bluuebunny/crossref_metadata_2025 huggingface dataset:
Learning, Teaching, and Standards
Authors: Dossey John A.
Published: December 2020

Learning, teaching, and learning teaching these prophetic words from George Polya's (1963) title for his address to the members of the Mathematical Association of America nearly twenty-five years ago echo the questions that face our profession today (Curcio 1987). Polya's legacy has not only touched our knowledge and abilities in mathematics and problem solving but has also had a significant effect on our thoughts about teaching.
Soundoff: Manipulative Algebra the Culprit or the Scapegoat?
Authors: Waits Bert K. , Demana Franklin
Published: December 2020

Is algebraic manipulation important for students' mathematical understanding? Will students be crippled in subsequent, more advanced courses in mathematics and science if most, or all, algebraic manipulation is removed from high school algebra courses? Champions of the full and unrestricted use of computers and calculators to do algebraic manipulation (e.g., MuMath, Macsyma, HP 28C) argue for a substantial reduction of the time spent teaching algebra manipulation. They would use this time to teach new, modern numerical topics and techniques that foreshadow the study of statistics and computer science. But is the old mathematics bad mathematics? Many conservative mathematicians and mathematics educators argue for the status quo. But have we no way to improve on the old mathematics with new technology? Is it fair to keep students in the dark about modern developments in the field of mathematics.
Secondary School Results for the Fourth NAEP Mathematics Assessment: Algebra, Geometry, Mathematical Methods, and Attitudes
Authors: Brown Catherine A. , Carpenter Thomas P. , Kouba Vicky L. , Lindquist Mary M. , Silver Edward A. , Swafford Jane O.
Published: December 2020

This article is the second of two articles reporting on the seventh-grade and eleventh-grade results of the fourth mathematics assessment of the National Assessment of Educational Progress (NAEP) administered in 1986. The first article (Brown et al. 1988) presented the background, methodology, and the results of students' performance on discrete mathematics, data organization and interpretation, number and operations, and measurement. This article reports students' performance on variables and relations, geometry, fundamental methods of mathematics, and attitudes. An analysis of eleventh-grade students' performance by mathematics course background was possible, and these data will be reported here where appropriate.
NCTM's Standards for School Mathematics, K – 12
Authors: Thompson Charles S. , Rathmell Edward C.
Published: December 2020

The National Council of Teachers of Mathematics is in the process of generating a set of Curriculum and Evaluation Standards for School Mathematics (Standards) (Commission on Standards of the NCTM 1987). NCTM has committed considerable resources to this project, anticipating that the Standards will have a pervasive effect on mathematics education during the next five to ten years. The expectation is that the Standards will influence curriculum writing at the state and local levels and that the resulting curricular changes will influence the content of textbooks adopted by states and school districts. Furthermore, the newly written curricula, together with the new Standards for the evaluation of mathematics learning, should influence the content and emphasis of local, state, and national tests.
The Wonder and Creativity in “Looking Back” at Problem Solutions
Authors: Taback Stanley F.
Published: December 2020

Mathematics educators have always viewed problem solving as a preferential objective of mathematics instruction. It was not, however, until the National Council of Teachers of Mathematics published its position paper An Agenda for Action: Recommendations for School Mathematics of the 1980s that problem solving truly came of age. As its very first recommendation, the Council (1980) directed that “problem solving be the focus of school mathematics in the 1980s” and proclaimed that “performance in problem solving will measure the effectiveness of our personal and national possession of mathematical competence.”
Applications: Mathematics of the Rainbow
Authors: Austin Joe Dan , Dunning F. Barry
Published: December 2020

Figure 1 shows a picture of a rainbow. Have you wondered what causes a rainbow, why it is brighter inside than out, or why more than one rainbow may appear? The answers to these questions involve optics, geometry, trigonometry, and calculus. This article presents some of the optics and mathematics related to these questions.
Fisher's Effect: Real Growth Is Not Interest Less Inflation
Authors: Nievergelt Yves
Published: December 2020

Students often add, subtract, and multiply percentages in a careless fashion, perhaps because teachers do not sufficiently emphasize the relative nature of percentages. This article explains why adding and subtracting percentages does not make sense and illustrates this phenomenon with an example from the Wall Street Journal: the real rate of growth of an investment.
Sharing Teaching Ideas
Authors: Moody Mally , Jordan James H. , Shultz Harris S.
Published: December 2020

Teachers of elementary algebra are forever in search of ways to help their students prepare to do word problems.
Sharing Teaching Ideas
Authors: Klein Paul A.
Published: December 2020

Few arithmetic skills have eluded as many students as those used in reducing fractions or in obtaining the least common denominator of two fractions. In the more popular instructional sequences the general concepts of greatest common factor (GCF) and least common multiple (LCM) are developed first Usually, these concepts involve the decomposition of the numbers into prime factors as shown in example 1.
Geometrical Inequalities via Bisectors
Authors: Hoehn Larry
Published: December 2020

The proofs of several theorems in secondary school mathematics that involve geometrical inequalities are more complicated than they really need to be. This article presents an easier-to-understand alternative to the usual proofs of inequalities in triangles. The only inequality with which we need to assume familiarity is the triangle inequality (i.e., the sum of any two sides of a plane triangle is greater than the third side).
Computers and the Rational-Root Theorem—Another View
Authors: Waits Bert K. , Demana Franklin
Published: December 2020

A recent article offers a 153-line computer program that numerically determines the rational roots of polynomial equations with degree n ≤ 10 (O'Donnell 1988). The program forms and systematically checks all possible rational roots. This article outlines another approach to finding the rational roots of polynomial equations based on computer graphing that is more general and integrates graphing with the purely algebraic approach. The method is easy and can be used with popular computer graphing utilities or with graphing calculators such as the Casio fx- 70000 or the Sharp EL-5200.
An Interactive Approach to Problem Solving: The Relay Format
Authors: Sundar Viji K.
Published: December 2020

This article illustrates one type of activity that can be used in the classroom to reteach previously learned concepts, to review a unit, to introduce a distinct vocabulary, to practice estimation, or to improve number sense. The exercise is adaptable to suit the needs of all, “including students with special needs and those who come from groups that have historically been underrepresented in upper-level mathematics classes” (California Mathematics Framework 1985, 2). It can be varied with the teacher playing the game with the entire class (format 1), the entire class playing with each other (format 2), or the class playing in teams of four to eight students (format 3). Students in grades 9–12 can play.
Nearly Nice Right Triangles
Authors: Reid Bob
Published: December 2020

Algebra and geometry students learn the very useful relationships that hold for the sides of the 45-45-90 and the 30-60-90 triangles. The discussion of “nice” right triangles generally ends there, although the question may arise, “Do other such triangles exist?”
Discovering Calculus
Authors: Decker Robert
Published: December 2020

What can you do in a mathematics class when a computer is available that you can't do otherwise? How do you actually incorporate the computer into a mathematics class without rewriting the textbooks? This article addresses these questions by relating my experience using computers in a mathematics class.
The Ultimate 180-Degree Rotation—from Spectators to Problem Solvers
Authors: Askins Larry E.
Published: December 2020

The major part of every meaningful life is the solution of problems; a considerable part of the professional life of technicians, engineers, scientists, etc. is the solution of mathematical problems. It is the duty of all teachers, teachers of mathematics in particular, to expose their students to problems much more than facts.
Soundoff: Geometry: A Remedy for the Malaise of Middle School Mathematics
Authors: Posamentier Alfred S.
Published: December 2020

Many mathematics educators perceive that the weakest part of the precollege mathematics curriculum is at the middle school level, more specifically, the years immediately preceding the study of algebra. It seems that in the middle grades the development of mathematics has been put into a “holding pattern.” A quick glance at the curriculum for seventh and eighth grades—or in some cases sixth and seventh gradesshows that much arithmetic is still being taught. Haven't we, or shouldn't we have, completed teaching arithmetic in the previous five or six years? Indeed, how much arithmetic teaching do we need to do in an age of ever-improving calculators (Heid 1988)? Very often students greet a unit in these grades with the now famous comment, “Oh, I had this already.” “Sure,” thinks the teacher, “you may have had it, but have you learned it?” It is clear to many educators that these middle grades are key to turning a student “on” to or “off” from mathematics.
Implementing The Standards: A Core Curriculum for Grades 9–12
Authors: Hirsch Christian R. , Schoen Harold L. , Schoen Harold L.
Published: December 2020

A three-year core curriculum is the most fundamental change proposed for grades 9–12 in the Curriculum and Evaluation Standards for School Mathematics (National Council of Teachers of Mathematics, Commission on Standards for School Mathematics 1989). The Standards document identifies a common body of mainstream mathematical topics that all high school students should have the opportunity to learn. Present curricula attempt to accommodate differences in students' backgrounds, interests, and educational goals through the selection of topics. Unfortunately, the narrow, trackable programs that evolve from this perspective restrict many students to arithmetic computation only and thus serve as an early critical filter to opportunity and careers. Within the proposed core curriculum, differentiation would occur primarily in the manner in which topics are treated. It would be based on the depth to which common topics are pursued, the degree of difficulty of exercises and applications, the level of abstraction at which ideas are discussed, and, of course, the pace of instruction.
A Chain of Influence in the Development of Geometry
Authors: Lightner James E.
Published: December 2020

Teachers of high school geometry are confronted daily with the problem of making the subject as interesting as possible for their students. We teachers, of course, find the challenge of producing a logical deductive argument enough to keep us interested and motivated, but, unfortunately, it may be insufficient for our students. Perhaps sharing with them some of the history of the development of geometry, especially introducing them to some of the great mathematicians, will get them to appreciate the fact that mathematics is the product of human minds, created in response to human needs. Let us look at some of these early “giants” of geometry to see their role in the development of this field of mathematics and to note how each had a tremendous influence on the next generation of geometers.
Trisecting an Angle—Almost, Part 2
Authors: Lamb John F.
Published: December 2020

When students study constructions in high school geometry, they usually make the following conjecture: Since it is easy to bisect a line segment, easy to bisect an angle, and not too difficult to trisect a line segment, there ought to be a way to trisect an angle. The students may even devise ways that they think will accomplish the task. The errors in these methods are sometimes hard to detect. This article highlights one way that has been discovered that comes very close to trisecting an angle with Euclidean tools. Several ways are discussed in one of the books in the Classics in Mathematics Education series published by the National Council of Teachers of Mathematics titled The Trisection Problem (Yates 1971). One of the methods discovered by d'Ocagne was examined in “Trisecting an Angle—Almost” (Lamb 1988). His method was very easy to do, but it was not as accurate as some of the other methods. One of those methods that is also relatively easy to do and gives a much better approximation than the d'Ocagne method for angles between 0 degrees and 90 degrees was discovered by Karajordanoff in 1928 (Yates 1971). The procedure is as follows.
Elevator Probabilities: Chances of Coincidences
Authors: Duncan David R. , Litwiller Bonnie H.
Published: December 2020

Teachers of probability and statistics are constantly on the lookout for real-world examples that can be used to promote or exemplify concepts. The question of “coincidence” furnishes a rich source of these examples, one of which is considered in this article.
Collaboration and Writing in the Mathematics Classroom
Authors: LeGere Adele
Published: December 2020

For twenty-five years I have taught mathematics using a great deal of lecture, much dialogue between students and myself, a limited amount of small-group work, and very little writing. As a mathematics teacher at both the high school and community college levels, I believed that clear presentations were my primary responsibility. It was difficult enough to find sufficient time to cover all prescribed content without taking more time for writing and group work.
Implementing the Standards: Developing Communication Skills in Mathematics for Students with Limited English Proficiency
Authors: Cuevas Gilbert J.
Published: December 2020

The Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) emphasizes the need to address communication skills. These skills, including reading, writing, listening, and speaking, enhance mathematical understanding and problem-solving ability. Moreover, to communicate effectively, one must be able to interpret and analyze mathematical ideas. The curriculum and evaluation standards recommend that opportunities be afforded students to “use language to communicate their mathematical ideas” (NCTM 1989, 78). Although these recommendations are valuable, teachers may find them difficult to implement with students who are not proficient in English.
Monty's Dilemma: Should You Stick or Switch?
Authors: Shaughnessy J. Michael , DICK THOMAS
Published: December 2020

Recently, a colleague in our department shared a version of this problem with us. Since then, we have had a great deal of fun with it, provoking considerable debate among students, teachers, and mathematicians alike. The problem involves introductory probability concepts, and we have found it appropriate for, and interesting to, students from middle school to college.
WITPO (What is the Probability of)
Authors: Ericksen Donna Bird , Frank Martha L. , Kelley Ryan
Published: December 2020

Games can be used in the classroom for many educational purposes. For example, they can be used to introduce and develop a concept. This purpose was illustrated by Noone (1988) with the game “Chuck-aluck: Learning Probability Concepts with Games of Chance.” Games can also be incorporated at the end of a unit to review material. Students enjoy studying probability because it often involves games. The following probability game, WITPO, was developed by Ryan Kelley, a preservice secondary school teacher.
Sharing Teaching Ideas be Your Own Substitute
Authors: Blomquist Ann , Steman Pamela Wheeler
Published: December 2020

All teachers know the difficulties inherent in their absence from school, even for a day. Too frequently when the teacher is absent the students end up doing busy work or the' substitute doesn't know how to explain the material, so the day is lost to academic instruction.
Activities: Line and Rotational Symmetry
Authors: Whitman Nancy
Published: December 2020

Introduction: The NCTM's Curriculum and Evaluation Standards (NCTM 1989) recom- mends that teachers connect mathematics to the world outside the classroom. Classroom activities should afford students the opportunity to work individually as well as in small- and large-group arrangements. Teachers should furnish experiences that enhance students' ability to communicate in mathematics and that integrate the students' cultural background into the learning of mathematics. The activities described in this article respond to these recommendations. They were designed for students in Hawaii, but all students can benefit from doing them.
Finding Points of Intersection of Polar-Coordinate Graphs
Authors: Esty Warren W.
Published: December 2020

Students studying polar coordinates may be required to find the points of intersection of the polar-coordinate graphs of two functions, f and g. Their experiences with rectangular coordinates may lead them to expect that all the points of intersection can be found by solving the equation f((J) = g (O). They may be distressed to discover that points of intersection of the graphs can occur that do not correspond to solutions of that equation.
Writing to Learn Mathematics
Authors: Miller L. Diane
Published: December 2020

The use of the writing-to-learn strategy in mathematics classes is one way teachers can implement “Mathematics as Communication,” the second standard in NCTM's (1989) Curriculum and Evaluation Standards for School Mathematics. For students in grades K-12, writing is suggested as a means through which students should be able to communicate their understanding of mathematics and its applications. Using writing in a content area has earned the support of many educators. Emig (1977) believes that writing in a content area can cause students to analyze, compare facts, and synthesize relevant material. Writing about a topic requires students to think about the topic, focus on and internalize important concepts, and make those concepts to some degree their owu (Kennedy 1980).
Activities: Make Your Own Problems- and Then Solve Them
Authors: Kimball Robert L.
Published: December 2020

The mathematics curriculum should include the refinement and extension of methods of mathematical problem solving so that all students can recognize and formulate problems from situations within and outside mathematics. (NCTM 1989, 137)

The mathematics curriculum should include the continued development of language and symbolism to communicate mathematical ideas so that all students can express mathematical ideas ora lly and in writing. (NCTM 1989, 140)

The teacher of mathematics should pose tasks that … call for problem formulation, problem solving, and mathematical reasoning [and) promote communication about mathematics. (NCTM 1991, 25)
Euclid and Descartes: A Partnership
Authors: Wasdovich Dorothy Hoy
Published: December 2020

Although Descartes developed the application of algebra to geometry over 400 years ago, his work has had little impact on the high school geometry course. Geometry and algebra are still taught as separate, unrelated subjects rather than as complementary approaches to mathematics. Any coordinate geometry that is included in a course in Euclidean geometry is apt to be placed in one chapter or unit with the implication that it is “optional” and the material covers theorems that have already been proved. If methods of proof are to be compared, why not do it the first time a theorem is encountered?
Editorial: The Mathematics Teacher Has a New Look
Authors: Johnson Jerry
Published: December 2020

“With this issue, The Mathematics Teacher is changed in form and in size. The new format enables us to give our readers more material and present it in a more desirable way. This is done by increasing the size of the page so as to …”
Soundoff: A Computer for All Students
Authors: Demana Franklin , Waits Bert K.
Published: December 2020

The views expressed in “Soundoff” do not necessarily reflect the views of the Editorial Panel of the Mathematics Teacher or the National Council of Teachers of Mathematics. Readers are encouraged to respond to this editorial by sending double-spaced letters to the Mathematics Teacher for possible publication in “Reader Reflections.” Editorials from readers are welcomed.
Implementing the Curriculum and Evaluation Standards: Using Technology in Graphing
Authors: Carter Claudia R.
Published: December 2020

Perhaps we know what happens when the x-values of a linear relationship are squared, but what happens when the coordinates of points that create a picture are squared? The answers to this question and others were the impetus for an algebra project for one of my classes.
A “Simple” Probability Problem
Authors: Chu David , Chu Joan
Published: December 2020

Probability has been suggested for inclusion in the high school or even junior high school curriculum (for example, probability and statistics is one of the strands in The Mathematics Framework for California Public Schools, Kindergarten through Grade Twelve [Sacramento: California State Department of Education, 1985]). The suggestion appeals to many because probability is viewed as a natural and intuitive subject manageable with very simple mathematics, It is also a good foundation for understanding statistics, which is in prevalent use in today's society. Many teachers are drawn to it because they see all the balls, cards, coins, and dice as ideal teaching tools to make the class interesting.
Soundoff: Does Everybody Need to Study Algebra?
Authors: Arthur Steen Lynn
Published: December 2020

The views expressed in “Soundoff” do not necessarily reflect the views of the Editorial Panel of the Mathematics Teacher or the National Council of Teacher of Mathematics. Readers are encouraged to respond to this editorial by sending double-spaced letters to the Mathematics Teacher for possible publication in “Reader Reflections.” Editorials from readers are welcomed.
Retooling the General-Mathematics Curriculum
Authors: Gerver Robert , Sgroi Richard
Published: December 2020

The climate in United States schools has changed considerably since the days when gumchewing and running in the halls were the major problems. Social, political, economic, and technological changes have dramatically altered the classroom environment, as well as the mathematics skills needed to succeed in an increasingly quantitative society. How have the general-mathematics programs in our high schools adapted to these changes? What can be done to retool the general-mathematics curriculum?
Teaching Mathematics to the At-Risk Secondary School Student
Authors: Vatter Terry
Published: December 2020

Nearly every middle and high school classroom has students who are failing, who seldom do homework, who are inattentive, or who seem unable to benefit from instruction. These students either drop out or end up in classrooms like mine in a school for at-risk youth.
Soundoff: Math-Moron Myths
Authors: Allen Paulos John
Published: December 2020

At the request of the Editorial Panel, John Paulos adapted his editorial, “Math Moron Myths” (New York Times, 24 April 1991), for readers of the Mathematics Teacher.—Ed.
Supplementing the Graphing Curriculum
Authors: Piston Calvin
Published: December 2020

Most students, both at the secondary school level and the university level, have a great deal of trouble understanding graphs. Even many of the best students seem to have inadequate intuition and understanding when confronted with a nonstandard problem involving interpretation of data represented graphically (see, e.g., Monk [1988]). Students need improved skill in interpretation of graphical data to respond better to information that assails them daily, both outside and inside the classroom. This article presents some suggestions for supplementing the traditional curriculum that serve to help develop the full power of graphs.
An Abundance of Solutions
Authors: Barry Donald
Published: December 2020

ever had our mathematics club begun with more excitement than this year.
What Is Left to Teach If Students Can Use Calculators?
Authors: Mercer Joseph
Published: December 2020

Let us begin at the beginning, say. first grade. We give every student a simple calculator that does all the basic arithmetic operations, including square roots and operations with fractions, We teach every student to use the calculator, that is, to find the answer to problems like 2 + 5, 3 – 7, −4 + 9 – (−8), 32.97 × 485.47, 2/3 + 3/4, and so on. Students do not have to memorize anything except how to use the calculator to find the value of an arithmetic expression. Of course, they have to know how to read and write the numbers and know which keys denote plus, minus, and so on, so as to be able to communicate verbally with the teacher or someone else, so some learning is necessary besides pressing keys. But then what?
Mathematical Modeling and the Presidential Election
Authors: Witkowski Joseph C.
Published: December 2020

In recent years, interest in problem solving and mathematical modeling has increased. In 1975, the Conference Board of the Mathematical Sciences issued its Overview and Analysis of School Mathematics K-12, which recommended the incorporation of mathematical applications and modeling into secondary schools. More recently the Curriculum and Evaluation Standards for School Mathematics (1989) formulated by the NCTM stressed the importance of mathematical modeling as a facet of problem solving. The purpose of this article is to look at an interesting mathematical-modeling problem regarding presidential elections.
Are Seven-Game Baseball Playoffs Fairer?
Authors: Lee May E.
Published: December 2020

Sports can be a useful vehicle for introducing the concept of mathematical modeling. I was reminded of that fact recently as I listened to a talk on the “climatically optimal baseball season” (Skeeter 1990). The speaker, a geographer, discussed whether the major-league baseball season should be shifted from its traditional early-April-through-October time period so that most of the season could be played in better weather. (He said that the season should start and end two weeks later than it now does.) He also remarked offhandedly that seven-game playoff series are fairer than five-game ones. This comment sparked in me some old prejudices, along with some new thoughts.
The Many Uses of Algebraic Variables
Authors: Philipp Randolph A.
Published: December 2020

The concept of variable is one of the most fundamental ideas in mathematics from elementary school through college (Davis 1964; Hirsch and Lappan 1989). This concept is so important that its invention constituted a turning point in the history of mathematics (Rajaratnam 1957). However, research indicates that students experience difficulty with the concept of variable, a difficulty that might partially be explained by the fact that within mathematics, variables can be used in many different ways (Rosnick 1981; Schoenfeld and Arcavi 1988; Wagner 1983).
Implementing the Curriculum and Evaluation Standards: Assessing Students' Dispositions: Using Journals to Improve Students' Performance
Authors: Bagley Theresa , Gallenberger Catarina
Published: December 2020

Since the NCTM developed the Curriculum and Evaluation Standards for School Mathematics in 1989, dramatic changes have been taking place in high school classrooms. The role of the teacher is changing. What was once a teacher-centered lecture environment is now a student-centered cooperative-Seaming environment. Evidence of this change can be seen in many classrooms as students work in teams to teach and help each other understand and develop concepts. Students are now communicating their ideas verbally as well as in writing. The teacher's role as a facilitator is emerging.
Alternative Curriculum and Assessment at an Alternative High School
Authors: Simon Marilyn K.
Published: December 2020

Academic decathlons and continuing education high schools seem an unlikely mixture, but not for students at Alta Vista High School, a continuing education school in Southern California for teenagers who have had difficulty in traditional high schools. The students made an impressive showing in the academic decathlon sponsored by the Orange County Department of Education. In keeping with the esprit de corps of the academic decathlon, teachers at Alta Vista High School created the Environmental Studies Academic Pentathlon (ESAP) for its students.
Tips for Beginners: Positive Classroom Management
Authors: Arnold Connie B. , Reynonds Sallee H. , Stellern Cathy D. , Bohannon Diane , Harmon Katherine , Hill Leanne , Mamantov Cathy , Reed Suzanne H.
Published: December 2020

A beginning teacher must have a classroom-management plan to maximize instruction and minimize distraction. Such a plan can help ensure that students are consistently engaged in learning mathematics.
A Quadrilateral Hierarchy to Facilitate Learning in Geometry
Authors: Craine Timothy V. , Rubenstein Rheta N.
Published: December 2020

The NCTM's Curriculum and Evaluation Standards (1989) challenges mathematics educators to rethink both content and instructional style. Specifically, it calls for “a shift in emphasis from a curriculum dominated by memorization of isolated facts and procedures … to one that emphasizes conceptual understandings, multiple representations, and connections …”(p. 125). We would like to illustrate such an approach for the study of quadrilaterals in a year-long high school geometry course. This strand is part of a complete course described in more detail in Craine (1985).
Bulgarian Solitaire
Authors: Nicholson Al
Published: December 2020

Bulgarian solitaire is a topic in discrete mathematics that is easily understood by both the most and least able students and challenges each at a different level. It is one of many mathematical recreations made popular by Martin Gardner (1983). It offers an opportunity for classes to find patterns, solve problems, and find further problems to study. NCTM's Curriculum and Eualuation Standards (1989) emphasizes the value of finding patterns as an aid to problem solving and problem posing. Bulgarian solitaire is openended, and lessons can be tailored to nearly any level, from elementary school to senior secondary school.
Star Numbers and Constellations
Authors: Francis Richard L.
Published: December 2020

The actual number of digits in a number's representation can sometimes fall into the same category as the number itself. Note, for example, that the number of digits in 5378 is an even integer, as is the number 5378; or that the number of digits in 108 is an exact square, as is the number 108 itself. Such comparisons lend a certain appeal to easily overlooked relationships and open the door to varied forms of generalization and learning by discovery, all of which conform perfectly to NCTM's curriculum standards. Our area of exploration is accordingly that of the integers and their digital representations. Brought into focus are fascinating connections, as well as an assortment of whole-number classifications. Let us begin with a challenging example and the more rigorous treatment afforded by a precise definition.
Mathematics as Current Events
Authors: Klein Mary F.
Published: December 2020

The Curriculum and Evaluation Standards for School Mathematics (NCTM1989) lists “learning to value mathematics” as one goal for students. The intent of this goal is “to focus attention on the need for student awareness of the interaction between mathematics and the historical situations from whkh it has developed and the impact that interaction has on our culture and our lives.” One way to reach this goal is to study the history of mathematics; another way is to study current events and see mathematics in the making
Tournaments and Geometric Sequences
Authors: Schielack Vincent P.
Published: December 2020

Many of our students are interested in sporting events or other games involving more than two teams or contestants in a single elimination- tournament format, in which one loss results in elimination. Their interest affords us an opportunity to introduce and promote the topic of geo- metric sequences, on which the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) calls for an increased emphasis within the discrete mathematics standard for grades 9-12. Besides having many other important applications, geometric sequences relate quite naturally to tournaments. This article shows that through the consideration of certain types of single-elimination tournaments, geometric sequences can be summed without resorting to the standard method involving telescoping sums. In the following discussion we take the convention that a team can consist of a single individual
Sharing Teaching Ideas
Authors: Hayek Linda , Telford William D.
Published: December 2020

Observing geometry outdoors can be a way of making mathematics become real for students. A treasure hunt is one activity that keeps students' interest level high as they solve problems cooperatively. Visualization of the problem is important as the students seek to locate specific points in the school yard
The “Method” of Archimedes
Authors: Grande John Del
Published: December 2020

Many great developments of modern mathematics find their origin in the work done some two thousand years ago by the ancient Greeks. An eminent Harvard geometer, James Lowell Collidge, used to say, “There were giants in the land then.” The greatest of all these giants was the famous Archimedes. Some of his exploits and achievements can be appreciated by students who have had little exposure to physics. Archimedes, with the limited means at his disposal, found the volume of a sphere using a method that gave birth to integral calculus, which was finally perfected by such great mathematicians as Kepler, Leibniz, and Newton. We begin with some famous stories about the accomplishments of Archimedes.
A Tale of Three Taxes
Authors: Cibes Margaret
Published: December 2020

U.S. Supreme Court Justice Louis Brandeis once referred to state governments as laboratories for experimentation in social theory. Their legislative chambers may also be laboratories for mathematics. One such laboratory in 1991 was Connecticut, which was faced with the need, some said, to restructure its revenue system. Many policymakers favored the introduction of a tax on total personal income as the first step toward generating a more reliable and equitable revenue stream than that yielded by the traditional sales-tax-based system.
Tips for Beginners: Attention Getters
Authors: Moldavan Carla
Published: December 2020

Before students can learn, their attention needs to be engaged (Gagne and Briggs 1988). Methods of gaining attention might include using colored chalk, moving closer to the class, changing the size of print on certain portions of a handout, and changing voice inflection. To this list I want to add personalizing mathematics and using humor.
The Birthday Problem Again?
Authors: Jones Kevin S.
Published: December 2020

Allow me to describe an event many of us have experienced. You are reading a newspaper or a magazine and stumble across an unfamiliar word. True to your elementary school teacher's wishes, you dutifully dust off your big dictionary and find the definition. Later, perhaps within a week, you stumble across the same word again and think, “What a coincidence!” Similarly, perhaps you recently came across the “birthday problem” and its intriguing, nonintwtive result: the probability that at least two people in twenty-three have the same birthday is greater than 1/2. Now, here in this issue of the Mathematics Teacher, you find the birthday problem mentioned again! Coincidence?
Implementing the “Professional Standards for Teaching Mathematics”: Becoming a Teacher of Mathematics: A Constructive, Interactive Process
Authors: Feldt Constance Curley , Farrell Margaret A.
Published: December 2020

In this, the final article in this department for the 1992-1993 academic year, the author addresses the standard Developing as a Teacher of Mathematics (NCTM 1991, 160-66). Teacher conceptions and teacher reftections have been alluded to by each of the previous three authors. In this article, the author places the spotlight directly on teacher reftection and its relation to classroom practice. Teachers who are looking for ways to reexamine their own teaching will find practical suggestions and the theoretical and research rationale on which these ideas are based.
Asymptotes and Other Limiting Functions
Authors: Mayes Robert
Published: December 2020

Mathematics and the way we teach it are undergoing dramatic changes. The NCTM's Curriculum and Evaluation Standards (1989) has determined a direction and focus for the future of mathematics education. The study of mathematics should be an active experience for students—one in which they explore the world of mathematics, make conjectures based on patterns discovered, and induce the mathematical concept. The idea of a guided-discovery approach for students in mathematics is hardly new. Socrates taught this way. However, he had the luxury of a lighter student load and no state or federal mandates on expected student outcomes. In short, Socrates had the time.
Humanize Your Classroom with the History of Mathematics
Authors: Bidwell James K.
Published: December 2020

In classrooms, we often treat mathematics as if we were learning on an island. We travel to that island once a day for mathematics and delve into a study that is pure, clean, and logically solid and has clear lines and no dirty corners. Students think that mathematics is closed, dead, emotionless, all discovered. It lies completely in a book or in the mind of the teacher, ready to be rolled out and absorbed. They probably have never read Augustus DeMorgan (Jones 1980), who said in 1831 that the teacher.
Back to the Present: Ruminations on an Old Arithmetic Text
Authors: Swetz Frank J.
Published: December 2020

A student of mine, Andy Ohrman, received a thoughtful but unusual gift from his sister. It is an old arithmetic book she had stumbled on at a New England yard sale. Although the pages are brittle and browned by time, their contents are quite legible. Knowing that her brother is studying to be a mathematics teacher, she purchased it for him. The book's front page bears the title The Columbian Arithmetician and declared the author's ambitious mission to present a “New System of Theoretical and Practical Arithmetic Performed in Simple and Compound Numbers adopted to the Commerce of the United States and Designed for the Instruction of Youth.” Its author remains anonymous, identifying himself pointedly as “an American”; however, the printer is clearly designated as William Allen of Haverhill, Massachusetts, and the year of publication 1811. Andy knew of my interest in old mathematics books and lent it to me to read.
Inflections on the Bedroom Floor
Authors: Weiner Jack L. , Chapman G. R.
Published: December 2020

When introducing mathematical concepts in the classroom, students' interest and appreciation are heightened when examples are drawn from everyday life. Recently, while discussing points of inflection with a calculus class, I was challenged to find an example. Surveying the lecture theater, I was unable to find a single example of this very simple geometric idea.
Some Surprising Probabilities from Bingo
Authors: Mercer Joseph O.
Published: December 2020

It is human nature to look for easy ways to get ahead. As teachers we would hope that studying probability would help our students understand why gambling is not such a way. Since bingo is a form of legalized gambling that is familiar to almost everyone, the game is worth spending some time to analyze. In fact, most of the fundamental concepts from discrete-probability theory can be covered by studying bingo
Connecting Research to Teaching: Affective Responses to Problem Solving
Authors: McLeod Douglas B.
Published: December 2020

The vision of the mathematics classroom that IS presented 1n the Natwnal Council of Teachers of Mathematics's Curriculum and Eualuation Standards for School Mathematics (1989) has inspired many of us to want to change the way in which we teach. We want to pose challenging problems, to see our students work cooperatively, and to have productive discussions with students about significant mathematical ideas. But as Ball and Schroeder have pointed out, that vision is “much more difficult to realize than to endorse” (1992, 69). We will encounter many difficulties as we move toward that ideal classroom of the future; getting students to respond positively to nonroutine problems or other tasks that require higher-orderthinking skills is one difficulty that teachers often face. Research suggests that students' affective reactions to nonroutine problems can be a source of both difficulty and support as we work to reform mathematics classrooms.
Recruitment Communication and Psychological Contracts in Start-Ups
Authors: Rasmussen Mia Thyregod
Published: December 2020

Recruitment communication presents a dilemma for organisations. When organisations hire, they often engage in branding themselves as employers (Backhaus & Tikoo, 2004) and rely on positive framing to present vacant positions in order to attract candidates. This leads to the ensuing challenge of living up to these promises for the candidates who are ultimately hired. Overpromising and underdelivering leads to a breach of the initial psychological contract. This balancing dilemma is especially pertinent for new and unknown companies, where concerns about the company’s legitimacy as an employer may cause potential candidates not to apply (Williamson, Cable, & Aldrich, 2002). On the one hand, start-ups need and want to attract the best, and on the other hand, they need to be wary of the impression they are creating of the job and the organisation as a place of work, as they would also like the candidates to stay once they are hired. I draw on interviews with managers and newcomers in Danish start-ups to give empirical examples of this challenge and its results, using the literature on psychological contracts (Rousseau, 1995) as an explanatory framework. I discuss what organisations might do to accomplish this balancing feat from theoretical and practical perspectives.
The Language of Destructive Cults
Authors: Palayon Raymund T. , Todd Richard Watson , Vungthong Sompatu
Published: December 2020

Destructive cults are the most notable damaging religious groups in society where leaders convince their followers to engage in destructive acts. Examples of such cults include Peoples Temple led by Jim Jones and Heaven’s Gate led by Marshall Applewhite who convinced their followers to commit mass suicide. Previous research into destructive cults has mainly focused on their social-psychological development. This research breaks new ground by examining the patterns of linguistic features in the sermons of destructive cults indicating the characteristics of their language using keyness analyses. The main data sets are the sermons of Jim Jones and the sermons of Marshall Applewhite in the period leading to mass suicide. As a benchmark, these sermons were compared to the sermons of Billy Graham and the sermons of Rick Warren, leaders of mainstream religious groups. The findings show that the language of destructive cults based on the sermons of the leaders upholds extreme non-religious ideologies that cannot be found in the sermons of mainstream religious groups. The styles of their language focus on othering, intensifying, elaborating, and negating with the aim of controlling their followers. The results may allow destructive cults to be identified before damaging events occur.
Mimicry and artistry
Authors: Bertilsson Jon , Rennstam Jens , Sullivan Katie
Published: December 2020

Many public organizations think of themselves as brands and engage in branding to increase their attractiveness. Often this is seen as a good practice, but this paper takes a more skeptical view and interrogates the value of place branding expertise for public organizations. Through observation of a place branding conference, drawing on some principles from ethnomethodology, this paper provides clues to what the place branding profession constructs as “good” and legitimate expertise and the norms that guide their work. We identified two levels of legitimate place branding expertise: Mimicry and Artistry. Mimicry signifies imitation of already institutionalized ideas and practices, and proficiency in supplying beautiful yet detached and superficial representations of cities. Artistry refers to place branding that stands out as brilliant and inventive, encompassing unexpected play with symbols and creative combination of branding models. We argue that both these levels of expertise may be at odds with civic values of city government such as inclusion and representativeness. Our study concludes that the ways in which the branders construct expertise risks deflecting attention from the core problems of a place or a city and separating place branding from city management practice.
A Socio-technical-cultural System Perspective to Rethinking Translation Technology in Intercultural Communication
Authors: Li Mei , Zhou Chunfang , Henriksen Lars Bo
Published: December 2020

As technology has radically changed language translation in the age of globalization, the research on translation technology should not only benefit current research on translation of languages but also have a long-term positive impact on technology in the sociocultural context. The focuses of this paper are twofold. Firstly, it discusses how translation technology drives the changes in intercultural communication that bring both bright and dark sides. Secondly, it explores how translation technology’s involvement and interaction with human translator in practice of language translation from a socio-technical-cultural system perspective. Based on the discussion, this paper particularly addresses human translator’s collaboration with translation technology should be regarded as a cultural mediator helping to realize successful intercultural communication; and meanwhile, the human translator’ s subjectivity should be highlighted, and translation technology’s cultural design should be explored in order to improve usability that further brings benefits to the future cultural mediator.
Doktryna cartel facilitator – początek oraz rozwój w praktyce decyzyjnej zagranicznych organów ochrony konkurencji oraz orzecznictwie sądów unijnych
Authors: Jelska Agnieszka
Published: December 2020

The purpose of the article is to present the issue of the ‘cartel facilitator’ doctrine based on the decision-making practice of the European competition authorities and the case-law of EU courts. The article is an attempt to approximate the behaviour of third parties, which have so far been assessed in terms of their compliance with competition law, as well as the criteria developed in EU judicial decisions, which must be met in order to be able to hold such entities liable for violations of competition law.