bluuebunny/crossref_metadata_2025 huggingface dataset:
Some implications of programmed instruction for the teaching of mathematics
Authors: Heimer Ralph T.
Published: January 2021

Can and should mathematics be taught by the use of programmed-instruction materials and machines?
Tips for Beginners: Finding illustrative examples and exercises
Authors: Hayden Dunstan
Published: January 2021

The alert teacher knows it is dangerous to construct examples and exercises purely at random. The results will frequently lack variety and interest and fail to manifest all the various possibilities. What is needed are formulas which will generate problems or examples in such a way that the teacher can predetermine the characteristics in question.
The game of Euclid
Authors: Albaugh A. Henry
Published: January 2021

Here's an idea for the mathematics club party.
Tips for Beginners: Let's say what we mean
Authors: Robinson Edith
Published: January 2021

All of us who teach mathematics hope that our students will learn to think more clearly, yet our own generalizations sometimes lack the precision which would best foster clarity of thought.
Formalization and sterilization
Authors: Gattegno Caleb
Published: January 2021

The intuitive content of mathematics is as much a reality for learners as for creators.
Nomography
Authors: Wylie C. R.
Published: January 2021

An interesting application of several important topics from algebra and analytic, geometry offers the students an opportunity to exercise their ingenuity.
Historically Speaking,—: Some uses of graphing before Descartes
Authors: Smith Thomas M.
Published: January 2021

During the first half of the seventeenth century at least four students of mathematics and natural philosophy were making use of a simple graphing technique.
Historically Speaking,—: Some reflections on Gulliver's Travels
Authors: Merrill C. F.
Published: January 2021

It has been stated that “Gulliver's Travels draws upon at least five traditions of world literature …, the literal travel account, realistic fiction, utopian fiction, symbolism, and the fantastic voyage [1].”* There is, however, another aspect of Gulliver's Travels which perhaps some of its readers have not considered.
Infinite decimals
Authors: Ringenberg Lawrence
Published: January 2021

Some mathematical ideas implicit in the meaning of infinite decimals
Solution of equations by means of rectangular circuits
Authors: Lundin Enor E.
Published: January 2021

An adaptation of a method of graphical statics presented by Cremona in 1888
Historically Speaking,—: Problems 1 to 6 of the Rhind Mathematical Papyrus
Authors: Gillings R. J.
Published: January 2021

When the Rhind Mathematical Papyrus was received by the British Museum in 1864, it was broken in some places, and portions of the brittle fragments were missing. Subsequently the papyrus roll was unrolled and mounted in two glass frames, the first portion, called the Recto (reading from right to left), having a jagged break because of these missing fragments. The whole papyrus was originally 18 ft. long and a little over a foot in width, the second half, called the Verso, having a blank space about 10 ft. long towards the left-hand end, upon which nothing is written.
Tips for Beginners: Using the overhead projector in an algebra class
Authors: Osborne Alan R.
Published: January 2021

Most teachers of mathematics could make better use of audiovisual materials in addition to the blackboard. Though it is hard to imagine teaching mathematics without a blackboard, one should be cognizant of the many audiovisual tools which exist and are of particular use in the mathematics classroom. The purpose of this article is to point out advantages derived from using an audiovisual tool called the overhead projector and to describe inexpensive accessories that are of use specifically in an algebra class.
Applications of finite arithmetic, III
Authors: Dubisch Roy
Published: January 2021

Some properties of lines in a finite geometry
Are students’ questions a valid criterion for evaluating creative teaching?
Authors: Adkins Julia
Published: January 2021

Students ask questions which lead to the discovery of new relationships.
Stressing the creative aspects of mathematics in teaching the gifted child
Authors: Wallen C. J.
Published: January 2021

The many rich and interesting methods which the inventor used remain hidden to the student unless the teacher uses techniques similar to those described in this article.
Continued fractions
Authors: Moore Charles G.
Published: January 2021

An opportunity for the teacher to set before his students situations which will stimulate creative thinking
A description of a summer institute for academically talented high-school juniors
Authors: Gutzman Wayne W.
Published: January 2021

The first institute at the State University of South Dakota was hold in 1958. It was created in response to a request by certain South Dakota high-school superintendents that high-school students of outstanding academic ability be provided an on-campus program to challenge their capabilities. The first institute was attended by eighteen students who paid most of their own expenses. The curriculum for the institute was planned by the Honors Committee of the university and included mathematics and English as the basic subjects.
Numerical analysis as a twelfth-year elective
Authors: Dodes Irving Allen
Published: January 2021

The Bronx High School of Science started a course in numerical analysis in 1955. The first such course featured the programming of an IBM 650, a stored-program electronic computer with drum storage. For this purpose, the facilities of the Watson Laboratories were used with the kind permission of the IBM Corporation.
The education of a mathematics teacher for the German Gymnasium
Authors: Fehr Howard F.
Published: January 2021

With the great surge in mathematics, both pure and applied, the scientific and industrial world has called for a reform, both in our school mathematics curriculum and in the training of teachers to teach a more modern program. Just how much mathematical and how much professional education training a teacher needs is a matter of conflicting philosophies today, between those who believe in a body of knowledge called professional education and those who constitute the set of pure scientists. Since schools and the purposes of school education in America are quite different from those in most European countries, a critical examination of programs in other countries may give some perspective by which some of our conflicts may be resolved.
A method of proof for high school geometry
Authors: Ness Harald M.
Published: January 2021

The traditional method of displaying proofs in two columns has some disadvantages as well as advantages. Here is a suggestion for another form of proof.
The Council's Drive to Improve School Mathematics—a Progress Report
Authors: Allen Frank B.
Published: January 2021

An account of numerous activities in which the Council is engaged
Mathematicians Looking Ahead
Authors: Vinogradov I.
Published: January 2021

IT IS said sometimes that mathematical talent is as rare as a good ear for music. However, that is debatable. First of all, the system of seeking and educating mathematically talented young people seems to be an important factor. The better the system and the wider the circle, the greater the number of talented people that will be discovered. This seems clear enough.
Invocation
Authors: Gold Aaron S.
Published: January 2021

O Lord, let our books be our gardens, and our studies a fragrant walk amidst the flowers of the spirit!
Reviews and Evaluations
Authors: Tinnappel Harold
Published: January 2021

Although this is a calculus text, the first two chapters might be of interest to teachers of eleventh- and twelfth-grade mathematics, both for the selection and the treatment of topics. For example, a definition is given for a “positive” real number which reflects the completeness property of the reals rather than the order property, and the section on mathematical induction includes definition by induction as well as proof by induction. The familiar “laws of exponents” are proved as theorems, using an induction argument; and some proofs are to be found in these chapters which make rather novel use of the decimal representation for a real number. Also of interest to the high school teacher is a section on infinite sequences of rational numbers and the chapter on functions.
Rotations, Angles, and Trigonometry
Authors: Troyer Robert
Published: January 2021

Can trigonometry be done without first assigning a measure to angles? Certainly a large part can; in particular, the definition of the trigonometric functions and the interrelating trigonometric identities do not depend on the measurement of angles. The assignment of a measure to an angle is important, but it is also important for the teacher to see the precise role which measurement plays. The main objective of this article is to give a brief development of trigonometry without using angular measurement. (The point of view given here is not necessarily the one which is pedagogically the best suited for the high school level.)
Experimental Programs: Revision of Secondary Mathematics in a Selected Number of Schools
Authors: Schaefer Sister Mary Geralda
Published: January 2021

Revision in the secondary mathematics curriculum is under way. For the past decade committees and study groups representing the thinking of prominent professional organizations have been active in discovering appropriate modifications of curriculum content and instructional procedures. Their recommendations have been subjected to extensive experimentation and evaluation and have now reached the mathematics classrooms through commercially published textbooks.
Square Root and Cube Root Extraction in 1788
Authors: Hansen Viggo P.
Published: January 2021

Recently I acquired a book entitled A Treatise of Arithmetic, in Theory and Practice, written by a John Gough and published in Philadelphia in 1788. It contains material on how to extract the square and cube root, explained in dialogue and poetry. The material runs as follows.
Reviews and Evaluations
Authors: Tinnappel Harold
Published: January 2021

A translation of the first venture into the history of mathematics by a professor of Greek at Lausanne University, this book seeks with only very limited success “to explain the birth of modern mathematics by describing the progress of mathematical thought at the time of Plato” and to convince the reader that “twenty-five years of thought and discussion in Plato's Academy sufficed to delimit the field of mathematics in all its breadth [and] saw the working-out of new ideas on which the whole edifice of modern mathematics rests.”
Experimental Programs: Student Attitudes and Achievement: A Comparison Between the Effects of Programmed Instruction and Conventional Classroom Approach in Teaching Algebra I
Authors: Devine Donald F.
Published: January 2021

It is certainly not news to persons directly involved in education that the entire educational process is in a state of upheaval today. Advocates of all sorts of new programs including such things as individualized instruction, modular scheduling, team teaching, computerized instruction, programmed materials, new content curricula—as well as some proponents for the return to the “good old days” in both content and techniques—are making their voices heard.
Have You Read?
Authors: Peak Philip
Published: January 2021

I am always interested in seeing new quirks of old ideas, especially when written by students. This article on “casting out nines” is a challenge to you and your students to sharpen up your number power. Symbolic and mathematical, “The cross-sum of a number is the sum of its digits.” Here is the format.
A New look at Old Committee Reports
Authors: Bidwell James K.
Published: January 2021

It is not unusual for us to read some quotation, remark how pertinent it is today, and find it was first said years ago. This applies to classical statements that have survived over the centuries. It also applies to statements in many academic discipilines, particularly discipilines that deal with learning. People have not changed greatly in their ways of learning. Particularly, the ways of learning in the public schools are slow to change.
Detection and Care of the Gifted in Mathematics
Authors: Srinivasan P. K.
Published: January 2021

We are in the golden age of mathematics. But has the world been made golden for the gifted? All along they have been functioning, in spite of educational systems and regardless of handicaps and obstacles placed in their way through mistreatment born out of misunderstanding.
Curves with Proper Names
Authors: Phillips J. P.
Published: January 2021

In the days before the invention of analytical geometry it is understandable that curves were sometimes named after their discoverers, particularly if they served some special purpose in mathematics or science.
A Proof of the Space-Separation Postulate
Authors: McComas Charles A.
Published: January 2021

The space-separation postulate stated in many of the recent high school geometry texts can be proved by use of the corresponding plane speration postulate and the plane-intersection postulate (together with the usual incidence and betweeness properties). Though it is a bit long, the proof is easy enough to be deduced by some of our better students, if suitable hints are provided.
A One-to-One Correspondence Formula
Authors: Bailey William T. , Musser Gary L.
Published: January 2021

When proving that there is a one-to-one correspondence between the set of natural number, N = {1, 2, 3, …}, and the set of positive rationals, Q+, the usual technique is to arrange the positive rationals in an array, draw some arrows, and say “etc.” Occasionally authors include a formula which appears so complicated that it surely must have been found by a most sophisticated computor. Surprisingly, however, we can establish a formula through discovery by making some simple observations.
Historically Speaking—: The History of Mathematics as a Part of the History of Mankind
Authors: Garder Arthur O.
Published: January 2021

Many conclusions useful for the future of mathematics might result from a sustained investigation into the relationship between the history of mathematics and the history of the societies in which mathematics has developed. For example, is it posible to reach some conclusion about the nature of the social, economic, and political milieu which is optimal for the development of science in general and of mathematics in particular?
Towards the Golden Jubilee Year-1970
Authors: Hlavaty Julius H.
Published: January 2021

THIRTY years ago, in Atlantic City, I attended my first meeting of the National Council of Teachers of Mathemataics. Everyone considered that meeting a great success, for there were 343 people present! Thirty years ago, the total membership of the Council was smaller than the number of those attending this present convention.
Classics in Mathematics Education: The Universality of Mathematics
Authors: Crosswhite F. Joe , Reeve William David
Published: January 2021

Arithmetic, eternal bugbear of the American youngster, is in for a drastic and sweeping revision, if plans made public yesterday at Teachers College, Columbia, are carried into effect. At least 85 percent more arithmetic is now being taught in our public schools than is required in “life situations,” according to the survey.
In-Service Education in Mathematics: Computers and In-Service Education
Authors: Andree Richard V.
Published: January 2021

The impact of computer science will probably not drastically change the secondary mathematics program, but it will most certainly change the emphasis on some concepts, ideas, and topics. The availability of computer technology demands more stress on the understanding of algorithms, basic definitions, and important numerical theorems.
A Christmas Tree for 1968
Authors: Groenke Lucille
Published: January 2021

Class activity during the few days prior to a school recess can be both fun and worthwhile. My students have enjoyed graphing this Christmas tree from directions given in set notation. Told only that these directions describe regions that are to be colored as indicated, the students eagerly compete to see who will “discover” the picture first. Sometimes enough enthusinsm is generated so that the next day students will bring in their own originals to be graphed.
Historically Speaking— Music and Mathematicians Since the Seventeenth Century
Authors: Brown J. D.
Published: January 2021

Mathematics and Music, the most sharply contrasted fields of intellectual activity which one can discover, are yet bound together supporting one another as if they would demonstrate the hidden bond which draws together all activities of our mind, and which in the revelations of artistic genius leads us to surmise unconscious expression of a mysteriously active intelligence [1, p.1].1
International Mathematical Education: Present-Day Problems of Mathematical Instruction
Authors: Servais Wily
Published: January 2021

Mathematics, since the dawn of civilization, has been considered as a rational discipline of excellence and as one of the master components of the intellectual equipment of mankind. From day to day it has amplified its scope over the theoretic plane, in the use of calculating machines, and in the domain of its applications.
Half Man, Half Myth
Authors: Fant Alfred B.
Published: January 2021

NO MAN gains immortality merely by living a life. It is only through his actions that his life is remembered and that he wins a niche in a corner of the world's memory. Pythagoras was such a man.
Message from the President
Authors: Hlavaty Julius H.
Published: January 2021

You have been hearing at meetings and have read in the Newsletter that the Board of Directors has been studying in depth the problem of streamlining the organization of Lhe NCTM for the vastly increased responsibilities it faces and the vastly increased services it now renders to its members. We have had committees of the Council studying various aspects of the problem, and we have even had professional assistance on management (from McManis Associates).
Points and Viewpoints: Announcing A New Department
Authors: Brune Irvin H. , Koetke Walter J.
Published: January 2021

“Computer-Oriented Mathematics” becomes a department in this issue. It enlarges the services rendered by The Mathematics Teacher. It meets a need that numerous readers have expressed for at least two years. It deals with an aspect of mathematics education that fascinates the neophyte, encourages the experimenter, and reassures the pioneer. We welcome the new department. We wish for it a steady supply of manuscripts of superior quality.
Lively Functions for Algebra One
Authors: McCreery Louis R.
Published: January 2021

The abstract ideas making up the concept of function can be made to come alive in the everyday experiences of a beginning algebra student. For instance, the gas pump at the corner service station produces a realistic example. The gallon indicator interlocked with the cost wheel presents a set of ordered pairs in a most rapid and vivid fashion. Because a given number of gallons produces only one possible price, this set is a function. The price per gallon is the constant ratio that defines the function, and it is right there, sometimes in letters a foot high!
Computer-Oriented Mathematics: Generating “Random” Numbers Using Modular Arithmetic
Authors: Koetke Walter , Indelicato Brother Arthur
Published: January 2021

A TRAFFIC engineer designs a busy intersection and wants to know whether the intersection will easily handle the innumerable traffic patterns that may occur throughout the day. A physicist wants to simulate the impinging of electrons on a light-sensitive surface. A photographer captures on his film the light patterns of a city ready for sleep. A student wants to flip a coin 1,000 times to see if he gets approximately 500 heads; to see if that probability is really close to one-half.
International Mathematical Education: Mathematical Education in Malaysia
Authors: Fehr Howard F. , Swetz Frank
Published: January 2021

The federation of fourteen states that comprises Malaysia was formed in 1962. A country of mountains and jungles where the blowpipe is as common as the television antenna, Malaysia is rapidly emerging as a modern nation of Southeast Asia. As with many of its neighbors, Malaysia's recent history has been marred by violence and social upheaval. Settled by the British, conquered by the Japanese, torn by internal strife in the form of “the Emergency”1 and still later threatened by confrontation with Indonesia—a self-identity has been difficult to achieve. Surmounting all these obstacles, Malaysia has emerged a one of the most progressive countries of the East. Thus it is interesting for us as mathematical educators to examine mathematics education in such a developing nation.
Reviews and Evaluations
Authors: Eakin Richard R.
Published: January 2021

In the 145 pages of the text, the author presents a variety of topics that are of real concern to today's elementary school teacher of arithmetic. The book is highly recommended for teachers whose training predates the “modern math” era. The approach is kept at the most elementary level and presupposes only a knowledge of traditional arithmetic.
Historically Speaking— The Witch of Agnesi—Exorcised
Authors: Kennedy Hubert C.
Published: January 2021

In the summer of 1965, at an informal luncheon attended by a prominent algebraist, the name of Maria Gaetana Agnesi and the curve known as the Witch of Agnesi were mentioned. The algebraist asked: “Wasn't this curve called the ‘Witch’ because Agnesi knew how to read and write when this was unusual for women? Wasn't she burned at the stake in the thirteenth century?” Now, Maria Gaetana Agnesi (1718-1799) died quietly in a home for the aged where she had been director of women for many years. Her tombstone was inscribed “noted for piety, learning and good works.” These facts are known to contemporary historians of mathematics. The gross errors of our algebraist are not likely to be repeated, but the legends concerning Agnesi and the Witch seem to die slowly. Perhaps this justifies the appearance of yet another note on the subject.
Summer Employment of Mathematics Teachers in Industry
Authors: Wilsted H. D. , Bryant P. J.
Published: January 2021

Lord Kelvin made this statement: “When you can measure what you are thinking about, or when you can express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind.” Having a satisfactory knowledge of what is going on in a large industrial organization means to be able to put numbers on manufacturing tolerances, standards of production or performance, control of processes, and other factors ad infinitum. The mathematician, for the foregoing reason, should be of value to any kind of business whether it involves design, analysis, manufacturing, insurance, research, or marketing.
An Analemma Construction for Right and Oblique Ascensions
Authors: Id Yusif
Published: January 2021

In CLASSICAL times and during the Middle Ages a variety of spherical astronomical problems were attacked by graphical methods which we would call descriptive geometry, and to which the name analemma was then attached. Historians of mathematics have given the topic little attention, although the close connection between the earliest spherical trigonometry and the analemma techniques has been pointed out.1
International Mathematical Education: Mathematical Structures and the Role of Algebra in School Mathematics
Authors: Fehr Howard F. , Lefebvre P.
Published: January 2021

Editor's Note.—In April 1966, at a meeting of ministers of education in the Arab states, held at Tripoli, a resolution was adopted requesting an updating of instruction in school subjects—particularly mathematics, pure and applied science, and foreign languages. In November 1966, the General Conference of UNESCO invited member states to undertake a major program for improvement of science and mathematics instruction and selected the Arab states as an initial place to start because of their April resolution. Mathematics was selected as the initial subject primarily because, worldwide, the reforms in education during the last fifteen years began with mathematics.
Reviews and Evaluations
Authors: Eakin Richard R.
Published: January 2021

Analytic Geometry is an extremely valuable contribution to the field. At the present time when more and more colleges are returning a separate course in analytic geometry to their curriculum and many high schools are offering the course as their fourth year of mathematics, this book should be seriously considered by teachers at both levels.
The Third U.S.A. Mathematical Olympiad
Authors: Greitzer Samuel L.
Published: January 2021

The U.S.A. Mathematical Olympiad, now in it third year, has become sufficiently well known for inquiries about it and interest in having students participate in it to be widespread. Participation is by invitation only. In 1974, invitations were sent to 155 students to take part in the Third U.S.A. Mathematical Olympiad. Most students were selected on the basis of their performance on the Annual High School Mathematics Examination. Students from Michigan and Wisconsin, which conduct their own statewide mathematics contests, were also invited, and a few students were invited on the basis of strong recommendation from school officials. Of these 155 invitations, 149 complete acceptances were received. (An acceptance is considered complete when the student agrees to participate and the school agrees to administer the test.)
Projective Space Walk for Kirkman's Schoolgirls
Authors: Ehrmann Rita (Cordia)
Published: January 2021

In 1847 T. P. Kirkman proposed his now classic fifteen schoolgirls problem: “A school-mistress was in the habit of taking her girls for a daily walk. The girls were fifteen in number, and were arranged in five rows of three each, so that each girl might have two companions. The problem is to dispose them so that for seven consecutive days no girl will walk with any of her school-fellows in any triplet more than once” (Ball and Coxeter 1960).
Some Interesting Mathematics about Prime Factorization
Authors: Rodgers Joe Tom
Published: January 2021

Throughout their school careers, students are taught how to find the prime factorization of numbers. Most students readily learn the technique and find the process interesting. Most teachers then teach the ways in which prime factorizations can aid in factoring polynomials and finding greatest common divisors as well as least common multiples.
Posters from “Mathematics in Use, as Seen on Postage Stamps”
Authors: Schaaf William L.
Published: January 2021

The delicate symmetry of the snowflakes, the spiral of the ammonite (fossilized nautilus), the crystal of cassiterite (tin dioxide, which crystallizes in the tetragonal system in low pyramids and prisms), and the honeycomb of the bee are here shown on stamps.
The Spirograph and the Greatest Common Factor
Authors: Cavanaugh William E.
Published: January 2021

One of the more interesting toys that has been developed recently is the Spirograph (a product of Kenner Products, Cincinnati, Ohio)—interesting not only for the designs that can be made from it but also for the mathematics that can be related to it. It consists of a collection of different-sized wheels; the larger wheels are rings with gears both inside and outside the rings, and the small wheels have only the outside gears, or teeth. In addition, the wheels have a series of holes in them so that a pen can be set at different locations.
Graphic Solution of Linear and Quadratic Inequalities with Absolute Values
Authors: Scott Wayne R.
Published: January 2021

Frequently I have heard, “That’s not the way we learned to do it. What our teacher did last year was much easier.” And after this easier method has been observed, the solution has either been incomplete or erroneous.
Posters from “Mathematics in Use, as Seen on Postage Stamps,”
Authors: Schaaf William L.
Published: January 2021

The Iranian stamp depicts finger reckoning; the Romanian stamp shows a Mannheim slide rule and a vernier caliper; the stamp of Denmark shows a comptometer type of calculator. The adhesives of Israel, East Germany, and the Netherlands recognize the advent of electronic computers and data-processing equipment.
Variations on the Closure of a Set
Authors: Francis Richard L.
Published: January 2021

The number concept is essential to any definition of the word “mathematics.” A step beyond this is the fundamental notion of a number system. Such a system necessitates three things: (1) a set of numbers, (2) operations defined on the set of numbers, and (3) rules governing the operations. This list of rules often contains a property called “closure,” intended to insure that results of an operation are elements of the underlying set. However, for most number system presentations, checking closure is redundant. Closure is implicit in the usual definition that a binary operation ⊕ on a set T associates with each ordered pair of elements of T a unique element of T.
Posters from “Mathematics in Use, as Seen on Postage Stamps,”
Authors: Schaaf William L.
Published: January 2021

The United States “Apprenticeship” commemorative pictures the micrometer caliper used in machine shop measurements. The Japanese stamp suggests that length, capacity, and mass are readily measurable. Metric units, the SI abbreviations, and conversion to metric are indicated on the stamps of Korea, Romania, and Tanzania-Kenya-Uganda.
The Game of Double Zero
Authors: Guckenheimer Gertrude
Published: January 2021

The three different versions of the game will be explained first by stating rules and then by simulated play.
Paper Folding as a Technique in Visualizing a Certain Class of Transformations
Authors: Faulkner J. Earl
Published: January 2021

Visual demonstrations are important as a supplement to the abstract treatment of transformations. The method described here models transformations that are circular and finite under one set of conditions and become rectangular and infinite when conditions change.
“The Pit and the Pendulum”: Source for a Creative Activity
Authors: Keese Earl E.
Published: January 2021

Are you looking for ideas to correlate mathematics with other subjects? To give your students an impetus to be creative? The author has good suggestions toward these ends.
Map Coloring
Authors: Roth Norman K.
Published: January 2021

Some interesting suggestions for classroom activity in exploring the problem of map coloring.
The Advanced Placement Program in Mathematics—Update 1975
Authors: Jones Chancey O. , Kenelly John W. , Kreider Donald L.
Published: January 2021

The recently released 1973 Calculus AB and BC examinations offer some challenging problems for your students.
Summary and Curricular Implications: An Outgrowth of Articles by Thom and Dieudonné
Authors: Stevens John G. , Garfunkel Robert
Published: January 2021

The reader should be intrigued by the thoughts expressed in this article, even though he may not agree with the authors. The questions raised cannot be ignored by anyone interested in the high school mathematics curriculum.
Three-Dimensional Discovery
Authors: Prichett Gordon D.
Published: January 2021

One of the primary problems in structuring a mathematics laboratory is the difficulty in finding projects that will yield answerable questions within the students' grasp. Most projects imply one or two such questions and then become inadequate for the needs of a high school situation. The following is the description of a project that can yield an abundance of questions all within the capability of an interested high school student.
Curve Stitching in an Elementary Calculus Course
Authors: Goldberg Kenneth P.
Published: January 2021

Elementary calculus students often need simple yet interesting calculus problems to whet their appetites for mathematics. One popular problem whose solution depends only on using the derivative to maximize a function comes directly from the method of geometric con-struction known as curve stitching.
More Discoveries in Linear Algebra
Authors: Wellen Darrell
Published: January 2021

In a recent issue of the Mathematics Teacher, Nicolai (1974) detailed several “discoveries” in linear algebra that were made by him and one of his classes. A look at their ideas yielded several discoveries showing that the relationships described by Nicolai are special cases of more general, interesting situations.
Eniac: The First Computer
Authors: Strauss Aaron
Published: January 2021

The electronic digital computer is a war baby. The high-speed calculating and data-processing machines have led to the development of new ideas and methods in pure mathematics; they have made possible technological progress leading to interplanetary space travel; and they administer activities of daily life from traffic control and department store sales to industrial payrolls and income tax collection. The computer is truly a major contribution of American mathematics, science, and technology. Yet the stimulus for the production of ENIAC—as so often has been true throughout history — was the demand for ever more sophisticated military methods.
New Products
Authors: Leake Lowell , Rowan Thomas E. , Shulte Albert P.
Published: January 2021

Introductory Note: The following article was reprinted with permission from Mathematics Teaching, March 1975. no. 70, pp. 42—43, copyright 1975 by the Association of Teachers of Mathematics [England]. It is a superb journal, and the cost is $8.00 a year for four issues, which includes membership in ATM. I have belonged to ATM for about six years and look forward to new copies of Mathematics Teaching as much as I do to publications of the NCTM. I urge readers to consider joining the ATM, or at least to have their schools subscribe to Mathematics Teaching. The address is Association of Teachers of Mathematics, Market Street, Chambers, Nelson, Lancashire, BB9 7LN, England.—LEAKE.
A Two-Square One-Square Puzzle: The Pythagorean Theorem Revisited
Authors: Engle Jessie Ann
Published: January 2021

Add this experiment to your geometry collection.
Extrasensory Probability
Authors: Loase John Frederick
Published: January 2021

Here are several extrasensory-perception activities that will intrigue students and stimulate enlightening debate about statistical proof or disproof at various levels of sophistication.
Secondary School Students' Perceptions of Math Teachers and Math Classes
Authors: Cooper Charles R. , Petrosky Anthony
Published: January 2021

Here is a summary of a recent, revealing nationwide survey of high school students perceptiollS of their teachers and the classroom atmosphere they create.
Real-Eee.… !
Authors: Cieply Joseph , Kevin McCoy J.
Published: January 2021

A Student and a teacher combine to compute e to 200 digits using the division algorithm.
The Tilted Probability Board
Authors: Austin Joe Dan
Published: January 2021

This “new angle” on a familiar probabilistic device provides an intriguing basis for experimental study of modeling.
Mathematical Reflections and Reflections on Other Isometries
Authors: Bishop Thomas D. , Fetters Judy Kay
Published: January 2021

Mirrors can be used to explore some new geometric relationships.
Algorithms and the Bicentennial
Authors: Walbesser Henry H.
Published: January 2021

“Practical examples not only show at once the object to be accomplished, but they greatly assist the imagination in unfolding the principle and discovering the operations requisite for the solution.”
Pascal's Triangle and the Binomial Probability Distribution
Authors: Killian C. Rodney , Kepner Henry S.
Published: January 2021

Relationships between both the elements and the overall characteristics of Pascal's triangle are explored.
Backtracking the Euclidean Algorithm
Authors: Kappel Michael W.
Published: January 2021

The algorithm tells its how to find the G.C.D. of two numbers a and b. It also tells us there are numbers r and s such that ra + sb = k. But how do we find r and s?
Circles, Chords, Secants, Tangents, and Quadratic Equations
Authors: Olson Alton T.
Published: January 2021

An interesting way to use geometry to solve quadratic equations, thus truly integrating algebra and geometry.
What’S in a Trick?
Authors: Bolduan Michael S.
Published: January 2021

It may be a trick but it contains a lot of mathematics. Try it out on your students.
President Garfield and The Pythagorean Theorem
Authors: Graham Malcolm
Published: January 2021

Since mathematics is an abstract subject, the recent growing interest in its history is easily understood. Historical references properly integrated into mathematics instruction can do much to stimulate interest.
Real Transformations From Complex Numbers
Authors: Alexander Robert D.
Published: January 2021

Advanced students of high school or two-year college mathematics should find a nice exposition of their view of complex numbers in this article. Graphs might he something of a surprise to them.
When We Will Meet Again? A Modified Answer
Authors: Metz Jim
Published: January 2021

Racers and planets lapping each other—all explained by simple equations in modular arithmetic.
A Strip Of Wallpaper
Authors: Troccolo Joseph A.
Published: January 2021

A good reminder that students can explore many symmetries “off the wall”.
Finding Chord Factors of Geodesic Domes
Authors: Blaisdell Fred , Indelicato Art
Published: January 2021

Buckminster Fuller really started something when he popularized the dome. Now here is a careful look at the mathematics you will encounter if you start to build a dome.
The Old Integer Game
Authors: Erickson Robert
Published: January 2021

As a break from class routine, students can brush up on their basic skills and enjoy it.
Cube Roots On A Calculator
Authors: Eimer Rebecca A.
Published: January 2021

Suppose you are given the problem of determining the radius of a given sphere, knowing the volume to be 1910.0928 cubic units.
Introductory Statistics: A Contemporary Approach
Authors: Emerson John D.
Published: January 2021

Three years of high school mathematics can enable students to explore an interesting collection of statistical concepts.
On Generating A Bivariate Distribution Having Any Desired Correlation
Authors: Travers Kenneth J.
Published: January 2021

Help in producing examples of patterned data for illustrations.
Celebrating the Birthday Problem
Authors: Spencer Neville
Published: January 2021

Do you have an intuitive guess of a probability that two people in a group have the same birthday? This article will prove that your guess is probably a large underestimate.
There are More Ways Than One to Bisect a Segment
Authors: Gibb Allan A.
Published: January 2021

Why not try some alternative construction methods? Here are some that many of us have heard about but never used.
Career Education and The Mathematics Classroom
Authors: Finn Peter , Brown James
Published: January 2021

Everybody's talking about career education: here are some ideas to try in your classes.
Testing and Grading: Procedures for Improved Student Motivation
Authors: Stover Donald W.
Published: January 2021

Some refreshing classroom test variations for grading.
Is the Graph of Temperature Variation a Sine Curve? an Application for Trigonometry Classes
Authors: Lando Barbara M. , Lando Clifton A.
Published: January 2021

Here's curve-filling without least-squares techniques. Minicalculators remove the drudgery from the task.
Demonstrating the Central Limit Theorem
Authors: Smith Scott G.
Published: January 2021

Try some simple experiments and notice the trends.
Student Projects in Geometry
Authors: Zucker Andrew A.
Published: January 2021

Looking for ideas and references to start or expand your list of student projects?
Precalculus Mathematics: A Look Through the Big End of the Telescope
Authors: Harvey John G.
Published: January 2021

A look at the content and emphasis in current calculus courses and what they imply. Also a list of performance objectives that should provoke a discussion in your department.
Crazy Roller Coasters
Authors: Bruce William J.
Published: January 2021

Follow the rules for developing a chain of numbers. Plot and connect the numbers on a graph and you will draw a roller coaster run.
Calculus and Capitalism— Adam Smith Revisits the Classroom
Authors: Brazier Gerald D.
Published: January 2021

A linear and a quadratic equation help determine how a manufacturer can survive in the market place.
A Low-Stress Algorithm for Fractions
Authors: Ruais Ronald W.
Published: January 2021

A process for students who are frustrated with conventional methods.
Using a Calculator to Find Rational Roots
Authors: Waits Bert K.
Published: January 2021

A recent article in the Mathematics Teacher (Eimer 1977) included a recursive method for finding the cube root of ten that used a square root algorithm. The method required only a simple four-function calculator with a square-root key and a memory. In this note we present a more direct method for approximating the nth root of any positive number that requires only a four-function calculator with a square-root key and repeat multiplication capability.
What are the Real Problems Involved in Getting Computers Into the High School?
Authors: Saunders James
Published: January 2021

The purpose of this paper is to consider some of the problems in getting a computer into a high school. The ideas are based on the experiences of teachers in a suburban Pittsburgh. Pennsylvania, school district. Upper St. Clair. Any consideration of the problems involved in getting computers into the high school will necessarily focus on the cost involved. Teachers, administrators. school directors, and concerned students will readily agree that the main problem is money.
Computers (And Students!) Need Explicit Notation
Authors: Ducharme Robert
Published: January 2021

Here is one possible solution for students who have difficulty with the order of operations.
Recycle Those Activities
Authors: Williams David E.
Published: January 2021

You will find many uses for these lively ideas in your algebra classes.
Medical Diagnosis: A Real Life Application of Logic
Authors: Goldberg Kenneth P.
Published: January 2021

Of all the mathematical topics taught at the high school and junior college level, logic must certainly be one of the most frustrating for both students and teachers. On the one hand, the student is told that logic is the necessary basis on which all rational reasoning and decision making rests. On the other hand, since few situations involving human reason are simple enough to permit the application of mathematical logic, it is extremely difficult to discuss and illustrate the use of logic in a believable and interesting manner.
Geodesic Domes in The Classroom
Authors: Lund Charles
Published: January 2021

Buckminster Fuller has created a myriad of ideas that are highly appropriate for study at various points in the mathematics curriculum. This article describes some practical, hands-on ways in which Fuller's ideas about geodesic domes are being used in the secondary school mathematics classrooms of the St. Paul public schools.
Activities: Discovery with Number Triangles
Authors: Ouellette Hugh
Published: January 2021

This section is designed to provide mathematical activities suitable for reproduction in worksheet and transparency form for classroom use. Appropriate laboratory experiences, discovery activities, and model constructions of mathematical significance drawn from the topics of the seventh, eighth, and nine grades are most welcome. Send your suggestions, in a four-page format similar to the “Activities” already published, to the managing editor for review.
A Physical Model for Operations with Integers
Authors: Kohn Judith B.
Published: January 2021

Several models for operations with integers are shown. The model for the product of two negatives was discovered by a sixth-grade student.
Using The Compass and Carpenter's Square: Construct 23
Authors: Westwood Jack R.
Published: January 2021

Mary Kay Smith and Sidney L. Rachlin wrote an excellent article in chapter 10 of NCTM's 1976 yearbook, “Measurement in School Mathematics,” advocating the use of the carpenter's square in geometric construction.
An Optimization Problem and Model
Authors: Arganbright Deane
Published: January 2021

Several ways to solve an intriguing problem that has a nice physical model.
Crossfactor Puzzle
Authors: Witczak Terry A.
Published: January 2021

Everyone likes crossword puzzles, so why not give your students a crossfactor puzzle? While covering factoring in first-year algebra, my students came to a point where they understood the concepts but needed some practice. To make their practice less painful. I gave it to them in the form of a crossword puzzle. The students enjoyed it and completed the needed drill.
Minimum Mathematical Skills: A Subjective Examination
Authors: Rauff James V.
Published: January 2021

An outline of a program based on applications of mathematical skills.
The Twentieth International Mathematical Olympiad
Authors: Greitzer Samuel L.
Published: January 2021

The Twentieth International Mathematical Olympiad was held in Bucharest, Romania, on 6 and 7 July 1978. A special thanks is due to the Office of Naval Research for supporting a three-week training session of the United States team at the Naval Academy, Annapolis, and to the Army Research Office for providing travel expenses.
Calculator Calculations
Authors: Russakoff Andrew
Published: January 2021

It is certainly true that there are many calculations that are difficult, if not practically impossible, without the aid of a calculator. The article “Calculations You Would Never Make without a Minicalculator” (Litwiller & Duncan 1977) illustrates a sample of these. All of those mentioned involve lengthy multiplication and division much in the fashion of problems for which a slide rule was helpful. This is to say that the results were obtainable with the use of logarithms, I have found that there is another class of problems for which logarithms are some help, but even the use of these is tedious. Here a scientific calculator is needed.
Fun with Repeating Decimals
Authors: Wagner Sue S.
Published: January 2021

A different look at repeating decimals.
Constructing and Trisecting Angles with Integer Angle Measures
Authors: Austin Joe Dan , Austin Kathleen Ann
Published: January 2021

A classic geometry problem made accessible to high school students.
Polygons, Spirals, and Shells
Authors: Wasdovich Dorothy Hoy
Published: January 2021

After thirty-two years of teaching, Dorothy Hoy Wasdovich retired in 1974. Most of those years were with the Harrisburg school system in which her assignments ranged from eighth-grade mathematics through matrix algebra and advanced placement calculus.
What Mathematics Skills Hiring Officials Want
Authors: Swadener Marc
Published: January 2021

Often in mathematics classes, teachers are faced with the question, Why do we have to study this? referring to some specific mathematical topic. Teachers typically respond to such a question in one of several ways. A common approach is to refer to an application of the topic either in other areas of mathematics or in some field outside mathematics.
Eliminating The xy Term
Authors: Roberti Joseph V.
Published: January 2021

Here's a mind stretcher for you and your best students on the classic problem of visualizing the general second-degree curves.
Intuitive Definitions for Division With Zero
Authors: Knifong J. Dan , Burton Grace M.
Published: January 2021

Five different approaches show there really is something to division by 0.
Mathematics and the Reading Process: a Practical Application of Theory
Authors: Henrichs Margaret , Sisson Tom
Published: January 2021

A report on how one junior high school mathematics department was involved in a reading program.
Isosceles Triangles With the Same Perimeter and Area
Authors: Bradley Mark E.
Published: January 2021

In the February 1974 issue of the Mathematics Teacher, Robert W. Prielipp found two noncongruent isosceles triangles with the same perimeter and area. This leads to the question: Are there more than two isosceles triangles with the same perimeter, P, as well as the same area, k? I will show that there can never be more than two isosceles triangles having the same perimeter and area.