bluuebunny/crossref_metadata_2025 huggingface dataset:
New Projects
Authors: Watson Charles D.
Published: December 2020

At Kent State University, a National Science Foundation project, Development of a Logo-based Geometry Curriculum, is currently re· conceptualizing the elementary school geometry curriculum. Based on the premise that computers will be widely available to students in the near future, the project is exploring ways that the Logo computer language can be used to enhance the learning of geometry
Archimedes' Pi—an Introduction to Iteration
Authors: Lotspeich Richard
Published: December 2020

One method (attributed to Archimedes) of approximating n: offers a simple, yet interesting introduction to one of the basic ideas of numerical analysis, an iteration sequence
Secondary School Results for the Fourth NAEP Mathematics Assessment: Discrete Mathematics, Data Organization and Interpretation, Measurement, Number and Operations
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 first 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 elementary school results appear in companion articles in the Arithmetic Teacher (Kouba et al. 1988a, 1988b). Secondary school data from previous national assessments have been reported in the Mathematics Teacher (see, e. g., Carpenter et al. [1980, 1983))
Primitive Quadruples for the Law of Cosines
Authors: Mettler Marll A.
Published: December 2020

One of the reasons I still find mathematics teaching so rewarding after many years in the classroom is the occasional serendipitous question, usually when it is least expected.
A Summer Program in Probability and Statistics for Inner-City Seventh Graders
Authors: Scheinok Perry A.
Published: December 2020

During the summers of 1983 and 1984, four faculty members from the New Jersey Institute of Technology, assisted by six college student-tutors or mentors, ran an Experimental Mathematical Science and Communications (EMSAC) program for seventh-grade inner-city youths. In 1984, the EMSAC program focused on mapping and surveying, observational astronomy, Apple Logo, communications, and probability and statistics. This article describes the probability and statistics component its day-to-day approaches and its successes and failures.
The Issue of Reform: What Have We Learned from Yesteryear?
Authors: Cooney Thomas J.
Published: December 2020

One of the central issues facing our profession is how the NCTM Curriculum and Evaluation Standards for School Mathematics (1987) can influence mathematics teacher education programs. This question and the issues in which the question is embedded not only are highly relevant to our professional interests but are, ultimately, primary to our professional activity. My experiences both as a teacher and as a teacher educator suggest that the task that lies ahead is both significant and awesome.
Adapting Test Construction for Mainstreamed Mathematics Students
Authors: Miederhoff Jennifer Wingo , Wood Judy W.
Published: December 2020

Recognizing the need for making adaptations for special students in regular classes, Project Train at Virginia Commonwealth University has developed a model for adapting the curriculum for mildly handicapped children (Wood 1985). The model is generic to all academic subjects and grades K-12. This article focuses on adapting the construction of teacher made mathematics tests for mildly handicapped children, that is, the educable mentally retarded, the emotionally handicapped, and the learning disabled, in the mainstream.
Number Sense
Authors: Ronau Robert N.
Published: December 2020

Good number sense is fundamental for success in estimation, approximation, and problem solving. We need to develop a sense of large numbers because newspaper and television news reports contain many references to large quantities. The federal budget is expressed in billions and trillions of dollars, space distances in millions of light years or trillions of miles, computer speeds in nanoseconds (1 nanosecond = 1 billionth of a second), computer storage memory in gigabytes (1 gigabyte = 1 billion bytes), world populations in millions and billions of individuals, and nuclear weapons in millions of tons of TNT. Clearly, the average citizen needs a well-developed sense of large numbers to understand many of the news items that invade the home. Large-number concepts a re appropriate for development in upper elementary school, high school, and beyond. This article presents some examples to enhance a sense of large numbers in middle and high school students.
Inductive Patterns, Finite Differences, and a Missing Region
Authors: Chinn Phyllis Zweig
Published: December 2020

The September 1985 Arithmetic Teacher contained an article entitled “On Patterns, Conjectures, and Proof: Developing Students' Mathematical Thinking.” In it, author Alba Thompson discusses some situations that can arise in exploring patterns as a problem-solving strategy. The third of her examples involves a fairly well known problem, sometimes referred to as the “problem of the missing region,” where the “obvious” pattern turns out to be incorrect (Gibbs 1973; Glenn 1968). Thompson mentions the problem and merely suggests it as an example to be used to illustrate the danger of assuming that patterns continue to hold just because they seem to be well established.
How Did “They” Get That?
Authors: Benton Joanne
Published: December 2020

The November 1985 issue of Student Math Notes contains an article by Gai I Burrill that presents the following formula for rank correlation: where r is the correlation coefficient, d is the difference between each pair of ratings, and n is the number of items ranked. Burrill stated that r would take on values from – 1 to 1
A Power Pattern
Authors: Schwartzman Steven
Published: December 2020

One of the curiosities of arithmetic is that when any integer is raised to the fifth power, the units digit of the result is always the same as the units digit of the original number. This fact is mentioned from time to time in books on recreational mathematics, often with an appreciation for the striking nature of the pattern, but without further explanation or generalization. Table 1 demonstrates the truth of the assertion simply by showing the ten possible cases.
National Council of Teachers of Mathematics
Authors: Dossey John A.
Published: December 2020

The questions posed by the title of this article are ones that all professional organizations should take time to consider on a regular basis. If one does not know the intent of the membership, nor their beliefs, then the organization may not be positioning itself to serve their interests as best it could. Further, without knowledge of the members' beliefs and desires, it is difficult to move ahead with vigor and have the conviction that the necessary support will materialize to strengthen the gains made.
Girls in the Mathematics Classroom: What's Happening with Our Best and Brightest?
Authors: Strauss Shiela M.
Published: December 2020

With the advent of both the women's movement and a heightened national nterest in educational excellence, scholary attention has been focused on the success of females in mathematics. Sells (1978) Identified mathematics as the “critical filter” that could limit the range of career choices available to those who do not take an adequate number of high school mathematics courses. Casserly and Rock (1979) looked at factors that differentiate girls who enroll in Advanced Placement mathematics courses from those who do not. Studies of Fennema and Sherman (1977) demonstrated that gender differences in mathematics achievement are substantially reduced when the number of mathematics courses is controlled. Benbow and Stanley (1982), however, found that sex differences in mathematics achievement did not reflect differential mathematics course taking. Their study involved students in the Study of Mathematically Precocious Youth (SMPY) who, as seventh and eighth graders, scored as well as a national sample of eleventh- and twelfth-grade females on the S.A.T. mathematics and verbal tests
A Number Game—Summing Consecutive Positive Integers
Authors: Cowles Mary Jane
Published: December 2020

One of the classroom activities that my students and I enjoy is finding expressions that relate the digits of the page number of an assignment. For example, when we are working on page 283, the students might notice that the power of 2 that produces 8 is 3. All arithmetic facts and relationships are fair to use. Phone numbers and license plates also furnish some interesting possibilities.
Investigations Involving Involutions
Authors: Wallace Edward C.
Published: December 2020

Many second-year algebra textbooks include a discussion of functions and their inverses prior to introducing logarithms and the inverse trigonometric functions. Since these functions and some others can be approached neatly as inverses of more familiar functions, a good understanding of the notion of inverse allows students to understand these topics better.
Integrating Spreadsheets into the Mathematics Classroom
Authors: McDonald Janet L.
Published: December 2020

Spreadsheets have become an integral part of computer literacy and business courses, allowing students to see the power of such utility software and use it to solve problems. But, the spreadsheet can also be an extremely effective tool in the mathematics classroom. There the spreadsheet can be used to help solve many real-world problems and, at the same time, promote students' understanding of important mathematical concepts and principles.
Let ABC Be Any Triangle
Authors: Schwarz B. , Bruckheimer M.
Published: December 2020

Once upon a time, when geometry—of the Euclidean variety— was still taught, Mr. Griffiths, an enlightened teacher, began the lesson as follows.
Teaching Logic with Logic Boxes
Authors: Sanders Walter J. , Antes Richard L.
Published: December 2020

Logic is an important but frequently neglected topic of the mathematics curriculum. Recognizing the need for better formal reasoning skills in preparing students to study mathematics and solve problems.
Explorative Writing and Learning Mathematics
Authors: Keith Sandra Z.
Published: December 2020

Short explorative writing assignments can transform the mathematics classroom into a dynamic and exciting learning laboratory. In explorative writing, students explore their knowledge about a topic by writing what they know about it in their own language. Then, to refine their ideas further, they share reactions with other students and the teacher. In spite of the significant benefits of using such writing assignments, the teacher may feel uncomfortable with the more interactive environment. with the challenge of designing writing assignments, and with figuring out what to do with the results. This article illustrates how explorative writing assignments can help to expose and identify learning problems and offers some assignments that address these problems. This article briefly mentions some techniques that I have found helpful for working with the results and comments on the benefits of working with these assignments.
The Fourth NAEP Mathematics Assessment: Performance Trends and Results and Trends for Instructional Indicators
Authors: Silver Edward A. , Lindquist Mary M. , Carpenter Thomas P. , Brown Catherine A. , Kouba Vicky L. , Swafford Jane O.
Published: December 2020

This article is the third one to appear in the Mathematics Teacher reporting the results and analysis compiled by the NCTM Interpretive Team for the Fourth Mathematics Assessment of the National Assessment of Educational Progress (NAEP). The first two articles reported on the performance of seventh- and eleventh-grade students in specific content areas (Brown et al. 1988a. 1988b). This article discusses trends in performance across the last three NAEP mathematics assessments and reports on indicators of instructional activity. A companion article on the performance of third- and seventh-grade students appears in this month's Arithmetic Teacher (Carpenter et al. 1988).
Helping Students Understand Factors and Terms
Authors: Martinez Joseph G. R.
Published: December 2020

Even though students are introduced to the concepts of factors and terms in elementary arithmetic, they are often confused when applying this understanding to algebra. To avoid confusion, teachers need to make more connections between previously learned information and newly learned information. This article offers an approach to teaching factors and terms that builds directly on students' knowledge of arithmetic.
The Power of Proportions in Problem Solving
Authors: Strickland James F. , Denitto John F.
Published: December 2020

Mathematical skills and concepts are often taught in a narrow context. Consequently, many students fail to understand the general applicability and power of the ideas they have learned. When given a problem in mathematics, they are unable to see a relationship between it and a previously learned mathematical principle that could be used to solve the problem. That students view mathematics as a collection of isolated facts and formulas with little utility beyond the settings in which they first encountered the ideas is not surprising.
Soundoff: Mathematics Teachers' Preparation: A Challenge
Authors: Borasi Rafaella , Brown Stephen I.
Published: December 2020

The 1982 “excell ence reports” on the dire state of education and the two influential proposals by the Carnegie Forum (1986) and the Holmes Group (1986) beckon mathematics educators to identify implications for our own field and in particular to reexamine the recurrent question. What should he included in a program for the preparation of mathematics teachers?
Interest in Mathematics—It's in the Cards
Authors: Mulligan Catherine Herr
Published: December 2020

Interest in mathematics can often be generated by seemingly nonmathematical items. Perhaps not surprisingly, students cited card tricks as their favorite activity in a course in recreational mathematics that I taught to middle school students in Miami University—Middletown's “Kids in College” program. When a card trick was shown to them, the students were at first inclined to believe that the trick was based on magic or sleight-of-hand manipulation, asking such questions as, “How did you do that?” I answered that I hadn't done anything … it was “in the cards,” or, more precisely, in the mathematics of cards. Our pursuit of finding the mathematics in a particular card trick involved several important mathematical activities: problem-solving analysis, making a conjecture and testing it, discovering, and verbalizing.
An Interest in Interest
Authors: Tisdale Joseph C.
Published: December 2020

Money and interest in the growth of money promote the study of compound interest and exponential functions. Using the formula A(n)= P(1+r)n we calculate the amount of money A(n) accumulated if P dollars are invested for n compound periods at r interest rate each period. But does the amount accumulated reflect real money growth given the effect of inflation and taxes? Examples of compound interest used in the classroom and in textbooks can be misleading. This article addresses the real growth of an investment in light of inflation and taxes.
Visualizing the Geometric Series
Authors: Bennett Albert B.
Published: December 2020

Mathematical proofs, a lthough a necessary part of mathematics, often leave the student unconvinced or without understanding of what has been proved.
Area Ratios of Quadrilaterals
Authors: Anderson David R. , Arcidiacono Michael J.
Published: December 2020

Given the current emphasis on problem solving in mathematics education, it is important to offer students opportunities to explore mathematical settings that promote investigation, hypothesis testing, and generalization. In earlier issues of the Mathematics Teacher, Hiatt (1972) and Kulm (1974) described such a setting in their investigations of area ratios of convex quadrilaterals.
Probability in Ancient Times; or, Shall I Go Down after the Philistines?
Authors: Milton Janet S.
Published: December 2020

This article explores an interesting application of combinations, geometric series, and elementary probability that arises in a Biblical setting. I have used this example in courses in finite mathematics, mathematical statistics, and elementary calculus. Its nontraditional nature seldom fails to arouse the curiosity of students.
Soundoff: Our New Colleagues
Authors: Krist Betty J.
Published: December 2020

With the resurgent popularity of mathematics teaching today, all of us who interact with student or beginning teachers have a special opportunity to make substantial changes in schools.
Fraction Patterns—Visual and Numerical
Authors: Bennett Albert B.
Published: December 2020

A major goal of mathematics instruction is the involvement of students in discovery. One neglected method of accomplishing this objective is through visual models. Such models often have symmetries or contain patterns arising from the relative sizes or arrangements of figures. These patterns will usually have corresponding numerical patterns.
The NCTM “Standards”—Challenges for All Classrooms
Authors: Frye Shirley M.
Published: December 2020

Just as the Curriculum and Evaluation Standards for School Mathematics generates excitement among our members and colleagues, the document also engenders questions from them. Educators and colleagues have asked questions about the standards and the implementing procedure at conferences and committee meetings, through the mail, on the telephone, and in personal conversations. From those submitted or posed, I have selected some typical questions to which to respond.
Using Problems from the History of Mathematics in Classroom Instruction
Authors: Swetz Frank J.
Published: December 2020

Many teachers believe that the history of mathematics, if incorporated into school lessons, can do much to enrich its teaching. If this enrichment is just the inclusion of more factual knowledge in an already crowded curriculum, the utility and appeal of historical materials for the classroom teacher is limited. Thus to include a historical note in a student's text on the life or work of a particular mathematician may shed a historical perspective on the content, but does it actually encourage learning or illuminate the concept being taught? The benefits of this practice can be debated.
Henry Wadsworth Longfellow, Poet Extraordinaire
Authors: Mitchell Charles E.
Published: December 2020

That Henry Wadsworth Longfellow was one of America's most outstanding poets is hardly a little-known fact. That he was also an accomplished mathematician is something few people know. One of the problems mathematics instructors face is dealing with mathematics anxiety and the myth that “some people have a math mind and some don't” (Kogelman and Warren 1978). Mathematics is often viewed as a “mystique accessible to few” (Buxton 1981); and even many educators will categorize a student as a non mathematics person, thus students “are steered away from precollege mathematics” and “not even given the chance to fail” (Tobias 1978). We forget that success in mathematics is dependent on interest as much as ability. and I am afraid that we often do not take the steps we could to reach individual students
Publications
Authors: Anderton David E. , Skow Donald P. , Truckson Erika , Stephany Edward . , Taylor Roger , Wiley Martha
Published: December 2020

Code Tj = Textbook junior high, Ts = Textbook senior high, Tt =Textbook two year college L = Library P= Protessronal: S =Supplementary student reading
How Symbolic Mathematical Systems Could and Should Affect Precollege Mathematics
Authors: Kathleen Heid M.
Published: December 2020

Computer programs are now available that perform most of the algebraic and numerical-manipulation procedures on which school mathematics now concentrates. The defining characteristic of these symbolic mathematical systems is that, unlike many of the popular computer languages, they can manipulate variables as well as numbers. They can perform rationalnumber arithmetic, solve equations, produce equivalent expressions of a variety of types, apply trigonometric identities, evaluate limits and sums, compute the algebraic form of derivatives and integrals, perform matrix manipulations, and produce accurate numerical answers-with hundreds of digits if appropriate. In short, they can perform with more dependable accuracy and greater speed most of the algebraic and numerical procedures on which students now spend most of their mathematical careers.
Beliefs and Their Influence on Mathematical Performance
Authors: Garofalo Joe
Published: December 2020

Recent research in mathematics education has shown that success or failure in solving mathematics problems often depends on much more than the knowledge of requisite mathematical content. Knowing appropriate facts, algorithms, and procedures is not sufficient to guarantee success. Other factors, such as the decisions one makes and the strategies one uses in connect ion with the control and regulation of one's actions (e.g., deciding to analyze the conditions of a problem, planning a course of action, assessing progress), the emotions one fee ls while working on a mathematical task (e.g., anxiety, frustration, enjoyment), and the beliefs one holds relevant to performance on mathematical tasks, influence the direction and outcome of one's performance (Garofalo and Lester 1985; Schoenfe ld 1985; McLeod 1988). These other factors, although not explicitly addressed in typical mathematics instruction, are nonetheless important aspects of mathematical behavior.
Math Class Logs
Authors: Socha Susan C.
Published: December 2020

“I found the log a great help when I was absent for seven days. It really helped me get back on track.”
Mathematical Modeling: Lemonade from Lemons
Authors: Daniels David S.
Published: December 2020

What teacher has not had the discouraging experience of grading a test and discovering the scores to be depressingly lower than expected? What teacher has not been concerned about the effect of low test scores on students' attitudes and motivation? When this situation happened in my second-year algebra class, I launched an impromptu lesson that captured students' interest and offered them a new opportunity for success. The lesson also gave the class insight into elementary mathematical modeling, review and practice of first-year algebra topics, and a forum for lively discussion about the fairness of scaling, or curving test scores.
Around the Sun in a Graphing Calculator
Authors: Demana Franklin , Waits Bert K.
Published: December 2020

The topics of polar equations and parametric equations are found in almost all precalculus and calculus textbooks. They furnish important a lgebraic representations of real-world problems. For example, polar equations are used to describe the e lliptical orbit of planets around the sun (Kepler's first law), and parametric equations are frequently used to describe the trajectory of moving objects. These topics are seldom given more than superficial treatment in high school and college precalculus and calculus courses because of the difficulty of obtaining appropriate graphical representations.
An Arithmetic Method for Converting Repeating Decimals to Fractions
Authors: Sherzer Laurence
Published: December 2020

Students can easily use calculators to convert fra ctions to decimal numbers. But students who have studied the algebraic method of changing a repeating decimal to a fraction often find the method tedious and tend to avoid it. This difficulty raises questions about students' under standing of the nature of repeating decimal numbers. Do they know that all repeating decima ls are shown as a pproximations on the ca lculator display?
Sharing Teaching Ideas: Guessing The Slope Function
Authors: Gallian Joseph A.
Published: December 2020

In the excellent film “Let Us Teach Guessing,” Pólya (1966) poses a geometry problem to a class and asks the students to guess the solution. Without saying whether these guesses are right, he proceeds to lead the class step by step to the point where they can guess the correct solution of the problem. As Poly a has said, “Everything is just a guess—concerning your job, your home, your family, even the laws of physics.”
Assembling a Mathematical Rectangle
Authors: Santulli Thomas
Published: December 2020

This activity gives students a chance to practice factoring quadratics. Students in several levels of first-year and second-year algebra have used this activity and enjoyed it because it is a good way to practice needed skills in a nonroutine way. Also, it allows for hands-on experience, and the students can all work at their own pace. You might wish to try cooperative groups in solving or creating such a puzzle.
When and How Can We Use Modeling?
Authors: Swetz Frank
Published: December 2020

Increasingly over the past ten years, national conferences and committees investigating the state of North American mathematics education have urged an increased instructional emphasis on problem solving and mathematical applications (CBMS 1975; NCTM 1989). But despite these repeated recommendations and exhortations, in general, little progress has been made on the introduction and use of mathematicalmodeling techniques in the secondary school classroom. In part, teachers are unsure about just what mathematical modeling is and why and how it should be incorporated into the curriculum. Let's examine each one of these issues separately.
Problem Posing in Geometry
Authors: Hoehn Larry
Published: December 2020

Problem solving received dramatic emphasis throughout the 1980s. However, its necessary counterpart, problem posing, has received scant attention. Some notable exceptions are the works of Brown and Walter (1983), Klamkin (1986), and, of course, Polya (1973). In this article a typical geometry theorem is used and problems are posed involving its application. The methods presented here work well to create geometry test questions, geometry contest problems, and calendar problems for the Mathematics Teacher. Although these methods are intended primarily for secondary school teachers, creative geometry students could use them for making their own mathematical discoveries.
Activities: Patterns in the Hundred Chart—Part 1
Authors: Ajose Sunday
Published: December 2020

Introduction: The NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) advocates the study of patterns and relationships, which “helps students develop mathematical power and instills in them an appreciation for the beauty of mathematics” (p. 98). The exploration of patterns and relationships is often so interesting that students sense the artistic quality of geometric patterns and enjoy just “playing with numbers.” Moreover, as students talk about their observations and discoveries, they become better at communicating mathematical ideas.
Soundoff: Technology in the Classroom: Beginnings and Endings
Authors: Matras Mary Ann
Published: December 2020

Two as-yet-unanswered questions raised by the NCTM's curriculum standards involve the appropriate use of technology in mathematics classrooms.
Keeping Current with Technology
Authors: Brady Mary L.
Published: December 2020

Many of today's students will be using calculators and computers in some way once they enter the work force. They should become proficient in their use.
A Fractal Excursion
Authors: Camp Dane R.
Published: December 2020

Recently, chaos theory and the related topic of fractal geometry have blossomed as creative fields of study in mathematics and physics. Fractals are shapes containing self-similarity on arbitrary magnification. One such object, the Koch curve, is generated by simple recursion on an equilateral triangle. The process used to produce the curve is a great way to introduce students to some concepts of fractal geometry.
An ABSOLUTE-ly VALUE-able Manipulative
Authors: Stallings-Roberts Virginia
Published: December 2020

One of the greatest disadvantages children have is being expected to perform symbolically before they are ready to do so. (Schultz 1986, 53)
Mathematics Education in Britain: An American Viewpoint
Authors: Meuser Mark H.
Published: December 2020

Before I embarked on my year of teaching in England under the auspices of the Fulbright Teacher Exchange Program, I had envisioned classes composed of English choirboys, colleagues teaching in academic gowns, and breaks in the day for morning coffee and afternoon tea. Although the tea break is still very much a part of the school day, “choirboy” students and academic robes are still to be found at only a very few schools. The reality was closer to Sidney Poi tier's experience in the movie To Sir with Love, with students challenging the teacher's authority at every possible opportunity. Nevertheless, I grew personally and developed professionally during my year of teaching abroad. My wife and I made new friends and explored a fascinating country. I was able to observe and become part of a foreign school system, and I tried to understand and analyze that system's approach to education. This article discusses some of the differences between the American and English systems and offers recommendations on the basis of my judgments about their relative merits.
Inscribing an “Approximate” Nonagon in a Circle
Authors: Lamb John F. , Aslan Farhad , Chance Ramona , Lowe Jerry D.
Published: December 2020

Throughout history even the best mathematicians have been challenged by certain problems. Indeed, some of these problems became famous because they defied solution. Many students of mathematics are familiar with the problems of squaring a circle, doubling the cube, and trisecting an angle. Some may have studied the problem of inscribing regular polygons with a given number of sides in a circle. A few of these polygons, such as those with three sides, four sides, six sides, and eight sides, are easily inscribed. Other such constructions are more involved but are still possible, such as those involving polygons with five sides and with seventeen sides. Still others turn out to be impossible to perform with Euclidean tools, such as those for polygons with seven sides and with nine sides.
Activities: Starring in Mathematics
Authors: Fairbairn Donald M.
Published: December 2020

Making conjectures, gathering evidence, and building an argument to support such notions are fundamental to doing mathematics. (NCTM 1989, 6)

In grades 5-8, the mathematics curriculum should include explorations of patterns and functions so that students can… describe, extend, analyze, and create a wide variety of patterns. (NCTM 1989, 98)
Teachers' Reasons for Instructional Decisions
Authors: Glidden Peter L.
Published: December 2020

One of the lessons learned from the “new math” reform movement of the sixties is that effecting lasting change requires more than developing curriculum materials at a national level for adoption at the local level (National Research Council 1989; NCTM 1989; Mumme and Weissglass 1989). Lasting reform also requires directly involving teachers in curriculum development so that they have “ownership” of the product (National Research Council 1989). This ownership is necessary because teachers act as curriculum filters (Holmes Group 1986; Porter et al. 1988; Romberg 1988).
Communicating Mathematics
Authors: Hatfield Mary M. , Bitter Gary G.
Published: December 2020

Affording more opportunities to engage students in thinking and communicating mathematically and integrating technology into mathematics education are clear trends in curricular reform. Recent recommendations emphasize adopting a more active, process-oriented approach to mathematics learning and teaching. The Mathematical Sciences Education Board's document Reshaping School Mathematics (1990) emphasizes that a person engaged in mathematics gathers, discovers, creates, and expresses facts and ideas about patterns. The National Council of Teachers of Mathematics in its Curriculum and Evaluation Standards for School Mathematics (1989) advocates mathematics teaching through activities that encourage students to explore mathematics, to gather evidence and make conjectures, and to reason and communicate mathematically as they discuss and write about ideas that use the language of mathematics. This vision of the classroom specifies a mathematics curriculum in which students are “doing and investigating” mathematics rather than just “knowing” mathematics.
Addition of Fractions—the Unrecognized Problem
Authors: Howard Arthur C.
Published: December 2020

Anyone who teaches mathematics in grades 4-10 is painfully aware of the difficulty students have with fractions in general and with addition of fractions in particular. This article discusses the tendency of many students to apply an “add the numerators and add the denominators” approach to adding fractions, why they do it, and what should be done about it.
A Diary of Two Problem Solvers
Authors: O'Shea Thomas
Published: December 2020

Recently, I taught a mathematics methods course for thirty-six preservice elementary school teachers. Taking heed of calls from mathematics educators to ensure that teachers have some understanding of the process of problem solving (e.g., Thompson [1985, 292]), I gave a lot of thought to what to do with such a large class. I wanted to go beyond just presenting the ideas of P6lya and others. I wanted the students to become actively involved and to feel what it means to solve problems.
Counting Squares
Authors: Pagni David L.
Published: December 2020

This wonderful problem affords students and teachers the opportunity to explore, hypothesize, and deduce a relationship between the size of a rectangle and the number of squares intersected by a diagonal (see fig. 1).
Changes in the SAT in 1994
Authors: Braswell James S.
Published: December 2020

This article was prepared at the invitation of the Editorial Panel to respond to the concerns addressed in Harold Baker's “Soundoff” in this issue. Reactions to this article are welcomed for consideration in “Reader Reflections.”
Geographic Constructions
Authors: Johnson Art , Boswell Laurie
Published: December 2020

Constructions are usually presented to students as mathematical exercises with little or no practical applications included. To help students appreciate the real-life application of standard constructions, this article presents two activities that apply to boundary lines in geography. By integrating mathematics with other disciplines, we can demonstrate that mathematics does not exist in a vacuum.
Let's Put the Mathematics in General Mathematics
Authors: Hadley William S.
Published: December 2020

As larger numbers of teenagers entered high school early in this century, many of these students were judged not able or willing to take traditional academic mathematics. Thus, a new course was introduced to reinforce the basic-arithmetic skills of this group of students. While changing little over the years, general mathematics has become one of the most taught and least examined mathematics courses offered at the secondary school level. Even as the world has become more technologically advanced, general-mathematics enrollment has remained relatively stable, and the curriculum is still rooted in the “basic skills.” Many high school students understandably think of the study of mathematics as both boring and as a set of “rules to follow and memorize” (IEA 1985, 81).
Tips for Beginners: Tips on Discipline
Authors: Sinicrope Rose
Published: December 2020

The first year I was given a lesson-plan book to use for teaching, I was bewildered. My plans could not possibly fit into those two-inch-by-three-inch rectangles. Besides, we were also required to submit weekly lesson plans, which included a list of performance objectives, activities, materials, and assignments. I decided to use the plan book as a type of appointment calendar in which I would note changes in the school day, special assemblies, fire drills, meetings with counselors, and conferences with parents. To keep my materials organized, I stored my weekly lesson plans in the back of the lesson-plan book. But extra space still remained. Then I discovered what, for me, became a very effective tool in classroom management.
Sharing Teaching Ideas: Finding Extrema Without Calculus
Authors: Touval Ayana
Published: December 2020

In my precalculus classes, questions about the points of extremum of a cubic usually arise. Students often think that the points are halfway between successive x-intercepts (when more than one is involved). For example, in figure 1 they think that the point of relative minimum is at x = 2 and that the point of relative maximum is at 0.5. Many interesting discussions ensue from this erroneous assumption. The following approach to find extremum moves from a study of the quadratic equation to cubics by using previously learned techniques. Calculus methods are not used.
Rates and Taxes
Authors: Esty Warren W.
Published: December 2020

Benjamin Franklin said, “Nothing is certain but death and taxes.” A mathematics class may not be the right place to discuss death, but it is a good place to discuss taxes. In the process, students can learn some important lessons about slopes, split functions, averages, rates, marginal rates, and percents (and taxes).
The Bug on the Box
Authors: Wallace William
Published: December 2020

Sometimes, in spite of my greatest efforts to be sparkling and fascinating, one of my mathematics classes gets bored. At such a time a change of pace may get things moving, and the “bug on the box” idea has worked beautifully for me on many occasions.
Decimals, Rounding, and Apportionment
Authors: Meeks Kay I.
Published: December 2020

With the end of each census year comes an opportunity to discuss current events and the relationship of mathematics to American history while reinforcing in general-mathematics classes the concepts of decimals, fractions, and rounding. This opportunity arises from the reapportioning of seats in the House of Representatives following each decade's census. In general, apportionment problems deal with the distribution of objects that cannot be subdivided into disjoint sets. Additional examples of such distributions may range from assigning seats in a student council to various classes to deciding on the basis of people's preferences how many pepperoni, mushroom, or deluxe pizzas to purchase for a party needing eleven pizzas.
Grading Cooperative Problem Solving
Authors: Lambdin Kroll Diana , Masingila Joanne O. , Tinstey Mau Sue
Published: December 2020

Do your students use cooperative-group work when they are involved in mathematical investigations and problem solving? Many teachers these days answer “yes” because they find that working together helps students become better problem solvers. Teachers who encourage their students to work problems cooperatively, however, frequently do not grade the results of those cooperative sessions, probably because using cooperative groups for classwork is a lot easier than assigning grades for that work.
Inequalities, Absolute Value, and Logical Connectives
Authors: Parish Charles R.
Published: December 2020

Algebra students often have difficulty in sorting out and correctly using the concepts involved in the solution of linear inequalities. The introduction of the concept of absolute value causes additional complications for many students. Two of the major problem areas are (1) the concept of absolute value as normally defined and (2) the recognition and appropriate use of the logical connectives and and or in the solution of problems that combine inequalities with absolute value. One way to alleviate students' confusion and promote their problem-solving ability in such a setting is to adopt an approach different from that found in current textbooks. This approach involves consideration of the function concept as the graph generated by associated ordered pairs of real numbers rather than functional values as obtained from the standard definition of absolute value.
Solution Revolution
Authors: Day Roger P.
Published: December 2020

In Consortium, the newsletter of the Consortium for Mathematics and its Applications, Froelich (1988) explored graphical strategies for solving the equation 2x = x10. He described his use of a function plotter as he sought a graphical solution for that equation. The NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) has proposed the increased use of function plotters and other technology to investigate and solve problems. Equations such as 2x = x10 can be used to illustrate the potential of that technology and to consider its implications for the teaching and learning of mathematics.
Multiple Solutions Involving Geoboard Problems
Authors: Smith Lyle R.
Published: December 2020

This article focuses on using various methods to determine perimeters and areas of polygons formed on the geoboard. Although the emphasis is on the use of various mathematical techniques, a possible by-product is that more mathematics teachers will use geoboards in their instruction. It should be noted that dot paper can be used instead of the geoboard. However, most teachers who have used both devices claim that their students prefer the geoboard when they are engaged in activities that involve the creation of figures; dot paper is usefuJ when students want to preserve particular figures for further analysis.
Can a Purchaser Save Money by Financing?
Authors: Cohen Donald
Published: December 2020

The prevalence of calculators in our mathematics classrooms makes it possible to use topics from the mathematics of finance to offer realistic appbcations of algebra to our students. The evaluation of complicated formulas from such topics as compound interest and present and future values of annuities (see Waits [1979] for the development of these formulas) is well within the reach of students studying a second course in algebra. Yet such applications will not be meaningful to students unless we take the time to present problems within the context of real-life situations. The following true story about my purchase of a new car offers a particularly appropriate contextual setting.
Tips For Beginners: Beyond The Curriculum
Authors: Arnold Connie B. , Reynolds Sallee H. , Stellern Cathy D. , Bohannon Diane , Harmon Katherine , Hill Leanne , Mamantov Cathy , Reed Suzanne H.
Published: December 2020

Teaching effectively is more than simple teaching your particular content area. Effective teaching requires hard work, communication, and dedication both inside and outside the classroom. It requires developing a familiarity with the school's procedures and resources, becoming acquainted with your colleagues, establishing a good working relationship with other mathematics teachers, and developing techniques for structuring your classroom that will minimize routine tasks and maximize your effectiveness. The end result is increased efficiency and effectiveness as a teacher that is well worth your initial time and effort.
The Farmer and the Goose—a Generalization
Authors: Gannon Gerald , Martelli Mario
Published: December 2020

Central to most problem-solving strategies is the desire to state and solve a problem in its most general form. Unfortunately, often the most general form of a problem is not easy to state. Moreover, the proof of the generalization is often beyond the abilities of many of our students. So like the idea of “synthesis” in test construction, generalizing often becomes that part of the problem-solving process that is admired from afar but seldom used.
Some Pleasures and Perils of Iteration
Authors: Cannon Lawrence O. , Elich Joe
Published: December 2020

In “A Lost Chord,” the author discovered a marvelous musical chord by accident while letting her fingers wander over the organ keyboard. The chord brought such peace to her soul that it pronlised solutions to all earthly problems. Unfortunately, she was never able to recreate the lost chord, despite years of searching, and the poem ends with the hope that perhaps in the eternities she may be able to discover and hear the chord again.
Conventional Cryptography
Authors: Wright Marie A.
Published: December 2020

Too many students find matrices to be abstract and without practical significance. The frustration expressed by these students is due, at least in part, to the insufficient number of practical examples used in the classroom. This article demonstrates a practical application of a linear transformation by means of conventional cryptography.
Gary O's Fence Question
Authors: Daniels David S.
Published: December 2020

Good modeling problems often occur when least expected. Take, for example, the other morning when I was checking into school at the sleepy hour of7:00 A.M. and our business teacher and baseball coach, Gary O'Sullivan, posed what he thought was Good modeling problems occur when least expected 252 an easy question for a mathematics teacher: “I want to put a fence around the outfield of our expansive baseball field to create a home-run challenge. I want it to be 325 feet down both the leftand right-field lines and 400 feet to straight-away center field. How much fencing do I need?” This question djdn't seem too difficult or involved at first, but, as we'll see, it's a wonderful modeling problem for mathematics classes at various levels.
From Algebra to Calculus—a Tonka® Toy Does the Trick
Authors: Borlaug Victoria A.
Published: December 2020

This classroom presentation is designed to introduce and interpret the graphical representation of a Tonka® toy truck's forward and backward motion. It can be used to illustrate an application of slope in an algebra class or to introduce the derivative in a calculus class. In the presentation, the teacher moves the Tonka® truck up and down on the chalkboard using chalk to record the motion, asks students questions about the motion, and encourages discussion. The class is asked to pretend that the toy truck has a speedometer. Unlike speedometers in real trucks, this toy speedometer has negative values to represent backward motion, in addmon to its usual positive and zero values. (Speed is the absolute value of velocity. In this situation, the speedometer might better be renamed “velocity-ometer,” but that term would introduce vocabulary unfamiliar to the student.) The presentation leads students to develop a graphical representation of the truck's one-dimensional motion, creates graphs representing constant velocity, leads students to a definition of average velocity, and introduces the concept of instantaneous velocity.
The Particle-Motion Problem
Authors: Demana Franklin , Waits Bert K.
Published: December 2020

Linear particle-motion problems constitute a rich and interesting source of real-world problems for the mathematics classroom. Furthermore, they can be simulated very nicely with computers using appropriate software or with modern graphing calculators that include built-in parametric-graphing utilities. Consider these applications that involve linear motion: (1) changing circular or curvilinear motion into linear motion and (2) linear particle accelerators in physics.
Young's Vision
Authors: Stein Sharyn L.
Published: December 2020

The boxed paragraph's objective can also be considered one of the fundamental concepts found in NCTM's Curriculum and Evaluation Standards for School Mathematics, published in 1989. Making the underlying assumption that a metamorphosis was drastically needed in our nation's mathematics curriculum, the standards document thoroughly examined and analyzed all aspects of the current methods by which mathematics is presented at all grade levels (K–12), as well as what is considered to be essential material that must be presented to students. (For an overview of the Curriculum and Evaluation Standards, see Thompson and Rathmell [1988].) However, the succinct statement below is not the product of the NCTM's standards; rather it is the wisdom of Jacob William Albe1t Young (1865–1948), a tum-of-thecentury pioneer in the field of educational pedagogy. (J. W. A. Young is not to be confused with John Wesley Young, 1879–1932, a professor of mathematics at Dartmouth College during the same era.)