bluuebunny/crossref_metadata_2025 huggingface dataset:
Probability and the Seating Chart
Authors: Ptak Dave
Published: December 2020

The problem-solving activities outlined here add to the collection of applications- at the middle school level and above- dealing with basic probability concepts. The context is an unusual and unexpected one: the classroom seating chart. Students need to know these concepts because they are at the heart of decision-making processes. Solving problems of the type that follow reinforces these concepts and gives students practice in the problem-solving techniques of counting, finding patterns, using diagrams, and generalizing.
Fun with Fibonacci; or, What to Do after the BC Calculus Exam
Authors: Gonzalez-Velasco Donna M.
Published: December 2020

After the AP exam in BC Calculus, I am always faced with the question of curriculum. To delve into new major topics one needs sufficient time, and we usually have only about two weeks. In my school, the AP exam is a surrogate final, and all enrolled in an AP course must take the corresponding exam. So I have to combat both the feeling of anticlimax and “senioritis.” I have, however, found an answer that works for me. I have devised a set of mini topics, each of which takes just two or three days to develop and which uses much of the mathematics that we have learned through the year. One such topic deals with the Fibonacci sequence.
Intuition and Logic
Authors: O'Regan Patrick J.
Published: December 2020

Most secondary schools do not have courses in formal logic. Teachers must teach this topic as it comes up naturally through class discussions in algebra, geometry, or general mathematics. Our students do present arguments to support their positions, and the strategies they use reveal the intuitions they have developed about logical relationships. It is important that we emphasize this connection between the rules of logic and our intuition. In fact, intuition is the underlying basis for our understanding of formal logic. This article shows how we can capitalize on our students' ways of thinking to lead them to a greater understanding of logical relationships.
Soundoff: Calculators on Tests—One Giant Step for Mathematics Education
Authors: Heid M. Kathleen
Published: December 2020

The proposal is a simple one—mathematics students should have access to calculators in all testing situations. What may sound to some like Jonathan Swift's “A Modest Proposal” is intended as a serious and necessary start on a solution to the present school mathematics dilemma.
An Application of Affine Geometry
Authors: Shilgalis Thomas W.
Published: December 2020

Recently a geometry problem from a mathematics contest (ICTM 1987) was presented to me. My first reaction was, “not enough information,” but I'd been down that road before and had usually ended up embarrassed.
The Silver Ratio: A Vehicle for Generalization
Authors: Coleman Donald B.
Published: December 2020

The golden ratio has a long and ri ch history. It is a fascinating and beautiful number. It appears in the regular pentagon and in the nautilus shell. Paintings and architecture abound in golden rectangles. The golden ratio has connections with the Fibonacci sequence. It is a source of interesting puzzles and a fascinating topic for students to explore.
Supersolids: Solids Having Finite Volume and Infinite Surfaces
Authors: Love William P.
Published: December 2020

A curious and interesting class of geometric solids exists that I have named supersolids. A supersolid is a bounded solid that has finite volume and infinite surface area. A bounded solid is one that may be contained inside a sphere having finite radius.
Applications: Fast Brakes!
Authors: Daniels David S.
Published: December 2020

As I was about to begin a new topic on square roots in my first-year algebra class, the question we have all heard popped up: “Is this stuff going to be good for anything?” Instead of some Pythagorean examples that are likely to be of more interest to a mathematics teacher than to ninth-grade students. I wanted a more compelling and forceful reply. Little did I know that help was on its way from the local police department and that I would soon discover enough about traffic-accident investigations to provide motivational applications of mathematics for my students.
The Odd-numbered Magic Square
Authors: Paterno James X.
Published: December 2020

Acommon exercise in elementary school arithmetic is the completion of magic squares, in which the pupil is to enter certain numerals so that the sum of any row, column, or diagonal is constant. The simplest form involves the use of numerals one through n2, where n2 represents the total number of boxes in a square configuration n boxes by n boxes. This article reviews a familar procedure for generating magic squares and points out a surprising pattern.
What Our Collegiate Business Students Need from Secondary Schools
Authors: D'augustine Charles H.
Published: December 2020

Not too long ago secondary school graduates could feel adequately prepared to undertake a college business curriculum after completing two years of high school algebra. However, rapidly changing requirements for students pursuing majors in collegiate business programs are placing new demands on the mathematical skill of students entering colleges of business.
Paradoxes in Averages
Authors: Mitchem John
Published: December 2020

Sometimes statistics yield surprising re-sults that can be misinterpreted. This article presents two major paradoxical examples of one of the simplest statistics, the “average.” These surprising and easily understood examples may be useful in your teaching as supplementary material to keep students intrigued with mathematics. Furthermore, they represent ideas that we should try to teach to all students.
How Many Games in a Tournament?
Authors: Woodward Ernest
Published: December 2020

Most mathematics teachers periodically look for real-world problems that they hope their students will find interesting and challenging. Since we are an athletically oriented society, situations with an athletic context are often appealing to students. Thus an ideal source for real- world problems is athletics. I found some interesting problems in the area of athletic tournaments.
Using Puzzles to Teach the Pythagorean Theorem
Authors: Beamer James E.
Published: December 2020

One of the aims of a mathematics prog- gram is to familiarize the students with the Pythagorean theorem. The result, stated algebraically, is c2 = a2 + b2. Stated geometrically, the Pythagorean theorem refers to squares drawn on the three sides of a right triangle. The theorem states that the square drawn on the longest side has exactly the same area as that of the other two squares combined. Is it possible systematically to dissect the two smaller squares into pieces that will cover the larger square? The answer to this question is the focus of this article, which offers sugges-tions about how the Pythagorean theorem can be introduced to students in the middle school years. Enrichment challenges in the form of proofs suitable for high school geometry students are also included. Finally, three proofs of the Pythagorean theorem based on careful analysis of the puzzles are discussed. Senior high school students can be asked to prove that the pieces actually fit and to use this tessellation as a starting point to provide an alternative proof of the theorem.
Area Formulas on Isometric Dot Paper
Authors: Litwiller Bonnie H. , Duncan David R.
Published: December 2020

Isometric graph paper can be a useful teaching aid when considering the concept of area. We shall demonstrate its use in the discovery of nonstandard area formulas. The enrichment activities described in this article involve the use of definitions; simple geometric argument; orderly record-keeping; and conjectures about, and investigations of, patterns. These valuable outcomes are among those that teachers wish to encourage in their students.
Square-Dance Numbers
Authors: Schwartzman Jan , Shultz Harris S.
Published: December 2020

Good problems are often simply stated and clearly resolved, and the decision about how to incorporate them into the curriculum is usually left to the teacher. Such problems can be used for enrichment, warmups, rainy-day activities, or challenges for colleagues. A problem of this type was presented by Carole Greenes of Boston University, the banquet speaker at the 1987 annual conference of the California Mathematics Council— Southern Section. Her topic included a myriad of challenges for the gifted and those who want to be gifted.
Reader Reflections
Authors: Bodine Charles H. , White Albert , Doubet Susan , Krueger Barbara Zimmanck , Yeshurun Shraga , Kessler James E. , Klamkin M. S. , Shapiro Jonathan , Hollingsworth Caroline , Winterink John A. , Nadji Taoufik , Mizrakhi Nissim , Redmond Don
Published: December 2020

Reactions to articles and points of view on teaching mathematics
Sharing Teaching Ideas: Making Mathematics Manageable
Authors: Baker Mary Beth
Published: December 2020

Perhaps most teachers would agree that homework is an essential component of any course in mathematics because students learn to solve mathematics problems by actually working them.
Sharing Teaching Ideas: A Device For Teaching Slopes
Authors: Price J. J.
Published: December 2020

Certain observations about slopes of lines often puzzle students and give rise to such questions as these: How is it that a line of slope 2 looks a lot steeper than a line of slope 1, but a line of slope 6 doesn't look much steeper than one of slope 5? Why isn't the slope of a 60-degree line twice the slope of a 30-degree line?
Metric Week—the Capitol Way
Authors: Diane Miller L. , Miller Jim
Published: December 2020

Metrication in the United States has been a long, slow process. In fact, some people continue to question the necessity for conversion. Why should instructional time and teachers' energies continue to be committed to improving our students' knowledge of metric measures? Because in addition to being more logically constructed and more convenient to use than our customary system of measurement, the metric system, known as Systeme International (SI), is the system of measurement being used in the world in which we are preparing our students to live. A lack of metric sense will, for some, be a handicap as they seek to participate in careers and leisure activities that involve international trade and travel. It is time to renew our commitment to helping our students understand and use this system of international measurement. This article illustrates how the faculty of Capitol High School in Baton Rouge, Louisiana, accepted the responsibility of helping students gain a better working knowledge of metric measures during the National Council of Teachers of Mathematics National Metric Week, 4-10 October 1987.
Retaining a Problem-solving Focus in the Technology Revolution
Authors: Duren Phillip E.
Published: December 2020

By the end of the eighth grade, students should be able to choose the most appropriate method for routine or nonroutine computations from among estimation, paper-and-pencil, calculator, or mental-computation procedures.
Sketching the Graph of 1/f(x)
Authors: Carpenter James E.
Published: December 2020

One class of functions that students find difficult to graph are the reciprocals of elementary functions. My students have found it helpful to use these five rules to sketch the graph of a function of the form 1/f(x).
A Constructive Proof of a Common Inequality
Authors: Ritter Richard C.
Published: December 2020

When students are first introduced to the axiomatic approach to mathematics, usually in geometry, they are often skeptical of the need for a deductive proof. To develop students' appreciation of the deductive approach, we need more examples of proofs that do furnish an enlightened connectiona point at which the student can say, “It has to be true, doesn't it?”
When Less Means More
Authors: Somers Kay B.
Published: December 2020

If you were offered a choice between having your salary raised $200 every six months or $1000 every year, which would you choose? Professor Hiram Bleeker (1983) posed a similar question as reported in the Chronicle of Higher Education. He stated that “many people would take the $1000 raise and so deprive themselves of the better deal,” In a later issue of the Chronicle (1983), several letters to the editor indicated nonbe lievers among the readers. We shall see that the answer depends on severa l factors.
Looking Ahead in Teacher Preparation: A Personal Perspective on NCTM—MAA Cooperation
Authors: Meserve Bruce E.
Published: December 2020

The preparation of teachers has been a major concern of many colleges and universities throughout their history. I shall use some of my personal experiences to place our present efforts in the perspective of activities over the last fifty years. This use of personal experiences restricts the scope of the examples but furnishes the basis for emphasizing a tested, practical philosophy—that cooperation between the National Council of Teachers of Mathematics (NCTM) and the Mathematical Association of America (MAA) exists and should be extended to gain increased influence on the entire mathematical community.
Four Labs to Introduce Quadratic Functions
Authors: Malone Jim
Published: December 2020

Most high school students get little hands-on experience with mathematics and rarely get an opportunity to see how the techniques they study really work. To remedy this difficulty, I have used four different “labs” in my algebra and precalculus classes.
Sharing Teaching Ideas: Designing Dreams In Mathematics
Authors: Powell Linda S.
Published: December 2020

To make the area of polygons and circles come alive for secondary school students, I like to give them a practical, interesting approach. Using some basic drafting skills and their knowledge of area formulas, the students find the area of preprinted floor plans. A variety of plans are used, preferably those that include the dimensions of each room. The students must answer questions about their particular house; for example, list the dimensions for specific rooms, then compute the area of each room. With this information they can find the area of “their” house. Using a hypothetical cost per square foot, they are astonished to find how much their house costs.
Sharing Teaching Ideas: Bisymmetric Matrices: Some Elementary New Problems
Authors: Councilman Samuel
Published: December 2020

In introductory linear algebra courses one continually seeks interesting sets of matrices that are closed under the operations of matrix addition, scalar multiplication, and if possible, matrix multiplication. Most texts mention symmetric and antisymmetric matrices and ask the reader to show that these sets are closed under matrix addition and scalar multiplication but fail to be closed under matrix multiplication. Few textbooks, if any, suggest an investigation of the set of matrices that are symmetric with respect to both diagonals, namely bisymmetric matrices. The following is a sequence of relatively straightforward problems that can be used as homework, class discussion, or even examination material in elementary linear algebra classes.
Thanks From The Editorial Panel
Authors: Channell Dwayne E. , Laing Robert A. , Watson Charles D. , Reeves Charles A.
Published: December 2020

The success of the Mathematics Teacher is very much dependent on the volunteer efforts of many mathematics educators. Those who serve as department editors, manuscript referees, and publication and courseware reviewers include junior and senior high school teachers, curriculum designers, college and university mathematicians, and teacher educators. The following are the names of those individuals who were active as of 7 July 1989. Their contributions are deeply appreciated. Anyone interested in serving as a referee of manuscripts should request an application form and guidelines from the National Council of Teachers of Mathematics.
Implementing the Standards: Bin Packing: A Mathematical Application
Authors: Malkevitch Joseph
Published: December 2020

Many recent ideas in mathematics are affecting the average person's life on almost a daily basis. The purpose of this article is to show how ideas, born of questions in operations research (the branch of mathematics designed to help businesses and governments operate more efficiently) can be taught at various levels K-12 in a manner that implements some of the recommendations made in the NCTM's Curriculum and Evaluation Standards for School Mathematics (Standards) (1989). For example, the problem we solve illustrates mathematical modeling, multistep problem solving, and the design of algorithms. Although the binpacking problem to be discussed involves simple ideas, the proofs of some of the results in this area are far from simple. The problem to be discussed has a variety of current, easy-to-follow applications.
Compositions of Positive Integers
Authors: Grassl Richard M. , Johnson Karen Anne , Vickrey Rachel
Published: December 2020

During the summer of 1988, the first author conducted an institute on discrete mathematics for sixteen New Mexico high school teachers. One of the presented topics, compositions of integers, proved to be a fruitful source of interesting problems.
Tips for Beginners: Graphing Inequalities with the y-Intercept
Authors: Morrell Linda A.
Published: December 2020

The confusion that often accompanies the topic of graphing inequalities came to my attention last spring as a result of my tutoring a student in basic algebra. Students are taught to shade to the left of the line, to the right of the line, and above or below the line. These techniques are valid methods for teaching graphing, but what is described as “to the right of the line” by one teacher may be referred to as “above the line” by another teacher. Strong first-year algebra students develop an intuitive approach to graphing and discern patterns in graphs and the relationship of the graph of the inequality to the graph of the equation having the same x- and y-intercepts. However, students who are not strong mathematics students may never see this relationship and develop only greater confusion!
Digits Count: Significant Digits and Calculators
Authors: Browne Joseph
Published: December 2020

Agroup of tourists visiting a museum in Washington, D.C., came to the moonrock exhibit. One rock is displayed out in the open so people can touch it, and a guard has been stationed nearby, presumably in case the rock gets homesick and tries to leave.
Implementing the Standards: Incorporating Mathematical Modeling into the Curriculum
Authors: Swetz Frank
Published: December 2020

In suggesting plans of action for the reform of mathematics education in North America, NCTM reports have focused strongly on the need to improve problem-solving skills and the need to “do” mathematics. Most recently, these goals have been reiterated and clarified in Curriculum and Evaluation Standards for School Mathematics (NCTM 1989). In discussing the impact of Standard 1: Mathematics as Problem Solving on students in grades 9-12, the report notes that students should be able to “apply the process of mathematical modeling to real-world problem situations” (p. 137). By using the phrase “apply the process of mathematical modeling,” the authors of this standard were most precise in their language. Mathematical modeling is a process and must be taught as a process. Certainly mathematical modeling involves problems, but it should not be considered as merely a collection of interesting problems and solution schemes. More important, modeling is a multistage process that evolves from the identification and mathematical articulation of a problem through its eventual solution and the testing of that solution in the original problem situation. The challenge for teachers is to understand this process of mathematical modeling and to apply it effectively in problem solving.
Estimating the Volumes of Solid Figures with Curved Surfaces
Authors: Cohen Donald
Published: December 2020

The problem of finding the volumes of solid figures can be used by calculus teachers to instill in their students the very useful habit of checking the plausibility of their answers against previously made intuitive estimates. Furthermore, the process used by students to arrive at their estimated answers might help them to develop the necessary insights for exact mathematical solutions.
The Sierra Curve—an Introduction to Periodic Concepts
Authors: Hughes Barnabas B. , M. . F. , Freier Lois Whitman
Published: December 2020

“Two important elements in trigonometry are the ideas of periodicity and periodic functions.” So began James E. Sconyers when he shared his teaching idea “A Simple Periodic Function” (Sconyers 1986). His clear exposition inspired what follows, although the technique used here differs radically from Sconyers's. The computer is used as a tool for making changes in an unusual periodic function-the Sierra curve, so called because it resembles a mountain chain in the western United States. By appropriate directed inputs, students transform the shape of the Sierra curve. In so doing, they investigate four characteristics of many periodic functions: period, amplitude, horizontal shift, and vertical shift. A listing of the program for the Apple II computer appears in the Appendix.
Implementing the Curriculum and Evaluation Standards: What Will Implementation Take?
Authors: Parker Ruth E.
Published: December 2020

A long history of traditions has grown up around what is meant by a good mathematics teacher and a good mathematics student. As many educators recognize, however, those traditions have little in common with mathematics in the world of the 1990s. Mathematics as it is used in the real world is not about the memorization of theorems or rote procedures for getting right answers. It is not about performing well on multiplechoice or short-answer tests under time constraints. “At the heart of mathematics is the search for sense and meaning, order and predictability. Mathematics is the study of patterns and relationships” (Richardson and Salkeld, in press). The challenge for mathematics educators is to align the culture of school mathematics with the culture of mathematics in the real world. With its publication of the Curriculum and Evaluation S tandards for School Mathematics (1989), the National Council of Teachers of Mathematics (NCTM) established the direction for such mathematics reform.
A Holistic Approach to Algebra
Authors: Barbeau Edward J.
Published: December 2020

When the school curriculum is subdivided into short units and the class moves quickly from one topic to the next without penetrating very deeply into any of them, students are in danger of coming away without any sense of purpose or context. Although facts and techniques may be covered, their significance and power may be unrealized, so that students quickly forget them.
A Brief Look at the History of Probability and Statistics
Authors: Lightner James E.
Published: December 2020

The NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) includes concepts from probability and statistics throughout the school program, K- 12. Indeed, these areas of mathematics are listed as topics for increased attention in the curriculum. Because of this emphasis, all teachers should become more aware of the historical development of these areas of mathematics so we can incorporate historical topics for motivation and extension of the content and so we can show the very human factor in the creation of mathematics. Let us, therefore, trace the development of the areas of probability and statistics.
Exhibiting Connections between Algebra and Geometry
Authors: Laing David R. , White Arthur T.
Published: December 2020

NCTM's Curriculum and Evaluation Standards for School Mathematics (1989, 81) encourages the exploration of interconnections among mathematical ideas and the development of an appreciation of “the pervasive use and power of reasoning as a part of mathematics.” This article exhibits some connections between algebra and geometry and presents deductive arguments appropriate for the secondary school mathematics classroom.
Activities: Recursion and the Central Polygonal Numbers
Authors: Miller William A.
Published: December 2020

During the middle years, the study of patterns and functions should focus on the analysis, representation, and generalization of functional relationships. (NCTM 1989, 98)

Sequences and series, currently topics in contemporary advanced algebra courses, should receive more attention, with a greater emphasis on their descriptions in terms of recurrence relations (formulas expressing each term as a function of one or more of the previous terms). (NTCM 1989, 177)
A Map-coloring Algorithm
Authors: Keeports David
Published: December 2020

Among the most tantalizing propositions of mathematics are those generalizations that are easily and concisely stated and are readily shown to be true for specific cases yet, despite their apparent simplicity, defy concise proof. Outstanding examples of such propositions include Fermat's last theorem (see Vanden Eynden [1989]), the Goldbach conjecture, and the four-color theorem. Because such propositions can be understood by students with almost no previous background in mathematics, they are easily introduced in mathematics courses intended for the liberal arts student.
Sharing Teaching Ideas: Very Special People, Mathematically
Authors: Albert Benita H.
Published: December 2020

For several years the Oak Ridge Schools have observed “Very Special Person (VSP) Week” during Valentine's Day week in February. This five-day period is specifically intended to remind administrators, teachers, and students that acts of kindness toward each other enhance the school environment. As a mathematics teacher, I think that the most special thing I can do for my own students is to offer continual challenges and excitement in mathematics. It occurred to me, however, that I meet less than 10 percent of the high school population in any year and that many other students and teachers may not share my enthusiasm for mathematics. Thus I was determined that my Very Special Person project for 1990 would reach every classroom — in fact, every teacher and administrator — in the school.
Cooperative Learning Works in Mathematics
Authors: Sutton Gail Oberholtzer
Published: December 2020

After fourteen years of adapting and refining my teaching methods, I judged hat I had become an above-average teacher. My expectations were high, clear, and attainable. My classroom was structured and well planned yet flexible. Discipline was rarely a problem.
Games for Developing Mathematical Strategy
Authors: Williford Harold
Published: December 2020

The three games described in this article are very interesting activities for developing abilities in critical thinking at the intermediate and upper-elementary-grade levels. As such, they go hand in hand with the focus on problem solving and reasoning skills that are general goals of mathematics instruction as recommended in Curriculum and Evaluation Standards for School Mathematics (NCTM 1989). The games prove to be “magnet activities,” that is, are most fascinating to students and attract repeated play. Each is a two-person game not very widely known. All offer a challenge yet are not so complicated that they stifle participation. Students feel that with just a bit more analysis, they “can figure out how to win every time,” The games rank as favorites of the students I worked with over a nine-year period in a summer enrichment program for academically talented middle school students.
Probability, Problem Solving, and “The Price Is Right”
Authors: Wood Eric
Published: December 2020

The document Academic Preparation in Mathematics (Kilpatrick 1985) sets out in some detail the mathematical topics that students entering college in the 1990s need to know. One point that is made rather forcefully is that students should have some grounding in statistics, probability, and computer simulation, a theme that is echoed in Standard 11 in the Curriculum and Evaluation Standards for School Mathematics (1989). Furnishing such experiences and at the same time ensuring that the material is both relevant and at an appropriate level of difficulty is often a challenge in the typical high school classroom. This article discusses the analysis of a decision-making problem faced by contestants on the popular game show “The Price Is Right.” An analysis of the original and related problems involves pattern searching, inductive reasoning, quadratric functions, and graphing, as well as presents an intrinsically interesting question in elementary probability that can be investigated by students at many different grade levels.
Drugs and Pollution in the Algebra Class
Authors: Sandefur James T.
Published: December 2020

This material is based on work supported by the National Science Foundation under grant number 8850450. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.
Patterns of Postage-Stamp Production
Authors: Kullman David E.
Published: December 2020

Perhaps this activity will win the stamp of approval of teachers who are looking for a fresh example of exponential growth. An editorial in Linn's Stamp News (Laurence 1988) observed that as of the end of 1988, the Scott Standard Postage Stamp Catalog (1989) listed more than 2400 different United States postage stamps. This number includes only regular and commemorative issues and excludes such items as airmail stamps, special-delivery stamps, and postal cards.
Graph Coloring Used to Model Traffic Lights
Authors: Williams John
Published: December 2020

Have you ever sat at a stop light and wondered how the tight pattern is determined? Who gets a green light at the same time? How many different light periods are necessary? Or have you ever wondered if final examinations could be scheduled so that no student would have a time conflict? Both these problems can be modeled with a graph and solved using graph coloring. In this short article, we shall first go through the basic definitions having to do with graphs and coloring, then describe a few conditions on coloring to give a feel for how to color a graph, and finally solve examples of the problems stated in the foregoing.
An Application of Matrix Theory
Authors: Glaister Paul
Published: December 2020

An ever-increasing proportion of class-preparation time for teachers is given over to ways in which mathematical ideas can be put across using practical examples. Even more important, teachers must consider how abstract concepts can be shown to be applicable by the use of a simple example. If the class can investigate the ideas for themselves, then so much the better.
Euler's Amazing Way to Solve Equations
Authors: Flusser Peter
Published: December 2020

Wouldn't it be terrible for a mathematics teacher to suggest that we teach our students an algorithm as though it were based on nothing but wishful thinking? And wouldn't it be even more horrible if we made it seem that this algorithm were based on assumptions that are not true? Please allow me to be both terrible and horrible. I want to demonstrate, in this terribly horrible manner, an algorithm for solving polynomial equations.
Infinite Sequences: A Logical (?) Extension
Authors: Duffy McDonald Wm.
Published: December 2020

It is difficult at the best of times to get students to consider the infinite in some coherent manner. The intent of this article is to show that great care must be taken in making the jump from the finite to the infinite and that, as the title suggests, what seems like a natural extension can lead to contradictions.
Implementing the Curriculum and Evaluation Standards: Making Connections: Beyond the Surface
Authors: Brutlag Dan , Maples Carole
Published: December 2020

In the past, most mathematics concepts were presented to students as separate ideas, each living in a different section of the textbook. Students were then expected either to make connections on their own or to wait until the ideas were used together in another course. The result was that for many students, the connections were never made, nor was the significance of the mathematics apparent.
Sharing Teaching Ideas: Puzzles, Puzzles,…
Authors: Isaacs Carol , Fisher Julie
Published: December 2020

… and more puzzles. Laura is trying to fit tangram pieces into the outline on a worksheet. Sergei is maneuvering a marble through a wooden maze while Brad is manipulating a ball using two poles in a game called “shoot the integer,” in which students try to score the highest number possible. Jessica is working on a jigsaw puzzle that has a plus sign on one side and a minus sign on the other. A classroom visitor would think that Fairfax County spends all its money on puzzles! Actually, all these puzzles were made by our students.
Insights from a Math Phobic
Authors: Wescott Dodd Anne
Published: December 2020

Many adults readily admit to having “math phobia.” I am one of them, and until recently I believed that math phobia was an incurable disease. However, what I discovered about learning and teaching mathematics as a result of taking a statistics course helped me realize that with a more personal and process-oriented approach, teachers can not only prevent math phobia but may also be able to turn some phobics into fans.
Popcorn: An Explosive Mixture of General Mathematics and General Science
Authors: Westerberg Judy , Whiting Jack
Published: December 2020

The linking of mathematics, science, and writing in an integrated, discovery-based environment builds a supportive classroom atmosphere in which students can learn to tackle the unfamiliar. Through the use of the laboratory approach, mathematical insight and intuition can be developed in a real-life situation.
Statistics in the Classroom: Written Projects Portraying Real-World Situations
Authors: Sommers Jonita
Published: December 2020

“Why do I have to do this math? This is not something I need to know. I will not use it when I get out of school!” exclaimed Jesse and some of his classmates. Have you ever heard these comments? In the past, my students were learning the concepts, hut they were not associating the importance of mathematics and its uses in the real world. This year, I have tried to show the students in my eighth-grade mathematics class how mathematics will apply to their lives, whether they work on a ranch, work in the oil fields, or get a higher education after high school.
Symmetries of Irregular Polygons
Authors: Shilgalis Thomas W.
Published: December 2020

The purpose of this article is to investigate bilateral symmetry in irregular convex polygons. Just how much symmetry can they have? This question arose in a “topics in geometry” course that I taught, and it provoked much fruitful discussion and conjecturing among the secondary school mathematics teachers enrolled.
Tin Cans Revisited
Authors: Verderber Nadine L.
Published: December 2020

Teachers of elementary calculus are familiar with problems requiring their students to maximize or minimize certain quantities subject to certain constraints, Representative of these problems is the problem of minimizing the surface area of a tin can that has a fixed volume. A noncalculus approach using a program written in BASIC has previously appeared in this journal (Inman and Clyde 1981).
Connecting Logic, Algebra, and Functions in Discrete Mathematics
Authors: Masat Francis , Mitchell Robert
Published: December 2020

When students are introduced to logic and truth tables in discrete-mathematics classes, they often ask, “Can we get the computer to do these problems?” The answer is yes, and we shall demonstrate an interesting, elegant, and easy way to do so. First, we shall combine some basic ideas from logic, algebra, and functions, and then we shall implement the results on a computer. The process is important for two basic reasons. First, practice in combining logic, algebra, and functions helps to replace rote symbolic manipulation with an understanding of the mathematics involved. Second, using the results with a computer emphasizes the fact that logic is reducible to computation.
Tips for Beginners: Begin Mathematics Class with Writing
Authors: Diane Miller L.
Published: December 2020

First impressions often influence the development of our attitudes toward, and decisions about, people, places, and things. The same can be said about the beginning of a mathematics class. The first few minutes of a class often influence the success of the class.
SSA and the Steiner-Lehmus Theorem
Authors: Beran David
Published: December 2020

Geometry students are regularly introduced to the SAS, ASA, SSS, and SAA congruence criteria for triangles. They may be tempted to apply an “SSA theorem.” Even though this type of correspondence does not guarantee congruence of triangles, this article explores conclusions that can be drawn from an SSA correspondence, as well as several applications, including a proof of the Steiner-Lehmus theorem.
Mathematics in Weighting
Authors: Francis Richard L.
Published: December 2020

The cycloid has proved the source of considerable curiosity, challenge, and controversy across the centuries from the late Renaissance to the modern era. Since its historical appearance in sixteenth-century literature, the cycloid (fig. 1) has rivaled the circle and the conic sections in classical charm. Such a curve admits a simple definition las simple perhaps as that of the circle itself). Precisely, it is the locus of a point that remains fixed on a circle as the circle rolls on a line. One of the intriguing cycloid questions that surfaced early was in the realm of enclosed areas and concerned the exact area beneath the cycloidal arch. It was to prove a weighty question in the developmental years of modern mathematics. This question's earliest disposition—that of areas and templates—has an appealing extension to other figures and proves a highly valuable activity in the classroom of today.
Extending the Number Line to Make Connections with Number Theory
Authors: Graviss Tom , Greaver Joanne
Published: December 2020

For many years we team taught a ninth-grade general mathematics class consisting of seventy students. We had to develop teaching methods and materials that were different from those that had not helped our students in the past. As part of our efforts, we developed a specially coded version of a standard number line to furnish representational and concrete experiences for concepts that are too abstract for many students.
Tips for Beginners: Grading
Authors: Arnold Connie B. , Reynolds Sallee H. , Stellern Cathy D. , Bohannon Diane , Harmon Katherine , Hill Leanne , Mamantov Cathy , Reed Suzanne H.
Published: December 2020

Students should have a complete understanding of how their grade is determined. On the first day of school it is desirable to hand out a typed copy of the grading scale, grading system, and grading requirements (table 1) to each student and discuss your expectations. The grading scale you use may be your own or a schoolwide or systemwide scale. Whichever scale is used, be sure that the students are clearly aware of it. Keep a copy of the grading scale posted in the classroom.
Implementing the Professional Standards for Teaching Mathematics: Sharing, Supporting, Risk Taking: First Steps to Instructional Reform
Authors: Leinwand Steven J.
Published: December 2020

For many of us, the Professional Standards for Teaching Mathematics (NCTM 1991) represents a much scarier and much more intimidating vision of school mathematics than its predecessor, the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989). Accordingly, implementing the teaching standards will require different strategies from those being used or proposed to implement the curriculum standards.
Families of Parabolas
Authors: Owens John E.
Published: December 2020

This article explores the nature of changes in the graphs of the equation y = ax2 + bx + c resulting from changes in the coefficients a, b, and c. The purpose is threefold: (1) to present an interesting extension of the normal second-year algebra treatment of parabolas; (2) to demonstrate the power of a function plotter as a classroom tool that allows students to make conjectures from visual patterns; and (3) to illustrate the use of parametric forms as a method of describing equations.
The Standards Applied: Teaching Data Visualization
Authors: van Reeuwijk Martin
Published: December 2020

As part of the project Design, Development and Assessment in Mathematics Education, I spent four weeks at Whitnall High School in Milwaukee, Wisconsin, testing a booklet on descriptive statistics called Data Visualization (Freudenthal Institute 1989). This textbook was developed in 1989 by the Freudenthal Institute's Research Group on Mathematics Education of the University of Utrecht in the Netherlands in cooperation with Gail Burrill of Whitnall High School. Especially written for this project, the booklet is based on the NCTM's Curriculum and Evaluation Standards (1989) and on the philosophy of “realistic mathematics education” developed by the research group of Utrecht University.
Combinatorics Connections: Playoff Series and Pascal's Triangle
Authors: Litwiller Bonnie H. , Duncan David R.
Published: December 2020

One major theme of the National Council of Teachers of Mathematic's Curriculum and Evaluation Standards far School Mathematics (1989) is the connection between mathematical ideas and their applications to real-world situations. We shall use concepts from discrete mathematics in describing the relationship between sports series and Pascal's triangle.
Sharing Teaching Ideas: Folding Perpendiculars and Counting Slope
Authors: Blomquist Ann
Published: December 2020

During the study of coordinate geometry, students are told that the slope of a line and the slope of a perpendicular to the line are negative reciprocals. The NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) calls for “active involvement of students in constructing and applying mathematical ideas.” This activity will encourage inductive thinking and allow students to discover this relationship for themselves. Students should have previously computed the slope of a segment by counting squares and using the rise-over-run format.
By Way of Introduction: Teachers Are under Pressure to Produce “Results”
Authors: Bush William S.
Published: December 2020

Today's teachers are under pressure to produce results. At the national level, Congress sanctioned the Educational Testing Service in 1988 to compare states using the National Assessment of Educational Progress (National Center for Educational Statistics 1991). Continuing this trend toward national accountability, a national test in mathematics is anticipated in the next few years. In my state, Kentucky, schools will be rewarded or sanctioned depending on their students' performance on various types of assessment and on such factors as attendance and drop-out rates.
Soundoff: How National Examinations Can Benefit Students and Teachers
Authors: Glidden Peter L.
Published: December 2020

Several reform groups, including the President's Education Policy Advisory Committee and Educate America, are calling for national examinations for high school students. Examination advocates claim the following benefits: the establishment of national standards, increased accountability, and increased motivation for students. Advocates also point out that the United States is the only major industrialized country without some sort of national examination. Critics claim that examinations restrict teachers' creativity; force teachers to teach for the examination; and promote improper comparisons among states, districts, schools, teachers, and students. (See, e.g., DeWitt [1991]).
A Joint Vision for Classroom Assessment
Authors: Cain Ralph W. , Kenney Patricia A.
Published: December 2020

The NCTM's Curriculum and Evaluation Standards for School Mathematics proposes a vision for assessment in the mathematics classroom that would “help teachers better understand what students know and make meaningful instructional decisions” (NCTM 1989, 189). For assessment to be truly aligned with the mathematics curriculum, teachers would give more emphasis to taking a holistic view of mathematics—using multiple assessment methods, including written, oral, and performance formats, and incorporating calculators, computers, and manipulatives as part of assessment. Thus, mathematics teachers would be empowered to trust their own abilities and judgments in the area of mathematics assessment (Clarke, Clarke, and Lovitt 1990; Cooney and Badger 1990; Schoen 1989; Webb and Briars 1990).
Subjective Grading
Authors: Stallings-Roberts Virginia
Published: December 2020

The Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) and the Professional Standards for Teaching Mathematics (NCTM 1991) advocate an array of alternative-assessment strategies that encourage modification of teachers' practices in measuring students' performance. Unfortunately, many teachers must operate within the constraints of the system under which they serve. Teachers feel bound to traditional testing by virtue of the adopted percent procedure established by their schools and simply because grading has been done in the traditional manner for years.
Integrating Transformation Geometry into Traditional High School Geometry
Authors: Okolica Steve , Macrina Georgette
Published: December 2020

The grades 9-12 section of NCTM's Curriculum and Evaluation Standards for School Mathematics defines transformation geometry as “the geometric counterpart of functions” (1989, 161). Further, the Standards document recognizes the importance of this topic to the high school mathematics curriculum by listing it among the “topics to receive increased attention” (p. 126). Also included on this list is the integration of geometry “across topics.”
Geometric Patterns for Exponents
Authors: Thompson Frances M.
Published: December 2020

NCTM's Professional Standards for Teaching Mathematics suggests that “tasks that require students to reason and to communicate mathematically are more likely to promote their ability to solve problems and to make connections” with other mathematical ideas (1991, 24). Yet too frequently our classroom introductions to mathematics concepts and theorems demand little reasoning from students, leaving them unconvinced or with minimal understanding. Concrete, visual, or geometric models are seldom offered as aids, particularly when studying new numerical relations (Suydam 1984, 27; Bennett 1989, 130), even though many people depend heavily on visual stimuli for their learning, The challenge to the teacher is to select appropriate tasks and materials that will stimulate students to visualize and think about new mathematical concepts, thereby allowing them to develop their own understanding.
A Generalized Area Formula
Authors: Usnick Virginia E. , Lamphere Patricia M. , Bright George W.
Published: December 2020

NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) states that mathematics instruction should reflect a constructvist's view of the learning process, that is, that learners are not passive absorbers of knowledge but rather active participants in the construction of knowledge. Such instruction would approach mathematics as interrelated concepts and skills to be understood rather than as independent concepts and skills to be memorized. It would also encourage learners to examine concepts and their relationships in such a way as to encourage pattern recognition, generalizing, hypothesizing, proving, and other higher-order thinking skills.