bluuebunny/crossref_metadata_2025 huggingface dataset:
Businessmen and Mathematics Teachers
Authors: Leichliter Van H.
Published: November 2020

In the late 1880's, a young German student of mathematics was to take his doctor's degree at the University of Breslau. His dissertation had been accepted and only minor formalities remained to complete his qualifications.
In-Service Education in Mathematics: Teachers Earn and Learn
Authors: Rickey Agnes Y.
Published: November 2020

Throughout this country, in the thousands of classrooms in which mathematics is being taught, an observer can find substantial numbers of teachers implementing—in some fashion-—the many curriculum innovations of the 1950's and 1960's. During these years, a large number of in-service teachers recognized the need for reeducation.
Historically Speaking—: Of Bees and Mathematicians
Authors: Siemens David F.
Published: November 2020

Observers have repeatedly noted that honeybees build hexagonal cells with trihedral bases. The astronomer Maraldi, nephew of the great Cassini, appears the first to propose that the base may be shaped for reasons of economy.
Mayan Arithmetic
Authors: Bidwell James K.
Published: November 2020

It is not unusual in the schools to study numeration systems other than our own decimal system. A good unit of this type can stimulate the students' interest in arithmetic and can be helpful in providing for practice in computation.
Operational Systems and Checker Games
Authors: Steiner Hans-Georg
Published: November 2020

In the teaching of graphs and elementary group theory at the secondary school level it is important to use intuitive and operative material in which group structure is involved. Working with materials can help to deepen the understanding or even lead the student to a first observation of the fundamental properties of a group.
Some Remarks on Discovery
Authors: Szabo Steven
Published: November 2020

It is quite common to hear from the proponents of the so-called new math that the student should he allowed to discover basic principles of, say, the real numbers. And, when they say these magic words, there is a general nodding of heads.
A Christmas Puzzle
Authors: Steiger Anne Agnes Von
Published: November 2020

Days before holiday seasons frequently are lost days, as far as the learning process is concerned. There is an atmosphere of excitement for both teachers and pupils that is not conducive to abstract mathematics. Inventing devices to keep the students mentally alert and happy is a challenge to the ablest teacher. Of no time is this fact more true than of the Christmas season.
The Magic of A Square
Authors: Freitag Herta T. , Freitag Arthur H.
Published: November 2020

Perhaps you have seen the beautiful woodcut, “Melancholia,” by the well known sixteenth century German artist, Albrecht Durer. Melancholy-depicted by Dürer not as a dejected spirit but as a creative genius of thought-muses over a problem. Amid the many symbols hanging on the wall is a square array of num bers. This is what it looks like (see fig. 1).
Continued Fractions and “Leap” Years
Authors: Rasof Bernard
Published: November 2020

The January 1968 issue of The Mathematics Teacher contained a fas cinating expository paper on “Continued Fractions.”1 It does not appear to be generally known that, until about 30 years ago, the subject called “continued fractions” was the province of a handful of “pure” mathematicians who could not abide the countless applications of their beloved discipline to science, engineering, and technology, and occupied themselves with continued fractions because they believed this to be the one area of mathematics which had no possible applications. The employment of continued fractions in other branches of mathematics was not considered to be applications; that there are many of these, however, can be seen from only a cursory glance through the National Bureau of Standards Handbook of Mathematical Functions.2
Have You Read …?
Authors: Peak Philip
Published: November 2020

The trend seems to be that of requiring mathematics as a prerequisite to outrance into most areas of study. This puts mathematicians in a position of having to change their methods of instruction.
Progress Report of the Joint Committee of the American Statistical Association and the National Council of Teachers of Mathematics
Authors: Mosteller Frederick
Published: November 2020

During the past two years the. statis tical community has put new effort into educational developments. Three main movements stand out.
Pythagorean Paper Folding: A Study in Tuning and Temperament
Authors: Mcclain Ernest G.
Published: November 2020

In Tuning the tones of a scale the ear wrestles with some fascinating prob lems concerning vibration ratios. This paper describes a convenient method for displaying to the eye the mathematical relationships involved.1
A Noncombinatorial Proof of the Number of All Subsets of a Finite Set
Authors: Arora Manmohan S. , Allen Jerry D.
Published: November 2020

Given a finite set Sn = {a1, a2, …, an}, we denote the set of all subsets of Sn by 2Sn, and prove by induction that #(2Sn) = 2n, where “#” denotes “the cardinality of.” If n = 1, S1 = {a1}, and clearly 2S1 = {ø, S1}.
Cubic Equations For High School
Authors: Ropes George H.
Published: November 2020

Methods of solving cubic equations are not generally included in Outrent secondary school curricula; nevertheless the subject is one that can be both instructive and interesting. Students with com-petence in intermediate algebra or beyond should be able to follow the material satisfactorily.
Reviews and Evaluations
Authors: Eakin Richard R.
Published: November 2020

This book is the first volume in a new series on the history of science, and is a clearly written, historically oriented introduction to the theory of numbers. It is written in a witty style especially suitable for high school students, and encourages its readers to think about numbers in an adventuresome, exploratory manner.
The Readability of Mathematics Textbooks Revisited
Authors: Kane Robert B.
Published: November 2020

All mathematics textbooks are written in more than one language. Each contains portions in a natural language, such as English, together with portions in one or more additional languages, such as HinduArabic numeration, various algebraic notational systems, the language of the sentential calculus, and the like. The mix of these languages varies greatly from book to book, topic to topic, and level to level. The natural language component often is replete with translations from one of the other symbol systems. For example, phrases such as if and only if, if … then, A and B are direct translations from the sentential calculus. Their precise meanings must be comprehended by the reader before he can hope to assign the author's intended meaning to the passage in which they occur.
A Psychedelic Approach to Conic Sections
Authors: Miller William A.
Published: November 2020

Conic sections have intrigued mathemat icians since antiquity. Apollonius wrote about them more than two thousand years ago. Although not always recognized as such, they are among the commonest curves in everyday life. They are the shadows of a circular object cast by the sun or other light source on a plane surface. The shadows cast on a wall by a lamp with a circular shade open at the top or bottom are conics. The paths of planets and satellites, both natural and artificial, relative to their parent bodies are conics. The paths of a thrown ball and water sprayed from a hose are other examples of conics. With some thought one may readily find numerous additional examples of conics in nature and industry.
Computer-oriented Mathematics: A Student Computer that Really Works
Authors: Leonard William A.
Published: November 2020

At Some time during his mathematics career, the typical high school student has been introduced to the topic of number bases and has probably changed many numbers from one base to anothet, almost always involving base two. The usual justification for this topic is that “it helps students to better understand our own number system” or that “computers use base two.” If you, as a teacher, feel guilt pangs upon offering such an uninspired rationale, you might find the demonstration described below a palatable substitute. Rather than talk about computers and their use of base two, build one and demonstrate base-two utilization! And do it with the minimum materials available in every mathematics classroom, namely, some students and a chalkboard.
Perfect Numbers, Abundant Numbers, and Deficient Numbers
Authors: Prielipp Robert W.
Published: November 2020

For several centuries students of mathematics have been fascinated by the system of positive integers and some of the remarkable properties that it possesses. Among the positive integers that have received special investigation are perfect numbers, abundant numbers, and deficient numbers.
Formulation of the Differentiation Algorithm
Authors: Kregg James F.
Published: November 2020

In an attempt to confirm the statement made by Bruner at the Woods Hole Conference of 1959 that any concept can be taught to a learner as long as it is presented in an intellectually honest form, I tried an experiment with the discovery method.
Mathematical Misteaks
Authors: Carman Robert A.
Published: November 2020

A Mathematical misteak is an incorrect operation that leads to a correct result.
A Form of Proof
Authors: Hallerberg Arthur E.
Published: November 2020

Teaching the concept of proof over a period of years to students with widely varying mathematical backgrounds has suggested several ideas to the writer.
A Note On The Teaching Of Trigonometry
Authors: Hamming R. W.
Published: November 2020

It Should Not be necessary to argue about the importance of compressing or deleting various parts of the traditional mathematics curriculum to make room for new (not necessarily recent) material. Of course something is lost in the process; all that is hoped is that the value of the new material exceeds the loss. Trigonometry seems to be a good example of material that may, with minor loss, be significantly compressed.
Helping Students to See the Patterns
Authors: Eagle J. Edwin
Published: November 2020

At many stages in the study of mathematics the theorem, problem, or topic under consideration is a generalization or further development of a specific theorem, problem, or topic previously studied. This generalization or further development, in turn, is often a narrower aspect or more specific treatment of even broader generalizations yet to come.
Empirical Evidence For The Central Limit Theorem: A Classroom Demonstration
Authors: Fischer Frederic E.
Published: November 2020

In almost every college course in elementary statistics there comes a time when the student is introduced to basic sampling theory. Because of its major importance, the central limit theorem is usually the main topic of discussion.
Sequences And Progressions
Authors: Ellis Wade
Published: November 2020

Let us consider a function whose domain is the set of positive integers and whose range is an arbitrary set. (Can you prove that the range must be a countable set?) The range of such a function is called a sequence. The nth term of a sequence is the range element corresponding to n.For convenience, the terms of a sequence are often indicated symbolically in the form S1, S2, S3, and so on.
A Factorial Curiosity
Authors: Roger Jerry L.
Published: November 2020

WOULD you believe that there are more possible seating arrangements in a classroom with thirty chairs than there are drops of water in all the oceans of the earth? Of course not! So let's prove there aren't.
The Dependence of Mathematics on Reality
Authors: Hansen David W.
Published: November 2020

MOST mathematicians today regard mathematics as an abstract structure built up from axioms by the use of logical principles. By altering these axioms, different types of mathematics can be developed; for example, Euclidean and non-Euclidean geometries differ by only one axiom—the parallel postulate—but this difference is quite sufficient to lead to very different conclusions in each system. This modern concept of mathematics stresses the idea that man invents his mathematics (and hence controls it to some extent.) rather than discovers it (the assumption here being that mathematics exists apart from and independent of man). Mathematics becomes an intellectual game. You select your axioms according to taste and work out the corresponding mathematics, using logical principles. If the results are aesthetically pleasing or elegant, this new system becomes a part of mathematics. Notice here that for the modern mathematician there is no mention of applicability to reality. The system stands alone on its consistency and beauty. If, for some reason, the system can be used in explaining certain properties of the real world, this is of no import to the mathematician. G. H. Hardy, in his book A Mathematician's Apology, expresses this view quite forcefully. It has been said by Huxley that six monkeys pecking unintelligently on typewriters would, after millions of millions of years, eventually write all the books in the British Museum (Sir James Jeans quotes Huxley's statement on page 4 of The Mysterious Universe). Similarly, mathematicians randomly selecting various axioms would eventually produce significant mathematical systems, but undoubtedly only after an enormous amount of time had been wasted.
A Theorem on Lines of Symmetry
Authors: Shilgalis Thomas W.
Published: November 2020

Through the influence of the trans-formation concept, in geometry, the topic of symmetry has been receiving renewed I attention. The UICSM's Motion Geometry, for example, makes extensive use of symmetry in the classification of triangle and quadrilaterals.
Discovering S=n(n+1)2
Authors: Schattschneider Doris J.
Published: November 2020

Every student who learns the method of proof by mathematical induction is familiar with the formula
A Generalization of Vux Triangles
Authors: Brumfiei Charles
Published: November 2020

In the may 1970 issue of the Mathematics Teacher Professor Fitch Cheney considered triangles having one angle the double of another, giving them the name of vux triangles.
On Stieltjes Integrals
Authors: Iacobacci Rora
Published: November 2020

There are few topics in elementary calculus that elicit as much interest and spirited activity as that of the Riemann integral and its applications. Even with a moderate proficiency in the subject, a student can sense and appreciate the power of the concept. Unless a student pursues advanced courses in analysis, however, this is usually the only type of integral to which he is exposed, and his formal mathematical studies are terminated without his having seen any of its possible modifications. There are, in fact, a great number of integrals derived through various and successive modifica-tions of the Riemann integral. Among them are stieltjes integrals.1
Superstitious?
Authors: Wagner John , McGinty Robert
Published: November 2020

FRIDAY the 13th!!! Those who cringe at the thought of Friday the 13th will be unhappy to know that there is at least one such Friday every year. As usual, proof by assertion or intimidation doesn't make a statement true. However, a statement such as this one should have some built-in motivation for students, and in trying to justify the statement you can bring in some ideas from number theory as well as some techniques of proof.
You Too Can Be a Computer, or Part of One
Authors: Henle James M.
Published: November 2020

In these days when technology triumphs over all, and when science answers almost more questions than we can ask, there are a nervous few who are not as impressed as they are frightened. As we are flooded by dials and digits, by machines and memory banks, by programs and print-outs, by dishwashers and DO-loops, by analogs and algorithms, and by submarines and subroutines, we frequently hear the anguished humanists crying out against this “madness,” this “dehumani-zation.” They protest that a man is no longer identified by his name, but by his number. They protest that society is robbing man of his humanity. They protest that thinking is being replaced by calculation. Worst of all, they claim, Human Beings are slowly being turned into Machines and Parts of Machines.
Some Techniques in Handling a Slow Class in Elementary Algebra
Authors: Garf David
Published: November 2020

THE first lesson of the College-Bound Algebra I class in a New York City high school turned out to be a fiasco for the teacher, Mr. B. It seemed to run the gamut of almost all undesirable classroom experiences. Twenty students were registered for the class. Some were late, others absent. Those who did come were apathetic, inattentive, unresponsive, stared out the windows, doodled, and communicated with their neighbors. When the period ended, they had not learned one iota of mathematics.
A Geometric Technique for Factoring Polynomials
Authors: Meconi L. J.
Published: November 2020

The task of factoring a first- or second-degree polynomial is not a difficult one; most students experience success after attempting several examples involving the various common procedures. Some of the usual procedures are applicable to higher degree polynomials, but in general, the task becomes much more difficult. In this paper we shall inspect an unusual approach to factoring polynomials of the form anxn + an-1xn-1 + · · · + a1x + ao, where the ai and n belong to Z, the set of integers, with n ≥ 0. The procedure has proved to be of interest to some students, and it lends itself very naturally to the beginning of independent research.
New Publications
Authors: Dodes Irving Allen , Fox Mary Lou , Bernice Munro H. , Peak Philip , Schaumberger Norman , Skeen Kenneth C.
Published: November 2020

This is a complete review of secondary school geometry but with the added features of suggestions for motivation, visual aids, teaching strategies, student and teacher activities, and enrichment materials. The historical materials are well done and appropriately located for optimum use. The nature of geometry as a discipline is made clear, and both modern and Euclidean geometries are investigated. The appendix includes tables. formulas, summaries of definitions, postulates, theorems, and reference lists of films, books, and periodicals.-PEAK.
Graphical Mathematics for the Preengineering and Science Student
Authors: Elkins Richard L. , Wockenfuss William A.
Published: November 2020

A Picture is worth a thousand words! All of us are familiar with this cliche, hut have we given any thought to applying it to our college preparatory mathematics courses? Graphical methods can certainly help foster an understanding of the physical concepts of mathematical theory and the application of the theory.
Mathematical Induction and a Programming Problem
Authors: McGuire Bill
Published: November 2020

An interesting application of nathematical induction to programming came up in the computer-science class at North Bend High School. The problem was to construct a FORTRAN program so order an arbitrary set of numbers in ascending sequence.
Collapsible Models of Isosceles Tetrahedrons
Authors: Trigg Charles W.
Published: November 2020

To most laymen, the word pyramid invokes an image of the Egyptian variety, with isosceles triangular lateral fares and a square base. Similarly, to many students a tetrahedron is the regular kind with four equilateral triangular faces. This restricted concept is fostered by the prevalent use of models of this type. Understanding can be increased by offering the student an opportunity to see, handle, and make models of other types of tetrahedrons.
Construction of a Slide Rule with Compass and Straightedge
Authors: Westwood Jack R.
Published: November 2020

An article in the may 1970 issue of the Mathematics Teacher, “The Rationale of Slide Rule Manipulation,” by Margaret Anne Rogers, motivated me to search for a method of using a compass and straightedge to construct the scale of a slide rule. This would eliminate the use of logarithms, which are needed in her method, and enable the student to approach the problem from a geometric point of view.
The Empirical Foundations of Probability Theory
Authors: Burdick David L.
Published: November 2020

It is desirable that scientifically oriented secondary and college students be exposed to the role played by mathematics in modeling physical theory and experiment. The nature of the probability model and the empirical evidence offered to support its applications can furnish a very effective counterpoint to the standard examples of the empirical evidence from astronomy and physics that justifies theoretical mechanics.
Problem Solving and Multiplication Of Rational Numbers—Three Old Devices
Authors: Clason Robert G.
Published: November 2020

This article examines three old devices for solving problems that require the mul-tiplication of rational numbers. These devices can be characterized as “rule” rather than “reason.” They are not in keeping with “modern” mathematics Whose abstractness and generality abrogate such concrete and special methods. They are, however, of interest, since they show how arithmetic has changed, and they might be used for the amusement of teachers and students and possibly for enrichment.
On The Occasional Incompatibility Of Algebra and Geometry
Authors: Farrell Margaret A. , Ranucci Ernest R.
Published: November 2020

With due apologies to Descartes, the marriage of algebra and geometry has its ups and downs.
Assignments: Yesterday, Today, and Tomorrow-Today
Authors: Laing Robert A. , Peterson John C.
Published: November 2020

Observers of mathematics teachers often comment how easy it is to predict what they will observe in a mathematics classroom. First the class will go over the homework. This includes checking answers and answering any questions. Then there will be a short lesson—normally, in spite of all the lip service given to “discovery” teaching, this will be a lecture. And finally the homework assignment will be given, and the students will begin working on it.
Let Your Students Write a Book
Authors: Montague Harriet F.
Published: November 2020

A book on matrix algebra authored by able students in grades 10-12 was the result of a teaching-learning experiment in an NSF Summer Institute in Mathematics for High School Teachers. The experiment might serve as a model for classroom and/or laboratory activities at any grade level and with different ability groups.
The Second U.S.A. Mathematical Olympiad
Authors: Greitzer Samuel L.
Published: November 2020

Because the U.S.A. Mathematical Olympiad is a new venture, a brief explanation is pertinent. The purpose of the Olympiad is to attempt to discover secondary school students with superior mathematical talent—students who possess creativity and inventiveness as well as computational skills. Participation is limited to about one-hundred students selected mainly from the Honor Roll of the Annual High School Mathematics Examination plus a few recommended students from the states that sponsor their own high school mathematics competitions. The Olympiad consists of five problems of the essay type requiring mathematical power on the part of the participants.
An Application of Newton’s Law Of Cooling
Authors: Hurley James F.
Published: November 2020

Many introductory calculus texts now include a section devoted to applications of the exponential function. The large variety of commonly encountered phenomena whose processes of growth or decay are exponential affords a rich source of interesting and appealing applications. Students who can easily compute the size a century hence of a world population growing continuously at a rate of 2 percent need little further in the way of convincing that calculus can be useful in the real world of today. Problems involving radioactive decay and growth of investments also strike most students as interesting, dealing as they do with questions that, if not familiar, have at least been heard of by most students.
Computer-Programming Problems Involving Place Value
Authors: Norris Fletcher R.
Published: November 2020

This article shows how a computer can solve problems and how programming a computer can reteach a concept such as place value in numerals.
Transformations in High School Geometry before 1970
Authors: Usiskin Zalman
Published: November 2020

A thorough discussion of early appearances of approaches using transformations in geometry textbooks intended for use in high schools.
The Golden Section and Conic Sections
Authors: Verno C. Ralph
Published: November 2020

The points of intersection of various conics are shown to involve the Golden Section.
Recreation: Traceable Houses
Authors: Spaulding Raymond E.
Published: November 2020

THE diagrams in figure 1 are floor plans for houses. A house is considered traceable when it is possible to construct a path that takes you through each door of the house exactly one time. Which of the given houses are traceable?
Area Ratios in Convex Polygons
Authors: Kulm Gerald
Published: November 2020

This article illustrates how a geometric relationship for quadrilaterals can be extended to a new generalization.
A Note on Mathematical Induction
Authors: Brumfiel Charles
Published: November 2020

Do your advanced students really understand the concept of mathematical induction? Try this phony “proof” on them.
Geometric Generalizations
Authors: Miller Leslie H. , Waits Bert K.
Published: November 2020

First a simple construction problem, then a classic solution, next a more novel approach, and finally a generalization leading to more difficult problems.
Extracting Fifth Roots—a Classroom Activity
Authors: King Irv
Published: November 2020

This article describes an activity that may be used to stimulate interest in work with powers and roots.
Equations of Geometric Figures
Authors: Johnson Carl S. , Ahuja M. M. , Palmer Leonard
Published: November 2020

A geometry student who wonders about ways to write equations for segments, triangles, and faces of a cube will find this article of interest.
Editorial
Authors: _ _
Published: November 2020

From time to time we have expressed our desire to receive manuscripts on topics of interest to our readers. A recent review of the manuscripts we receive as compared with the interests of our members indicates that more material is needed in a variety of areas. These areas are not ranked in order of importance.
Mathematical Olympiad Competitions in the People's Republic of China
Authors: Becker Jerry P. , Hsi Kathy C.
Published: November 2020

A delegation of mathematics teachers sponsored by the National Council of Teachers of Mathematics (NCTM) made a study visit to the People's Republic of China 21 October-6 November 1977.* During that visit the delegation learned of Chinese plans to initiate national mathematical olympiad competitions and university admissions tests. The nature and content of these competitions seem to be assuming considerable importance in China.
Clubs: Pascal's Triangle: Some Recent References from the Mathematics Teacher
Authors: Grinstein Louise S.
Published: November 2020

Pascal's triangle appears in many different contexts in mathematics. A study of its properties and applications can make a fascinating mathematics club presentation. The following annotated list of some pertinent references was compiled from issues of the Mathematics Teacher since 1973. An asterisk (*) indicates items for students interested in programming.
Help!!! What Computer Should I Buy???
Authors: Braun Ludwig
Published: November 2020

The plea in the title of this article has been made to me hundreds of times in the past three years by educators and administrators all over the United States. These people are bewildered by the technology and its capabilities, by the large number of companies marketing microcomputers, by the rapid rate of new developments in microcomputer technology, and by the wide range of models and peripherals available.
Computer Literacy—What Should It be?
Authors: Luehrmann Arthur
Published: November 2020

The way things are is not necessarily the way they should be.
A Partridge in a Pear Tree, a Stack of Cubes, and Four Buckets of Balls
Authors: Vance Irvin E.
Published: November 2020

A variety of seemingly unrelated problems have similar solutions.
The Tenth U.S.A. Mathematical Olympiad
Authors: Greitzer Samuel L.
Published: November 2020

Although it might be expected that after nine years of the U.S.A. Mathematical Olympiad (USAMO) procedures would be standardized, it is a fact that there are always sufficient differences and surprises that lend uniqueness to the contest each year. This year was no exception.
Probability, Computer Simulation, and Mathematics
Authors: Walton Karen Doyle
Published: November 2020

Many students fail to recognize the interrelationships among computers, pure mathematics, and the real world. The subject of probability can motivate students to explore various methods of problem solving, combining empirical data, simulations, and theoretical solutions. Probabilistic statements encountered in everyday life (e.g., weather predictions, state lotteries, financial forecasting) are frequently derived from the integration of observation and theoretical analysis.
Sharing Teaching Ideas:Simplifying Radicals On A Geoboard
Authors: Coltharp Forrest L.
Published: November 2020

Geoboards are commonly used to introduce geometric concepts. With the help of the Pythagorean theorem, the geoboard can also be used to help students understand the simplification of radicals.
Octagons at Monticello
Authors: Wielenberg Peggy
Published: November 2020

Having to conduct my grandson through his course of mathematics, I have resumed that study with great avidity. It was ever my favorite one. We have no theories there, no uncertainties remain on the mind; all is demonstration and satisfaction.
Editorial: A Focus on Data Analysis
Authors: Burrill Gail
Published: November 2020

“Few Women Educators Are at the Top in Waukesha County, Survey Shows”; “Alarming Stats on Assaults Lead to Assembly Bill 431”; “Big-Time Athletes Lose in Class, Study Finds.” These headlines, all from one edition of a large metropolitan newspaper, are typical of the increasing importance of data analysis in today's society. We are confronted with quantitative information in every phase of our lives. Decisions on health, citizenship, parenthood, employment, and sports are made using charts, graphs, percentages, probabilities, averages, and forecasts. Workers must read charts and graphs to determine whether a product meets the company's standards. Business personnel are given sheets of data on the basis of which they must decide which company to close, which product to sell, or which drug has dangerous side effects.
Implementing the Standards: Uses of Technology in Prealgebra and Beginning Algebra
Authors: Heid M. Kathleen
Published: November 2020

The NCTM's Curriculum and Evaluation Standards for School Mathematics (Stan dards) (1989) designates four standards that apply to all students at all grade levels: mathematics as problem solving, mathematics as communication, mathematics as reasoning, and mathematical connections. These and NCTM's other standards are embedded in a vision of technologically rich school mathematics classrooms in which students and teachers have constant access to appropriate computing devices and in which students use computers and calculators as tools for the investigation and exploration of problems.
Why is the SsA Triangle-Congruence Theorem not Included in Textbooks?
Authors: Hirschhorn Daniel B.
Published: November 2020

An SSA congruence theorem does exist. The theorem has been the subject of two articles in the Mathematics Teacher since 1981 (Litwiller and Duncan [1981] and Yeshurun and Kay [1983]). These articles, it appears, have had no effect on geometry textbooks. Although current textbooks correctly point out that SSA in its most general form is not enough to prove triangles congruent (Hirsch et al. 1987; Jurgensen, Brown, and Jurgensen 1988; Nichols et al. 1986; Travers et al. 1987; Sobel et al., 1986), the SSA theorem discussed in these articles can be used to prove triangles congruent.
Applications: Linear Function Saves Carpenter's Time
Authors: Crouse Richard J.
Published: November 2020

Recently, a carpenter asked me a very interesting question. He was building roofs that were 30 feet long at the base (fig. 1). The pitch of each roof rises 4 inches for every horizontal foot. His job was to put in vertical supports every 16 inches, forcing him to climb up the ladder, measure 16 inches horizontally, and then measure the vertical height at that point. Then he climbed down the ladder, sawed the support, and climbed back up to put it in place. This procedure was repeated for each support. He asked me if a formula existed that he could use to determine the lengths of the supports in advance so that he wouldn't have to climb up the ladder each time to measure.
Sharing Teaching Ideas: Princess Di, Spaghettios, and M & Ms
Authors: Isaacs Carol M. , Fisher Julie
Published: November 2020

What do the items in the title have in common? Well, read on.
The Graphics Calculator: A Tool for Critical Thinking
Authors: Dion Gloria
Published: November 2020

Several years after the introduction of graphics calculators, no consensus has been reached on how these calculators should be used in secondary school and undergraduate mathematics. In this article I explore some issues that arise in integrating calculators into the traditional precalculus and calculus curricula, including a discussion of perils and pitfalls awaiting the unsuspecting, unthinking user, presented as a Stress Test for Calculators at the end of this article.
PiDay
Authors: Waldner Bruce C.
Published: November 2020

If locking for an interesting way to encourage students and foster creativity and to help students express themselves mathematically in imaginative ways, try a fun day of mathematical contests and activities. Pi Day, in honor of the number with approximate value 3.14, is held on March 14, otherwise known as 3/14. I have made the day's activities into a set of contests that are specifically designed to help students recognize that they are able to participate in mathematics even if they have had limited classroom success with the subject.
Sharing Teaching Ideas: Geometry and Poetry
Authors: Thompson Betty B.
Published: November 2020

In my geometry classes, I encourage students to notice how we are influenced by geometry and how common it is in our everyday lives. We also try to explore ways to integrate geometry into other subject areas. This year we did a geometry-in-poetry project. After spending a week studying Carl Sandburg at the North Carolina Center for the Advancement of Teaching, I chose him as our poet to study. As a poet of the people, Sandburg used realistic images and themes. He wrote about ordinary things-the places and people my students know and understand. This quality made his poetry especially good for thi project
Using Logarithms to Explore Power and Exponential Functions
Authors: Rahn James R. , Berndes Barry A.
Published: November 2020

Power functions and exponential functions often describe the relationship between variables in physical phenomena. Power functions are equations of the form y = kxn (see fig. 1), where k is a nonzero real number and n is a nonzero real number not equal to 1. Exponential functions are equations of the form y = kbx (see fig. 2), where k is a nonzero real number and b is a positive real number. Students should be able visually to recognize these functions so that they can easily identify their appearance when experimental data are graphed. When physical phenomena appear to describe exponential and power functions, logarithms can be used to locate approximate functions that represent the phenomena.
Connecting Research to Teaching: Problem Solving: Managing It All
Authors: Fernandez Maria L. , Hadaway Nelda , Wilson James W. , Graeber Anna O.
Published: November 2020

Teachers and researchers have found that students presented with mathematics problems often begin performing calculations without giving much thought to the problem (e.g., Artzt and Armour-Thomas [1990]; Lester [1985]; Schoenfeld [1983, 1985a, 1985b, 1987]). Students often attend primarily to such surface features of the problem as context or type and the size of the given numbers. Such students often pursue unfruitful directions without monitoring or assessing their knowledge or actions, only to become frustrated and withdraw from the problem-solving activity. Schoenfeld (1983, 1985a, 1985b, 1987) presents examples of students' Jack of monitoring or managing their actions or knowledge during problem-solving episodes, which result in their pursuit of “wild mathematical geese.” Kilpatrick (1985) indicates that school instruction often reinforces students' impulsivity. Typically, students are not given much time to think about and make sense of the mathematics concepts they are learning. They are not encouraged to invoke managelial processes to monitor, regulate, or assess their understanding and actions. This article includes a blief review of research dealing with the individual's managelial processes during problem solving and discusses the implications of this research for problem-solving instruction.
Using Graphical Misrepresentation to Stimulate Student Interest
Authors: Hansen Will
Published: November 2020

The graphing calculator is a wonderful tool that can be used to generate visual representations to help develop a deeper understanding of mathematical relationships. In addition, the graphing calculator can be used to generate startling misrepresentations that can be thought-provoking. Whenever we can surprise our students with counterintuitive results—Why did that happen?—their increased interest level presents an exceptional learning opportunity. This article gives a few examples of how misrepresentations of periodic fun ctions on the Texas Instruments TI-81 graphing calculator can be used to enhance the understanding of periodicity and the discrete nature of the graphing screen. Although the illustrations use the TI-81, similar results can be obtained with other graphing calculators or computer-based graphing utilities.
A.I.M.S. in the Classroom
Authors: McLachlan Ian D. , Ryan David J.
Published: November 2020

Hilton College is an independent school for boys in South Africa with strong traditions and a well-respected academic reputation. Three years ago, although Hilton's examination results were good and the vast majority of finalyear students (the equivalent of grade 12 in the United States) were transferring to universities, the mathematics department was nevertheless uneasy. This uneasiness rested in the fact that, in our opinion, schooling and education are not necessarily synonymous.
Soundoff: Four Teacher-Friendly Postulates for Thriving in a Sea of Change
Authors: Leinwand Steven
Published: November 2020

Many of us chose mathematics teaching because it was always so neat and clean. We felt an affinity toward teaching and learning mathematics because it was orderly and logical. Almost always, we arrived at only one numerical answer by using one right procedure that could be easily graded either right or wrong. We knew that with our beloved mathematics, we suffered none of the gray areas that plague the disciplines of language arts and social studies. And we knew that we would be rewarded for teaching mathematics the way we ourselves were taught. But, oh, how things have changed!
Teaching Relationships between Area and Perimeter with The Geometer's Sketchpad
Authors: Stone Michael E.
Published: November 2020

The question “What shape will have the largest area for a given perimeter?” identifies an important relationship between area and perimeter that has long been intuitively realized in many cultures. Zaslavsky points out that numerous cultures make practical use of this relationship between area and perimeter: cylindrical houses in West Africa, beehive houses of the Chagga people of Mount Kilimanjaro, and hemispherical igloos of the Eskimo people. While struggling to cope with their environments, all these cultures used available materials to build dwellings with shapes that maximized floor space. Since perimeter is limited by the available materials, determining what shape will have the largest area is very important to people who find materials difficult to work with or hard to obtain. Mathematics is integrally related to social studies and architecture (Zaslavsky 1989, 19). The purpose of this activity is to go beyond intuition and instill a conceptual understanding of relationships between area and perimeter.
Connecting Research to Teaching: Geometry and Proof
Authors: Battista Michael T. , Clements Douglas H.
Published: November 2020

Educators continue to debate the relative emphasis that formal proof should play in high school geometry. Some argue that we should continue the traditional focus on axiomatic systems and proof Some believe that we should abandon proof for a less formal investigation of geometric ideas. Others believe that students should move gradually from an informal investigation of geometry to a more proof-oriented focus.
Sharing Teaching Ideas: Starship
Authors: Camp Dane R.
Published: November 2020

Every year in my precalculus classes, I introduce polar coordinates. Though students seem to have little trouble with the concept of using a different coordinate system, they do have difficulty with the fact that a location has multiple representations. For instance, the point (2, 30†) can also be expressed as (2, −330†), (−2, 210†), (−2, −150†), (2, 390†), and so on. Knowing how to manipulate these alternative forms is crucial to understanding polar graphs.
The Volume of a Sphere: A Chinese Derivation
Authors: Swetz Frank J.
Published: November 2020

The corpus of mathematical knowledge possessed by the ancient Chinese can primarily be found in one book, Jiuzhang suanshu, or Nine Chapters on the Mathematical Art. This book was compiled about 200 B.C. as a compendium of the mathematics known and used in China. It was a bureaucratic handbook and contained all the mathematics believed necessary for a court official to run the empire; for example, the book possessed sections devoted to computing tax rates and salaries, finding the volumes of geometric solids needed in the construction of fortifications, and undertaking land surveys (Swetz 1972). In the presence of such an officially sanctioned handbook, mathematical scholarship was limited to copying this work and perhaps commenting on its text. Through this process, the contents of the Nine Chapters were both perpetuated and clarified. The crude mathematical approximations it contained were refined, and actual mistakes in its formulas and procedures were eventually corrected. One such error to undergo this evolution was the determination of the volume of a sphere with a known radius. Examining just how ancient Chinese mathematicians attempted to resolve this problem lends valuable insights into early mathematical thinking and problem solving
A Pôlya Sampler
Authors: Sinicrope Rose
Published: November 2020

On 1 June 1990, the Mathematical Association of America dedicated the Pôlya Building. In her account of the dedication ceremony, Maureen A. Callanan wrote, “[T]he participants demonstrated that Polya’s influence and reputation have extended not only to high school mathematics teachers and established NSF [National Science Foundation] scientists, but also to the mathematics students of the 1990s” (1990, 2). Who was George Polya that he has been honored by mathematicians and has influenced scientists, high school teachers, and students of mathematics?
Connecting Research to Teaching: Guidelines for Teaching Plane Isometries in Secondary School
Authors: Jaime Adela , Gutiérrez Angel
Published: November 2020

Two key concepts strongly supported by the NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) are (1) that geometry should be taught from multiple perspectives and (2) that it is necessary to make mathematical connections. Both concepts can be implemented by appropriately selecting topics that can be presented to students from several points of view in different environments and that can also link different branches of mathematics or mathematics and other sciences.
Implementing the “Professional Standards for Teaching Mathematics”: What Is It Worth?
Authors: Alper Lynne , Fendel Dan , Fraser Sherry , Resek Diane
Published: November 2020

It used to be easy. You followed the book, and all books were very much alike in terms of content and approach. You occasionally supplemented the content with an activity of your own, but it was clear what was expected. A curriculum had been established, and everyone taught it in pretty much the same way. The idea was to train students to do specific types of problems.
Teaching Sensible Mathematics in Sense-Making Ways with the CPMP
Authors: Hirsch Christian R. , Coxford Arthur F. , Fey James T. , Schoen Harold L.
Published: November 2020

Current policy reports addressing mathematics education in American schools, such as Everybody Counts (NRC 1989), Curriculum and Evaluation Standards for School Mathematics (NCTM 1989), Professional Standards for Teaching Mathematics (NCTM 1991), and Assessment Standards for School Mathematics (NCTM 1995), call for sweeping reform in curricular, instructional, and assessment practices. Implementing the proposed reforms poses new opportunities and challenges for school districts, mathematics departments, and classroom teachers.
The Football Coach's Dilemma: “Should We Go for 1 or 2 Points First?”
Authors: Schielack Vincent P.
Published: November 2020

Situations arise in many everyday endeavors that can be analyzed using various mathematical techniques. These situations give mathematics educators many opportunities to connect real-world problem-solving situations with appropriate mathematical models, as recommended in the NCTM's Curriculum and Evaluation Standards for School Mathematics (1989). The mathematics topic here involves applying elementary concepts of probability to a hotly debated question arising in football. h will be assumed throughout that a team values a win significantly more than a tie and also values a tie considerably more than a loss.
Technology Tips: Classroom Technology: Tool for, or Focus of, Learning?
Authors: Day Roger
Published: November 2020

The appropriate use of technology is an important component of mathematics learning, teaching, and assessment as we strive to realize the vision of the NCTM's Standards documents (1989, 1991, 1995) and as we anticipate the arrival of the next millennium. Computers and calculators can make mathematics an active laboratory experience. The following four scenarios exemplify technology use in support of these goals.
Folded Paper, Dynamic Geometry, and Proof: A Three-Tier Approach to the Conics
Authors: Scher Daniel P.
Published: November 2020

Some of the most lasting impressions that we retain from the study of circles come from our informal exposure to them in daily life. It may have been years since we learned, and perhaps forgot, the equation x2 + y2 = r2, but we can still summon our hands-on experiences to remember the essential features of a circle. The notion of a circle's having a center that is equidistant from all its points gets reinforced whenever we use a compass or twirl a rope or cord with our fingers. The symmetry of a circle becomes apparent each time we turn a knob or struggle to loosen a lid on a jar.
Two Car-Buying Strategies: An Old Problem Revisited with a New Analysis
Authors: Sen Tapan , Raiszadeh Azar D. , Raiszadeh Farhad M. E.
Published: November 2020

This article is an extension of one published in this journal by Bland and Givan (1983). The objective of that piece was “to present a problem in applied mathematics that can be used as a study project by a secondary school student.” The problem addressed applications of sequences and series. It touched on a variety of topics, such as beginning algebra, the theory of interest, and the theory of sums of series of finite length. This article expands the scope of the previous work and demonstrates the application of spreadsheet analysis in solving equations beyond quadratics.