bluuebunny/crossref_metadata_2025 huggingface dataset:
Points and Viewpoints: A Fitting Tribute
Authors: Brune Irvin H.
Published: November 2020

A column of unofficial comment
Dr. Hopkins’ Proof of the Angle Bisector Problem
Authors: Constantia Mary
Published: November 2020

Paucity of proofs? Yes, if you mean direct proofs.
Tips For Beginners: Motivating students with projects and teaching aids
Authors: Wright Frank
Published: November 2020

As teachers, we all appreciate the value of any motivational device, and we recognize the value of teaching aids to stimulate interest. I have found the assignment of teaching aids and projects a useful way to motivate students.
A world of hope-helping slow learners enjoy mathematics
Authors: Proctor Amelia D.
Published: November 2020

How pupils can enjoy mathematics in two senses of the word.
Historically Speaking,—: How Ptolemy constructed trigonometry tables
Authors: Brendan Brother T.
Published: November 2020

Historians point out that, while in economics Gresham's Law holds (“bad currency drives out good currency”), in mathematics a kind of reverse law holds: Good books drive out bad or inferior books. Thus, historians can offer evidence that Euclid's Elements drove out of circulation the geometry texts of his predecessors, and, in a less well-known instance, they can show that Ptolemy's Almagest made the trigonometric writings of Archimedes, Hipparchus, and perhaps others disappear. And in Ptolemy, just as in the case of Euclid, there are some beautiful gems of mathematical literature available —even if his system of astronomy has been superseded.
Connectionism and the teaching machine
Authors: Manheim Jerome H. , Manheim Sylvia R.
Published: November 2020

Is it time to reassess the victories?
On the meaning of structure in mathematics
Authors: Taylor Howard E. , Wade Thomas L.
Published: November 2020

How should we endeavor to teach structure?
The secondary teacher and elementary school mathematics
Authors: Tredway Dan
Published: November 2020

A challenge lies at our own back door.
Historically Speaking,—: The mathematics of the honeycomb
Authors: Siemens David F.
Published: November 2020

Bees have long interested man. The sometimes accurate, sometimes fanciful, reports of ancient writers, including Aristotle and Vergil [1],* demonstrate this interest. But the first known man to comment on a mathematical aspect of bees was Pappus, the fourth-century Alexandrian. He noted that bees are wise enough to use the most economical pattern for their cells (see Table 1) [2], With this, the documented observations of bees relevant to mathematics ceased for some thirteen centuries.
Of “prime” concern: What domain?
Authors: Brown Stephen I.
Published: November 2020

Comparisons can be made among different domains in solving number theoretic problems.
A portrait of √2
Authors: Ringenberg Lawrence A.
Published: November 2020

A careful look at the most frequently cited irrational number
Addition and subtraction on the soroban
Authors: Haas Victor E.
Published: November 2020

Teachers of pupils who see well and teachers of visually handicapped pupils can all learn from this discussion
Archimedes
Authors: Pixley Toren W.
Published: November 2020

It is my contention that the dry bones of history can be rattled to produce a merrier tune than the mournful dirge so often associated with this subject. Tho following is an attempt to take the life of a famous mathematician, Archimedes, and to present the accepted facts of his life in a way that may titillate the high school pupil's interest and thus be conducive to developing in the pupil an interest in the history of mathematics so that he will do more reading in this area. The facts for this article have been garnered from histories of mathematics by Howard Eves, Vera Sanford, E. T. Bell, and others.
A Sieve for Primes p > 5
Authors: Shogren Merle , Yates Richard
Published: November 2020

Although there are many listings of primes available for the person who is interested in the primes per se, the classroom teacher is often more concerned with a procedure for making a list of primes than with the list itself. Probably the best-known method is what is commonly called the Sieve of Eratosthenes, which, paradoxically, is probably the least-used method. It is cumbersome in that so many composite integers must be written and then eliminated.
Designs for Future Explorations in Programed Instruction
Authors: Heimer Ralph T.
Published: November 2020

There are evidences within the educational profession of a growing disenchantment with what is popularly known as programed instruction. It is not difficult to find reasons for this turn of events: (1) very few teachers have learned to make really effective use of programed materials; (2) extensive experience with existing programed materials has convinced many users that the teaching effectiveness of these materials has fallen short of expectations; and (3) there is widespread feeling— backed by considerable research support —that the fundamental principles upon which most programed materials have been based are in need of considerable reexamination.
Historically Speaking—: A Book Printed in the Year VII
Authors: Read Cecil B.
Published: November 2020

When I tell one of my colleagues that our library contains a book printed in the year Seven, there is usually a somewhat incredulous and startled look. Many do not believe the statement; very few ask the question: “According to what calendar?”
Omar Khayyam
Authors: Kennedy Evelyn
Published: November 2020

Omar ibn Ibrahim was born around the middle of the eleventh century near Nishapur, Persia. He worked at his father's trade of tentmaking until the time might come when he could devote his energies to study.
International Mathematics Education: Mathematics Teaching in Afghanistan, 1964
Authors: Tranberg Carl C. , Kayhan Hussein
Published: November 2020

Editor's Note—Most readers of the Mathematics Teacher are interested in forward movements of the developed countries. Here, however, is a description of the status of mathematics in a developing country and some of the problems that are encountered in adapting a modern program to the schools of a country relatively poor financially, but rich in its endeavor to improve educational opportunity.—Howard F. Fehr.
Aba Daba Daba
Authors: Plunkett Betty
Published: November 2020

From a song popular in the 50's we heard “'Aba daba daba' in monkey-talk means 'Chimp, I love you true.'“ This may well be, but, in this paper, I would like to assign somewhat different usages to the terms and examine the results of such assignments.
A Note Concerning Teaching Machines and Theories of Learning
Authors: Fejfar James L.
Published: November 2020

The March 1965 issue of the Mathematics Teach eh contains a cogent argument to the effect that current interest in programmed instruction has brought about a revival of an outmoded method of teaching based upon the learning theory known as connectionism [1].* This revival has occurred at a time when mathematics teaching has advanced beyond a stimulusresponse- reinforcement type of teaching into an era in which we focus on concept formations through discovery, understanding of generalizations, cognitive structures, and other characteristics which are more compatible with other theories of learning. The article raises some pertinent pedagogical questions and suggests that the answers will not automatically come from the compounding of connectionism and programmed instruction. The purpose of this note is to extend and analyze some of the theoretical issues raised in the original art icle by pointing out that programmed materials based upon other theories can be, and have been, produced.
International Mathematical Education: What Does The Secondary-School Teacher Of Mathematics Expect From The University?
Authors: Delessert André
Published: November 2020

Again the editor gives his sincere thanks to Dr. and Mrs. Julius H. Hlavaty for making a translation from the French of the original manuscript.
A Proposal for the High School Mathematics Curriculum—What Does It Mean?
Authors: Zant James H.
Published: November 2020

The significant thing about Dr. Kline's paper is the question it raises in the mind of the reader. Is Dr. Kline finally coining around to the point of proposing a new mathematics curriculum for the high school and doing something about it? He will certainly not make the mistake, made by many of us in the past ten years, of calling it “modern mathematics.” The idea for the first-year algebra is here; and, though he says, “The content … is the traditional one,” this is not entirely true; certainly its organization is not traditional. If he would now find what mathematics is being taught in grades 1-8 and obtain the help of some good high school teachers who know how students of this age level learn, he could write a book, and then all of us could find out what he is talking about.
Experimental Programs: Prediction of Achievement in College Mathematics
Authors: Wampler Joe F.
Published: November 2020

Most previous studies predicting success in college mathematics have used combinations of measures of intelligence and previous knowledge of mathematics as predictors of grades in mathematics courses. While the prediction formulas derived in these studies were useful for the purposes for which they were developed, in none of these studies was more than about 50 percent of the total variations in the criterion variables attributed to their relationship with the prediction variables.
International Mathematical Education: The Teaching of Vectors in The German Gymnasium, I
Authors: Athen Herman
Published: November 2020

Editor's note: Only recently has the teaching of vectors in secondary school mathematics been given serious attention in the United States. As yet, it is generally true that the subject is taught not at all in the mathematics classroom and only to a limited extent in the physics classroom. Here is a genuinely modern branch of mathematics which has not only great value as a unifier of geometry and algebra but also many applications of real significance in other branches of knowledge. In Germany this topic has been an integral part of the mathematics instruction for university-preparing students since the beginning of this century. We can well profit, then, from the success this teaching has had in the German Gymnasium. The editor expresses his thanks to Dr. Athen for his excellent summary, which is being presented in two parts, the second to appear in the next issue of this journal.—Howard F. Fehr.
Factorization in Integral Domains
Authors: Briggs W. E.
Published: November 2020

The noun “factor” in mathematics quite generally denotes an element appearing in an expression that indicates that the element is to be multiplied by another element. Similarly, the transitive verb “to factor” means to express a given element as a product of factors.
Experimental Programs: The Effects of High School Calculus on Students’ FirstSemester Calculus Grades at the University of Virginia
Authors: Mckillip William D.
Published: November 2020

The author wishes to acknowledge with gratitude the guidance of his adviser, Dr. William C. Lowry, in the preparation of the dissertation on which this article is based.
The Hidden Potential of Low Achievers
Authors: Fremont Herbert , Ehrenberg Neal
Published: November 2020

A NINTH-GRADE class is considering a topic in algebra. Tables of information on the blackboard seem to refer to the results of some experiment in science (Figure 1).
Experimental Programs: Project D: Program for Talented Students in Mathematics in Secondary Schools in Hawaii
Authors: Whitman Nancy C. , Nichols Eugene D.
Published: November 2020

This project was a cooperative venture between the State Department of Education of Hawaii and the University of Hawaii. Its primary purpose was to provide appropriate materials and curriculum experiences for mathematically talented students in the three-year sequences (1) grades seven through nine and (2) grades ten through twelve.
Developing Spatial Perception Via the Experimental Technique
Authors: Whitman Nancy C. , Oda Linda
Published: November 2020

Generally, the experimental technique of teaching mathematics has been relegated to the elementary and junior high school levels. However, this technique can he used advantageously in helping high school students perceive spatial relationships. The following experiment on cycloids was performed by a group of high school students studying analytical geometry. The purpose of the experiment was to help the students visualize the locus of a moving point—something that students have found difficult to do.
Concrete Examples of Commutative Nonassociative Systems
Authors: Scandura Joseph M.
Published: November 2020

There are few examples of commutative nonassociative systems that are readily comprehended and meaningful to junior high school students. Some examples of such systems are outlined in this short article.
Classics in Mathematics Education: Romance of Mathematics
Authors: Slaughi H. E.
Published: November 2020

It May seem unwarranted to speak of romance in connection with a subject commonly supposed to be as dry and prosaic as mathematics. The dictionary defines a romance as a “fictitious and wonderful tale.” The tale which I am about to relate is indeed wonderful but it is not fictitious—it is true and it is the kind of truth that is stranger than fiction.
Editorial
Authors: Smith Eugene R. , Gaylord Harry D. , Babb Maurice J. , Breckenridge William E.
Published: November 2020

In this number of the magazine there appears a preliminary report of the Committee on Testing the Results of Geometry Teaching. This committee has undertaken one of the most difficult investigations under way in this country. Scientific investigation of the results obtained from the study of various subjects is a comparatively new branch of education, and if its conclusions prove to be dependable it is likely to revolutionize curricula and methods of teaching.
The Comprehensive Examination in the Teaching of Secondary Mathematics
Authors: Hart Howard F.
Published: November 2020

The problem of this paper is to trace the changes in the subject matter and method of secondary mathematics which would seem to be required by an examination which is to test the student’s ability to use his mathematics as tools to accomplish the tasks set him.
What Mathematical Subjects Should be Included in the College Curriculum?
Authors: Holder F. J.
Published: November 2020

In the forenoon of this twentieth century, when democracy is asserting itself, no less in the political realm than against the old aristocracy of learning, we who are voluntarily yoked to the common load of teaching mathematics must realize its present state of unrest. Possibly this is not more noticeable in mathematics than in many other branches of the curriculum, and it is probably a bit less conspicuous in the colleges and universities than in the secondary schools; and furthermore, this unsettled condition is by no means confined within the walls of our American institutions, but its constant throbbing is felt in the educational pulse of every progressive country in the world. There seems to be an ever-present desire for a change without first counting the cost of the move; a mere effort to have things different, with no well-defined plan of having them better.
Mathematics as a Means to Culture and Discipline
Authors: Yocum A. Duncan
Published: November 2020

Perhaps the strongest obstacle to the determination of the disciplinary value of mathematics has been the common assumption that its disciplinary value is not open to question. While the new psychology long since shattered the theory of formal discipline in the sense of the training of generally useful mental faculties, the mass of thinkers still look upon mathematics and the languages as formal disciplines,—as pre-eminently adapted to a mental training that results as a matter of course and is carried over without specific instruction into every field of experience.
An Alternative Definition for Equivalence Relations
Authors: Buck Charles
Published: November 2020

It has become standard to define a relation of equivalence as a relation having the R.S.T. properties (reflexivity, symmetry, transitivity). There is an equivalent definition which is sometimes easier to use, and equally easy to remember. In essence, it is much older than the now standard definition, for it is derived from the axiom which Euclid quoted as Common Notion 1.
A New UICSM Approach to Fractions for the Junior High School
Authors: Braunfeld Peter , Dilley Clyde , Rucker Walter
Published: November 2020

Traditionally, a good part of the junior high school mathematics curriculum is devoted to a grand review of fractions. Unfortunately, despite the best efforts of teachers, this review often fails to achieve its aims, and many students still graduate from junior high school without a respectable working knowledge of fractions.
Using the Analytic Method to Encourage Discovery
Authors: Marks John L. , Smart James R.
Published: November 2020

Discovery of ideas, theorems, and methods of proof by pupils is certainly an objective of the mathematics program at every grade. The importance of this objective is revealed as one reads the prefaces of textbooks or articles in professional magazines or listens to speeches at meetings. If asked, most teachers would ordinarily say, “Sure, I use the discovery method.”
A Mathematical Abstraction
Authors: Musser Gary L.
Published: November 2020

When proving the theorem, “For any natural numbers a, b, c, if a|b and b|c, then a|c,” in a class for elementary teachers, I somehow felt that I had made this same proof earlier in the course.
Euclid's Algorithm Revisited
Authors: Foster B. L.
Published: November 2020

Since integer division reduces to repeated subtraction, Euclid's algorithm for finding the greatest common divisor may be recast in terms of subtraction. This is done, for example, in Trakhtenbrot,1 for automatic machine computation.
Mathematics Word Search
Authors: Reys Robert E.
Published: November 2020

The square array of letters below contains at least 56 words that are associated with mathematics. These words are arranged horizontally, vertically, or diagonally, and may be spelled frontwards or backwards.
Sight Versus Insight
Authors: Sitomer Harry
Published: November 2020

Think twice before you say to your high school geometry students, “A conclusion based on observation of a figure is not admissible in a proof.” Let us consider some pedagogic implications of this statement as we examine a proof.
The Revolution at Sputnik-Plus-Ten
Authors: Mueller Francis J.
Published: November 2020

To an outsider, the title of my presentation this evening may suggest a speech on foreign affairs. But to members of the National Council of Teachers of Mathematics, there is little question as to what the revolution refers to: it is our revolution in mathematics education.
Experimental Programs: Effects of Programmed Instruction in Remedial College Algebra Classes
Authors: Nagel Thomas S.
Published: November 2020

The department of mathematics at Oregon State University considers the work in Math 50, Intermediate Algebra (4 credits) to be remedial. For several terms, a questionnaire concerned with student background was completed by each person registered in Math 50. The information tabulated from these questionnaires indicated that the work to be done in the course had already been covered in high school for many students but apparently had not been learned very well.
Grids, Integers, and Number Bases
Authors: Mehl William G.
Published: November 2020

In an earlier article1 the writer suggested a way in which to present a drill exercise for signed numbers. After presenting the material to many junior high school classes, he offers what he considers to be some major improvements over the original suggestion.
Classics in Mathematics Education: The Jabberwocky was A Lark or “The Mathematician Takes a Holiday”
Authors: Jablonower Joseph
Published: November 2020

When Humpty Dumpty turns his perch on the wall into a lecturer's platform and proceeds to popularize the highly technical “Jabberwocky,” he initiates his auditor into only those most obvious of the mathematician's tricks that are exemplified in the poem.
Experimental Programs: Discovery: Its Effect on Critical Thinking and Achievement in Mathematics
Authors: Price Jack
Published: November 2020

For over twenty-five years educators have tried to determine whether certain subject-matter areas promoted thinking which could transfer directly to other learning situations that the student might meet.
In-Service Education in Mathematics: Secondary In-Service Education Program in Bergen County, New Jersey
Authors: Cotter Margaret J.
Published: November 2020

If Large city or county school systems have unique problems with in-service training in mathematics at the secondary level, at least the search for their solutions can be relatively self-contained and self-determined. The problems facing smaller communities are no less serious, although they differ in size and scope. It is this difference in size which frequently places any consideration of local solution beyond the limited resources of individual school systems. Clearly, the problem of in-service education for all teachers in large or small districts is and will continue to be a matter of great concern to all in mathematics education.
The Expression ‘0°’
Authors: Vaughan Herbert E.
Published: November 2020

The expression ‘0°’ is usually not defined and, in fact, a slight fuss is fre quently made about the “impossibility” of assigning a value to this expression. Since definitions are always a matter of con venience, and are forbidden only if they are inconsistent with earlier definitions, it is clearly nonsense to say that it is impossible to give a meaning to ‘0°.’ The purpose of this note is to point out that the value 1 is frequently (usually, but not always, implicitly) assigned to ‘0°,’ and to argue for the explicit recognition of this evaluation.
Have You Read …?
Authors: Peak Philip
Published: November 2020

If you wonder why more progress has not been made in mathematics education, you should read t this article. The author is concerned with questions that he feels should he met, such as: How is mathematics related to science and reality? Why should mathematics be taught? How does a creative mathematician generate mathematical ideas? The article proceeds to discuss some of the positions taken by various people and groups who have worked on the im provement of mathematics education. The following questions were considered: Do leaders in one group tend to take over groups other than their own and thus confuse the issues? Is experimentation being defined in too many ways to make it realistic? Are too many rnathematical concepts left to intuition? Is there somewhere intuition plays little if any part? Is it necessary to re-create mathematics in order to understand it? If mathematics is tied too closely to the real world, are we endangering the attitude of inquiry for its own sake? Have we too long ignored the real goals of mathematics and have we failed to provide an interaction between content and method? I think you and your students will be interested in reading this reaction from Professor Brown.
International Mathematical Education: Arithmetic in the Fifth Class
Authors: Glaymann Maurice
Published: November 2020

The set of multiples of a number; Euclidean division of a natural number by a natural number.
Experimental Programs: High School Mathematics Background of College Freshmen before and after Sputnik
Authors: Kruglak Haym
Published: November 2020

The launching of Sputnik 1 stimulated numerous heated debates and reforms in the teaching of science and mathematics. There are few published studies which compare two generations of high school students with respect to mathematics background during this critical period.
Ave atque vale
Authors: Brune Irvin H.
Published: November 2020

The old order has changed—and given place to the new. In action taken one year ago, the Board of Directors of the National Council of Teachers of Mathematics created a new position and eliminated an old one. As of July 15, 1969, Miss Carol V. McCamman became Managing Editor of The Mathematics Teacher. Miss McCamman came to her new work with a rich experience in teaching mathematics at Coolidge High School, Washington, D.C. She also edited the invaluable Cumulative Index of The Mathematics Teacher, volumes 1– 58, 1908 through 1965. As of the date of this issue, the present editor completes his term.
An Interesting Relationship among the Roots of a Cubic Equation
Authors: Mandelbaum Joseph , Schild Albert
Published: November 2020

There are many instances in mathematics where students apply results that are true in special cases to situations in which these results are not true. We are going to discuss one such example.
Developing a finite Geometry: A Math Club Approach
Authors: Bundrick Charles M. , Frazier Robert C. , Gerber Homer C.
Published: November 2020

High School students who have had successful experience with elementary algebra, naive set theory, plane geometry, and mathematical systems form the audience for whom this paper was written. It is hoped, however, that others, too, will find it appropriate and enjoyable. It provides an example of a finite geometry; the axioms for a nine-point geometry are presented and used to prove ten theorems in a mathematically sound development. It illustrates the use of intuition (particularly sketches) in developing a rigorous proof. It strengthens the concept of mathematical system; to solve a complicated problem, the students in the story develop a mathematical system that describes the problem situation. Readers will undoubtedly discover other uses.
Computer-oriented Mathematics: Predicting the Outcome of the World Series
Authors: Brown Richard
Published: November 2020

The much-publicized odds for the “underdog” New York Mets of last year's World Series raised several questions in my probability class.
Multitudinous Kinds of Counting Numbers and Their Generating Functions
Authors: Zwier Paul J.
Published: November 2020

The distinguished mathematician Herman Weyl, in attempting to characterize the “life center” of mathematics, has called mathematics “the science of the infinite.”1 If this was true in 1937, it is even more true today. Thus it is common today not only to consider such infinite sets as the set of all counting numbers, or the set of all real numbers, or the set of subsets of the reals, but to associate numbers with such infinite sets and to perform arithmetic operations with them. Early in the training of any serious student of modern mathematics the terminology of the infinite cardinal numbers and ordinal numbers and their transfinite arithmetic becomes rather standard and commonplace.
Girolamo Cardano: A Defense of His Character
Authors: Bidwell James K. , Lange Bernard K.
Published: November 2020

If What he heard is correct, Girolamo Cardano was born much to the dismay of his parents, who were not married. They attempted several unsuccessful abortions before his birth on 24 September 1501. Girolamo's father, Fazio, although he practiced law in Milan, found more time for other interests, such as medicine, mathematics, and occult lore. Fazio was highly esteemed as a scholar by his contemporaries in Milan. He apparently, for example, consulted with Leonardo da Vinci about flying machines.
Zip the Strip
Authors: Long Clifford A.
Published: November 2020

Most of us have been intrigued at some time or other by the MÖbius strip— a one-sided, one-edged, bounded surface.1
National Assessment in Mathematics
Authors: Foreman Dale I. , Mehrens William A.
Published: November 2020

The development of the mathematics area of National Assessment is now in the final stages before the assessment. The actual field administration is scheduled to begin in the fall of 1972. This article is a report of the objective and exercise development in the mathematics area that has taken place between 1964 and the present date.
Developing an Algebra by Discovery
Authors: Henry Boyd
Published: November 2020

For many students, discovery makes mathematics interesting and fun. Therefore we should watch for chances to give students small research problems appropriate to their abilities and interests. This article points to a number of questions that can be explored by the able high school student in a topic with which he has some slight experience but which he is unlikely to have explored in any detail.
Mathematics on a Pool Table
Authors: Grant Nicholas
Published: November 2020

For centuries billiards—or pool, as pocket billiards came to be known—has held the interest of many people. It is not different with today's high school and junior high school students. Young people with little skill at approximating correct answers to algebra problems will persist in their attempts at drawing the correct path of a pool ball over a table until it drops into a pocket.
Mathematical Models and Applications as an lntegral Part of a High School Algebra Class
Authors: Bell Max S.
Published: November 2020

The problems of implementing widely supported recommendations for change in school ptactice are formidable and not well understood. Nowhere is this better illustrated than with the recommendations, repeated since at least 1900, that we do more in schools with the uses of mathematics. This article will first give some brief background relevant to this problem and then report in some detail an effort to deal with it in a classroom setting. This effort can be considered an attempted existence proof that mathematical models and contemporary uses of mathematics can be made a useful part of a standard school course1.
The Evaluation of ∫abdxx2 and ∫abdxx
Authors: Schaumberger Norman
Published: November 2020

Prior to the introduction of the fundamental theorem of calculus, it is a Common practice to use the definition of the definite integral to evaluate integrals of the form . These calculations are usually limited to cases in which k is a positive integer but are rarely extended to values larger than 3.
Gaining Independence in Mathematics Learning
Authors: Wilson Kay B.
Published: November 2020

The purpose of this article is to share ray experiences about two methods of teaching seventh-grade mathematics to high-ability students. Seventh graders are often restless and impatient in mathematics classes; this seemed especially true of a high-ability class I taught last year, For most of the year I used mainly a question-discussion method: using examples, asking questions; and trying to elicit summaries of the concepts from the students. I varied this by allowing time for individual or team work and using various audiovisual devices.
Adolescent Attitudes Toward Mathematics
Authors: Callahan Walter J.
Published: November 2020

A CONSCIENTIOUS and dedicated teacher must have the interests of his students in mind at all times. A teacher must constantly search for ways to help students succeed at their learning tasks. As a middle school mathematics teacher I feel a strong need to look for ways to help my students succeed in and enjoy mathematics.
New Products
Authors: Hansen Viggo P. , Johnson Randall E. , Koetke Walter J.
Published: November 2020

This is an intriguing game of linguistics wherein players use inductive reasoning to learn scientific methods of inquiry. It is designed to develop the basic skill of asking good questions. Players try to detect the properties of a generative language that one of the players has secretly defined by building a set of basic sentences and replacement rules. Thr opponent aims to construct good queries, and to do this he must become a good theorist. These processes are parallel to the methods used in science to probe thr laws of nature. The game can be played by two to four players and is for teen-agers and adults.-Sheila K. Berman, Patrick Henry Junior High School, Granada Hills, Calif.
A Physical Model for Factoring Quadratic Polynomials
Authors: Bidwell James K.
Published: November 2020

It is common practice to approach the actoring of polynomials by use of the listributive law.
‘Tain't Necessarily So, or Minus Minus a Equals A
Authors: Engen Henry van
Published: November 2020

Teachers sincerely concerned with instructional poblems in mathematics sometimes take inflexible positions, thereby ruling put alternate positions that have a solid base. This occasional inflexibility la nicely illustrated by those who contend that it is wrong to read —(— A) as “minus minus A.” They contend that the correct reading is the “inverse of the inverse of A.” However, the pre-SMSG courses in first-year algebra cannot be faulted for how the symbol —(—A) was read, but these courses can be faulted for what they left unsaid. Usually these courses did not teach the concept of an additive inverse.
The Forum: Individualization of Instruction: The Use of Computers to Individualize Instruction
Authors: Jerman Max
Published: November 2020

Webster's dictionary defines the word “individualized” as meaning (1) to make individual; mark as different from other persons or things; and (2) to consider individually; particularize. Others, including the writer, are more inclined to define the term in an educational context as meaning (1) that each student should be permitted to proceed as rapidly as his own ability will permit and (2) that instructional variables such as reading level or the level of difficulty of exercises be adapted to the achievement level of each individual as much as possible, wherever Possible. There are subtle differences between these two definitions which are ob-vious when one hears adherents of one or the other position cite his rationale for employing computer-assisted instruction (CAI) as one means of individualizing instruction.
Individualizing Instruction with Homemade Filmstrips and Audiotapes
Authors: Neatrour Charles R.
Published: November 2020

For many years teachers have found filmstrips useful in their classrooms. Now, through the commercial availability of “U” Film, the teacher can design and prepare his own filmstrips; he is no longer limited to catalog listings. Accompanied by an audiotape, a homemade filmstrip can enhance both group presentations and individualized instruction; it offers endless opportunities for enriching learning.
Individualized Mathematics Instruction through a System of Continuous Progress
Authors: Osmundson Arnold
Published: November 2020

The word in secondary education today is to “individualize” instruction. Attend any education convention and check the sectionals dealing with the topics of “continuous progress” or “individualized instruction” and my observation will probably be borne out. At Kennedy High School in Bloomington, we have done things along these lines that I believe are in part unique and that may prove informative and error-saving to other teachers considering this type of an individualized program.
Modulo 7 Arithmetic—A Perfect Example of Field Properties
Authors: Henry Boyd
Published: November 2020

MANY students find it difficult to get very excited about formal proofs of theorems that they already believe intuitively. Properly developing the field axioms and the usual theorems that follow requires several class meetings. To the more mathematically mature, the process of justifying many operations that have previously been taken for granted is one of the most satisfying of all activities in mathematics. The care with which the field axioms were chosen is elegantly demonstrated as proofs are developed for theorems that have previously been accepted because they appealed to the intuition. Many of our students, however, fail to appreciate the beauty of these proofs. Perhaps one of the reasons that they remain unmoved is that they see little application of the field properties other than to facts that seem intuitively obvious to them.
Problem Solving in Geometry
Authors: Hiatt Arthur A.
Published: November 2020

One of the most important tasks of education is to develop the decision-making process of each individual. If mathematics is to aid in this process, its presentation must elicit from the student creative, problem-solving abilities. A strictly axiomatic approach to the teaching of mathematics often stifles this creative thinking. Although axiomatics and structure are important, they are secondary considerations in problem solving.
A Transformation Proof of the Collinearity of the Circumcenter, Orthocenter, and Centroid of a Triangle
Authors: Shilgalis Thomas W.
Published: November 2020

ONE of the arguments in support of a transformation approach to high school geometry is that mapping techniques often greatly simplify the proofs of important theorems. The well-known result contained in the title of this article is an excellent example of a theorem that lends itself to a proof by transformations.
A Classroom Discovery in High School Calculus
Authors: Fabricant Mona
Published: November 2020

Many times we, as teachers, get too involved with the concepts that we think are important for the student to grasp and forget that an important part of learning mathematics is exploring theories and ideas that crop up incidentally.
Teacher Perceptions of Basic Mathematics Skill Needs in Secondary Vocational Education
Authors: Long Thomas E. , Herr Edwin L.
Published: November 2020

Mathematics teachers, as well as specialists in other subject-matter areas, are confronted by the need to assist students in understanding the importance of their subject to different areas of application. It is necessary, therefore, to articulate learning within the objectives of a particular course sequence to other courses, educational experiences, or vocational tasks to which the learning is a requisite. This helps the learning of specific content become something more than an isolated academic exercise in “getting the correct answers.”
Spirolaterals
Authors: Odds Frank C.
Published: November 2020

The square, the two-by-one rectangle, and the radioactivity symbol shown in figure 1 are familiar sights to most students. What has not been previously shown, however, is that all three figures are members of an infinite series of geometric patterns called spirolaterals.
Some Investigations of N-Dimensional Geometries
Authors: Abbas Sallie W.
Published: November 2020

There are certain regularities in the extension of a geometry from n dimensions to n + 1 dimensions, and thence to n + 2, and so on. Our intuition is of great aid in the transitions from zero-dimensions to oue- to two- to three-dimensional geometries, and the patterns exhibited in these low-level jumps continue on to higher dimensions. Some of these regularities will be explored in this paper.
Some Investigations for Students of Mathematics
Authors: Henry Boyd
Published: November 2020

Sometimes a rather simple mathematical question can open a Pandora's box of other questions … each of these in turn leading to still more questions for investigation. This, of course, is how new knowledge is built. Consider the following question and some of the many possible problems that subsequently arise.
A Geometric Interpretation of the Simplex Method of Linear Programming
Authors: Nadler Maurice
Published: November 2020

Concepts and techniques of matrix algebra, sometimes relatively sophisticated ones, are the mathematical basis of the simplex method of solving linear programming problems. The procedures, however, can be learned without advanced mathematical knowledge and can to an extent be explained on the basis of elementary mathematics. Simple analytic geometry of the straight line will be used for an explanation of the strategy of the simplex method.
Teaching Mathematics to the Disadvantaged, Israeli Style
Authors: Weiss Sol
Published: November 2020

A class of twelfth-grade “culturally deprived” students sit in a spacious, cheerful room in the finest academic high school in Jerusalem discussing the derivative of a function and its geometric interpretation. These students are in their fourth year of an experimental study designed to test the hypothesis that an adolescent whose intelligence has been impaired by adverse social conditions can acquire, through suitable teaching methods, intellectual equality with the ordinary student in an academic high school.
Functions, Mappings, and Mapping Diagrams
Authors: Brieske Thomas J.
Published: November 2020

Mathematicians, mathematics educators, and educational psychologists agree that “unifying concepts” are important in learning mathematics. More emphatically, mathematicians and mathematics educators realize that the concept of function, or mapping (in this paper the terms are synonymous), has great unifying power. Thus mapping should not remain a rather isolated component of the high school mathematics curriculum but should permeate the entire mathematical experience of our students. The purpose at this paper is to point to a few topics in school mathematics in which the use of mappings and mapping diagrams could be nade explicit.
The Organization, The Goals, and Some Activities of The National Council of Teachers of Mathematics
Authors: Price H. Vernon
Published: November 2020

The great watchword of the French Revolution was Liberty, Equality, Fraternity. Although a great oversimplification, it has been said that France exemplifies liberty, Great Britain equality, and the United States fraternity. Without attempting to apportion these virtues among the nations of the world, I should like to dwell for a few moments on fraternity as it applies in the United States to the National Council of Teachers of Mathematics, I believe it is in this domain that we have developed into the largest mathematical organization in the world and—we should like to think—one of the most influential.
A Computational Confidence Builder
Authors: Ulrich Robert C.
Published: November 2020

There is a “game” students have been playing for which a winning strategy can be devised by performing a few simple mental arithmetical computations.
Big Bad Basic Skills—Or, What One School System Is Doing To Help Low Achievers
Authors: Hestwood Diana L. , Taylor Ross
Published: November 2020

Things were starting to look up for low achievers in the Minneapolis secondary schools in January 1970. After all, as a result of teacher recommendations a computer-generated testing system was now available through a terminal in each school.
Mathematics in Use, as Seen on Postage Stamps
Authors: Schaaf William L.
Published: November 2020

VARIOUS uses of mathematics have been exhibited effectively in pictures, as attested by the assortment of postage stamps reproduced herewith. For convenience we have arbitrarily chosen several aspects of “mathematics in action” and then selected outstanding stamps that are typical of each category.
Some Experiences with Slow Learners
Authors: Lane George L.
Published: November 2020

AT THE check-out counter of a college cafeteria, one girl runs a keyboard machine and passes the slip of adding-machine tape on to another girl who collects the money. On a particular day, two friends and I decided that the “special” was to our liking. We ordered three, and I informed the first checker that I would pay for three specials. Specials, it turned out, were 95ȼ apiece, plus tax. The girl rang up 95ȼ plus tax (4ȼ) for my special, a 99ȼ total. She then rang the other two specials on the same ticket and came up with a figure of $1.97 for the two. I presumed the adjustment was in the tax. She had cleverly avoided having to multiply 95ȼ plus tax by three. The tapes were passed on to the second girl, who looked at me rather helplessly and asked, “Are you paying for all three?”
An Application of Volume and Surface Area
Authors: Mercaldl Robert W.
Published: November 2020

REINFORCEMENT of skills to find surface area and volume in grades seven, eight, and nine can be effected by using a game requiring seven wooden blocks.
Learning Mathematics in a Group Situation
Authors: Davidson Dennis
Published: November 2020

Mathematics teachers are often frustrated when trying to give help to many individual students in a short class period. The teacher is outnumbered, and even with a small class of sixteen to twenty-five students, the class hour is too short to give every student the help he wants and deserves. Often the conscientious, aggressive student monopolizes the teacher’s time while the poor, introverted student may remain slouched at his desk virtually unnoticed day after day. When a student returns from an absence of even one day, the frustration is multiplied. This student must make up the missed lessons, but the teacher is already overwhelmed just trying to help students with the current lesson.
Uffon’s Needle Problem: An Exciting Application of Many Mathematical Concepts
Authors: Schroeder Lee L.
Published: November 2020

While teaching integral calculus, I have often looked for interesting applications of the calculus that are relatively easy for the students to master but yet not trite. One such problem is known as ‘Buffon’s Needle Problem.” A consideration of this problem involves the solution of a simple definite integral and requires a basic introduction to probability theory. Let’s consider the “Needle Problem.”
The Permutations Triangle
Authors: Toschi Larry M.
Published: November 2020

During the past few years it has become popular to explore Pascal's triangle by a discovery approach similar to that which George Polya presents in Mathematical Discovery (1960), especially pages 70-73 and 83-84. (The Bibliography at the end of this article will give other references indicating the rich variety of relationships found in this triangle.) The simple beauty of these relationships can be used to experience the joy of discovery and at the same time learn an ever more general method of problem solving. Some teachers have already begun incorporating such material into the high school curriculum, and I believe this practice should be increased because of its motivational value. Furthermore, the student who has derived enjoyment and profit from discovering the relationships of Pascal's triangle needs an opportunity to expand his work in this area. This article presents what I believe to be a natural extension suitable to the senior high school curriculum.
Something New on Number Lines
Authors: Sconyers James M.
Published: November 2020

This new and interesting use of number lines to multiply numbers and determine inverses of numbers should prove of interest to junior and senior high school students who are familiar with the idea of number line. It would make a nice presentation at a mathematics club meeting.
A Simple Sorting Sequence
Authors: Duncan David R. , Litwiller Bonnie H.
Published: November 2020

The authors answer conclusively the question of how certain regular polygons can be “fitted together” around a point in the plane. Proofs lead to the establishment of seventeen cases.
Mixture Problems by Some “Old Math”
Authors: Maxfield Margaret
Published: November 2020

The author describes briefly some methods and techniques used to solve problems in years gone by.
Population Explosion: An Activity Lesson
Authors: Piele Donald T.
Published: November 2020

Here is an activity lesson that deals with a current problem, namely, population. It also illustrates an interdisciplinary lesson.
Factoring Polynomials and Place Value
Authors: Olson Alton T.
Published: November 2020

A way of relating whole numbers and polynomials provides the reader with a new look at the concept of factoring.
The Move to Metric: Some Considerations
Authors: Firl Donald H.
Published: November 2020

Following a brief historical review, the author discusses SI, various units, and some of the problems involved in changing to a different system of measurement.
Problem Solving in the General Mathematics Classroom
Authors: Troutman Andria Price , Lichtenberg Betty Plunkett
Published: November 2020

The teachers of general mathematics will find in this article (1) descriptions of seven specific abilities related to solving problems and (2) appropriate problem-solving activities that help to develop these specific abilities.
Some Changes in Shop Mathematics Due to Metrication
Authors: Hawkins Vincent J.
Published: November 2020

Some implications of conversion to the metric system for shop or applied mathematics courses.