bluuebunny/crossref_metadata_2025 huggingface dataset:
The Euclidean Algorithm as a Matrix Transformation
Authors: Ibrahim Aziz , Gucker Edward
Published: November 2020

The algorithm of Euclid for finding the greatest common divisor of two positive integers is based on repeated application of the division algorithm.
The Schwarz Paradox: An Interesting Problem for the First-Year Calculus Student
Authors: Atneosen Gail H.
Published: November 2020

The following result is well known in at it frequently appears tucked away in problem list at the end of a beginning calculus text.
Garbage Collection, Sunday Strolls, and Soldering Problems
Authors: Meyer Walter
Published: November 2020

Is there an area of mathematics that deals with garbage collection, Sunday strolls, and soldering problems all at once? Indeed there is, and it is called graph theory, a subject that considers the properties of configurations consisting of points and connecting lines such as the configuration shown in figure 2. (There is another meaning for the word graph, as in bar graph or graph of a function, which is not meant here.) The practical applications of graph theory are so widespread that this theory has become one of the most important and rapidly growing areas of applied mathematics in recent years. What is especially unique about it, however, is the extreme simplicity of the basic ideas. Because of this dual nature of practicality and simplicity, graphs have been creeping into the high school curriculum lately, often in the form of optional topics.
A = P(l + r/n)nt, or How to Gain Some Interest in the Classroom
Authors: Goldberg Dorothy
Published: November 2020

In the study of logarithms it is common practice to teach students to evaluate the compound-interest formula using logarithms.
Investigating the Nature of Periodic Decimals
Authors: Hutchinson Margaret R.
Published: November 2020

Every junior high school student knows that a rational number has a decimal representation that can be found by dividing the numerator by the denominator, and he has learned by experience that these decimals follow a pattern. Some-times the division “comes out even”; that is, after a number of places he gets a remainder of zero. If this does not happen, he lims observed that the decimal will eventually repeat a block of its digits indefinitely.
An Analysis of Elementary Functions for High School Students
Authors: Boersig Teresa M.
Published: November 2020

Probably no subject in the curriculum begins more concretely and ends more abstractly than does mathematics as it travels the brain waves of students from K to 12. From using colored beads and flannel boards to finding zeros of higher-order polynomials or determining whether a given system is a group or a field is a giant step that few complete. While it is not to be expected that all or even most students will persevere through twelve years of mathematics, many capable students quit the ranks too soon.
A Preservice Institute for Teachers of Secondary Mathematics
Authors: Retzer Kenneth A.
Published: November 2020

Undergraduates, while student teaching, often have little contact with their campus except for visits from their college supervisor. They have no convenient way of sharing their experiences with their peers, of asking a mathematics professor ahout a troublesome topic in subject matter, of becoming acquainted with alternatives in textbooks, supplementary materials, and teaching aids, or of becoming involved in the production or field testing of experimental materials. Some have never taken a mathematics-education course, and others do not realize the relevance of what they have studied to teaching mathematics until they begin to wrestle, with problems in teaching mathematics.
Another Geometric Introduction to Mathematical Generalization
Authors: Kung H. L.
Published: November 2020

This article presents a geometric model that, through Various arrangements, can also exhibit the results of the above sums.
The Forum: Individualization of Instruction: A Profile of Individualized Instruction
Authors: White Virginia T.
Published: November 2020

Individualized instruction means that the student has been matched to an instructional system such that he is working at his own speed, learning style, and ability level on appropriate materials in keeping with his goals, supported by adequate assistance in a suitable learning environment. Individualized instruction is not synonymous with independent study, although independent study might be a proper subset of the total methods used.
Ways of Varying Classroom Procedures
Authors: Zamboni Floyd F.
Published: November 2020

The desire to individualize instruction as class size increases is a growing problem to many teachers. An ever increasing challenge to teachers is varying classroom procedures and presentations, while keeping the daily routine interesting, exciting, and relevant to the individual student.
A Comparison of Approaches for Teaching Remedial Courses at the College Level
Authors: Bittinger Marvin L.
Published: November 2020

When arithmetic, elementary algebra, and intermediate algebra are taught at the college level, they are considered remedial. At some colleges pre-calculus courses (al-gebra and trigonometry, elementary functions, and so on) are also considered remedial. This article will describe several approaches to teaching such courses and will point out pros and cons of each. The views are pragmatic, based on experiences and reactions of students and instructors. While the experiences on which the article is based are at the college level, the courses dealt with are at the high school level. It is felt that what is said applies both at the high school and at the college level.
Discoveries in Mathematics—Pythagorean Triples
Authors: Arpaia Pasquale J.
Published: November 2020

We start with a very innocent assertion.
The Wheel of Aristotle
Authors: Ballew David W.
Published: November 2020

PARADOXES have often been used to add interest to logic and proof. One which has often been used is known as the Wheel of Aristotle. We are given two circular wheels A and B with different radii and with the smaller wheel B fixed to A so that the centers coincide (see fig. 1).
The Forum: Accountability: A Useful Idea Whose Time Has Come
Authors: Wells David W.
Published: November 2020

THERE seems to be little doubt that the demand for accountability in education is one important consequence of the economic pressures on schools. To keep pace with salaries in other professions, teachers' salaries have more than doubled since 1960. The cost of physical plants and educational materials and the cost of other nonteaching services required for operating schools have also increased at a rapid rate. The financial needs of many school systems both large and small have exceeded their ability to produce revenues. The public is now demanding that schools demonstrate that the monies provided arc used efficiently and that the effectiveness of the schools increase in proportion to the increase in financial support of the schools.
Mathematics of Musical Scales
Authors: Malcom Paul S.
Published: November 2020

THERE are many musical scales. Although it is not apparent to the casual listener, the scale of tones of the piano differs from that of the violin. Middle Eastern music can only be approximated on most Western instruments because their scales are different. These scales and their differences can best be explained through mathematics.
The Three-Point Problem
Authors: Bradley A. Day
Published: November 2020

This problem may be of good use for (he due Scituation of Sands or Rocks, that are within sight of three Places upon Land, whose distances are well known; or for Chorographical Uses &e. Especially now there is a method of observing Angles nicely accurate by ayde of the Telescope; and was therefore thought fit to be now published. though it be a competent time since it was delivered in writing.
In Search of the Perfect Scalene Triangle
Authors: Raphael Brother L.
Published: November 2020

Geometry seems to afford more surprises than other branches of mathematics, since we have so many more spatial intuitions than numerical ones. We find it easy to deal with figures and drawings, as opposed to the rather more difficult business of juggling numbers mentally. However, many times our intuitions about geometrical figures lead to curious errors. We shall now look at an extremely simple ease and show how a small amount of orderly thinking can give a clear, general result in a situation where our natural feelings seem to go wrong.
Mathematical Induction-Indirectly
Authors: Conklin June
Published: November 2020

Finding myself with an infinitely long ladder in one hand and in the other a class whose disbelief seemed as unending as the ladder, I began to consider an indirect approach to mathematical induction. We had worked with the negations of existential and universal quantifiers which would be essential for this different approach. Also we had developed the nec-essary background in summation notation to maintain some degree of clarity in the proof.
The National Council of Teachers of Mathematics: Nominees for 1974 Election
Authors: Tolson Juanita , Begnaud Lurnice , Brydegaard Marguerite , Ferguson W. Eugene , Greenholz Sarah , Heinke Clarence H. , Hlavaty Julius H. , McNabb William K. , Smith Lchi T.
Published: November 2020

The Nominations Committee needs your help in selecting nominees for the 1974 election. At that time, four persons will be elected to serve on the Board of Directors. Please send your suggestions of NCTM members whose records of service and leadership in the field of mathematics education qualify them for consideration to Mrs. Juanita S. Tolson, 3003 Van Ness Street, NW, Apt. W727, Washington, D.C. 20008.
An Algebraic Treatment of Magic Squares
Authors: Munger Ralph
Published: November 2020

Magic squares have always served as a good motivation device for drill in elementary and junior high mathematics. This article will consider an algebraic treatment of the 3×3 magic square to show some algebraic activities that can be carried out by the beginning or the advanced algebra student.
The Binary Adder: A Flow Chart for the Addition of Binary Numbers
Authors: Ellison Alfred
Published: November 2020

Flowcharting has proved extremely useful in activities ranging from programmed instruction to computer programming. Flowcharting binary addition offers students a mathematical setting for an excellent initial experience in flow charting.
The First U.S.A. Mathematical Olympiad
Authors: Greitzer Samuel L.
Published: November 2020

At ITS meeting on 1 September 1971, the Mathematical Association of America agreed to sponsor a U.S.A. Mathematical Olympiad in addition to the Annual High School Mathematics Examination. The purpose of tlie Olympiad was to attempt to discover secondary school students with superior mathematical talent, students who possessed mathematical creativity and inventiveness as well as competence in computational techniques. Participation was to be limited to about 100 students selected from the Honor Roll on the High School Mathematics Examination, plus a few students of superior ability selected from those states that did not participate in the High School Mathematics Examination. The Olympiad itself was to consist of five essay-type problems requiring mathematical power on the part of the participants.
Remarks on the Neglected Mean
Authors: Ercolano Joseph L.
Published: November 2020

The harmonic mean of two positive, real numbers was known to early Greek mathematicians. In fact, it is alleged that “Pythagoras learned in Mesopotamia of three means—the arithmetic, the geometric, and the subcontrary (later called the harmonic)—and of the ‘golden proportion’ relating two of these: the first of two numbers is to their arithmetic mean as their harmonic mean is to the second of the numbers” (Boyer 1968). Archytas, a disciple of Pythagoras (whose most important contribution to mathematics may very well have been his intervention with Dionysius to save the life of his friend, Plato), wrote on the application of these three means to music, and is possibly the one who is responsible for renaming the suhcontrary mean the harmonic mean (Boyer 1968).
Paths on a Grid
Authors: Willcutt Robert
Published: November 2020

The classic street problem can be used to present new challenges for your students.
Fixed Point Theorems in Euclidean Geometry
Authors: Clemens Stanley R.
Published: November 2020

The direction of future high school geometry courses is currently the subject of much discussion. One frequent suggestion is that high school geometry should be presented with transformation theory as the unifying theme. In support of this new direction, we shall illustrate that transformations can be employed to bring theorems from classical synthetic geometry into the so-called mainstream of modern mathematics. The thread tying these two points of view together will be the application of fixed point theorems.
A Transcendental Machine
Authors: Musser Gary L.
Published: November 2020

Often we attempt to encourage a student's creativity by emphasizing that there can be many correct, yet different, solutions to a single problem.
Providing Advantage to the Disadvantaged
Authors: Donald M. Peck , Stanley M. Jencks
Published: November 2020

Disadvantaged students entering a university are considered remedial and “on trial.” The belief exists that they must be well grounded in computational procedures before being exposed to the demands of creative problem solving. They are, in light of these judgments, given a diet of technically explained (but not understood) processes and endless drill on the fundamentals to prepare them for in-depth study. The authors have found this is not necessary. Many of these people are intelligent, curious, creative, and resourceful. Furthermore, difficulties with the basic processes clear up as the students find opportunity to exercise their reasoning powers.
Flow Charts For Equation Solving and Maintenance of Skills
Authors: Morrow Lorna J.
Published: November 2020

As Mathematics teachers, we all look for ways to make drill interesting or to clarify an idea such as equation solving. The use of flow charts as a “gimmick” for both purposes has proved to be very effective with seventh-grade students of all abilities as well as with eighth- and ninth-grade students of lower ability.
Algebra in Card Tricks
Authors: Stern Burton L.
Published: November 2020

The “magic” of card tricks often can be exposed through algebraic manipulation, and card tricks appear to have particular attraction for students. This attraction can be put to good use by helping students discover the mathematical basis for the apparent “magic.”
Two-Thirds Of The Most Successful …
Authors: Booth Ada
Published: November 2020

For many years psychologists have been interested in the relationship between position in the family and success in later life. Numerous published studies, both in professional journals and in popular media, have speculated on the reasons for the disproportionate number of first-born sons who became successful in various fields. (Girls were generally ignored in the studies in those pre-Women's Lib days.) The psychologists' theories on the reasons for the oldest son's greater achievement (this, of course, includes the only child) hinge on various aspects of the family interaction, particularly the child-parent relationship. It is clear that the oldest son has more undivided attention from his parents, and especially his father, at an early age. The differences in his environment might reasonably he expected to show up in many later personality differences, (In recent years many investigations have also been done on such aspects as anxiety, peer relationships, affiliation, and other personality factors of psychological interest.)
Selecting Goals For An Introductory Computer Programming Course
Authors: Moursund David
Published: November 2020

The instructional use of computers in the secondary schools is growing rapidly, One can divide instructional uses of Computers into two categories—those that require student knowledge of computer programming and those that do not. In the latter category, one typically finds com-puter-managed instruction, various forms of computer-assisted instruction (such as drill and practice, tutorial, and gaming and simulation), and use of canned programs to do the computations involved in solving particular problems. Computer programming, on the other hand, is a fundamental tool in general problem solving. That is, the computer is an essential tool to many people who attempt to apply mathematics to “real life” problems. Thus it is natural that computing should come into the mathematics classroom, and that mathematics teachers should get involved in the teaching of computer programming.
Recreation: Tac Tix
Authors: Spaulding Raymond E.
Published: November 2020

Tac Tix is a challenging board game for two players. Like so many similar games, it has a mathematical optimum strategy for play that run he discovered by application of some poweful problem-solving techniques.
Categorization of Questions That Mathematics Teachers Ask
Authors: Schmalz Rosemary
Published: November 2020

In a time of rapid technological advances, when there is need for frequent adaptation of skills, it is generally agreed that teachers must do more than just teach students a certain body of facts. They are responsible for teaching students processes of thinking and learning so that they have the ability to discover something new through consideration and reorganization of the known. One indication of how well a teacher does this is the kinds of questions he asks. From these one can judge how active a part the students are encouraged to take in the learning process.
A General Algorithm For Factorization
Authors: Draim N. A.
Published: November 2020

The determination of the primeness or compositeness of an integer is useful in arithmetic for simplifying rational fractions. obtaining an understanding of the arithmetical structure of particular integers, and aiding in calculations relating to the theory of numbers. Numerous special tests have been devised for determining divisibility by special divisors such as 2, 3, 6, 7, 9, and 11. However, a general algorithm for testing any number for its factors, in ascending order of magnitude, does exist and is the subject of this paper. The algorithm is a repeated division process with the successive divisions linked together by the rules of the algorithm. It is amenable to programming for computer factorization and, in fact, has been so used. It reduces the size of the number under test to one-nth its original size after division by n, thus making a significant reduction in the amount of calculation involved and imposing less burden on the storage banks of the computer.
Ripley’s Believe it or Not— a Source of Motivational Incentives
Authors: Crouse Richard
Published: November 2020

Probably one of the weaker areas in the instructional sequence of activities that mathematics teachers perform is the introduction of a topic or unit. These introductions often take the following form, or some variation thereof: “Today, we are going to study functions, A function is …”; or, “The last chapter that we studied was on exponents. Now we are going to study logarithms and we will see that logarithms are related to exponents.” However, one of the purposes of an introduction to a topic or unit is to catch the attention and interest of the students. The introductions cited above do not seem to fulfill this purpose.
The Converses of a Familiar Isosceles Triangle Theorem
Authors: Moore F. Nicholson , Byrkit Donald R.
Published: November 2020

A THEOREM commonly found in current high school geometry texts states, in some form, that the segments that bisect the base angles of an isosceles triangle are congruent.
Network Theory—An Enrichment Topic
Authors: Reeves Charles A.
Published: November 2020

The last decade has ushered in numerous changes in the teaching of high school geometry—changes due primarily to the impetus of curriculum committees such as the Commission on Mathematics. One trend is that of integrating plane and solid geometry “topicwise” into one course; that is, as each topic is studied in two dimensions, the concept is immediately extended to three dimensions if possible. This pedagogical technique undoubtedly helps students grasp related ideas more readily than if those ideas were presented as separate entities.
“Thought Starters” for the Circular Geoboard
Authors: Jencks Stanley M. , Peck Donald M.
Published: November 2020

Using a circular geoboard, the authors illustrate a sequence of investigations that reveal significant ideas about angles and arcs of circles.
Making Sense out of Nonsense
Authors: Allen H. A. J.
Published: November 2020

First the author provides an example of obtaining the correct answers to a problem through a so-called nonsense solution. He then makes an in-depth study of the reasoning behind this solution and further generalizes for the reader.
Functional Equations in Secondary Mathematics
Authors: Maurer Stephen B.
Published: November 2020

A different look at functions, via functional equations, provides a vehicle for typing together many topics that often seem to have no relation to each other.
Teaching the Square Root Algorithm by the Discovery Approach
Authors: Tarte Edward C.
Published: November 2020

The procedure discussed in this article leads the student to discover step by step the square root algorithm.
The Smallest Prime Factor of a Natural Number
Authors: Gullen George
Published: November 2020

An interesting way to find the smallest prime factor of a given natural number. A computer program for this procedure is supplied.
Using a Game as a Teaching Device
Authors: Brumfiel Charles
Published: November 2020

Here is indeed a learning game. It should provide an exciting setting for learning some mathematical ideas and strategies.
Some Uncommon Sense about Individualized Instruction, Intuition, and Applications
Authors: Fremont Herbert
Published: November 2020

What “common sense” tells us is the way to teach may not be the way to individualize instruction and capitalize on applications and student intuition.
The Area of a Parallelogram IS the Product of Its Sides
Authors: Lepowsky William L.
Published: November 2020

A convincing experiment for junior or senior high school level geometry students to get them to see why the area of the parallelogram is not equal to the product of the lengths of two adjacent sides.
Some Methods for Constructing the Parabola
Authors: Ciotti Joseph E.
Published: November 2020

Four methods for sketching and constructing parabolas are suggested for your high school students.
Alternatives in Learning: Video-Based Individualized Systems
Authors: Gormley Tyrone D.
Published: November 2020

This article suggests significant alternatives in learning and describes the technology used to make these alternatives possible.
Summing Arithmetic Series on the Geoboard
Authors: Edwards Ronald R.
Published: November 2020

This article illustrates how formulas for sums of sequences can be visualized and discovered.
Periodic Decimals
Authors: Anderson John T.
Published: November 2020

How students may start with the investigation of some basic questions about periodic decimals and progress to more general questions and other topics.
“One Small Jump”—into Repeating Decimals and Prime Numbers
Authors: Alexander F. D.
Published: November 2020

High school and junior college students should find this article an interesting supplement to their study of decimals and prime numbers.
Recreation: Binary Grids
Authors: Dunn James A.
Published: November 2020

A problem that arises in one area of mathematics is often found to be related to another area or to be translatable into the language of another area. If this sort of translation is feasible or can be discovered, then it often has the effect of providing insights and symbolic clarity. Very often the problem arises from the need to translate an essentially geometric idea into some form of nonpictorial record.
New Conic Graph Paper
Authors: Rose Kenneth
Published: November 2020

An easier way for students to draw families of conics.
Problem Number 10
Authors: Lenchner George
Published: November 2020

A sample problem for the good geometry student and his teacher.
The Sum of n Polygonal Numbers
Authors: Strangman Kathryn Besic
Published: November 2020

Ways of finding the sums of some interesting sequences of numbers are discussed in this article. Teaching techniques using discovery approaches are suggested.
Recreation: Hit, Miss, Bull's-Eye
Authors: Stewart Ward R.
Published: November 2020

HIT, MISS, BULL'S-EYE is a contest of chance and strategy in which two players try to discover each other's secret number. With very simple rules, the game offers strategic challenge to students of widely differing mathematical experience.
A System to Analyze Geometry Teachers' Questions
Authors: Friedman Morton
Published: November 2020

Each teacher can improve his or her teaching by systematically analyzing it. Here is one way to analyze teaching.
A Key Theorem in Transformation Geometry
Authors: Pedoe Daniel
Published: November 2020

Attention is focused on a fundamental theorem involving the composition of rotations. This theorem has wide applications in solving problems in transformation geometry.
Let's put computers into the Mathematics Curriculum
Authors: Norris Donald O.
Published: November 2020

There is plenty of room in the curriculum if we just drop the study of geometry.
Snowfalls and Elephants, Pop Bottles and π
Authors: Boas Ralph P.
Published: November 2020

You must have noticed that we've had some remarkably extreme weather lately. The snowfall in Chicago in the winter of 1977- 78, for example, set a record for the years since 1871 (as far back as data exist). The '78- '79 snowfall surpassed even that! Some of us who live near Chicago were surprised. Should we have been? Nobody, after aU, is greatly surprised at a new rec- ord for the mile run.
Is Exponentiation Commutative?
Authors: Manhard Warren B.
Published: November 2020

An ideal problem to excite students and teachers— and it's open-ended.
Analysis of a Truck Driver's Square Root Algorithm
Authors: Carmony Lowell
Published: November 2020

An interesting study of the accuracy of an approximation method for finding roots.
A Measured Metric Statement
Authors: Gaughan Edward D. , Wtsner Robert J.
Published: November 2020

Some insights into how we can balance the pressures for total conversion to the metric system and those forces that prefer the customary system.
The Problem of Galaxa: Infinite Area versus Finite Volume
Authors: McKim James
Published: November 2020

A dilemma with infinity in the twenty-second century.
Monte Carlo, Probability, Algebra, and Pi
Authors: Hinders Duane C.
Published: November 2020

Monte Carlo techniques are those that use a random number generator to solve a problem. Often, but certainly not always, these are problems that might be difficult or even impossible to solve analytically. A random number generator can be a coin, a die or dice, a computer, a table of random numbers, or some other device.
Measuring Squares to Prepare for Pi
Authors: Ryoti Don E.
Published: November 2020

The ratio of the circumference of a circle to its diameter does not depend on the size of the circle. This can be one of the most interesting facts that middle school students learn in mathematics. Because of the importance of this relationship, the groundwork for understanding it must be very carefully laid. The purpose of this article is to outline some preparatory work using squares.
Sighting the Value of π
Authors: Srinivasan P. K.
Published: November 2020

Using parallel lines to find pi.
Math Anxiety: Teaching to Avoid It
Authors: Morris Janet
Published: November 2020

Consider your fear of heights or an IRS audit and you can begin to appreciate how others bear their mathematics anxiety.
Sharing Teaching Ideas: Detective Stories in Junior High School Mathematics
Authors: Woodward Ernest , Tolleson J. Sam , Metz James , Goldstein Eleanor M.
Published: November 2020

One of our goals should be to present students with challenging and motivating problem-solving activities. The detective story format is one setting that seems to capture the interest of our eighth-grade students. In creating these stories one of our purposes was to construct clues to reinforce previously developed concepts and skills. Usually we presented individual students with mimeographed copies of the detective stories, along with directions for solving the problems. Occasionally we presented the clues, one by one, to the entire class. We used detective stories as part of regular assignments, but they can also be used as problem-of-the-week items or as bonus problems on tests.
Do All Graphs Have Points with Integral Coordinates?
Authors: Dossey John A.
Published: November 2020

The development of the general equation of a line (ax + by = c) is usually based on geometric intuition and the concept of slope. Students are given a point, (a, b), whose coordinates are integers, and a slope, m, and asked to write the equation of the line and graph it.
Mathematizing “Frogs”: Heuristics, Proof, and Generalization in the Context of a Recreational Problem
Authors: Higginson William
Published: November 2020

A problem-solving experience: Using mathematics /o model and analyze tire chip-switch game.
Sharing Teaching Ideas: Keeping Calculators Ready
Authors: Gourdouze Robbyn , Kapp Lehmann E. , Kluepfel Charles
Published: November 2020

When our mathematics department recently purchased thirty Texas Instruments SR-40 calculators, we thought that all our problems were solved. The purchase guaranteed that every student in a class had access to a calculator during class time, but there were some unexpected difficulties. One was how to make sure that all calculators were charged and ready for use. Another problem was where to store and charge the calculators.
Editorial: The Computer in Mathematics Education
Authors: Greenes Carole
Published: November 2020

Just five years ago, only a very small segment of the population had any direct experience with computers. Computers were generally considered remote and mysterious, and were known about only through their impersonal roles in billing or their highly publicized roles in space exploration. However, it has been projected that by the year 1990, over 50 percent of the working population will interact directly with computers in some way in their jobs.
Button's Needle Problem on a Microcomputer
Authors: Carlson Ronald J.
Published: November 2020

There are two main categories of problems that I assign to my computer classes. Exercises of counting, looping, branching, and so on are assigned while the student is acquiring the skills required in programming. Subsequently, they attempt programs that emphasize equal amounts of mathematical problem solving as well as computer programming. These problems can be challenging exercises, practical applications, or realistic simulations. The needle drop problem of Count de Buff on is such a problem (Schroeder 1974).
Introducing Computer Simulation into the High School: An Applied Mathematics Curriculum
Authors: Roberts Nancy
Published: November 2020

The foci for mathematics education in the 1980s have been clearly stated by the public, as reported in the Priorities in School Mathematics project (Suydam and Higgins 1980). Two areas high on the agenda are problem solving and integrating computers into the curriculum. New secondary level materials, Introduction to Simulation: The System Dynamics Approach, funded by a grant from the U.S. Office of Education (Grant #G007903439), address both of these issues and are currently being negotiated with commercial publishers.
Clubs: A Student-Presented Mathematics Club Program—Infinities of Numbers
Authors: Nemecek Paul M.
Published: November 2020

There is widespread use of the words infinite or infinitely many by students; yet there seldom occurs a sound mathematical discussion of the topic. The proof of the existence of infinity, accomplished in the last hundred years, allows an important opportunity to discuss the concepts of number, cardinal number, infinite set, and one-to-one correspondence.
Implementing The Standards: The Role Of Technology in Teaching Mathematics
Authors: Demana Franklin , Waits Bert K.
Published: November 2020

Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) assumes that in grades 9–12 all students will have access to graphing calculators, that every mathematics classroom will have a demonstration computer available at all times, and that all students will have the opportunity to use computers in mathematics. Teachers must start now to implement the many technologies currently available and prepare for the explosion of technology to come in this decade. In this article we address some questions about technology that are sure to arise as we begin to implement the spirit and vision of the Standards. Our examples are drawn primarily from the use of graphing calculators in advanced algebra and beyond, but the questions are typical of those that arise with other technologies and other mathematical content.
Beyond Exploratory Data Analysis: The Randomization Test
Authors: Barbella Peter , Denby Lorraine , Landwehr James M.
Published: November 2020

Suppose a company has created two types of knee braces designed to prevent knee injuries to football players. One knee brace might be more effective than the other, or both knee-brace designs might be equally effective, and thus the company could choose to market one knee brace over the other on the basis of cost or other considerations. Part of the decision will likely involve having football players wear each type of knee brace and then analyzing the number of knee injuries that occur.
The Geometry Proof Tutor: an “Intelligent” Computer-Based Tutor in The Classroom
Authors: Wertheimer Richard
Published: November 2020

I magine a standard-track geometry class in which it is common for students to come to class early to work on constructing proofs, often continuing to work after the class is dismissed. Imagine a class in which you can spend significant amounts of time working one-on-one with students while the rest of the class is actively engaged in constructing challenging proofs. A fantasy? This scenario actually happened in my geometry classes throughout six months of regular use of a computer-based, “artificially intelligent” tutoring system in a Pittsburgh Public School classroom (Schofield and Verban, in press).
STAR Experimental Geometry: Working with Mathematically Gifted Middle School Students
Authors: Talsma Gary , Hersberger Jim
Published: November 2020

Issues concerning the proper mathematics content for gifted students have been addressed by many researchers, including Harpel (1983), Hersberger and Wheatley (1980), Stanley (1980), Wavrik (1980), and Wheatley (1983). One area of agreement is that geometry is an essential and insufficiently covered area of mathematics content for gifted students. In this article, we describe a course for mathematically gifted middle school students, including the instructional approaches used, along with some exemplary materials.
The Sierpinski Triangle: Deterministic versus Random Models
Authors: Cibes Margaret
Published: November 2020

Recently a book on mathematics—Gleick's Chaos: Making a New Science (1987)—hit the best-seller lists and bookclub catalogs. Graphics displays on the subject have found their way into art galleries and magazines.
Casting Out Nines: An Explanation and Extensions
Authors: Lauber Murray
Published: November 2020

The method of casting out 9s has been used for centuries, perhaps even as long as a millenium, for checking computations with integers involving the four mathematical operations. According to Eves (1980, 160–68) it was used by Hindu-Arabic scholars in the Middle Ages. It appears to have been imbibed by Western culture as a part of the decimal system of representation of numbers. With the invention of the electronic calculator its practical value has diminished. However, it is still an intriguing application of modular arithmetic that can be generalized to arithmetic in other bases. This article explains how casting out 9s is done, examines some reasons for including it as a topic for exploration in the mathematics curriculum, and uses modular arithmetic to explore its mathematical basis and its generalizability to computations in bases other than ten. A method of detecting errors in the transmission of computer code with some affinities to its analogues is also explored.
A Mathematics Magazine and Fair
Authors: Spieler Robert R.
Published: November 2020

An elective mathematics course titled “Math Research” has been developed at my high school. It is open to incoming freshmen and sophomores interested in pursuing a career in mathematics or science. The students' mathematics and reading skills range from below average to above average.
Student-authored Manuals as Semester Projects
Authors: Hurwitz Marsha
Published: November 2020

Consider a typical mathematics-classroom scenario: The teacher asks if anyone can state the Pythagorean theorem. Several students respond, “a2 + b2 = c2.” In another class, the teacher asks how to find the vertex of a parabola and a student quickly responds “−b/2a.” In yet another class, the question is how to find the slope of a line to which the students shout “m.”
Expected Value and the Wheel of Fortune Game
Authors: Woodward Ernest , Woodward Marilyn
Published: November 2020

The popular television game show “Wheel of Fortune” offers a nice application of expected value in probability. In this game, the player must determine a message, given only the position of the letters and a category. For example, in a recent show, the players were told that the category for the configuration in figure Ia was a thing. The solution to the puzzle appears in figure lb.
Using a Surface Triangle to Explore Curfvature
Authors: Casey James
Published: November 2020

Although substantial improvements have been made during recent years in the standard high school geometry curriculum, certain weaknesses still exist. Few graduating students are aware of modern geometrical concepts and their fundamental significance for physics and engineering. Thus, for example, the notion of surface curvature, which has had a profound influence on both modern mathematics and physics, is rarely discussed in high school geometry. Of course, one cannot approach this topic in an analytical manner at the high school level, but one can certainly introduce the concept of curvature in an intuitive and physical fashion. Indeed, we are surrounded in our daily life by a wide variety of objects that are curved in the most fascinating ways. For instance, bowls, bottles, and bananas all possess interesting and variable curvature, whereas the globe has curvature that is constant over its surface.
Sharing Teaching Ideas: If the Product of Two Numbers Is Zero…
Authors: Forringer Richard
Published: November 2020

My first-year-algebra class has just finished the topic of factoring polynomials. The groundwork has been laid for problem solving with quadratic equations, one of the real eye-openers in the course. I look forward to teaching this topic with the excitement and anticipation of knowing what is to come. My students sense my excitement but do not fully understand it and have not experienced it for themselves. As Confucius once observed, “Everything has its beauty, but not everyone sees it!” Many students have no way of knowing that this part of algebra is incredibly significant. A short, simple statement, “If the product of two numbers is zero, then one of those numbers must be zero,” seems too easy, too self-evident, and too obvious to be so important!
Applications: Using Technology to Understand the Jury Decision-making Process
Authors: Goldberg Kenneth P.
Published: November 2020

Two of the recommendations of the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) are to use technology to enhance teaching and learning mathematics and to relate school mathematics to the world in which the students Jive through developing and interpreting mathematical models. This article demonstrates how computer or graphing-calculator technology can be used to help students develop and interpret three increasingly realistic models of jwy behavior and explore the potential effect of such decisions as changing jury size. The only mathematics required is an understanding of simple binomial probabilities and the use of sigma, or summation, notation
Teaching Probablity through Modeling Real Problems
Authors: Konold Clifford
Published: November 2020

In an attempt to reduce the growth of its population, China has instituted a policy that limits a family to one child. This policy has been particularly unpopular among rural Chinese, who have suggested revising the policy to limit families to one son. Suppose you were among those in the government considering the implications of adopting this proposal.
Investigating Circles and Spirals with a Graphing Calculator
Authors: Moskowitz Stuart
Published: November 2020

Graphing will never be the same again. New technology let us see graphs we could never have imagined. The features and simplicity of graphing calculators bring the power of parametric equations within the reach of everyone.
Implementing the “Professional Standards for Teaching Mathematics”: Let the Mathematics-Science Connection Break the Mold in Teacher Preparation
Authors: House Peggy A. , Farrell Margaret A.
Published: November 2020

In this third article of the 1993-94 series, the author addresses the issue of how to get mathematics majors to reexamine the mathematics they think they already know. The context of science becomes the catalyst for reflection in both content and pedagogy.
Implementing the Curriculum and Evaluation Standards: Portfolio Assessment: Making It Work for the First Time
Authors: Kuhs Therese M.
Published: November 2020

The search to find better ways to assess students' learning is supported by teachers who for years have been saying, “Tests don't show what my students have learned.” Portfolio assessment is being used in many state assessment programs (Aschbacher 1990) and appears to focus on strategies that are consistent with the NCTM's Curriculum and Evaluation Standards (1989). Of course, the practice of having students keep a notebook or folder of work over time is not new, but “assessment portfolio” is more than a new name for an old classroom practice. Recognizing the difference will insure that portfolios do not become “fat grade books,” which are inefficient burdens for both teachers and students to manage.
Sharing Teaching Ideas: A Review Game
Authors: Goldstein MaryAnn
Published: November 2020

This article describes a mathematics game that calls for the active participation of each and every class member throughout the activity. I play this game with all my classes, including my advanced groups, and I can honestly say that they love it. Generally I use it as a review the day before a test or quiz. The game takes approximately the same amount of time as working and discussing a chapter review. However, it is much more interesting, more fun for the students, and more beneficial in reviewing material and evaluating students' understanding of a topic than the traditional approach of assigning chapter-review tests.
Editorial: Join Our Eighty-Six-Year-Old Network
Authors: Dickey Edwin M.
Published: November 2020

As the National council of Teachers of Mathematics celebrates its Diamond Jubilee, the Mathematics Teacher begins its eighty-seventh year of publication. In this issue we highlight three new sections: “Technology Tips,” “Media Clips,” and—introduced by the article “What Gets Graded is What Gets Valued”-“lmplementing the Assessment Standards.” Also, beginning with this issue the journal's grade-level focus changes from a concentration on grades 7 through 14 to an emphasis on grades 8 through 14.
Civic Mathematics: A Real-Life General Mathematics Course
Authors: Vatter Terry
Published: November 2020

Our hopes for education coincide with those for our children: we want to help them grow into adults who will make the world an even better place for future generations. This goal may be a lofty one for a high school mathematics teacher, but it is important to keep our ideals clearly in mind as we develop ways to equip our students to live full lives and to make positive contributions to society. As educators we need to be responsible for teaching the skills that will enable students to (1) enter into a career or pursue their mathematics education and (2) become informed participants in a democratic society. The civic mathematics curriculum described in this article was driven by those objectives. It was developed over a summer and partially funded by a stipend from my school district
What Gets Graded Is What Gets Valued
Authors: Wilson Linda
Published: November 2020

In countless high schools across the United States and Canada, mathematics teachers are struggling to make sense of, and to implement, the reforms advocated by the National Council of Teachers of Mathematics. The fall 1993 publication of the working draft of the NCTM's Assessment Standards for School Mathematics has placed assessment practices at the forefront of reform. Particularly in high schools, where grades have an elevated importance, changing the way we assess what our students know is a daunting challenge. I would like to relate one story of a teacher who believes very strongly in reform but whose efforts were largely thwarted by her grading system. This fictionalized story is based on a case-study research project conducted by the author during three months in spring 1992.
National Council of Teachers of Mathematics: Linking Yesterday to Tomorrow
Authors: Lindquist Mary M.
Published: November 2020

“Linkages,” our theme for this year's annual meeting, represents more than linking yesterday to tomorrow. It is about linking today. We need to link in many ways-with members, with each of the Affiliated Groups, with our committees and task forces, with the Headquarters staff, and with other professional groups. Most important, we need to link with our students, who are facing a much different world from the one that many of us experienced as students. We need to help them link ideas within mathematics and between mathematical topics and link mathematics to real-world problems. We need to strengthen many of our links, forge new ones, and sever some links to the past.
Soundoff: Future Classrooms: A Personal Vision
Authors: Kalman Richard
Published: November 2020

It is 21 February 2004. Mathematics teacher Mark Downe enters his high school classroom and checks its layout. The fifteen computers set against the walls and the two round tables in the center create an open, spacious look. His twenty-eight tenth graders follow him in, heading for the computers m pairs. Downe quickly settles them down to work. As usual, Patrick and Patricia need special urging.
Folding n-pointed Stars and Snowflakes
Authors: Dutch Steven I.
Published: November 2020

Creating snowflakes is a popular seasonal activity in many classrooms. My interest in this subject was kindled when my wife asked if I knew any ways to make six-pointed snowflakes. As it happens, I did. I also knew of a method for folding five-pointed stars that did not work, and so I began looking for good ways of folding stars and snowflakes with other numbers of points. The geometric connection with this subject lies in devising simple ways of folding angles with measmes that are rational fractions of 180 degrees; the star or snowflake is created by cutting designs in the folded paper.
Publications and Products
Authors: Handley David M.
Published: November 2020

Prices on software? books, and materials are subject to change. Consult the supplters for the current prices.
Mathematical Ties That Bind
Authors: House Peggy A.
Published: November 2020

Father's Day was coming and the Sunday newspaper was filled with glossy advertisements for neckties produced in many colors and designs. Nothing in the advertisements seemed likely to inspire mathematical investigations. Or was I wrong?