not specifically taught because of its integration with calculus, I found Mr. Nadler's article, on the whole, without logical persuasion and more than a little misleading.
Putting Reality into Mathematics
Authors: Ruchlis Hy
Published: November 2020

As is well known, people differ widely in abilities because of a combination of genetic and environmental factors. One can detect marked differences in such abilities at an early age; and for whatever reason, genetic or environmental, wide variations in the ability to work with mathematical abstractions are observed at an early age. In the upper grades these variations become extremely marked.
Some Mathematical Models Used In Plastic Surgery
Authors: Williams Horace E.
Published: November 2020

Teachers of mathematics courses, particularly at the secondary level, are frequently on the lookout for interesting and varied applications of their subject in order to infuse feelings of real-world relevancy into the courses, even very simple applications, especially if they are novel or varied in their scope, are a strong aid to a teacher in maintaining interest among students who do not foresee careers in mathematics, engineering, or physical science. Several interesting and very easily followed applications of plane geometry and plane trigonometry are used by practicing plastic and reconstructive surgeons in the course of their work. I would like to pass on a few “scar revision” techniques that have come to my attention in the hope that the reader will find them an interesting and unique addition to his repertoire of motivational materials used in the mathematics classroom. Since medicine is not at all my field, I won't attempt to discuss any of the cosmetic or therapeutic implications of this work, but I suggest that an interesting mathematics club speaker would be a plastic surgeon who is willing to show appropriately selected clinical photographs of some of his work and to answer technical questions regarding his clinic experiences. The only necessary background for following the procedures presented is an understanding of right-triangle trigonometry.
Distribution Of The Mean
Authors: Leake Lowell
Published: November 2020

The place of statistics and probability in the secondary school curriculum has still not been determined. A recent article by Mosteller described what some New Jersey teachers are doing with probability and statistics (Mosteller 1967), and an even more recent article by Simon and Holmes showed how probability can be taught by using Monte Carlo methods (Simon and Holmes 1969), A few excellent textbooks devoted to these subjects for high school students have appeared (Mosteller, Rourke, and Thomas 1961; Wil-loughby 1968). However, relatively few schools teach so much as a semester of statistics or probability, so that most students pick up some limited information about these subjects in a very random way, usually as brief units included in other courses. This is unfortunate for the students, since statistics and probability offer some engrossing mathematics with numerous important practical applications, and all this can be learned with limited mathematical prerequisites. Most high school students might be interested to know that statistics and probability have become required subjects in college in such diverse fields as medicine, business administration, engineering, psychology, sociology, and education-to say nothing of the sciences and mathematics itself. Almost any college-bound student can profit from the study of these subjects.
Using Televised And Aural Materials For Mathematics Teachers
Authors: Byrkit Donald R.
Published: November 2020

In recent years there have been many changes in the mathematics course of study from the kindergarten through the twelfth grade. As a consequence, it has become important for the teacher of mathematics to keep up with the changes. He must often learn new material and revise his thinking on current subject matter if he is to remain effective as a teacher.
General Finite Geometries
Authors: Heath Steven H.
Published: November 2020

In recent years the subject of miniature or finite geometries has aroused some interest. Most studies to date have been concerned with the study of finite projective and affine (Euclidean) geometries, Many teachers have found this material simple enough to present to students studying geometry and algebra in high school. This paper proposes to describe axiomatically a large class of finite systems called finite Bolyai-Lobachevsky geometries. The approach will be geometric instead of algebraic, although to pursue the subject in depth algebraic techniques are needed. The Bolyai-Lobachevsky geometries can be presented in somewhat the same manner as the studies of finite projective geometries.
Should We Grade on “Improvement” in Mathematics?
Authors: Pack Elbert C.
Published: November 2020

In Order to give students some in-centive to improve their work, and some-times in order to reward those whose increased efforts have resulted in improved test scores, some teachers base student grades on the amount of improvement demonstrated by the student. That is, if one student shows improvement from one test to the next and another shows none, the first is favored when grades are determined.
Space-Filling in Two Dimensions
Authors: Ranucci Ernest R.
Published: November 2020

It is a well-known fact that a plane can be completely covered by repetition of either equilateral triangles, squares, or regular hexagons. These are example of pure tessellation. In such coverings of the plane one and only one type of regular polygon is used, as we see in figures 1, 2, and 3.
Is X · 0 = 0 Always True?
Authors: Hunkler Richard , Heatherly Henry
Published: November 2020

There are many mathematical systems that arise in a natural fashion but that do not have all the usual properties of arithmetic. For example, it is well known that two functions under the operation of composition need not be commutative. Not widely known, however, is that the operation of composition can also be used to illustrate a mathematical system that is only distributive from one side and in which the very familiar equation X · 0 = 0 is not true.
Chinese Mathematics Revision in Accordance with the Teachings of Mao Tse-Tung
Authors: Swetz Frank
Published: November 2020

From 1962 until the present, the Great Cultural Revolution has been transforming the thoughts and lives of the citizens of the People's Republic of China. Confucian wisdom has been replaced by the philosophy of Chairman Mao. Reports reaching the West indicate how diligent study of the “Little Red Book” has assisted its readers in “increasing the speed of rotation of my cotton spinning machine from 9,000 r.p.m. to 13,000 r.p.m.” or “becoming world table tennis champion” or even “curing the children at our school of short-sightedness.”1 As Confucian teachings formerly did, so now the doctrines of Maoism form the foundations of the Chinese educational system. The most vociferant attacks of the Cultural Revolution have been directed at education, with the result, that every subject taught, even mathematics, is considered in the framework of Mao's teachings.
The Forum: The Utility of Behavioral Objectives: CON A Source of Dangers and Difficulties
Authors: Forbes Jack E.
Published: November 2020

All I have learned in the past nine years about teaching, writing and learning has served to strengthen that point of view.
A Tutoring Program for Prospective Secondary Mathematics Teachers
Authors: Bidwell James K.
Published: November 2020

Relevance is one of the current key words in undergraduate methods classes for prospective teachers. We have long realized that “talking about” teaching does not provide a sufficient basis even for student teaching experiences. Generally, contact with public school students is desirable prior to student teaching. However, it is often difficult to make the necessary administrative arrangements to provide additional exposure time for college students. As a partial answer to this problem, an experimental tutoring program was instituted at the University of Michigan. What follows is an account of the origin of this program and its development during the school years 1969-70 and 1970-71.
Maximum Rectangle Inscribed in a Triangle
Authors: Bird M. T.
Published: November 2020

In a paper entitled “Some Inequality Problems,” Professor L. H. Lange considered the problem of finding the largest rectangle that can be inscribed in a given triangle (Mathematics Teacher 56 [November 1963]:490-94). He limited his search to the collection of all such rectangles that have a side lying on a side of the given triangle and proved that, in these circumstances, the largest rectangle will always have an area equal to half the area of the enclosing triangle. He stated, but left unanswered, the question of whether there are cases where a larger rectangle can be placed in a triangle when it is not required that a side of the considered rectangle must lie on a side of the triangle. The answer to this question is no!
Analysis of Calculus Problems
Authors: Flanders Harley
Published: November 2020

The student preparing for science or engineering should enter college with the technical skills needed in calculus. Particularly, he must know how to set up and solve problems and get accurate, answers. Now the typical realistic calculus problem involves only a little of the ideas of calculus per se but a lot of high school algebra and geometry.
Building a Conceptual Computer
Authors: Bell Frederick H.
Published: November 2020

Providing students with an introduction to the rapidly expanding field of computer science can be somewhat of a problem for the majority of teacher who do not have easy access to a computer facility. The seemingly inherent facination that computers hold for many children can be subverted if a teacher's first lesson about computers does not provide proper motivation for students. Lengthly discussions of the history of computers, binary and hexadecimal arithmetic, computer hardware, and computer progmmming can best be presented after the stage has been set by an initial informative and motivational lesson providing opportunity for student participation.
Convergence of Some Iteration Formulas and Solution of Recurrence Formulas
Authors: Pinker Aron
Published: November 2020

An Article by P. H. Nygaard (1964) for the MAThematlcs Teacher concerned solving quadratic equations by means of iterations.
Parametric Construction of the Conics
Authors: Shwarger Michael
Published: November 2020

A method of constructing an ellipse based on a set of parametric equations (x = a cos θ, y = b sin θ) is widely known and appears in many texts on analytic geometry.
Geometric Solutions to Quadratic and Cubic Equations
Authors: Henning Harley B.
Published: November 2020

Among the various solutions to quadratic and cubic equations, some possess an intuitive appeal arising from direct identification with geometric analogs. One analog, discussed here, shows that the quadratic solution known as “completing the square” actually conforms to the pictorial situation of fitting rectangles together in the shape of a square figure. A second analog likewise deals with the cubic equation and demonstrates the correspondence between graphical and analytical versions of a solution that might be appropriately called “completing the cube.”
Some Novel Möbius Strips
Authors: Matthews Charles Joseph
Published: November 2020

The möbhis strip was named for its inventor, August Ferdinand Möbius (1790-1868). It can easily be made by joining a strip of paper in the form of a ring after giving one of its ends a twist of 180° (fig. 1).
Five Questions and Answers about Advanced General Mathematics
Authors: Watman Michael X.
Published: November 2020

A surprisingly large response to the author's article “Some Trends in Advanced General Mathematics” (1970) has indicated the need for a more detailed examination of an advanced course for low achievers in mathematics, this time emphasizing general goals, techniques, and further sources of materials. Since hard facts seem particularly lacking in this area, the expression of opinions is not out of place and can, indeed, serve as a framework for further discussion, research, and experimentation. My thanks go to Mrs. Shirley Severance of the Brattleboro Union High School in Brattleboro, Vermont, who suggested some of the questions that I shall pose and who provided the impetus for committing their answers to paper.