bluuebunny/crossref_metadata_2025 huggingface dataset:
Historically Speaking,—: A history of computers, I
Authors: Larrivee Jules A.
Published: January 2021

Computing devices became necessary when men realized the need to count and measure. These two methods of computing form the basis for classifying computers as digital or analog. The digital computer operates on numbers expressed in some form of digital notation, and counts in discrete steps. The analog computer represents numbers in terms of some physical quantity such as length, voltage, pressure, etc., and operates continuously to obtain measurements which, when adjusted for proper scale factors, give the required result.
Points and Viewpoints: The school mathematics study group
Authors: Begle E. G.
Published: January 2021

A column of unofficial comment
Testing Time: Grouping, standardized tests, the Education Act: a miscellany of information
Authors: Kalin Robert
Published: January 2021

Many things went “up” along with the first Russian Sputnik in the fall of 1957. Dozens of newspaper articles soon reflected a sudden rise of interest in education among the American public, with mathematics and science courses receiving a major share of attention. In the first full academic year since Sputnik I, many teachers are already reporting enrollment increases in nonrequired mathematics courses.
Intuition and fluid mechanics
Authors: Scheid Francis
Published: January 2021

Some important ideas concerning the role of intuition and common sense in mathematical developments.
Some related theorems on triangles and circles
Authors: Wiseman John D
Published: January 2021

The words are different but the theorems are about the same configurations of lines and points.
Points and Viewpoints: As another sees us
Authors: Riske Marcus
Published: January 2021

A column of unofficial comment
Tips for Beginners: Introducing the slide rule early
Authors: Fandreyer Earnest E.
Published: January 2021

The slide rule is usually introduced after the topic of logarithms. It seems reasonable that the student must know logarithms before he can understand a logarithmic scale such as the slide rule. This need not be the case, however. There are, in the writer's opinion, good arguments for an early introduction to the slide rule, and also good rebuttals to those who argue the slide rule can be taught properly only after a treatment of logarithms.
If not solid geometry, then what?
Authors: Schuster Seymour
Published: January 2021

Is the answer a study of vectors and a vector approach to geometry?
Mathematics from the modern viewpoint
Authors: Botts Truman , Pikaart Leonard
Published: January 2021

The axiomatic approach used in plane geometry can also he applied to numbers, and, in fact, to all of mathematics.
Mathematical bingo
Authors: Harris Patricia Ann
Published: January 2021

An interesting and challenging game for a mathematics club is an adaptation of the familiar bingo. Prepare a series of thirty cards each 5 × 6 inches from show card or some other reasonably stiff material.
Improving the program in mathematics in Oklahoma schools
Authors: Zant James H.
Published: January 2021

As elsewhere, improvement in mathematics teaching is taking place in Oklahoma.
The logic of nonsense
Authors: Buchalter Barbara Elpern
Published: January 2021

“You may call it nonsense if you like,” she said, “but I've heard nonsense, compared with which that would be sensible as a dictionary!”
The exclusive disjunction
Authors: Wiseman John D.
Published: January 2021

The study of algebra and geometry as formal mathematical systems introduces various kinds of statements. One of the most basic statements of all is the if-then sentence. Among other ways, statements of mathematics are related to each other by means of this pattern. If-then sentences, in turn, are related to each other to build the structure of mathematics and to formulate proofs.
Great challenges of mathematics education
Authors: Rosenbeug Herman
Published: January 2021

A discussion of key issues and problems in contemporary mathematics education.
Points And Viewpoints: Is your college giving proper training for teachers of secondary school mathematics?
Authors: Gager William A.
Published: January 2021

In most colleges and universitie, the department of mat hematics in the college of arts and sciences teaches the subject matter pertinent for the training of high school mathematics teachers. During the past few years, it has been most encouraging to observe the modernization of subject matter made by alert chairmen of many of our university mathematics departments.
II. Basic properties
Authors: Feldmann Richard W.
Published: January 2021

After Cayley published his expository articles on the theory of matrices, many mathematicians published papers expanding Cayley's ideas and developing new concepts, notations, and terminology. Some of these were fruitful and are still used, while others never turned up in print again.
The advantages of decimal notation
Authors: Shuster Carln
Published: January 2021

In a College recently visited, the professor or, in a class in the teaching of ari thmetic, listed on the board eleven reasons for claiming that decimal fractions were superior to common fractions.
Facilities for Secondary School Mathematics
Authors: Frame J. S.
Published: January 2021

Do you covet a pleasant and challenging environment for your pupils?
Annual Financial Report
Authors: Gates James D.
Published: January 2021

The following report gives a brief summary of the receipts and expenditures of the National Council of Teachers of Mathematics for the 1963-64 fiscal year, which ended May 31, 1964. Actual receipts were approximately $22,000 more than the expected receipts, while the expenditures were approximately $29,000 less than was budgeted.
Letter to the Editor
Authors: Spalic John
Published: January 2021

It is simply not possible to express the results of contemporary mathematical research in nonmathematical terms.
Historically Speaking— Computers: Their Past, Present, and Future
Authors: Spencer Donald D.
Published: January 2021

Everyone knows about computers. They have been discussed in many newspapers, magazines, books, and television programs. They have made headlines at election time and during missile launchings. These headlines, as exciting as they are, are only the highlights of a rather new development that has begun to impinge on the daily lives of practically everyone.
International Mathematical Education: Mathematics Education in Kenya
Authors: Hollingshead Irving
Published: January 2021

Educators in a developing country who are interested in improving their methods and curricula must, of necessity, build upon the work that has been done in other parts of the world. At the same time it is important that they revise and adapt foreign material to their own needs and culture. The present article describes progress in mathematics education in Kenya, which is one of the more developed countries of East Africa. Many of the problems of Kenya are problems of all East Africa, but mathematics teachers need hardly be warned about the danger of making unwarranted generalizations. Contrary to the impression held by some people in the United States, Africa contains a diversity of people with very different cultures and problems about which sweeping generalization is impossible. The reader is also warned that the following remarks are the impressions of an American in the country on a two-year contract, not those of a native Kenyan. This may have resulted in some biases—but these, I hope, are few.
What Shall We Teach in High School Geometry?
Authors: Adler Irving
Published: January 2021

The teaching of geometry, dominated by Euclid for two thousand years, has been undergoing searching criticism and significant change during the last sixty years. To understand why, we must consider the question, “What's wrong with Euclid?”
Binomial Coefficient Formulas by General Reasoning
Authors: Elkin Jack M.
Published: January 2021

The binomial coefficients are an almost endless source of formulas for the summation of series. A reference to the “Problems for Solution” pages of the American Mathematical Monthly or to an advanced collection of mathematical formulas will convince anyone who has not yet discovered this for himself. Some of these series summations can be derived with relative ease with the help of the binomial theorem or Pascal's triangle; many require a high degree of virtuosity in algebraic manipulation and, often, advanced methods of analysis. A number of them can be obtained simply by reasoning logically about the meaning of certain combinatorial expressions, with recourse to only a minimum of algebra or to none at all. These, naturally, have a special appeal of their own, and it is the purpose of this article to illustrate several such derivations.
The Wheel of Aristotle and French Consideration of Galileo's Arguments
Authors: Costabel Pierre
Published: January 2021

The paradox of Aristotle's wheel has been the subject of so many scholarly debates that it seems opportune to make these debates the substance of a new study. Galileo saved the subject from oblivion by discusing it on the first day of his Discorsi [1].* Philosophers consider the problem in term of infinitesimal calculus and are grateful to Galileo for having illuslrated the dangers of depending upon intuitive notions. Physical scientists, on the other hand, consider the problem elementary and rarely condescend to examine it seriously. We shall consider the texts and history of this argument, since these areas have been neglected by scientists and philosophers.
A Study of N-Dimensional Equations, Modulo Two
Authors: Green Thomas M. , Hamberg Charles L.
Published: January 2021

We feel that sometimes the study of a simple system can provide a student with many varied and meaningful exercises, developing a backlog of experiences to help support him in the more abstract aspects of modern mathematics.
Historically Speaking— A Geometry Capsule Concerning the Five Platonic Solids
Authors: Eves Howard
Published: January 2021

A polyhedron is said to be “regular” if its faces are congruent regular polygons and its polyhedral angles are all congruent. While there are regular polygons of all orders, it is surprising that there are only five different regular polyhedra. These regular polyhedra have been named according to the number of faces each possesses. Thus there is the tetrahedron with four triangular faces, the hexahedron (cube) with six square faces, the octahedron with eight triangular faces, the dodecahedron with twelve pentagonal faces, and the icosahedron with twenty triangular faces. See the accompanying figure.
Have You Read …?
Authors: Peak Philip
Published: January 2021

The Newsom Report was produced in England and covered the place of mathematics and science in the schools. This article is a reaction to the mathematics chapter of that report.
International Mathematical Education: The Chambery Plan— Stages and Perspectives in the Reform of Mathematics Instruction
Authors: Fehr Howard F.
Published: January 2021

For a number of years, the Association of Professors of Public Instruction (APMEP) has studied the problems posed by the evolution of the sciences, especially that aspect of education which concerns instruction in mathematics. Through the auspices of its Commission on Research and Reform, which has taken the initiative to bring things together, contacts have been made with professors of other disciplines, with researchers, with psychologists, and with engineers. The action of the APMEP is not unknown to the membership of the Ministerial Commission for Mathematics Instruction, headed by Professor Lichnerowicz, whose first report, published in March 1967, marked an important change in subsequent procedures.
Systems in Nonmathematical Disciplines
Authors: White Louise G. , Baker Virginia H.
Published: January 2021

HIGH school students “know,” because they have been told so often, that mathematics will help them to think logically; but they do not see how to apply this training to other subjects. Concerned about this inability to correlate mathematics with other disciplines, teachers representiog the English, history, and mathematics departments at Laurel School instituted a course in 1965 designed to emphasize the universality of the principles of logical reasoning learned in mathematics. This program is predicated on the premise that it is as important to be aware of the structure of the system (definitions, assumptions, functions, and relations) requisite to the presentation of an unambiguous argument in nonscience areas as it is to be aware of the structure of the mathematical system in which a theorem is to be proved or a computation performed; further, most students need explicit instruction in the application of this concept to areas other than mathematics.
Signed Numbers: A “Product” of Misconceptions
Authors: Brown Stephen I.
Published: January 2021

One of the first encounters with a mathematical idea that jars a student's intuition takes place when he is asked to accept that the product of two negative numbers is positive. There are, of course, many different explanations at various levels of abstraction that have been used to persuade the neophyte of the reasonableness of the assertion. One can offer an argument based upon a model, and this is done in such programs as UICSM and Madison Project.1 One can offer a plausibility argument based upon the behavior of positive numbers and faith in some sort of continuity principle, as suggested by Dubisch.2 One can (quite ironically, given the context that “modern” mathematics is an attempt to replace rules by explanations and justifications) assert, as Courant and Robbins do, that, though appeal may be made to various channels for psychological reasons, ultimately the conclusion is a matter of definition.3 The prime psychological motivation would be the desire to preserve principles already existing in the set of positive reals, as is done by SMSG.4
Experimental Programs: An Action Program in Mathematics for High School Dropouts
Authors: Kneitz Margaret N. , Creswell John L.
Published: January 2021

ONE of the most pressing problems today is what to do with high school dropouts, especially those who come from disadvantaged homes.
A New Way to Teach Probability Statistics
Authors: Simon Julian L. , Holmes Allen
Published: January 2021

The purpose of studying probability and statistics is to learn how to solve the problems these tools can help solve. Such problems are found in the practice of science, of course. But many of the problems of daily living can also be better understood with the aid of probability and statistics. It is particularly for the latter reason that one may consider teaching these ideas to high school or even pre-high-school students.
The Readability of Junior High School Mathematics Textbooks
Authors: Smith Frank
Published: January 2021

Choosing a textbook is a very difficult and trying task. This is true whether it is done by an individual or a committee, or for a large school system with many students or a small system with only a few. The same shoe fits only a small number; the same menu satisfies the taste of only a small minority.
Historically Speaking— The Remarkable Bernoulli Family
Authors: Eves Howard , Lick Dale W.
Published: January 2021

This past year a young man completed his Doctor of Philosophy degree in mathematics at the ripe old age of 18. This achievement made the news stories everywhere. This was indeed an outstanding educational feat and starts one wondering about precedents of such accomplishments. With a little historical investigation, one finds a case where this advanced degree was awarded to an individual at age 16. Even more astounding is the occurrence at age 13! These two individuals coincidently have the same surname, Bernoulli. This is not, however, a mere coincidence. It is only a minor episode in the history of the remarkable Bernoulli family. What follows sketches some of the exceptional and exciting history of this famous family.
A Simple “7” Divisibility Rule
Authors: Matthews E. Rebecca
Published: January 2021

There's magic in the number 1,001, which is as old as the tales of the Arabian Nights. A little-used algorithm makes use of that magic to reveal divisibility by those mystic primes, 7, 11, and 13, almost as easily as rubbing a magic lantern. Unlike Aladdin, we shall probably not be content to use the scheme unless we know why it works. The key to understanding depends basically on the fact that 7 × 11 × 13 = 1,001 and on a little elementary number theory.
The Area of a Pythagorean Triangle and the Number Six
Authors: Prielipp Robert W.
Published: January 2021

A TRIANGLE, the sides of which have measures, x, y, and z, is a Pythagorean triangle if and only if x, y, and z are positive integers and x2+y2 = z2. Several interesting facts concerning Pythagorean triangles have been established, and some of these are relatively unknown.
International Mathematical Education: Program of Mathematics for the First Year of Study in the Junior High School (Seventh School Year) in Belgium
Authors: Fehr Howard F.
Published: January 2021

The program below is the official program for all seventh grade students in Belgium, beginning September, 1968.
Homomorphism: A Unifying Concept
Authors: Krause Eugene F.
Published: January 2021

A CURRENT trend in mathematics education is a quest for unification. One manifestation of this trend is the growing number of unified courses; e.g., unified algebra and trigonometry, unified plane and solid geometry, unified calculus and analytic geometry, …. Even the traditionally distinct areas of algebra and geometry are being brought together in experimental courses in vector geometry, vector algebra, and transformation geometry.
A Simple Algorithm for Solving the n-Dimensional linear Diophantine Equation
Authors: Ibrahim Aziz , Gucker Edward J.
Published: January 2021

In an earlier paper, we showed that the Euclidean algorithm could be written as a matrix transformation to obtain the solution of a two-dimensional Diophantine equation (Ibrahim and Gucker, 1972). In the present paper, we generalize this approach to n-dimensional and also simplify the algorithm that produces the transformation matrix. We also show that the final matrix contains the most general solution of the given Diophantine equation.
On the Proof-Making Task
Authors: Kane Robert B.
Published: January 2021

We begin with a question that concerns mathematics teachers today even as it has for many years: How does a secondary school mathematics student become a proof-maker? “Proof-making” refers to the activities undertaken to create a sequence of statements, each one following inescapably from the preceding ones, with the final statement asserting that which was to be proved. The terms deductive thinking or necessary inference signify the proof-making task.
Partitioning of the Plane by Lines
Authors: Hoffman Nathan
Published: January 2021

The purpose of this article is to determine the maximum number of regions into which n lines partition the plane—a problem that could be appropriate for some secondary school students. The solution, as presented here, draws on some basic geometric notions, some elementary counting techniques, and Euder’s formula for plane figures—that is, V — E + R = 1, where V is the number of vertices, E is the number of edges, and R is the number of regions.
Applications of Groups and Isomorphic Groups to Topics in the Standard Curriculum, Grades 9-11: Part II
Authors: Usiskin Zalman
Published: January 2021

It is possible to form a set in an infinite additive group by beginning with any nonzero real a and adding it to itself over and over again, then including zero and the opposites of all numbers in the set. We call such a set the set of integral multiples of a. If a = 3, then here is such a set.
Of Shoes—and Ships—and Sealing Wax—of Barber Poles and Things
Authors: Ranucci Ernest R.
Published: January 2021

Give Lewis Carroll credit for the first part of the title. The barber pole is my own idea. It’s not the barber pole that intrigues me as much as the spiral wrapping on the pole. It’s in the form of a helix. The easiest way to construct a helix is to begin with a rectangular sheet of paper— lined paper is quite appropriate (fig. la).
A Finite Field as a Facilitator in Algebra and Geometry Classes
Authors: Swadener Marc
Published: January 2021

In an article entitled “Taxicab Geometry” in the December 1973 issue of the Mathematics Teacher (p. 695), Eugene Krause mentioned that finite fields can be used to assist in the understanding of the field properties, just as the base eight number system can be used to assist in the understanding of the base ten system of numeration and non-Euclidean geometries can be used to assist in the understanding of Euclidean geometries (1973, p. 695). The vastness of the topic of finite fields and their applicability to school mathematics has surprised me. This article will bring forth some definite ideas on how teachers of mathematics can use a finite field to exemplify many concepts in algebra and geometry as well as some general concepts in mathematics.
A Different Look at Word Problems
Authors: Rogers Margaret Anne
Published: January 2021

Suppose one were to judge the activities of contemporary American men and women solely on the basis of the word problems appearing in current second-year algebra texts. What impression would one get? To answer this question, the verbal problems in a number of recently published books were studied and catalogued. (See the list of “Textbooks Used” at the end of the article. In the remainder of this article each text will be referred to by the code letter used in the list.)
Balancing the Equation on CBTE
Authors: House Peggy
Published: January 2021

What are the issues and consequences regarding competency-based teacher education?
Your Council at Work
Authors: Gibb E. Glenadine
Published: January 2021

You may recall that one of the societal forces upon us at the NCTM 52d Annual Meeting in Atlantic City was the wave of attacks that had been made throughout that year on the mathematics programs in our schools. Among the questions we asked were, “Have we been responsible for a ‘lost generation’ in mathematics? Where have we failed? What should be our next steps?” There was cause for reflection, assessment, and renewed commitment.
A Note on Perfect Numbers
Authors: Francis Richard L.
Published: January 2021

This short article illustrates one of the surprising properties of perfect numbers.
Applications of Transformations to Topics in Elementary Geometry: Part II
Authors: Jackson Stanley B.
Published: January 2021

Some simple, intuitively appealing transformations form the basis for applications that may be presented at different levels of sophistication.
Self-Pacing with Constraints— A Modular Approach to the Teaching of Precalculus Mathematics
Authors: Chatterley Louis J.
Published: January 2021

A university mathematics department plans and tests a precalculus course.
Trisection Triangle Problems
Authors: Bicknell Marjorie
Published: January 2021

Students of geometry should find some interes ting and even surprising results falling out of dividing sides of a triangle into congment segments.
On the Steepness of Cones
Authors: Ranucci Ernest R.
Published: January 2021

Some observations about cones as functions of sectors of a circle.
The Odds Meet the Great Martingale
Authors: Andrew Archer J.
Published: January 2021

A system to beat the odds in roulette? Yes. and here is a complete analysis of “the system”; the casinos' defense against “the system”: and your chances when restricted by the house limit.
1776
Authors: Cunningham Helen
Published: January 2021

Celebrate 1776 in your mathematics class!
NACOME: Implications for Teacher Education
Authors: Trimble Harold
Published: January 2021

My First reaction to NACOME was one of unqualified delight. I still feel grateful to the committee. It was no small task to ferret out data to describe the ‘“healthy but not untroubled teen-ager” (p. 149), the phrase that the committee used to characterize school mathematics in 1975. It took remarkable restraint to resist temptations to overgeneralize from biased samples; to become dogmatic about issues of special personal significance to committee members: to attack critics; and so on. NACOME has given our profession a balanced, thoughtful “benchmark 1975.” Clearly, the most important recommendation that 1 can make is to read the report. I assume that you will do this before you read further.
Consistent Classification of Geometric Figures
Authors: Smith Joe K.
Published: January 2021

The author suggests a consistent scheme for classifying geometric objects. Teachers of geometry can make good use of the ideas presented here when discussing definitions of geometric terms.
An Approach to Absolute value Problems
Authors: Ahuja Mangho
Published: January 2021

Are you tired of the Case 1, 2, 3 approach to handling equations with absolute value? Consider using this method of working with distance between points.
A Wheel Going Forward and Backward at the Same Time!
Authors: Cavanaugh William E.
Published: January 2021

Have you ever watched a movie in which a stagecoach was speeding toward its destination, but the wheels seemed to stand still or even rotate backward? The article will give you some insight into this phenomenon.
Sum Squares On A Geoboard
Authors: Comella James J. , Watson James D.
Published: January 2021

Introduction, difference tables, discovery of formulas— all motivated by this intriguing use of the geoboard.
The Fifth U.S.A. Mathematical Olympiad
Authors: Greitzer Samuel L.
Published: January 2021

The Fifth U.S.A. Mathematical Olympiad was held on 4 May 1976. The examination consisted of five questions requiring mathematical ingenuity as well as a thorough knowledge of subject matter.
A Way to Individualize Instruction in First-Year Algebra
Authors: Straley H. W.
Published: January 2021

Here is a tried and proved variation on individualized instruction.
Word Chains: A Simulation of Proof
Authors: Bowling J. Michael
Published: January 2021

Here's a word game to be played before you introduce your students to proofs, and it just might help them avoid the usual first proof frustrations.
Hypercubes, Hyperwindows, and Hyperstars
Authors: Priest Dean B.
Published: January 2021

An excursion into several dimensions by students and their teacher.
SSA and The Law of Cosines
Authors: Cohen Martin P. , Goodman Terry A. , Bernard John
Published: January 2021

A clever combination of algebra and trigonometry to illuminate a fact of geometry!
The Question of Minimum Competency As Viewed From the Schools
Authors: Taylor Ross
Published: January 2021

Requiring minimum competence can pay off; it can also backfire.
Geometry Word Search
Authors: Conway Margaret M.
Published: January 2021

The answers to the following questions are hidden in the puzzle. They can be read in eight different directions: vertically, horizontally, on both diagonals, forward, and backward. Answers can be found on page 304.
Calculators in Secondary School Mathematics
Authors: Bell Max S.
Published: January 2021

About fifteen years ago R. W. Hamming directed our attention to the notion that “an order of magnitude” change (roughly a factor of ten) in a technology commonly produces fundamentally new effects. As an illustration he offered the tenfold increase in travel speed afforded by automobiles over horse and wagon and the profound consequences of that in our society.
Should Your School Get A Microcomputer?
Authors: Folk Mike
Published: January 2021

It has been recognized for some time that some level of “computer literacy” would be a valuable part of most high school programs. Unfortunately, factors such as high costs and lack of knowledge about computing hardware have made it difficult for most schools to provide students with access to computers. Now these difficulties are being overcome, however, and we should soon see a revolution in the use of computers in schools just as dramatic as that caused by the recent widespread availability of pocket calculators. The reason: microcomputers.
Promoting Discovery in Algebra
Authors: Borenson Henry
Published: January 2021

How many examples do you use before deciding if an operation is associative?
The Golf Tee Problem
Authors: Duncan Richard B.
Published: January 2021

A computer solution to a well-known and vexing board puzzle.
Teaching Mathematics via APL (A Programming Language)
Authors: Peelle Howard
Published: January 2021

Here is a rich source of ideas on how a computer can enhance the teaching of mathematics.
Statistics for The Non-College-Bound Student
Authors: Gallagher Joan
Published: January 2021

An alternative for the junior or senior year of high school might well include a course in statistics.
A Logarithm Problem and How it Grew
Authors: Lowerre George F.
Published: January 2021

Student errors lead to exploring the conditions under which logax = x.
Dunnwood Squares
Authors: Dunn R. G.
Published: January 2021

A way to disguise drill on basic skills and provide an incentive for doing homework.
Metrication AIDS Education—and Vice Versa
Authors: Bemer R. W.
Published: January 2021

All you ever wanted to know about why we're going metric—hut were afraid to ask.
Regardless of Sex
Authors: Burton Grace M.
Published: January 2021

An overview of where we have been and where we should be going in our efforts to assist young women in their study of mathematics.
Smart Soap Bubbles Can Do Calculus
Authors: Hoffman Dale T.
Published: January 2021

The problems of finding minimum distances, times, and costs are solved with soap bubble models.
GCM + TC + PR—A Formula for Survival in A Nonacademic Mathematics Class
Authors: Turner Linda M.
Published: January 2021

Students who are not self-motivated require a few extras from the teacher.
The Eighth U.S.A. Mathematical Olympiad
Authors: Greitzer Samuel L.
Published: January 2021

The Eighth U.S.A. Mathematical Olympiad (USAMO) was held on 1 May 1979. Participation was by invitation only and was based almost entirely on the students' performance on the Annual High School Mathematics Examination (HSME) given on 6 March. All participants (101) who had scored 118 points or better were invited to take part in the Olympiad. Ninety-eight participated.
Remedial Work in High School Mathematics
Authors: Whitcraft L. H.
Published: January 2021

Teachers of high school mathematics are confronted with the fact that there are more failures in the mathematics of the secondary school than in any other subject in the secondary school curriculum. These failures may be traced to some one of the following factors; (1) the materials of mathematics, consisting of the textbook, practice exercises, and special devices; (2) the teacher's method of instruction and manner of presenting the subject matter to the pupils; or (3) the methods and processes of the pupils themselves. Now that the teachers of mathematics realize that there is a great amount of criticism due the department of mathematics what are they going to do about it? The answer should be the same as the elementary teachers have given to the criticisms which have come to them- give remedial work.
If the Squares Don't Get You—the Circles Will
Authors: Swetz Frank
Published: January 2021

Magic from China—with circles instead of squares for your Friday recreation period.
The Pythagorean Theorem on an Isometric Geoboard
Authors: Hirstein James J. , Rachlin Sidney L.
Published: January 2021

Area measurement is used to establish the Pythagorean theorem.
Johnny Got a Zero Today
Authors: Allinger Glenn D.
Published: January 2021

Nothing is really something!
Results of The Second NAEP Mathematics Assessment: Secondary School
Authors: Carpenter Thomas P. , Corbitt Mary Kay , Kepner Henry S. , Lindquist Mary Montgomery , Reys Robert
Published: January 2021

The National Assessment of Educational Progress (NAEP) completed its second mathematics assessment during the 197778 school year. The two major goals of the assessment are to make available comprehensive data on specific educational attainments of young Americans and to measure change in their educational attainments.
The Method of Centroids in Plane Geometry
Authors: Pinker Aron
Published: January 2021

A weighty alternative for proving some familiar theorems. and some not so familiar.
Mathematics in the Newspaper
Authors: Czepiel James , Esty Edward
Published: January 2021

More than basic skills are needed to read a newspaper.
1979 National Middle School Mathematics Olympiads in the People’s Republic of China
Authors: Becker Jerry P.
Published: January 2021

Mathematics competitions were orga- nized in four major cities of China in 1956-Beijing, Shanghai, Tianjin, and Wuhan. Thereafter the competitions, whose objective was to identify mathe- matically talented middle school students, grew and eventually came to involve middle school and university classroom teachers on a fairly large scale. (The Chinese middle school [5 years] is analogous to the American secondary school [6 years]. Students begin middle school at (approximately] age II.) However, the competitions were terminated in I964 just prior to the Great Proletarian Cultural Revolution in China. Subsequently, with changes taking place in educational policy in the mid-1970s, the competitions were reinstituted and the Chinese organized and carried out competitions at the municipal, provincial, and national levels in 1978. (For fuller information on the 1978 olympiads, see the Becker article in the Bibliography.)
Competition, Praise, and Testing Problems in General Mathematics
Authors: Hutcheson James W.
Published: January 2021

We have recently concluded a series of activities that replaced frowns with smiles on the faces of both students and teachers. The activities centered on mathematical competition, teachers praising students, and learning about test taking. Several features of our school system and state have a bearing on the origin and success of our program.
Clubs: Happy Birthday, Lewis
Authors: Goldberg Dorothy , Butcher John
Published: January 2021

On a “frabjous” day in 1832, the Reverend Charles Lutwidge Dodgson was born. It is fitting, 150 years later, for mathematics teachers to celebrate the birthday of the Oxford University mathematician and teacher who is best known as Lewis Carroll, the author of Alice's Adventures in Wonderland.
Clubs: Mu Alpha Theta Based On A Points System
Authors: Gutzler Martha
Published: January 2021

St. Joseph’s Academy has a chapter of Mu Alpha Theta, a national honorary mathematics club (see “What Is Mu Alpha Theta?” October 1980). It strives to give students an opportunity to learn more mathematics and to have fun at the same time. Our club is open to those who have ability and inte rest in mathematics. The members must be either juniors or seniors who have been enrolled in an honors mathematics course. These restrictions keep the membership to about thirty-five students.
A Strategy For Avoiding A Hidden Trap In Trigonometry
Authors: Herrington James P.
Published: January 2021

The solution of an oblique triangle, given two sides and the included angle, can contain a hidden trap often overlooked by both students and textbooks. If one of the unknown angles is obtuse, it is possible to obtain an erroneous solution for the triangle.
A Shortage Of Mathematics Teachers In New York City
Authors: Posamentier Alfred S. , Stepelman Jay
Published: January 2021

There is a growing professional literature that has documented the shortage of mathematics teachers in different portions of the United States. To examine the extent of this problem in the New York City area, we gathered data from a sample of 51 of the 108 high schools in New York City and 18 of the 45 colleges in New York City and vicinity. Some of these colleges did not participate because they did not have programs in mathematics or mathematics education.
Changing Postulates Can Provide Variety and Meaningful Learning
Authors: Mahaffy Donald
Published: January 2021

The objective of this activity is to teach students to reduce a fraction to lowest terms by looking for the greatest common factor (GCF) of the numerator and denominator. This activity also encourages teamwork and cooperation.
Proof by Analogy: The Case of the Pythagorean Theorem
Authors: Levine Deborah R.
Published: January 2021

In the article “Problem Posing and Problem Solving: An Illustration of Their Interdependence” (Walter and Brown 1977), the following generalization of the Pythagorean theorem was proved.
The Corner Reflector
Authors: Harris Whitney S.
Published: January 2021

In July of 1969. the Apollo astronauts placed an array of corner reflectors on the moon. The array was used to reflect an earth-originating pulsed laser beam back to earth. The total travel time of a pulse (about 2.5 seconds) yielded a more accurate determination (to within 6 inches) of the distance from the earth to the moon (approximate mean value, 239000 miles).
A Useful Old Theorem
Authors: Underhill W. Vance
Published: January 2021

Claudius Ptolemy (second century A.D.) was one of the pioneers of trigonometry as well as a major contributor to astronomy, geography, and even astrology. His principal work. Mathematical Syntaxis, known generally as Almagest*, contained an extensive table of lengths of chords in a circle in terms of the angles subtended by the chords.
A Visual Approach to Absolute Value
Authors: Arcidiacono Michael J.
Published: January 2021

The study of absolute value is often one of the earliest encounters that students have with a very important and difficult mathematical process: analyzing a problem by breaking it down into cases. Given the current emphasis on problem solving, it may be that giving an experience involving this process is, in fact, a major reason for studying the concept.
Microcomputer-assisted Discoveries: Random Numbers
Authors: Kimberling Clark
Published: January 2021

Other articles in the “Microcomputer-assisted Discoveries” series present programs about prime numbers, the Euclidean algorithm, and number bases. These programs offer students a vast number of patterns from which to make discoveries. This article offers quite different grounds for exploration by students, since the patterns provided by random numbers are probabilistic and statistical, rather than number-theoretic and deterministic.
1089: An Example of Generating Problems
Authors: Blake Rick N.
Published: January 2021

Emphasis on problem solving in mathematics has gained considerable attention in the last few years. A joint position paper on basic mathematical skills by the National Council of Teachers of Mathematics and the National Council of Supervisors of Mathematics, in the February 1978 Mathematics Teacher, stated that “learning to solve problems is the principal reason for studying mathematics.”
Babylonian Quadratics
Authors: Mcmillan Robert D.
Published: January 2021

Although most mathematics teachers believe that teaching mathematical history is an important objective, they find it hard to incorporate into teaching algebra. One idea that seems to fit in the classroom and that students find interesting, however, is the method used by the ancient Babylonians to solve quadratic equations.
An Activity for Predicting Performances in the 1984 Summer Olympics
Authors: Henningsen Jacqueline
Published: January 2021

As the 1984 summer Olympic Games in Los Angeles approach, predictions in the media about possible record-breaking performances will abound. In addition, the current increase in interest in women's athletics will focus attention on the question of whether the performance of women is approaching that of men in specific events.
Two Meanings of Algorithmic Mathematics
Authors: Maurer Stephen B.
Published: January 2021

In recent discussions with mathematicians and mathematics educators, I have become increasingly aware that the phrase “algorithmic mathematics” has two radically different meanings. Most people (I was one) are not even aware that other people might be using a meaning different than their own.
Personal Computers, p-Values, and Hypothesis Testing
Authors: Morgan Larry A. , Morgan Frank W.
Published: January 2021

The increasing availability of microcomputers in schools throughout the nation enables today's mathematics teacher to introduce students to the ideas of probability and statistics in creative and challenging ways.
Cryptanalysis in World War II —and Mathematics Education
Authors: Hilton Peter
Published: January 2021

One tragic aspect of the First World War (1914-18), so far as Britain was concerned, was that no effective use was made of the skills and talents of young British scientists, mathematicians, engineers, and experts in other relevant areas. Such people simply entered the armed services, usually the army, and fought in the infantry and other combat battalions.
The Relativity of Mathematics
Authors: Kleiner Israel , Avital Shmuel
Published: January 2021

Mathematics is entirely free in its development and its concepts are restricted only by the necessity of being noncontradictory…. The essence of mathematics lies in its freedom.
Microcomputer-assisted discoveries: Bertrand's Box Paradox
Authors: Flusser Peter
Published: January 2021

Whenever I plan to introduce a new topic to one of my classes I try to recall how I felt when I encountered the ideas under consideration for the first time.
Making a Divergent Series Converge
Authors: Cuoco Albert A.
Published: January 2021

In a letter to the Mathematics Teacher (December 1982, p. 730), James Metz points out the necessity of checking the convergence of a series before looking for its limit.
A Property of Inversion in Polar Coordinates
Authors: Boyd James N.
Published: January 2021

One of the nicest rewards for teaching several different courses in secondary school mathematics is to learn something in one course that can be used in another. This link recently occurred between my ninth-grade geometry and my advanced placement calculus class. I had just come upon the very elegant monograph Lobachevskian Geometry by the Russian mathematician A. S. Smogorzhevsky (1976) and was reading through the theorems on inversion with respect to a circle in the Euclidean plane. Of course, this work was not suitable for my whole ninth-grade geometry class, but I was in search of material for projects for any of my students who might be interested.
Twelfth-Grade Mathematics in U.S. High Schools: A Report from the Second International Mathematics Study
Authors: McKnight Curtis C. , Travers Kenneth J. , Dossey John A.
Published: January 2021

What mathematics is taught to twelfth grade students in high schools in the U.S. who are enrolled in at least their fourth year of college preparatory mathematics? What are the teachers like who provide this instruction? How do they spend their time? How do the students spend their time? How well do the students do? What are their attitudes toward mathematics? Do they gain much in mathematics achievement during the year? How does their achievement compare with that of students at the end of secondary schooling in other countries? How do they compare with twelfth-grade college preparatory mathematics students of twenty years ago?
Is Discrete Mathematics the New Math of the Eighties?
Authors: Hart Eric W.
Published: January 2021

In the past thirty years a number of trends in mathematics curricula have been observed. We've seen new math, basic skills, problem solving, and, most recently, discrete mathematics. The aim here is to look more closely at discrete mathematics. It is being touted by its proponents as a much needed revolution in mathematics education. With this claim in mind it seems useful to consider simultaneously the last revolution in mathematics education the so-called new math looking for parallels and any possible lessons to be learned.
Microcomputer-assisted mathematics: From Simple Interest to e
Authors: Kimberling Clark
Published: January 2021

When students first meet e, they are usually told that this number appears in many important applications that can only be mentioned briefly at that time. Among these are the normal probability curve and differential equations. One application, however, can fit right into a first encounter with this fascinating number.
Sharing Teaching Ideas
Authors: Willy W. Edward , Fay Claudia , Jentho Bonnie , Willcutt Bob
Published: January 2021

“Sharing Teaching Ideas” offers practical tips on the teaching of topics related to the secondary school curriculum. We hope to include classroom-tested approaches that offer new slants on familiar subjects for the beginning and the experienced teacher. See the masthead page for details on submitting manuscripts for review.
Using the Discriminant for Problems Involving Extrema
Authors: Nagase Goro
Published: January 2021

Various methods for solving problems involving maxima and minima without calculus have been elegantly presented by Niven (1981). However, the method of using the discriminant of a quadratic equation is not emphasized in that book. The discriminant plays an important role in many areas of algebra, precalculus, calculus, analysis, and even statistics.
Problem Solving with Magic Rectangles
Authors: Reiter Harold B.
Published: January 2021

Magic squares have intrigued students of mathematics of all ages for many years. A variety of interesting techniques have been devised for constructing magic squares of various sizes, A sample 3-by-3 magic square is shown in figure 1. All the rows and columns and the two diagonals have the same sum. Recently, while discussing this topic with sophomores and juniors in my course on discrete mathematics, a student asked whether the notion of a magic rectangle was worthy of study. This article answers that question and provides a source of problems suitable for secondary school students.
Several Incarnations of McKay
Authors: Bruckheimer M. , Markovits Z.
Published: January 2021

Finding word problems is not the easiest task of mathematics teachers. Basic practical questions often degenerate into boring exercises, and whimsical puzzles are sometimes deemed too outlandish to pursue. Sometimes, however, a real situation arises that stimulates discussion beyond the limits of the textbook so that problem-solving techniques are sharpened, and the end of class comes all too soon.
A Geometric Figure Relating the Golden Ratio and Pi
Authors: Seitz Donald T.
Published: January 2021

Twenty-three centuries ago Euclid wrote proposition 11 of book 2 of the Elements, which proved that a given line segment can be divided into two parts so that the area of the rectangle whose length is the whole line segment and whose width is the shorter of the two parts is equal to the area of the square whose side is the other part (fig. 1).
Applications: Queues and Simulations for Secondary School Students
Authors: Copes Wayne , Sacco William , Jameson John W. , Sloyer Clifford W.
Published: January 2021

“Totally new, challenging, realistic, useful”—these were among the comments of a majority of sixty secondary school students expressing their reactions to a self-teaching module on queues, developed recently under an NSF grant. Most of the module deals with the use of random numbers in the simulation of realistic queueing or waiting-line situations. This article summarizes the mathematical content and pedagogy of the module.
Mathematical Firsts— Who Done It?
Authors: Williams Richard H. , Mazzagatti Roy D.
Published: January 2021

In mathematics and other scientific disciplines a common practice is to name a theory, an equation, and other discoveries in honor of the scientist who pioneered the investigation. Some examples of such expressions are Galois theory, Fahrenheit scale, Freudian psychoanalysis, pasteurization, Zorn's lemma, Planck's constant, Linnaean system of botanical classification, Hilbert space, Darwin's theory of evolution, Halley's comet, Keynesian economic theory, Mendelian genetics, and so on.
252/7: A Divisibility Pattern
Authors: Schatzman Gary
Published: January 2021

Very often we present mathematics to our students as a complete body of work without exposing them to the dynamics that went into discovering it. They are not given an opportunity actually to do mathematics: to develop new conjectures from old by changing initial conditions, to form hypotheses and test them by computation or measurement, to prove their conjectures, and to present their results in a clear and concise report.