bluuebunny/crossref_metadata_2025 huggingface dataset:
Visiting a Computation Laboratory
Authors: Gruenberger Fred
Published: November 2020

Numerous articles are appearing about automatic high speed computing machines. The art of machine calculation is still in its infancy, but is having a tremendous effect on the field of mathematics, both pure and applied.
What is Going on in Your School?
Authors: Brown John A. , Karnes Houston T.
Published: November 2020

This article is being written by a group of mathematics teachers who double as “grass-roots” curriculum workers. It is an attempt to encourage mathematics teachers who are not satisfied with their present curriculum to do something about it. We hope that people who are contemplating a curriculum revision program in mathematics may profit from this explanation and analytical evaluation of a mathematics curriculum revision program that was started in the thinking of its instigators some five years ago. The program, as it has developed, has been in practice in the classroom for the last three school years.
Research in mathematics education—1953
Authors: Brown Kenneth E.
Published: November 2020

A report on research in mathematics education for 1953. This should answer some of your questions, for example, what are the best predictors of college success?
Devices for the Mathematics Classroom: Simple paper models of the conic sections
Authors: Saupe Ethel
Published: November 2020

In second year algebra courses students are normally introduced to the conic sections by being told that certain curves which may be represented by quadratic equations in two unknowns are, in fact, equations of conic sections. In solid geometry the same students are told that a section of a right circular cone made by a plane is a conic section. Various kinds of models are occasionally employed to illustrate what the conic sections look like in the geometric sense, but ordinarily little effort is made to tie up the fact that a relation exists between the introduction given in algebra and the models presented in solid geometry.
Developmental mathematics in the secondary schools
Authors: Bold Benjamin
Published: November 2020

Teachers are not only responsible for introducing students to new ideas but they must deepen old insights. Insights into the properties of the number system can be deepened in the algebra after the pupil has studied operations with polynomials.
Take a number
Authors: Weiner L. M.
Published: November 2020

A little spice to whet the appetite of the algebra class.
Mathematical Miscellanea: Alligation—a mathematical relic
Authors: Eveland Kenneth C.
Published: November 2020

Mr. Gibbins’ article, “Historical Extra Credit,” in the November 1954 issue of The Mathematics Teacher presented the following problem: “A grocer has coffee worth 8¢, 16¢ and 24¢ per pound respectively. How much of each kind must he use, to a cask holding 240 lbs., that shall be worth 20¢ a pound?”
What is Going on in Your School?: Number Cartoons
Authors: Janicki George
Published: November 2020

Illinois took great interest in making original number cartoons. The class had great fun finding the hidden numbers and their sum. Try your luck.
Historically Speaking,— The Earliest Symbol in Mathematical Logic
Authors: Eves Howard
Published: November 2020

AlLhougb gottfried wilhelm leibniz (1646-1716) is generally regarded as the fou11der of symbolic logic, some mathematical symbols of a logical, rather than of a,n operational, nature were employed before him. Thus Pierre Herigone published, in Paris in 1634, a six-volume work on mathematics called Cursus mathematicns, in which he employed an extensive set of mathematical symbols of both the logical and the operational type. But his logical symbols were merely abbreviations of certain useful phrases.
Miniature geometries
Authors: Jones Burton W.
Published: November 2020

Recent interest in emphasizing “patterns” in high school mathematics may lead some teachers to consider miniature geometries as suitable for a unit for gifted students.
Freshman mathematics in the junior college
Authors: Quick Theodora R.
Published: November 2020

America's educational system provides a variety of educational programs of which the city junior college is a significant unit. This article describes the mathematics program and presents the educational philosophy of one junior college.
The nature and content of geometry in the high schools
Authors: Hlavaty Julius H.
Published: November 2020

The nature of the geometry course is being scrutinized carefully by the Secondary School Mathematics Group. Furthermore, the Commission on Mathematics has made its recommendation on geometry. What will the geometry program be in 1969?
Normative decision models
Authors: Raiffa Howard
Published: November 2020

Decision making and game theory are two of a number of exciting new fields in mathematics. This paper gives a few simple illustrations of the newer developments and raises some thoughtful questions for the consideration of high-school teachers.
Mathematics in the Junior High School: Stimulating interest in junior high mathematics
Authors: Willerding Margaret F.
Published: November 2020

The job of the junior high school mathematics teacher is a very difficult one. His students range in achievement from primary level through senior high level and above. To keep all his students interested—and the main task of any teacher is to make his subject interesting—the junior high mathematics teacher must have materials at hand for all of these achievement levels.
Points and Viewpoints: Thanks a million
Authors: Van Engen H.
Published: November 2020

A column of unofficial comment
The evolution of a mathematics contest
Authors: Shoemaker Richard W.
Published: November 2020

Mathematics contests sponsored nationally and locally are gaining in popularity and frequency. Here is a description of how one local mathematics council sponsored a contest.
Reason vs. reflex in the teaching of mathematics
Authors: Lister William G.
Published: November 2020

“A language in which mathematical statements can be made concisely and clearly does not have to be invented. There is a very satisfactory one in use among those whose business it is to create and communicate new mathematics, but it is largely ignored or misinterpreted in elementary work.”
The unique factorization theorem
Authors: Greenleaf Newcomb , Wisner Robert J.
Published: November 2020

The unique factorization theorem seems obvious to the beginner, but it is really a deep theorem. Examples presented here should help the teacher show the importance of the unique factorization theorem.
Solving simultaneous linear equations
Authors: Wendt Arnold
Published: November 2020

A method of solving sets of linear equations, especially useful if there are more than three variables.
Tips for Beginners: Graphing pictures
Authors: Cross Frances
Published: November 2020

The idea of graphing contained in the article by Carver in the January 1959 issue of The Mathematics Teacher* can be further used to allow for practice in drawing graphs of linear functions. I have found that my students enjoy the idea and they will work independently for long periods when their graphs produce “pictures.”
Right triangle construction
Authors: Gray Nelson S.
Published: November 2020

A helpful table for the teacher who wants to use problems about right triangles and still avoid the familiar 3, 4, 5 right triangle.
An interesting problem involving indeterminate equations
Authors: Lewis Jesse C.
Published: November 2020

In this simple but intriguing example the domain of the variables is the set of positive integers and zero.
Problems in presenting non-Euclidean geometries to high school teachers
Authors: Kattsoff Louis O.
Published: November 2020

What is the place of non-Euclidean geometries and miniature systems in the secondary school curriculum, and in the curriculum for prospective secondary school teachers of mathematics?
Tips for Beginners: Using the library in junior high school mathematics classes
Authors: Schmidt Roland L.
Published: November 2020

Many students possess the idea that mathematics is a hook full of rules by which one gets answers to certain kinds of problems. They need to be shown that it is more than just that. Often the beauty of the orderly structure by which numbers behave is left hidden; each new topic learned is a trick apart.
Encouraging creativity in the mathematics classroom
Authors: Hirschi L. Edwin
Published: November 2020

The teacher often finds himself struggling for survival when he should be pushing back frontiers, opening doors, and evidencing excitement over creative thought.
The mathematics laboratory—a meaningful approach to mathematics instruction
Authors: Kluttz Marguerite
Published: November 2020

A Laboratory, according to Webster, was originally the workroom of a chemist. A more comprehensive definition describes it as a place devoted to experimental study in any science.
Survey of Ohio college opinions with reference to high school mathematics programs
Authors: Brown Robert S.
Published: November 2020

From the time of the appearance of the Report of the Commision on Mathematics in 1959, the mathematics teachers in Marion, Ohio, have been concerned about the contents of our program for collrge-bound high school student.
Historically Speaking,— Can present-day students work problems of seven or eight generations ago?
Authors: Read Cecil B.
Published: November 2020

Students or teachers having access to textbooks their parents or grandparents used arc often intrigued by those problems which are “out of the ordinary,” perhaps reflecting economic conditions as well as changing concepts of what should be included in a particular course. Less attention may be paid to the difficulty of the problems, or the question of what present-day students might do with some of the problems.
The logic of absolute value inequalities
Authors: Chachere Marvin L.
Published: November 2020

Teachers motivated by the need to express the distance of any point on the real line from zero find the definition of absolute value useful and interesting. It is used in elementary algebra to define addition and multiplication of “directed” numbers and to describe the graphs of broken lines in one and two dimensions. Recent emphasis on inequalities and the inclusion in newer textbooks of absolute value inequalities raise several questions that cannot be handled in a perfunctory manner.
Geometric intuition and ab≤a+b2
Authors: Harris Edward M.
Published: November 2020

A plausibility argument leading to
A look at discovery
Authors: Bolding James
Published: November 2020

Although there are several discovery methods in use they all have a common feature. The student discovers by finding the desired conclusion rather than having the conclusion pointed out for his acceptance. If the teacher asks, “What hypothesis does this suggest?” or “What generalization can we make?” the student's mind starts hunting.
A bibliography on the changing curriculum in secondary school mathematics
Authors: McCamman Carol V. , Hill Jane M.
Published: November 2020

Some important articles and books concerning the changing mathematics curriculum
Repeating Decimals and Tests for Divisibility
Authors: Elkin Jack M.
Published: November 2020

Some tests of divisibility dependent upon base ten notation
Vectors in algebra and geometry
Authors: Glicksman A. M.
Published: November 2020

Do you deplore “watertight compartments”? Here is a good way to ignore them.
Historically Speaking,—: Applied mathematics in eleventh-century Iran: Abū Ja'far's determination of the solar parameters
Authors: Kennedy E. S. , Hamadanizadeh Javad
Published: November 2020

To ask people who are familiar with algebraic symbols and decimal fractions to adopt completely the attitude and approach of scientists who had neither, is to request the impossible. Nevertheless, if one seeks insight into the way mathematics developed, it is useful to make the attempt.
Tips for Beginners: Factoring using a square array
Authors: Payne Joseph N. , Maccia Alex
Published: November 2020

There is still a need to teach factoring in algebra. Students must be able to factor trinomials for work on rational expressions and to solve quadratic equations. The purpose of this article is to explain a simple method of testing trinomials to see if they can be factored. The method described is designed with the average and below-average student in mind.
Finding nonlinear factors of polynomials by modular methods
Authors: Lloyd Daniel B.
Published: November 2020

An extension of factoring via modular domains
Historically Speaking—: Recent evidences of primeval mathematics
Authors: Lloyd Daniel B.
Published: November 2020

Our attention here is directed to a site named Tel Harmal, near the east bank of the Tigris River, where that fabulous medieval city of Baghdad was erected over 1,200 years ago. The author had the rare privilege of spending the academic year 1962-63 in Baghad as visiting professor of mathematics at the Al-Hikma University. While there he was the guest of several archaeological field groups engaged in exploring some of the ancient sites in Iraq.
International Mathematical Education: Mathematical Reforms in English Secondary Schools
Authors: Thwaites Byran
Published: November 2020

Editor's note.—Professor Byran Thwaites of Southampton, along with Professor Cyril Hope of Worchester, is among the prime movers of reform in mathematical education in England. This paper gives not only a general review of the reform movement but also an overall philosophy of English education in language that is clear and altogether charming.—Howard F. Fehr.
Algebraic Proof of an Old Number Trick
Authors: Berry William F.
Published: November 2020

NUMBER games appeal to young and old alike. One of the oldest so-called tricks with numbers involves a simple process of reversing digits. The reader has no doubt discovered that if he takes a three-digit number, reverses the digits and subtracts the smaller from the larger, then reverses the digits once more and adds this value to the original difference, the sum will equal 1,089.
A Straight Line Model for Multiplication
Authors: Fukuda Donald , Mookini Edwin , Siu James K. M.
Published: November 2020

IN THE abundant material of modern elementary and secondary mathematics, there appears little to comfort teachers who are unfamiliar with the modern approach and students who have difficulty mastering the highly axiomatic development of subject matter. To overcome these handicaps, this unit of study is presented with the hope of offering structurally sound mathematics at a level which is intuitively discoverable and comprehensible by “average” students. (As this unit was being written, the material was tried in several classrooms by the writers.) This approach allows these students to discover some basic concepts of mathematics through linear relations.
Classics In Mathematics Education: Importance of Mathematics in General Education
Authors: Breslich E. R.
Published: November 2020

Mathematics, like any other subject, is important to the extent to which it supports and contributes to the purposes of general education. For children are sent to school in order that they become useful members of the community and contribute to the welfare of society. The mathematics referred to in this paper designates that subject matter and training which is important to the entire school population, as distinguished from the special mathematics needed for professional and technical education. It is the mathematics which aids in the preparation of pupils for efficient participation in the activities in which they engage later as adults, and for the assumption of their share of social responsibility.
The Method of Equivalence or How to Cure a Cold
Authors: Allendoerfer Carl B.
Published: November 2020

To Set the stage for my remarks to follow, I wish to remind you of a sure-fire method of curing a cold. We all know that the usual cold remedies such as aspirin, hot lemonade, whiskey, and bed rest do little more than relieve the symptoms of a cold, and that the disease itself just has to run its course. According to the theory I am suggesting, there is a better way! You should go out in the rain and get cold and wet, sleep in a draft, and otherwise expose yourself to the elements as much as possible. Then your cold will develop into pneumonia. Now you are in fine shape, for penicillin will cure pneumonia!
A Mathematician's Progress
Authors: Alfred Brother U.
Published: November 2020

Students often note that mathematics in its full dress suit of postulates, definitions, theorems, and formal proofs is more of a fossil than a living organism. The reader who scans such material has no idea of the hours and days of cogitation spent on even minor points, the misdirections that led to blind alleys, the sudden inspirations, and all the various thrills and disappointments that make mathematics a truly human experience. Furthermore, with no particular idea of just how the result was achieved, the student loses a great deal of useful information regarding methods of procedure—and it is precisely such information that would prove most valuable to him and others who are trying to penetrate the domains of mathematics by personal thought.
The Comprehensive Examination as an Entrance Test
Authors: Wilson A. H.
Published: November 2020

A recent political essay begins: “There is one great basic fact that underlies all the questions that are discussed on the political platform at this present moment. That singular fact is that nothing is done in this country as it was done twenty years ago.”
What Mathemathical Knowledge and Ability May Reasonably be Expected of the Student Entering College?
Authors: Hart James N.
Published: November 2020

If we contemplate the numerous committee reports of this and other societies upon the teaching of arithmetic, algebra, and geometry, together with the excellent syllabi published by them, it would seem that our question has been so fully and definitely answered, that further discussion is uncalled for, if not almost impertinent. If, on the other hand, we turn to the series of papers presented before the Association of Teachers of Mathematics for the Middle States and Maryland last year and printed in the MATHEMATICS TEACHER for March and June of the present year upon the question, “What Mathematical Subjects Should Be Included in the Curriculum of the Secondary School,” it is very evident that the doctors still disagree. At one extreme we find a speaker, a college graduate, but not, according to his own confession, a mathematician, arguing that so few students are “mathematically minded” that it would be more profitable to limit high school mathematics to arithmetic and allow the few who reach college to elect algebra and geometry — if they have any remaining curiosity regarding mathematics. At the other extreme, a high school principal presents an “outline of mathematical work that should be required of every student in a general high school course,” including not only the traditional work in algebra and plane geometry, but solid geometry, trigonometry and applications of algebra to mechanics, science, economics, statistics, shop mathematics and the slide rule. And again, when we consider the propositions of the committee on articulation of the N. E. A. and those of the commissioner of education of the state of Massachusetts we realize that the question is not finally settled, and that we mathematicians will not be allowed to settle it by ourselves.
How Shall We Define Angle?
Authors: Sitomer Harry , Fehr Howard F.
Published: November 2020

In Publications of the School Mathematics Study Group, an angle is defined as the union of two noncollinear rays having the same origin. This definition is now widely used in the United States, and has found its way into many contemporary textbooks.
Revolution, Rigor, and Rigor Mortis
Authors: Willoughby Stephen S.
Published: November 2020

When I first heard the recent changes in mathematics education described as a Revolution, my feeling was that the word was too harsh and implied too much violence I preferred to omit the “R.” Perhaps this is because the word “Revolution” always brings to mind the French Revolution and Charles Dickens’ Tale of Two Cities in which, for some reason, I always identified myself with Sidney Carton (the one who lost his head over a girl).
Classics In Mathematics Education: Mathematics in the Training for Citizenship
Authors: Smith David Eugene
Published: November 2020

It is the purpose of this paper to set forth certain facts to be kept in mind in the teaching of mathematirs for citizenship. It is proposed to consider in the limited space available what there is in mathematics that the working man needs and what of this science the woman requires in dirceting the education of her children and in managing her home; how mathematics trains the mind and how its poetry affects human life; what potency the subject has in the uplift of your soul and mine; how it has linked itself with all humanity in all times; and how we should go about to make all this real in our everyday teaching.
What's Between Q and R
Authors: Fey James
Published: November 2020

Among the objectives of school mathematics instruction, one of the most important is to develop understanding of the structure, properties, and evolution of the number systems. The student who knows the need for, and the technique of, each extension from the natural numbers through the complex numbers has a valuable insight into mathematics. Of the steps in the development, that from the rational numbers to the real numbers is the trickiest.
Successful Practices in Teaching Mathematics to Low Achievers in Senior High School
Authors: Greenholz Sarah
Published: November 2020

The NCTM's war on low achievement in mathematics somewhat parallels the President's war on poverty, and, as a matter of fact, the two are closely related. The low achiever is indeed worthy of consideration by teachers because this attention has been unduly delayed in the preceding decade. Especially has this focus been lacking in the senior high school, where there have been no courses in mathematics which the under-achiever could elect.
Pascal's Triangle: A Different Approach to Subsets
Authors: Sgroi James T.
Published: November 2020

At the junior high or senior high level, the usual way of determining the number of subsets of a set is by the choice method. In constructing a subset by this method there are two alternatives for each element of the original set: to place the element in the subset, or not to place the element in the subset.
Historically Speaking—: Poland's Contribution to Mathematics
Authors: Grace Mary
Published: November 2020

Written at any time, a series of articles on cultural developments in Poland would be incomplete without a special consideration of her many achievements in the field of mathematics. Such a consideration is particularly fitting on the occasion of a Commemoration of Poland's Millennium because, in consequence of her acceptance of Christianity in A.D. 966, Poland entered the orbit of influence of the Roman Catholic Church and Western Civilization, an effect which both eased and quickened the development of culture in Poland.
Evolution of Old Secondary-School Arithmetic Textbooks
Authors: Nietz John A.
Published: November 2020

Today arithmetic, except possibly business arithmetic, is commonly considered to be only an elementary-school subject. This was not true in early American schools. The only public schools in America during the colonial period were the “reading and writing schools” and some Latin grammar schools. Neither type commonly taught arithmetic during that period. This means that the earliest arithmetics were mostly used in private schools and academies.
The Basis of Music-Mathematics
Authors: Lawlis Frank
Published: November 2020

Modern man has become accustomed to thinking of music as representing the arts, and of mathematics as inescapably tied to the sciences. Thus, many of today's students fail to see relationships that were readily apparent to early mathematicians and musicians alike. Pythagoras, a Greek mathematician and philosopher who lived about five hundred years before the birth of Christ, investigated the acoustical basis of music through mathematical ratios. Observing the vibrations of a string, he divided its length and found consonant musical notes whose relationships could be expressed in simple number ratios.
The SMSG Angle is a Good One
Authors: Wright Frank
Published: November 2020

The article “How Shall We Define Angle?” by Dr. Harry Sitomer and Dr. Howard F. Fehr in the January 1967 issue of The Mathematics Teacher did not convince me that the SMSG definition of angle should be changed. In fact, after some thought I became convinced that the SMSG definition of angle is better than that proposed by Drs. Sitomer and Fehr.
Experimental Programs: Mathematics For the Learning Laboratory to Teach Basic Skills to Tenth, Eleventh, and Twelfth Graders in a Culturally Deprived Area
Authors: Borata Nicholas H. , Veitch Gladys M.
Published: November 2020

The Learning Laboratory was established to provide basic education for culturally deprived students preparatory to entrance into vocational training or employment.
Historically Speaking—: Rhetoric, Anyone?
Authors: Hughes Barnabas B.
Published: November 2020

A Century and a quarter ago, Nes selmann conveniently trisected “the for-mal representation of algebraic operations and equations.”1 From what he considered an historical viewpoint, he distinguished between rhetorical, syncopated, and sym bolic algebras.
The Rationale of Slide Rule Manipulation
Authors: Rogers Margaret Anne
Published: November 2020

Too often the slide rule is presented in a framework completely devoid of mathematics. To multiply 6 by 2, for example, a student is likely to be given a purely manipulative process such as, “Try setting the left index of the C scale on 6 of the D scale. Note that the slide projects so far to the right of the rule that the D scale no longer exists below the 2. When this happens, place the right index of the C scale on the 6 of the D scale. Find 2 on the C scale and below it read the product on the D scale.”
Polygon Sequences—an Example of a Mathematical Exploration Starting with an Elementary Theorem
Authors: Mann John E.
Published: November 2020

One way to bring out the mathematician in a student is to occasionally give him a topic to explore on his own. A teacher can offer himself as a resource for problemsolving hints, but should discourage the student from looking up relevant material in books. 1 Problems or topics used for this purpose must be simple enough to be within the reach of the student and should be presented in an indefinite enough manner so that the exploring individual can choose his own path of investigation. Ideally, the topic chosen should be rich in potential or implied mathematics so that the student may find the topic responsive to his efforts. 2
A Progress Report on the Implementation of the Recommendations of the Commission on Mathematics
Authors: Williams S. Irene
Published: November 2020

A Little over ten years ago the Commission on Mathematics was formed by the College Board at the request of its Committee of Examiners in Mathematics. Although the Commission's primary concern has been the mathematics preparation of college-bound students, it has nevertheless been influential in shaping the efforts of the many groups of mathematicians and educators to reinvigorate the teaching of mathematics throughout the schools.
Elliptic Parallels
Authors: Gridgeman N. T.
Published: November 2020

Although we habitually think of parallelism in terms of line segments, the concept is readily extensible to curves. The topic of parallel curves is old and presently unfashionable, but rich in interest. Various methods of study can be brought to it— those of differential geometry are particularly rewarding—but attention will here be confined to elementary methods, with emphasis on the ellipse as the primary curve.
That Area Problem
Authors: Greenberg Benjamin
Published: November 2020

The following problem has created interest in various high schools and even in some colleges. The proof presented here is designed to give the serious student of plane geometry some insight into techniques that can be used in other originals.
Induction: Fallible but Valuable
Authors: Graening Jay
Published: November 2020

Geometry teachers often face the problem of getting their students to see the power of inductive reasoning. To find ways of pointing up the limitations of induction without leading students to reject its strengths is not an easy task. Situations in which students will leap to incorrect inductive generalizations are common, and frequently such experiences are used to illustrate the need for proof and deductive methods.
A Problem, A Solution, and Some Commentary
Authors: Gross Herbert I.
Published: November 2020

Some time ago the following problem appeared in a rather popular journal: What is the smallest number with the property that when the digit furthest to the left is moved furthest to the right, the resulting number is 50 percent more than the original number?
Using flow Charts with General Mathematics Classes
Authors: Krulik Stephen
Published: November 2020

One of the problems facing teachers of mathematics in inner-city schools is to find materials with which to begin a course in general mathematics. Since many of the textbooks just repeat material that has led to previous failure, many teachers find that their main source of material has been taken away. To use this same material as a starting point will lead slow learners to ignore the teacher.
Probability, Integers, and Pi
Authors: Fey James T.
Published: November 2020

One of the most beautiful, yet puzzling, features of mathematics is the occurrence, in apparently unrelated situations, of important numerical constants and patterns.
A Triangle from Trigonometry
Authors: Tracy Melvin R.
Published: November 2020

Some interesting patterns develop when one considers the triangular array formed from the coefficients of sin (nθ) + cos (nθ) expressed in terms of powers of sin θ. Certain of these patterns are similar to those of Pascal's triangle.
A Transformation Approach to Tenth-Grade Geometry
Authors: Usiskin Zalman P. , Coxford Arthur F.
Published: November 2020

There are many individuals who have called for the insertion of transformations into the tenth-grade geometry course; for example, Jeger (1966, preface), Adler (1968, p. 226), and Betz (1933, p. 110). For some time, texts have been appearing with a section or chapter devoted to reflections, rotations, or other kinds of transformation (Kelly and Ladd 1965; Fischer and Hayden 1965; Fehr and Carnahan 1961; and Jurgensen; Donnelly; and Dolciani 1969).
New Publications
Authors: Dodes Irving Allen , Peak Philip , Price Jack , Schaumberger Norman
Published: November 2020

The intent of this book was to provide a beginning algebra text that is easily readable.
Preparation for Effective Mathematics Teaching in the Inner City
Authors: Feinstein Irwin K.
Published: November 2020

Little is known about inner-city youth or how they learn mathematics, the characteristics of good teachers in urban centers, and related educational problems. Prospective teachers for inner-city schools enter universities and schools of education with widely diverse abilities, most likely the result of their experiences in high school and their differing biological makeups. These future teachers of innercity youth differ in their ability to recall and understand the mathematics they have already learned, the speed and accuracy with which they learn new mathematics, the strategics of inquiry that they have formed, and most importantly, their desire to learn new mathematics (Shulman 1970).
Commutativity of Linear Functions
Authors: Ehrenpreis Walter , Iannone Michael
Published: November 2020

Quite early in a student's mathematical study, he is introduced to the idea of commutativity. Initially the concern is primarily with the arithmetic operations, and in most instances the study of this property terminates there. In this article, we incorporate some ideas typically taught at the high school level (linear functions and composition of functions) to investigate the notion of commutativity in yet another setting. Of perhaps even greater interest than the main result of this paper is the strategy used to arrive at the original conjecture. Polya has written extensively about such strategies1, and the authors believe that the material can be easily presented to senior high school students with a view to illutrating how certain s trategies are used in the cretttion of a mathematical theorem.
Happy Integers
Authors: Duncan Donald C.
Published: November 2020

A GREAT deal of the time in school mathematics is necessarily spent in learning processes and techniques. There is usually very little time for students to develop their own ideas and make their own conjectures in a field not already known. This paper grew out of the author's investigation of what he thought was an entirely virgin field of mathematics, a field that provided for him a taste of mathematical research. Recently several references to the topic were discovered; these are included in the notes at the end of the article.
Back to 2
Authors: Rothbart Andrea
Published: November 2020

In courses dealing with the real number system, it is fairly common to prove that is irrational. Indeed, as Harris (MATHEMATICS TEACHER, January 1971) pointed out, there are several ways of doing so. Unfortunately, many of the proofs cited cannot be easily adapted to establishing that numbers such as , , and are also irrational. The proof offered in this paper that is irrational can be generalized to a proof that numbers such as those listed above are also irrational.
Topology—Through the Alphabet
Authors: Ranucci Ernest R.
Published: November 2020

Visual aspects of topology rate rather low in the lexicon of the topologists. In fact, visual aspects of anything rate rather low with the mathematician. It is, however, the very naïveté of a visual approach that fascinates the young student of mathematics. An imaginative teacher can, from time to time, enrich the teaching of geometry with topological gleanings.
An Action Seminar in Mathematics Education
Authors: Shafer Dale M.
Published: November 2020

It is the responsibility of a university to provide for the individuals enrolled in its teacher-education program a place where they can observe, participate, and learn. This obligation was fulfilled for many years by the campus laboratory school. However, the campus laboratory school is becoming as extinct as the whooping crane.
The Results of a Longitudinal Study in Mathematics in New Hampshire
Authors: Austin Gilbert R. , Prevost Fernand
Published: November 2020

A series of longitudinal studies of mathematical computational ability at grades eight and ten in the state of New Hampshire produced the following results. In the fall of 1963. using the Metropolitan Achievement Test and the Arithmetic Computational subtest of that battery, the median raw score for eighth graders in New Hampshire equalled a grade equivalent of 8.8.
Computer Programming for the Mathematics Laboratory
Authors: Kieren Thomas E.
Published: November 2020

One type of worthwhile activity for a mathematics laboratory is the use of a computer facility to solve problems or even do some elementary “research.” To do this the student needs to know a simple computer language like BASIC and needs to have computer facilities available for running programs. This use of a computer is illustrated by a seventh-grade class studying prime numbers. They wanted to write programs that would result in the computer's printing out all prime numbers between 1 and 100. Since there was no set program to follow, each student or group of students wrote his own. The result was several different programs, some of which worked and Others that did not work. In the process of writing programs, executing and debugging them, and studying program output, the students learned a great deal of mathematics about prime numbers.
Mathematics Education: The People's Republic of China
Authors: Swetz Frank
Published: November 2020

The People's Republic of China has attempted to use education as the main avenue for ideological, sociological, and technological advancement. In the educational campaigns that have taken place on mainland China since 1949, two disciplines have always received primary attention: the Chinese language and mathematics. Improved language proficiency was necessary for the processes of indoctrination, while mathematics provided a base for the scientific studies so vital to an industrial competency.
If Pythagoras had a Geoboard …
Authors: Ewbank William A.
Published: November 2020

The theorem of Pythagoras is an old favorite of junior and senior high mathematics teachers, and it lends itself to much interesting discovery. This article describes an investigation into the famous theorem, using geoboards.
Teaching Concepts Incorrectly
Authors: Bompart Bill
Published: November 2020

Teachers are inevitably responsible for students' acquiring some incorrect or misleading information. Even the most capable teacher will, on occasion, make an incorrect statement in class innocently believing it to be correct or will misplace an emphasis in an otherwise correct statement and thereby mislead his students. And who among us can avoid the slip of the tongue that confuses even the best student? This type of incident is certainly unfortunate; but at the same time it is, to a great extent, unavoidable.
Instructional Strategies in Mathematics Education
Authors: Hernandez Norma G.
Published: November 2020

The Search for the perfect teaching method has persisted for many years, In the meantime, many effective strategies have been identified. A strategy is defined, here, as a plan of relatively structured procedures by which specific objectives are achieved. Although many plans have been developed, no one strategy has been shown effective under all conditions. The most promising direction toward effective mathematics teaching at this time appears to be the use of a variety of strategies tailored to the objectives of the lesson and student needs at the time of instruction.
Multiple-Choice Mathematics Questions—How Students Attempt to Solve Them
Authors: Williams S. Irene , Jones Chancey O.
Published: November 2020

AT THE request of the College Board Committee of Examiners for the Mathematics Achievement Tests (Level I and Level II), members of the Mathematics Department at Educational Testing Service, including the authors, carried out a small-scale study in an attempt to determine whether certain kinds of information that could not be obtained from regular statistical analysis could be obtained by interviewing students.
A New Application for Base Four
Authors: Blanton F. L.
Published: November 2020

A different approach to the use of base four may well be applicable at the secondary level when different number bases are discuued.
Recreation: Permutation Puzzle
Authors: Spaulding Raymond E.
Published: November 2020

Analysis of a difficult problem is often eased by prior solution of a simpler related challenge. The following permutation puzzle provides useful insight into a rich family of “sliding marker” games.
Figurate Numbers
Authors: Weaver Cloman
Published: November 2020

You will want to try some of these interesting ideas on patterns of figurate numbers with your junior or senior high school students.
A Geometry lesson from National Assessment
Authors: Kerr Donald R.
Published: November 2020

The geometry exercises administered during the 1977-78 mathematics assessment of the National Assessment of Educational Progress (N AEP) focused primarily on concepts of informal geometry rather than the more formal deductive geometry typically taught in a high school course. Results did show that students who studied formal geometry performed better.
Clubs: Skits in the Mathematics Class
Authors: Perry Sally
Published: November 2020

To add interest and variety to my mathematics classes and to stress a particular concept, I have students perform skits. Sometimes it is one that I have written, and sometimes groups of students write and perform them.
Randomness, Normality, and Hypothesis Testing: Experiences for the Statistics Class
Authors: Duncan David R. , Litwiller Bonnie H.
Published: November 2020

Teachers of statistics are always interested in numerical examples concerning basic statistical questions. Among many such questions, we have selected the following three for discussion:
Applications: Rating College Passers
Authors: Jamski William D.
Published: November 2020

In 1980 there were many good passers in college football, but how would you rate them? The National Collegiate Athletic Association (NCAA), the major governing body for intercollegiate athletics, proceeds by maintaining statistical data on the performance of the passers in a number of categories. This information provides the bases for making objective comparisons among the passers. For example, in table 1, the four leading passers for NCAA Division I (major college football) in 1980 are listed alphabetically with their respective passing statistics. From these data, differences in the performance of the passers can be noted.
Applications: Tracking Dinosaurs With Trigonometry
Authors: Hersberger Jim , Farlow James O.
Published: November 2020

If you and your trigonometry students grow weary of fictitious forest rangers and contrived scenarios involving ships that may or may not pass in the night, then the following real-life application will probably appeal to you. It demonstrates the usefulness of several trigonometric concepts and identities and can be used in a variety of classroom formats.
Reviewing an Old Subject: Math Anxiety
Authors: Skiba Aurelia E.
Published: November 2020

We are all too familiar with the flushed and fidgety student who is frustrated daily by the tortuous rigors of a mathematics class. For forty to forty-five minutes each day, this student cannot seem to do anything right. Even simple calculations are incorrect; everything becomes a jumble; her mind goes blank, and she is convinced that there is absolutely no way she can pass this mathematics test! Furthermore, the students with the most serious problems are those for whom their “math anxiousness” pervades other classes to the extent that they are failing two or three courses and are put on academic probation by the school.
Sharing Teaching Ideas: Angle Hunt
Authors: Reger Cheryl
Published: November 2020

The concept of angle is basic in the study of geometry. It is either directly or indirectly a part of the mathematics curriculum at all levels. The activity presented here allows students to estimate angle measures and actually to measure angles. It is easy enough for K–8 students and yet interesting enough to motivate ninth- and tenth-grade students.
A Problem-solving Approach to Teaching Second-Year Algebra
Authors: Kerekes Vera
Published: November 2020

Problem-solving strategies are important parts of our middle school curriculum. Teaching strategies is an excellent way to help students attack mathematical, as well as other, problems. Such strategies include guessing and checking, simplifying the problem, building a model, developing a systematic list or a chart, working backward, drawing a picture, and looking for a pattern. Our students spend an entire school year in the eighth grade to learn to use these problem- solving strategies to solve problems that would otherwise require sophisticated mathematical tools if they could be solved at all by mathematical methods. This experience promotes the development of intuition and number sense in young students.
Implementing The Standards: Using Calculators in The Standardized Testing of Mathematics
Authors: Kenelly John
Published: November 2020

Can standardized examinations in mathematics survive in the brave new world envisioned in the NCTM's Curriculum and Evaluation Standards for School Mathematics (Standards) (1989)? The answer must be yes.
Sharing Teaching Ideas: An Assessment Example
Authors: Schloemer Cathy G.
Published: November 2020

I wanted to create a vehicle for assessing students' ability to communicate to the class their attempts at problem solving. In the past, homework problems had been presented at the chalkboard by volunteers, usually the same few students each day. To afford everyone the opportunity to participate and to allow students to prepare their problem solutions in advance, I selected homework problems that I judged to be most troublesome and assigned them for presentation by drawing the students' names from a hat.
Can You Graph x5 + xy = x3 + y3?
Authors: Stover Donald W.
Published: November 2020

This problem is not “natural” for a graphing calculator nor for typical computer-graphing software. The difficulty is algebraic: neither y or x can be isolated in a form suitable for graphing separate functions. (Some symbolic computing packages, such as MAPLEV Release 2, do have the ability to graph implicit functions.)
When Is a Quadrilateral a Parallelogram?
Authors: Toumasis Charalmpos
Published: November 2020

In most high school geometry textbooks the subject of parallelograms is usually treated in the following manner. First the definition is given: “A quadrilateral in which the opposite sides are parallel is called a parallelogram.” After this definition, some properties of parallelograms are mentioned that can serve as criteria for determining when a quadrilateral is a parallelogram.
Projects
Authors: Long Vena M. , Gay Susan , Lawrence David P.
Published: November 2020

During the workshop, the teachers received training on the operation of Texas Instruments, Casio, and Sharp graphing calculators. Copies of the Quick and Easy series of books for using the calculators were used during the training. These books are authored by David P. Lawrence and distributed by Pencil Point Press. In addition to training on different types of calculators, the workshop participants received a Casio 7000 calculator. The teachers mentioned that the opportunity to become experienced with different types of calculators was a valuable experience that would help them when working with their students.
A Discrete Analysis of “Final Jeopardy”
Authors: Floyd Jeffrey K.
Published: November 2020

Jeopardy has long been a popular fixture among television game shows. Most Americans have watched it at least once, but the author will summarize the game for those who have not.
Trigonometry for Non-Trigonometry Students
Authors: Boyes G. R.
Published: November 2020

This team-oriented student activity is designed to study selected trigonometric functions by constructing tables of values. These tables are then put to practical use. Familiarity with trigonometry is not needed, since all pertinent information is included. The activities, which could take up to three days, are aimed at ordinary middle school students or any students desiring an introduction to trigonometry.
Implementing the Professional Standards for Teaching Mathematics: The Evaluation of Teaching: Challenge and Opportunity
Authors: Grouws Douglas A.
Published: November 2020

Teachers are under enormous pressure to produce results, and, directly or indirectly, teaching evaluations are part of that pressure. In Kentucky, for example, schools are rewarded or sanctioned, depending on their students' performance on various types of assessments (Bush 1992). To channel reform energy into productive effort, a change must occur in the current preoccupation with finding scapegoats for poor student performance and on holding teachers singularly accountable for shortcomings in the educational system. Shifting the focus of teaching evaluations from teacher accountability to improving instruction is a step in the right direction because it will increase the usefulness of these evaluations as teachers work to increase students' learning in mathematics. Any reoriented teaching-evaluation process must be comprehensive in nature and involve a cyclic process of teaching assessment, professional development, and instructional change, as advocated in the Professional Standards for Teaching Mathematics (NCTM 1991).