bluuebunny/crossref_metadata_2025 huggingface dataset:
The need for a new notional policy and program in secondary mathematics
Authors: Reeve William David
Published: November 2020

A forceful plea for action on the present “chaotic conditions” in mathematics from one who has had a long and rich experience with the mathematics program in the high school.
Seminars—an integrating force in a program of concentration
Authors: Corona Mary
Published: November 2020

In these days of changing mathematics courses and mathematical curriculums, it is good to see a description of an undergraduate program for mathematics majors.
Mathematics—a language
Authors: Seidel George R.
Published: November 2020

From industry's point of view, the readers of The Mathematics Teacher are told about some of the primary responsibilities of the mathematics teacher. What does it take to train a student to reason clearly?
Where do eligible mathematics teachers go?
Authors: Maul Ray C.
Published: November 2020

A timely report on a most pressing problem. Teachers can help by publicizing the. facts presented in this article.
Devices for the Mathematics Classroom: Parabola device
Authors: Olmsted Hugh
Published: November 2020

This device consists of a pair of crossed wooden sticks joined together at their intersection with a single screw. Thus either stick may be rotated toward or away from the other. Along each stick short brads should be driven at equal distances, and rubber bands stretched between those having alternate positions on the sides of each of the angles.
What is Going on in Your School?: Originality in arithmetic
Authors: Beeman William E.
Published: November 2020

Any teacher of seventh-grade arithmetic is always trying to find some way to stimulate the pupils’ interest as well as aid his learning of the subject. Last fall I was very pleased with the reaction of one seventh-grade section, in particular the work of one student, when some of the classic word-digit problems were presented.
Tips for Beginners: Pupil discovery in junior high school mathematics
Authors: Lowry William C.
Published: November 2020

Some teachers feel that the method of pupil discovery in teaching mathematics is too slow, too time consuming. Often, however, when the teacher merely “covers” the material of a course, it is found that what he thinks has been taught remains hazy and only partially revealed to the pupil.
The cardioid
Authors: Yates Robert C.
Published: November 2020

An investigation of the properties of a curve well known to mathematics teachers.
“Mathletes”
Authors: Lenchner George
Published: November 2020

Here is a report of the activities of the Nassau County Interscholastic Mathematics League.
How I came across the extraction of “Nth” roots
Authors: Shapiro David
Published: November 2020

An example of how a seventh-grade student's sense of pattern led him to generalize the method for finding the “Nth” root of a number.
Points and Viewpoints: How much mathematics should teachers know?
Authors: Fehr Howard F.
Published: November 2020

A column of unofficial comment
Reviewing by viewing promotes learning while relearning
Authors: Dinkel Robert E.
Published: November 2020

An analysis of the educational uses of projects in mathematics teaching.
Notes from the Washington Office: New developments in the NCTM Office
Authors: Ahrendt M. H.
Published: November 2020

As members of the NCTM would expect, the continuing and apparently increasing interest in the problems of mathematics education requires continuing adjustments and adaptations in the NCTM Office. We have made a number of changes recently which may be of interest to you.
Points and Viewpoints: Arithmetic and teacher preparation
Authors: Meder Albert E. , Hlavaty Julius H.
Published: November 2020

A column of unofficial comment
Values of mathematics for the modern world
Authors: Rosenberg Herman
Published: November 2020

What are the values that a student derives from the study of mathematics?
High school algebra via television
Authors: Andrews John J.
Published: November 2020

Mathematics has been and is being taught by television. In this study both teachers and students reported that they liked mathematics via television.
Historically Speaking,—: John Napier and his logarithms
Authors: Read C. B.
Published: November 2020

It is doubtful that any teacher would today try to introduce the subject of logarithms other than by means of some use of exponents. It is, then, indeed hard to realize that Napier constructed his logarithms before the concept of exponents as we know them was developed.
Determinants whose values are zero
Authors: Wheeler Norman E.
Published: November 2020

How can students develop imagination in mathematics unless they get opportunities to exercise their imagination like the one described here?
Experimental Programs: The “Madison Project” of Syracuse University
Authors: Davis Robert B.
Published: November 2020

Although it operates in many schools— we do not, in fact, know how many—the Syracuse University “Madison Project” is probably one of the least well known of all of the “new” mathematics programs. This unintended secrecy derives, perhaps, from the nature of the project itself. As we shall see presently, the project is represented by neither a large collection of sample textbooks, nor a small collection of incisive slogans.
Application of the cosine function
Authors: Wildermuth Karl P.
Published: November 2020

A project for the trigonometry class
Experimental Programs: High speed computer programming in the junior high school
Authors: Sweet Raymond
Published: November 2020

Each time that a mathematics project has been assigned here in the past few years, several students have indicated an interest in computers. Until recently none of the teachers had any training in the computer's use and the school library had no books pertaining to the subject. Last year, within four months after the arrival of a text describing the construction of several homemade computers, boys in an accelerated geometry class turned in three small computers for their term projects.
The importance of elementary operations
Authors: Alfred Brother U.
Published: November 2020

Why a student who is 90 percent accurate in doing operations will get only 65 percent on a computation test
Historically Speaking,—: The evolution of group theory
Authors: Miller G. H.
Published: November 2020

Now that the value of group theory in theoretical physics, chemistry, electrical engineering, and electronic computing machines is being realized [8],† it is of interest to look at the origins of group theory. This paper will be concerned only with a résumé of the significant contributions of earlier investigators through the nineteenth century. There are disagreements among the historical authorities as to who originated certain specific topics in group theory, but for the most part there is a uniformity of opinion.
The Equivalence of a given Collection of Statements
Authors: Dupree Daniel E.
Published: November 2020

An interesting use of finite induction
Sum Equals Product
Authors: Feser Victor
Published: November 2020

A little puzzle leads to some interesting ideas.
Mathematics in the Community Junior College
Authors: Pixley Loren W.
Published: November 2020

What should be the mathematics offerings of the community junior college?
The missing half of our technical potential: Can we motivate the girls?
Authors: Peden Irene C.
Published: November 2020

Some tips on advising talented girls
Ten mathematical refreshments
Authors: Duncan Dewey C.
Published: November 2020

It is the perennial delight of the student of mathematics to discover unexpected relationships among sundry mathematical topics.
Opinions of college teachers of mathematics regarding content of the twelfth-year course in mathematics
Authors: Buchanan O. Lexton
Published: November 2020

Helpful information for high school teachers
Gaspard Monge and descriptive geometry
Authors: Gafney Leo
Published: November 2020

Meandering about a second-hand bookstore some time ago, I came upon a dusty volume entitled Géométric Descriptive by G. Monge, Paris, 1827 (see Fig. 1). Out of curiosity, I bought the old tome, but gave neither subject nor author another thought until the following summer when I found the name “Monge” sprinkled through a course in differential geometry.
Remarks on some elementary volume relations between familiar solids
Authors: Loeb A. L.
Published: November 2020

Geometrical formulas are frequently based on an arbitrary system of units chosen for convenience in an orthogonal Cartesian coordinate system. Mechanical use of such formulas unnecessarily complicates work in other coordinates, and obscures such simple relations as the volume ratio of octahedra and tetrahedra of equal edge lengths, and of a parallelepiped and its inscribed tetrahedron.
The NCTM: Its growth and growing pains
Authors: Meserve Bruce E.
Published: November 2020

A report on the Council's services to its members
A mathematical diversion
Authors: Bishop David C.
Published: November 2020

Another way to test your word power
Abstract Algebra From Axiomatic Geometry
Authors: Macdonald I. D.
Published: November 2020

This article is entirely expository. It is devoted to the derivation of an abstract algebraic structure from a certain projective geometry. Though this piece of mathematics is already well known to the experts, it is so remarkable and so beautiful that it deserves to he much better known than it is to schoolteachers and to those whose business it is to modernize school mathematics.
An Optical Method for Demonstrating the Point of Inflecton of Curves
Authors: Hirsch A. Adler
Published: November 2020

An Inflection point, p, of a plane curve, AB, in the diagram, may be demonstrated optically by noting the disappearance of the faint secondary reflection (ghost image) from the lower front surface of an ordinary glass mirror, M, which straddles the curve perpendicularly. Perpendicularity is attained when the main reflected image, i, appears to blend smoothly at the bottom of the minor with the preceding arc. At all other points, such as r1 and r2, secondary images, s1 and s2, form in the mirror. These are always displaced from the primary images, i1 and i2, in a direction toward the concavity of the arc.
Multiplication, Addition, and Duality
Authors: Brown Stephen I.
Published: November 2020

For some theorems involving multiplication (addition) of real numbers, the analogous statement for addition (multiplication) is true, while for others it is false.
Divisibility and Factorization of Gaussian Integers
Authors: Willerding Margaret F.
Published: November 2020

Numbers of the general form z = a + bi, where a and b are integers, are called “Gaussian integers.” This set of numbers forms a commutative group with respect to addition since closure holds, the commutative and associative laws hold, there exists an identity element (0 + 0i), and each element has an inverse (− a − bi).
More About 1965 and 1966
Authors: Steinen Ramon F.
Published: November 2020

For several days I have been working on the intriguing puzzle described by Jerome Niebaum in the December 1965 issue of The Mathematics Teacher (“A Digital Problem for 1965,” pp. 713-14), and have found it possible to simplify parts of the solution offered in his article.
Are Particular Abilities Necessary for the Pupil To Gain an Understanding of the Elementary and Secondary Mathematics as Usually Given at the Present Time?
Authors: Babb Maurice J.
Published: November 2020

The director of one of our most prominent laboratories of anatomical research told me during this last month that he has been having a frequently recurring and vivid dream for a year or more, to the effect always, that somehow he had failed to make good in mathematics, and that discovery was both imminent and inevitable. He confessed the sense of guilt and horror of exposure was so real as to be depressing even after he awoke. I have been wondering whether he is a sort of vicarious sufferer for the sins of his profession in particular and of this age in general. It is good to find some evidence of the arousal of a mathematical conscience. Simultaneously the much-maligned appendix vermiformis has been found to have unique and valuable functions, to the great consternation of numerous and fashionable people. All this is personally particularly interesting, as my first visit to the University of Pennsylvania was as a prospective medical student and naturally I drifted into the laboratory of the late Professor John Rider, whom you doubtless remember was rejected in his youth as unable to appreciate and absorb the knowledge as then dispensed by the adjacent public schools. He told me that he recently had been a student along with the sophomores in calculus in order to gain sufficient knowledge of the mathematics to be able to express the cellular development of the ear from the brain in embryo mice. This year a student of my calculus class comes in the same way that he may gain knowledge which he may apply to his researches on muscle fibers. These men were said not to have had mathematical minds in their school days. Professor David Eugene Smith has recently given the New England branch of this Society a list of seemingly hopeless mathematical outcasts which on further examination has turned out to be a list of the great men of the science.
Trigonometric Excursions and Side Trips
Authors: Wegner Kenneth W.
Published: November 2020

Polynomial equation theory has been one of the casualties of recent curricular revisions. College teachers often assume that freshmen are familiar with the remainder and factor theorems, the rational root theorem, the relationship between roots and coefficients, etc., whereas these topics are not always found in even a four-year high school mathematics sequence.
Classics in Mathematics Education: Reality of Mathematical Processes
Authors: Hedrick E. R.
Published: November 2020

The attempt to appraise values in all our curricula is timely enough. Moreover, it is vital because it will correct the faults of our system, both present and future. The purpose of this chapter is to direct attention to phases of mathematical values that are overlooked in many surveys of this field
Auxiliary Lines And Ratios
Authors: Stover Donald W.
Published: November 2020

This depressing query, dashing the hopes engendered by eagerly waving hand and bright expression, is just one of those small, recurring punches teachers learn to roll with. Its occurrence may be forecast with confidence when geometry students encounter their first proof in which the figure contains auxiliary lines. In this instance, the answer is also predictable. Few teachers, taking into account the pressures and time limitations of a testing situation, dare require their students to select from the infinitude of mathematical existence an unrehearsed assertion of the form “There exists a line….”
Alternatives to Research
Authors: Christie Dan E. , Wells James H.
Published: November 2020

A Formidable challenge to excellence is posed by two trends in collegiate mathematics education: one is the admirable flow of numerous students into fields requiring mathematics, with a consequent demand for good college teachers; the other is the lamentable tendency for undergraduate mathematics faculties to be understaffed and overworked. The two trends together generate a challenge which the CUPM Panel on College Teacher Preparation now confronts: How can undergraduate faculty members achieve and maintain high quality in mathematics teaching?
An Application of Base-Three Numeration
Authors: Ginther John L.
Published: November 2020

From a pedagogical point of view, it is desirable to be able to show a subtle relationship between two pieces of mat hematics that are seemingly unrelated. When such a relationship can be shown at a rather elementary level and when the “marriage” helps to analyze still another interesting question, the potential benefits to the teacher are great. Such a situation will be described in this paper. It involves the use of the base-three positional numeration system in analyzing some topics from function theory.
International Mathematical Education: A Description of The Department of Mathematics, The Royal Danish School of Educational Studies
Authors: Christiansen Bent
Published: November 2020

The historical development of the Royal Danish School of Educational Studies, which dates back to 1856, will not be Considered here. It will only be empha sized that a strong tradition exists in Den-mark among the teachers with regard to the necessity of obtaining further knowl edge than that provided by the training colleges. Through the years the body of teachers has shown a very active interest in the organization of in-service training. Such further training has been given by the Royal Danish School of Educational Studies in the various phases of its de velopment and under varying conditions. The favorable status of the institution to day could hardly have been reached with out the strong and continuous support rendered by the teachers.
A Geometric Introduction to Mathematical Induction
Authors: Wiscamb Margaret
Published: November 2020

No explanation is given as to the source of these rather mysterious sums;he is merely presented with the equation as a problem to be proved by the method of mathematical induction and told to get on with it.
Improving the Witch
Authors: Oakwood Elliott F.
Published: November 2020

In order to aid in understanding the more general form of this curve which is presented in this paper, it is necessary to describe the method by which the locus is developed for the classical version.
Mathematics in Secondary School Vocational Education Programs
Authors: Fishman Joseph
Published: November 2020

Current investigation by mathematics educators into the problem of the low achiever invites further inquiry into the question of whether we are meeting our responsibility to the fullest extent. Research regarding the problem of the low achiever has shown that a significantly greater number of students are capable of learning much more mathematics than present programs now succeed in teaching.
An Unexpected Field
Authors: Goff Gerald K. , Johnston Hiram D.
Published: November 2020

A Current problem in high school mathematics education is what mathematical topics ought to be taught to twelfth-grade high school students. Woodby (1965), in a survey of emerging twelfth-grade programs, concluded that no particular program seems to be the most appropriate one at the present time; however, he noted that both acceleration and enrichment were included in what he termed “strong mathematics programs.” Many writers feel that acceleration is often overemphasized. For example, Grossman (1962) argues that too often enrichment is slighted in favor of acceleration.
A Pattern in Number Theory: Example→ Generalization→ Proof
Authors: Duncan David R. , Litwiller Bonnie H.
Published: November 2020

The study of patterns is an integral part of the study of mathematics. As we teach mathematics, we must point out how to search for patterns and how patterns may aid us in problem solving. The following problem is one that combines patterns, ideas from number theory, and mathematical induction: “Prove that it is possible to pay, without requiring change, any whole number of rubles (greater than 7) with banknotes of value 3 rubles and 5 rubles” (Sominskii 1964, p. 19).
On What Day Were You Born?
Authors: Stick Marvin E.
Published: November 2020

Oftentimes a person may know he date of a specific event but be curiouss to the day on which it occurred. If the vent is taking place during the present car, one need only look at the nearest alendar. However, other year present a problem unless a systematic file of calndars, both past and future, are availble.
Triangular Arrangements of Numbered Disks
Authors: Trigg Charles W.
Published: November 2020

Ten equal disks can be arranged in triangular form with A, B, C at the vertices and D in the center, as in figure 1.
The Forum: What Should Become of the High School Geometry Course?: An Improved Year of Geometry
Authors: Meserve Bruce E.
Published: November 2020

Geometry in secondary schools has evolved during the last few years so that the use of coordinates has become respectable and the use of vectors is beginning to gain acceptance. There still remains the basic problem of finding sufficient time to consider the topics from the traditional full year of plane geometry, an introduction to the geometry of three space, an introduction to coordinate methods of proof, and an introduction to vector methods of proof. There are also suggestions that many traditional topics of geometry be jettisoned and that other new approaches to geometry be included.
Modulo Systems: One More Step
Authors: Mann Nathaniel
Published: November 2020

The study of modulo or clock arithmetic has become popular enough to be included in many recent textbooks. A chapter on the topic is included in the NCTM's yearbook entitled Enrichment Mathematics for the Grades. But I have found that pupils can bo taken further Unto the study of these systems than is suggested in most textbooks and articles.
The nth Polygonal Number
Authors: Smith Joe K.
Published: November 2020

Many mathematics books include a unit on mathematical induction. Most of these units have examples and exercises involving the sum of the terms of an arithmetic sequence. Usually the reader is supplied with the first three or four terms together with the nth term. No indication is given as to how one might find this nth term or the expression for the sum if they had not been supplied.
Additive Inverses and Negatives—Two Different Concepts
Authors: Musser Gary L.
Published: November 2020

Certainly one of the best-known statements in arithmetic is the statement “The product of two negatives is positive.” This expression is generally associated with the study of the ring of integers, but it also applies in the fields of rational and real numbers. Usually a ring-theoretic proof of the fact (—a) (—b) = ab is provided to prove the statement quoted above; however, the equation [−(−5)] (−3) = (−5)3 is an instance of (— a) (— b) = ab, but certainly the equation [−(−5)](−3) = (−5)3 is not described by the sentence “The product of two negatives is positive.” Sometimes a and b are restricted to be positive integers; thus (— a) and (— b) are negative integers and all is well. Also, (—x) is sometimes referred to as “the negative of x,” and then, in this context, (— a) (— b) = ab is considered to be an instance of “The product of two negatives is positive.” However, it. seems that more needs to be said. It is the purpose of this article to clarify the distinction between the two statements “The product of two negatives is positive” and “The product of the additive inverses of a and b is ab.” I begin by establishing the fundamental notions that form the foundation of my discussion.
A “Head Start” Program in Student Teaching in Mathematics
Authors: Sowell Katye O. , Hodgin Katharine W.
Published: November 2020

Undergraduate students who are preparing to teach mathematics need as many opportunities as possible to engage in simulated or actual teaching. Planning lessons, constructing situations to facilitate learning, and communicating with learners are skills that are best learned through practice. At East Carolina University, the students enrolled in a course entitled “The Teaching of Mathematics” are given two quite different opportunities to teach under supervision during the quarter preceding their student teaching in a secondary school.
What Every Mathematics Teacher Ought to Read (Seventeen Opinions)
Authors: Leake Lowell
Published: November 2020

THE current wealth of good literature in mathematics education is a great blessing for mathematics teachers at all levels —high school, elementary school, and college. For many teachers, especially the beginners, this horn of plenty presents a perplexing problem; with limited time or financial resources, the mathematics teacher needs guidance as to what literature would be most beneficial for him to read and, it is, hoped to own.
A Simulation Game for General Mathematics
Authors: Watman Michael X.
Published: November 2020

Since the use of simulation games in social studies classes recently became quite widespread, I have searched for such a teaching aid that could be of value in a mathematics class—in particular a second-year course in general mathematics. I decided to experiment with a game based on the stock market, since at the very least, my students would have to do a great deal of arithmetical computation in order to participate. The results of my experiment indicated that far more desirable outcomes occurred than had been anticipated. The purpose of this article is to relate these outcomes and indicate some waye in which they might be expanded.
Mathematics as “Process”
Authors: Morley Arthur
Published: November 2020

When observing mathematics lessons, I often find myself reflecting on how powerfully the hind of image of mathematics the teacher has affects what he says and does. Lee Shulman puts, this very clearly in the excellent chapter “Psychology and Mathematics Education,” which he contributed to the NSSE Yearbook Mathematics Education.
Auxiliary Lines-a Testing Problem
Authors: Alpart Bruce Jay
Published: November 2020

Did you ever present a topic to one of your classes, confident that their grasp of the material was satisfactory, then test them only to discover that they failed to understand some important principles? Consider this problem that occurred in my tenth-year mathematics class.
Curriculum Construction with Student Input
Authors: Davidson Neil , McKeen Ronald , Eisenberg Theodore
Published: November 2020

Several educators, including Zacharias and White (1964) and Davis (1965), have stated that a curriculum should reflect the interests and aptitudes of the students for whom the curriculum is intended. The research of Mager and Clark (1963). Mager and McCann (1961), and Kaplan (1964) indicates that many students seek information in an order that is different from the order suggested by teachers, textbooks, or traditional programs. Their research findings also indicate that all students have some relevant knowledge concerning the subject matter at hand, no matter how naïve the students claim to be. Hence, it is suggested that student needs, interests, and aptitudes be given consideration in the construction of curriculum.
The Equivalence of Euler's and Pick's Theorems
Authors: DeTemple Duane , Robertson Jack M.
Published: November 2020

An illustration of creative mathematics at an elementary level.
An Open-Ended Problem on the Geoboard
Authors: Masalski William J.
Published: November 2020

An exploration on the geoboard that results in finding some surprise and some sophisticated geometry for the geometry class. It illustrates how versatile the geoboard is for discovering mathematical relations.
Unary Operations
Authors: Hess Adrien L.
Published: November 2020

Unary operations as discussed in this article become as important as the well-known binary operations. Perhaps more emphasis on unary operations should be given in school mathematics.
Using Mathematical Structures to Generate Artistic Designs
Authors: Forseth Sonia , Troutman Andria Price
Published: November 2020

This article shows how to add artistic designs to the study of mathematical systems. It is a fascinating way to illustrate the varied applications of mathematics.
A Mod Transfusion for Second-Year Algebra
Authors: Smith William D.
Published: November 2020

This article gives concrete examples of the use of a finite number system to illustrate a mathematical system. It shows how to emphasize structure in the second-year algebra course.
An Application of Parametric Equations to Weather Forecasting
Authors: Palmaccio Richard J.
Published: November 2020

Forecasting the path of a storm using high school mathematics.
The Volume of the Regular Octahedron
Authors: Trigg Charles W.
Published: November 2020

Five methods of computing the volume of a regular octahedron will interest geometry teachers.
Area = Perimeter
Authors: Markowitz Lee
Published: November 2020

When is the area of a triangle equal to its perimeter? I uncovered this question through a mistake I made when tutoring a student in geometry.
Magic Cubes: A Total Experience
Authors: Aviv Cherie A. , Rachlin Sid
Published: November 2020

Use the cubes for drill or have the students make them.
Graphing Polynomials with Computer Assistance
Authors: Kennedy Jane B.
Published: November 2020

An updated treatment of an old topic-the polynomial graph.
Computer Literacy and School Mathematics
Authors: Gawronski Jane D.
Published: November 2020

Computer literacy has been of concern to the mathematics education community for several years. The National Council of Supervisors of Mathematics (NCSM) in their Position Paper on Basic Mathematics Skills (1978, p. 150) stated the following: It is important for all citizens to understand what computers can and cannot do. Students should be aware of the many uses of computers in society, such as their use in teaching/learning, financial transactions and information storage and retrieval.
Computer Bibliography
Authors: Kosel Marge
Published: November 2020

The number of computer books available for secondary schools has increased exponentially in the past several years. The following books are currently being used in schools. Many new ones are specifically designed for a particular microcomputer.
Beneath the Surface of the Mathematics Teacher Shortage
Authors: Williams Robert T.
Published: November 2020

Although the shortage of mathematics teachers has been well documented for many years, there seems to be no serious effort to alleviate it. We cannot even assume that the shortage is widely known. Just before last November's election, I had occasion to mention to a candidate for statewide office that I was doing a study on this shortage. His reply was, “Is there a shortage?” Even our leaders are not always informed.
An Intuitive Approach in Teaching Linear Programming in High School
Authors: Ulep Soledad A.
Published: November 2020

Second-year algebra students usually study a complete unit on solving systems of linear inequalities graphically. At this point, resourceful teachers can present all students, not just honors or advanced ones, with the process for solving problems involving linear programming.
Graphics Software Side by Side
Authors: Difazio Martha Hunt
Published: November 2020

NCTM's Curriculum and Evaluation Standards for School Mathematics (Standards) (1989) makes some very specific recommendations concerning the use of microcomputers in secondary school mathematics. This document recommends that “a computer will be available at all times in every classroom for demonstration purposes, and all students will have access to computers for individual and group work.” It strongly advocates the use of graphics calculators and interactive graphics computer packages as a principal means for investigating the behavior of functions. Recently, Waits and Demana (1988a) have presented convincing arguments for the use of graphics technology as a supplement to traditional mathematics instruction. Interactive graphics software can be used to give quick, accurate plots of functions, conic sections, and parametric equations. The use of such software can foster a geometric approach to problem solving and is especially valuable for those students for whom visual images work better than symbolic manipulations. Estimation skills can be sharpened as students begin to search for roots and critical points visually even before they are taught the symbolic manipulations needed for an algebraic solution to their problem.
Equilateral Triangles on an Isometric Grid
Authors: Spikell Mark A.
Published: November 2020

The problem in figure 1 appeared on the cover of the Virginia Mathematics Teacher (Fall 1985). An interesting related problem is “How many equilateral triangles of different sizes can be constructed on the isometric grid shown” (using the points as vertices)?
Probability, Problem Formulation, and Two-Player Games
Authors: Buckhiester Philip G.
Published: November 2020

The Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) states in Standard 11 for grades 9 through 12 that students should have opportunities to “use experimental and theoretical probabilities to represent and solve problems involving uncertainty.” Standard 1 emphasizes the importance of students' learning to “formulate problems from situations within and outside mathematics.” This article discusses a simply stated problem involving uncertainty that students can investigate experimentally or theoretically. The problem places students in the role of problem formulator by giving them opportunities to generate various interesting problems of their own on the basis of a given situation. By changing certain characteristics of the original problem, students can be introduced to some fundamental concepts of decision making in two-player games.
Technology Reviews
Authors: Flanders Jim
Published: November 2020

Prices on software, books, and materials are subject to change. Consult the suppliers for the current prices.
Report of the Executive Director
Authors: Gates James D.
Published: November 2020

During the five years since the release of the Curriculum and Evaluation Standards for School Mathematics (1989), NCTM membership has increased from 77000 (54000 individuals) to 106000 (79000 individuals). These totals include both individual and institutional members. The most recent issues of the Arithmetic Teacher, Mathematics Teacher, and Journal for Research in Mathematics Education had circulations of 67000, 61000, and 12000, respectively.
Implementing the “Professional Standards for Teaching Mathenatics”: The Teacher and Evaluation
Authors: Koss Roberta , Marks Rick
Published: November 2020

Evaluation of teaching is usually something done to a teacher. In contrast, the Professional Standards for Teaching Mathematics (NCTM 1991) supports evaluation of teaching done by and for a teacher. That is, the teacher participates actively in designing and carrying out any evaluation, and the process contributes substantially to the teacher's professional growth. This vision dramatically changes the roles and responsibi)jties of the teacher as well as others involved in evaluating teaching. To begin moving effectively in this direction, both teachers and supervisors need to understand the rationale for change and to see how this change could occur.
Implementing the “Professional Standards for Teaching Mathematics”: Empowering Teachers through the Evaluation Process
Authors: Leiva Miriam A. , Prevost Fernand J.
Published: November 2020

Mr. Hille had been student teaching for three weeks when I made an announced observation in his second-year-algebra class. His lesson on ellipses had been carefully planned, and he was ready with models, string, calculators, overhead transparencies, and a lesson plan. Somewhere at the beginning of his enthusiastic presentation, he slipped from equations of ellipses to hyperbolas, while naming and graphing them as ellipses! Suddenly be realized what he was doing and looked at me in horror, but he went on for a few more minutes, “keeping his cool” as he would explain later. He asked questions and assigned a few problems from a previous section. The period ended. To an observer with little or no mathematics background, Mr. Hille's lesson appeared to be excellent.
For The Editorial Panel: Share the Wealth, Share the Wisdom
Authors: Rubenstein Rheta
Published: November 2020

Flying magazine has a column called “I Learned about Flying from That.” Each month a pilot tells a personal flying story involving a “close call” from which he or she learned an important safety lesson. The brief, firsthand, suspenseful stories are a great way to share learning about safety. Do we as teachers not have the same types of stories—critical incidents that affected our growth significantly? Life is too short for us each to learn everything firsthand. We, like pilots, increase our wisdom when we learn from one another. The Editorial Panel of the Mathematics Teacher believes that we all have stories to tell, whether they stem from the throes of the early years of teaching, the results of carefully orchestrated lessons, or the unexpected surprises of students' insights. We have stories that tell how we learned from our students: how a student's comment, question, or response to an! activity taught us an important lesson about learning, mathematics, or teaching. Each of us has stories for which we can say, “I learned about teaching from that!” We invite you to share yours.
Reading and Note-Taking Prior to Instruction
Authors: Maida Paula
Published: November 2020

How can students achieve mathematics literacy? One of the goals set by the NCTM to develop such literacy in students is the implementation of mathematical communication. Mathematical communication allows students to express their thought processes and is fundamental to the comprehension of mathematics. Mathematical communkation also forces students to be active learners as opposed to passive learners who simply accept and memorize procedures. For students to learn to communicate mathematically, NCTM's Curriculum and Evaluation Standards for School Mathematics recommends that teachers foster a classroom environment that encourages reading, writing, and verbal communication of students' thoughts. “Ideas are discussed, discoveries shared, conjectures confirmed, and knowledge acquired through talking, writing, speaking, listening, and reading” (NCTM 1989, 214).
A Bird in the Hand
Authors: Richbart Carolyn , Richbart Lynn
Published: November 2020

The expression “A bird in the hand is better than two in the bush” expresses the disposition of the average person when faced with a choice between two or more alternatives involving uncertainty. As far back as 1738, one of our most famous mathematicians, Daniel Bernoulli, studied how people think of gains and losses subjectively rather than in terms of mathematical expectation (see Bernoulli [1954]). In this article, we discuss a comparison between the decisions people make in risk situations and mathematical expectation.
Mathematics Contests
Authors: Koch Ernest H.
Published: November 2020

From time immemorial games have been a necessary complement to the more austere forms of intellectual and social development. We have historic records of contests which enable us to obtain a familiar acquaintance with the manners and customs of nations as well as their achievements.
The Order of Teaching the Parts of the Calculus
Authors: Lamond John K. , Grove C. C. , Marriott. Ross W.
Published: November 2020

When the average student reaches the calculus his ideas of a function, a variable, a variable approaching a limit, etc., are so vague, and oftentimes so entirely wrong, that it would seem wise, since the foundations of the calculus rest so largely on the theory of limits, to spend enough time at the beginning of the course to develop the theory of limits in a careful and thorough manner.
Euclid, Fibonacci, Sketchpad
Authors: Litchfield Daniel C. , Goldenheim David A. , Dietrich Charles H.
Published: November 2020

In late June 1995 I began teaching geometry and trigonometry to David Goldenheim and Daniel Litchfield at Greens Farms Academy's summer session. Their reputation in mathematics had reached my ears long before I worked with them, so I was prepared for a fascinating and challenging time. I had. however, no idea of the adventure on which we were to embark.
Soundoff! Common Sense: The Most Important Standard
Authors: Saul Mark
Published: November 2020

It is 1977. I am teaching in a very good school with a very good adminjstration. My students are workinng on a project in small groups. Attracted by the noise, the principal comes into the classroom. I tell him what I have done with the class. It is long before cooperative learning will become a recognized method, but a colleague and I have worked on this sort of teaching for several months. Nonetheless, the principal asks me why I am not teaching. What will he tell parents when they telephone? They have already complained that I do not teach and that the students have to learn from one another.
Reader Reflections
Authors: Rachofsky Ted
Published: November 2020

The secret to meaningful communication is that it must go two ways. Last week during class, I was once again reminded how easily this precept can be forgotten.
Equivalent Representations in the Learning of Algebra
Authors: Dyke Frances Van , Craine Timothy V.
Published: November 2020

In the past few years, much discussion has taken place about what direction the teaching of algebra should take in view of the NCTM's Standards documents (1989, 1991, 1995), the increased availability of graphing technology, and the movement toward “algebra for all.” Because we view algebra as a language for expressing relationships between quantities and for exploring patterns, we support a functional approach to the subject in which all ideas are put in a context and graphing calculators are used extensively. Within tills framework, however, students who are trying to learn algebra experience subtle difficulties. On the one hand, students fail to recognize the underlying equivalence when the same set of points is represented by a graph or an equation or a table. On the other hand, when given an algebraic representation for a particular function, students fail to see that they can use the rules of algebra to produce an equivalent expression. In both situations, they tend to see changes in form as producing unrelated representations.
Math-Abused Students: Are We Prepared to Teach Them?
Authors: Fiore Greg
Published: November 2020

This article deals with students whose math anxiety resulted from past verbal or physical abuse, in particular, abuse by a teacher or parent while doing mathematics. Let us define math abuse as any negative experience related to an individual's doing mathematics.
Projects: Project CLIMB
Authors: Kasten Peggy
Published: November 2020

Project CLIMB (Creating Links in Math and Business) is a teacherdeveloped project that was designed to help answer the students' question, When are we ever going to use this? The project allows precalculus students to communicate with people in the business world by using e-mail. Students are put into groups of three or four and assigned a business contact. The students determine from this contact person exactly what the company does, how teams are used in the company, and how specific mathematics topics are used by the contact person on the job. The student project includes six e-mail requests for information during a semester. The information requested centers on the precalculus topics of matrices, statistics, linear programming, logarithms, trigonometry, and probability. These broad topics are used by people in a variety of fields. The business contact uses e-mail to respond.
Including Leap Year in the Canonical Birthday Problem
Authors: Nandor M. J.
Published: November 2020

The solution to the canonical birthday problem is taught at all levels of high school mathematics from algebra to discrete mathematics. Although many excellent articles and applets have been written on the subject, I am surprised that the a priori assumption that only 365 days are in a year is ubiquitous; leap year is rarely—if ever—included in the calculation. In this article, I show how to include leap years, and I examine some of the consequences of doing so.
Exploring Nonconvex, Crossed, and Degenerate Polygons
Authors: Contreras José N.
Published: November 2020

How interactive software can be used to extend mathematical conjectures and theorems to non–convex, crossed, and degenerate polygons. The author demonstrates investigating Napoleon's Theorem with Geometer's Sketchpad.
Constructing Cooperative Logic Problems
Authors: Munakata Mika
Published: November 2020

As teachers of mathematics, we spend hours searching for good examples of mathematics problems to use in the classroom. We often seek problems that will grab students' attention, that are relevant to their interests, and that demonstrate the mathematical concepts that we hope to teach. Although we rely on many sources for these problems—such as textbooks, workbooks, journals, colleagues, and the Internet—we also spend a good deal of time constructing our own problems, whether they be for lectures, class work, homework, or tests. Designing problems is typically a task for teachers; however, this process can also be a valuable experience for our students. In this article, I describe the complexities involved in writing mathematics problems and suggest a cooperative learning activity that gives students the challenge of writing mathematical logic problems for their peers.
Delving Deeper: Revisiting Fibonacci and Related Sequences
Authors: Benjamin Arthur T. , Quinn Jennifer J.
Published: November 2020

We fully concur with Richard Askey's February 2004 “Delving Deeper” column. Discovering and proving identities containing Fibonacci numbers can be satisfying for students and teachers alike. His article touched on multiple strategies including induction, linear algebra, and a hefty dose of algebraic manipulation to derive many interesting identities. However, a single method can be employed to explain all of these identities more concretely, leading to deeper understanding and intuition. We are referring to the method of combinatorial proof.
Hands-On Fractals and the Unexpected in Mathematics
Authors: Gluchoff Alan
Published: November 2020

Within the subject of fractal geometry exist many examples of what we might call the unexpected in mathematics. Although there are many articles on fractal–building projects, one aspect of this material that provides such an example, the Contraction Mapping theorem, appears to be underrepresented. The purpose of this article is to provide a hands–on activity that illustrates this topic. We produce fractal images by iterating a set of simple rules to yield a sequence of images that in the limit consists of a fractal form. In the process, students are confronted with a surprising outcome that they actively participate in generating and that motivates a search for an explanation.
ON THE DEFLECTION OF A BEAM OF VARIABLE CROSS SECTION WITH FIXED SPAN
Authors: HO SHAN-YUH  ,   中国科学院数学研究所
Published: November 2020

The buckling problem of various beams of variable cross sections has been fully considered by many authors, and the eigenfunctions and eigenvalues of the buckling problem of those beams are available in books and papers on this subject. Firstly, this paper shows that the deflection of these beams under the combined action of lateral load and axial tensile force can be expanded in a series of buckling eigenfunctions and that the coefficients in such an expansion can be simply expressed in terms of the known eigenfunctions, eigenvalues, the axial tensile force and the lateral load. Furthermore making use of the eigenfunction expansion, an approximation method is given to determine the axial tensile force acted on the supports of a beam of fixed span.
APPLICATION OF γ RAYS EXCITED BY HEAVY CHARGED PARTICLES IN THE DETERMINATION OF ISOTOPIC ABUNDANCES
Authors: YEH MING-HAN  , LI CHENG-WU  ,   中国科学院物理研究所
Published: November 2020

Possibilities of using the γ rays excited by bombardment of an element with heavy charged particles to determine its isotopic abundances are discussed. With necessary facilities, this method, essentially a microanalytical or ultramicroanalytical method, would be convenient, especially if. a sample containing definite isotopic abundances is used as a standard for comparison. Nuclear reactions and γ rays suitable for the determination of isotopic abundances of Li, B, C and N are enumerated.