bluuebunny/crossref_metadata_2025 huggingface dataset:
A New Approach to the Council's Yearbooks
Authors: Hatfield Larry L.
Published: November 2020

Members of the NCTM are certainly familiar with the excellent volume produced as the Council's yearbook. These yearbooks continue to serve as significant references throughout the varied facets of mathematical education. The dedicated leadership of the yearbook editors and the unselfish contributions of many author have assured the high quality we have come to expect as each of the thirty-seven yearbooks has been released. Long and meticulous planning and production have often resulted in a considerable time lag between the inception of a theme and the presentation to the Council by the editor at an annual meeting. At the same time, the costs of production have increased, driving up the price paid by members. Acting under its charge, the Publication Committee has kept a close watch on these developments.
Partitions of Positive Integers: An Elementary Topic of Number Theory
Authors: Lenz Gerald E.
Published: November 2020

The author presents some interesting aspects of number theory and graphing that should be appealing to students.
The Percent Game
Authors: Pascale Marie
Published: November 2020

Learning the equivalents of percents by playing a game.
Mathematics and Miseducation: Toward the Next School Curricula
Authors: McGarvey Craig
Published: November 2020

First from a student's perspective and then as a teacher, the author reflects on his experiences with the school mathematics curriculum and what it means for the 1980s.
A New Look, Pythagoras!
Authors: Thornton Carol A.
Published: November 2020

Let's enlarge Pythagoras's ideas into 3-space and see what happens.
The Mathematical Preparation for Engineering
Authors: Petroski Henry J.
Published: November 2020

Why is mathematics identified as a prerequisite to engineering? It is true that engineers deal with tangible things like concrete, machine parts, and steel beams and that a good working knowledge of basic mathematics is essential to manipulate these building blocks into dams, machines, and structures. But there is a great deal more to engineering than playing with blocks.
The Mathematics Teacher Shortage—Some Solutions
Authors: Paul Clyde
Published: November 2020

Evidence continues to accumulate that mathematics education in the United States is facing a serious peril caused by the increasing shortage of qualified teachers. Dunathan (1979) surveyed school superintendents in nine midwestern states about this topic. Approximately 70 percent of those administrators who responded thought a shortage of qualified mathematics teacher applicants existed. Less than 5 percent indicated that they believed there was a surplus. A 1977 survey conducted by the federal government discovered that approximately one-fourth of the nation's school districts had at least one opening in some field for which no qualified applicant could be found (Jacobson, 1979). Data from other sources suggest that many of those unstaffed positions were in mathematics classrooms. Reporting figures prepared by the National Center for Education Statistics, Dessart (1979b) states “… 1100 mathematics teacher positions were unfilled in the secondary schools of the United States during the fall of 1977.” The November 26 issue of Education USA (“Teacher Shortage,” 1979) quotes the Houston, Texas, superintendent of schools as estimating “… that more than 5,000 of the school district's students have 'no certified mathematics or science teacher at a time when we are emphasizing those subjects.' “That same article reports that Dallas had 150 current vacancies, most of which were in mathematics, science, industrial arts, and special education at the secondary level.
Creativity in General Mathematics
Authors: Zeddies Melvin L.
Published: November 2020

Our students may be using some ingenious, valid methods to solve problems.
Applications: Relative Velocity: Vectors with a Difference
Authors: Knill George
Published: November 2020

Applications involving the addition of vectors are usually easy to find. The following is an application of the subtraction of vectors, suggested by Peter Dumbrille, Commander, RCN, Halifax, Nova Scotia. First, let's review a typical problem involving the addition of vectors.
When Are Logarithms Used?
Authors: Kluepfel Charles
Published: November 2020

With the widespread use of pocket calculators, no one would use a table of logarithms to multiply or raise a number to a power. Of course, the calculator raises numbers to arbitrary powers by using logarithms internally, but why should students learn logarithms if they don't plan to design calculators? Certainly the presence of the log key on many calculators will do away entirely with the need for tables of logarithms. But when is the log key to be used?
“Metric” Paper
Authors: Kissane Barry V.
Published: November 2020

There are some good reasons why the dimensions of metric paper seem unusual.
Reporting College Placement Scores and Grades to High Schools
Authors: Thompson Richard B.
Published: November 2020

As teachers of mathematics, we all like to know how well our students perform after they leave our classrooms. Such information allows us to evaluate the effectiveness of our programs and to provide the most relevant instruction that is possible for our current students. Unfortunately, the only follow-up information that we often get about our students consists of vague complaints from the public or from teachers at the next level of instruction. In many cases, it is impossible to tell whether or not these reports are founded in fact, whether or not they apply to our own students.
Sharing Teaching Ideas
Authors: May Beverly A. , Hartman Janet , Mercer Gene B.
Published: November 2020

Many textbooks present decimal division using the concept of multiplying the dividend and the divisor by 10, 100, 1000, and so on to make the divisor a whole number. For my junior high students, I have developed an alternate method of decimal division emphasizing the concept that division is the opposite of multiplication. In the following discussion, decimal places refers to the number of digits to the right of the decimal.
Applications: Baseball and the Midway
Authors: Knill George
Published: November 2020

The September 1980 issue of Inside Sports contained an article entitled “What You Don't Know about Baseball and Never Dreamed You'd Find Out,” by Thomas Boswell. In the article, the author describes the Big Bang Theory of Killer In- nings. The theory states that “in a majority of games, the winning team scores more runs in one inning than the loser does in nine innings” (Boswell 1980, p. 106).
Baseball Monte Carlo Style
Authors: Houser Larry L.
Published: November 2020

Baseball announcers and sportswriters often indicate that a team is in a “hot streak” or that a hitter is experiencing a “batting slump.” They imply that unusually good or poor performance is responsible for the streak or slump with little or no suggestion that the streaks or slumps may be explainable by chance.
What Are the Chances of Your Students Knowing Probability?
Authors: Carpenter Thomas P. , Corbitt Mary Kay , Kepner Henry S. , Lindquist Mary Montgomery , Reys Robert E.
Published: November 2020

Current recommendations for the mathematics curriculum identify probability as a critical basic skill area (National Council of Supervisors of Mathematics 1978; National Council of Teachers of Mathematics 1980). The results of the second mathematics assessment of the National Assessment of Educational Progress (NAEP) suggest that this area has been neglected in mathematics instruction. In an effort to summarize the findings, we will discuss the probability exercises within three key concepts: probability of an event, probability of independent events, and probability of compound events. There were seven, three, and five exercises, respectively, within these key concepts, but only results from a few of them will be mentioned. These results reflect the performance of 13- and 17-year-olds. The sampling procedures included between twenty-two hundred and twenty-four hundred students at each age and provide data representative of both age groups nationwide.
Even Perfect Numbers—An Update
Authors: Bezuszka Stanley J.
Published: November 2020

Do you have students who are computer buffs, always looking for a new problem to program efficiently? Do you have students who do independent study projects? If so, motivate them with this topic that is rich in the history of mathematics and number theory—perfect numbers. They provide an excellent resource for theoretical as well as computerized problem solving.
Sex-related Differences in Mathematics: Results from National Assessment
Authors: Fennema Elizabeth , Carpenter Thomas P.
Published: November 2020

The second mathematics assessment of the National Assessment of Educational Progress provides new insight into the problems of sex-related differences in mathematics. Information about course taking and achievement in specific content areas and at different cognitive levels is available from a representative national sample of over 70 000 9-, 13-, and 17-year-olds. The purpose of this article is to report the sex-related differences that were found in this assessment and to explore the significance of these differences.
Minorities and Mathematics: Results from the National Assessment of Educational Progress
Authors: Anick Constance Martin , Carpenter Thomas P. , Smith Carol
Published: November 2020

A fundamental goal of schooling in the United States is to provide equity in education for all students. There are at least two ways that equity might be defined: (1) in terms of opportunities provided by schools or (2) in terms of student outcomes. From the second perspective, one can argue that if any population of students is achieving significantly below their peers in the nation as a whole or is participating in fewer advanced level mathematics courses, then equity in education is not being provided. This does not imply that schools are totally responsible for differences in achievement patterns of different population groups. There are many other factors within our social structure that affect students' performance in school. But it is a matter of utmost concern when large segments of the population score significantly below their peers on achievement tests and fail to develop the academic skills required to enter many occupations. The elimination of these deficiencies should be a primary goal of education.
Getting Started in a High School: A Case Study
Authors: Gesshel-Green Herb
Published: November 2020

Many teachers today are asking bow they can get started teaching students about the computer, how they can motivate the students, and what kinds of activities they can try. In this article, I will relate my personal experiences in these areas.
The Second National Assessment in Mathematics: Area and Volume
Authors: Hirstein James J.
Published: November 2020

Measurement concepts are important in school mathematics not only for providing applications of arithmetic content during the middle years but also for offering a common source of illustrations in algebra and geometry. Even more important, measurement concepts form the basis for all judgment with respect to the size of objects that we encounter daily. One of the topics included in the NAEP second mathematics assessment deals with the ability to use measurement concepts. The results of the items relating to area and volume suggest several common misconceptions about measurement among middle school and high school students.
Real-World Mathematics
Authors: Mitchell Charles E.
Published: November 2020

For the past three years I have participated in Title II projects that have focused on the needs of business and industry and the preparation of our high school graduates to meet these needs. In the course of these projects, in-service mathematics teachers from fourth through twelfth grade were placed in business and industry to study how mathematics is used to solve everyday problems. The settings ranged from the local hospital and real estate and insurance firms to such national corporations as Holiday Inn and E. F. Hutton. Government agencies, such as the local sheriff's office and the U.S. Space Institute, also participated. The idea of better preparing our graduates to meet the needs of business and industry finds ready support in the business world, and local newspapers were quick to publicize the cooperative efforts of those participating in the project.
Sharing Teaching Ideas: Mathematics Pictionary: The Sounds of Silence
Authors: Thiry Jean M.
Published: November 2020

“It's a bird! It's a plane! No, it's Hero's formula!” This kind of dialogue was heard in geometry class one day while students were playing “mathematics Pictionary,” a spin-off of the popular game Pictionary applicable to the mathematics classroom. The object of the game is to identify a mathematical term or concept from sketches. The activity leads to group cooperation, the understanding of mathematical terms, and a lot of fun for the class!
Soundoff: Nctm's Standards, Curriculum Reform, And High School Mathematics Teachers
Authors: Hadley William S.
Published: November 2020

With the release of NCTM's curriculum standards in the spring of 1989, emphasis on mathematics-curriculum reform at all levels K-12 has been renewed. Reform is obviously long overdue, but one major obstacle stands in the way of its achievement: the classroom teachers. Given t he proper opportunities, however, teachers can become the key to success.
Soundoff: How Can We Best Help ESL Students?
Authors: Kimball Mary Holt
Published: November 2020

Give me your tired, your poor, Your huddled masses yearning to breathe free … Send these, the homeless, tempest-tost, to me. I lift my lamp beside the golden door!
Implementing the Curriculum and Evaluation Standards: Making Connections by Using Molecular Models in Geometry
Authors: Pacyga Robert
Published: November 2020

As instructors of mathematics, we need to demonstrate to our students the use and value of connections between mathematics and other disciplines. The authors of the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) state that the mathematics curriculum should include investigations of the connections and interplay among various mathematical topics and their applications so that all students can apply mathematical thinking and modeling to solve problems that arise in other disciplines.
The Magic of Mathematics
Authors: Morgan John L. , Ginther John L.
Published: November 2020

As a mathematics teacher, have you ever dreamed of your students asking, “How did you do that?” after you demonstrated some mathematical procedure? Probably.
Check-Digit Schemes
Authors: Wheeler Mary L.
Published: November 2020

Identification codes have been used for many years. We've seen a large increase in the use of scanners and computers in the last two decades. As a result, the need for identification codes increased tremendously. These codes are an assignment of numbers to an item for identification of information about that item. They are very useful with computers. information can be stored in a condensed format, and with the use of algebraic algorithms the accuracy of that information can be checked by using what is called a check-digit scheme. Identification codes are also very useful for secwity information. The use of a simple or complicated mathematical algorithm allows the information to be disguised.
The Sidesplitting Story of the Midpoint Polygon
Authors: Gau Y. David , Tartre Lindsay A.
Published: November 2020

Understanding what area means and learning ways to calculate the area of various figures are important objectives in geometry. Students as adults will use concepts related to finding areas of polygons in many contexts, such as finding the area of their backyard or knowing how much wallpaper is needed to cover a wall in their dining room. One context for exploring area relationships is comparing the area of a polygon to the area of its associated midpoint polygon, formed by joining the midpoints of consecutive sides of the original polygon. This article describes activities that examine the patterns and relationships between the areas of polygons and those of their associated midpoint polygons for triangles, quadrilaterals, pentagons, and other polygons. We shall also look at the pattern for regular polygons.
Seventy-Fifth-Anniversary Retrospective: What Does “Everyman” Really Need from School Mathematics?
Authors: Bell Max S.
Published: November 2020

Charting new directions for mathematical literacy and competency for our future citizens. A “must” reading for every mathematics teacher and curriculum developer.
Application: More on Apportionment
Authors: Flusser Peter
Published: November 2020

Kay Meeks (1992) describes how various American statesman first proposed that saets in the house of Representatives be apportioned to states. This proposal brings up two question that students might wish to explore: “Are any of these methods fair?” and “Which, if any, is the fairest?”
Technology Tips: Dynamic Geometry Environments: What's the Point?
Authors: Hoyles Celia , Noss Richard
Published: November 2020

The September and October columns of this department described features of dynamic geometry environments. This month's column is concerned with the distinction between drawingand constructing in these environments, a theme that will be continued in later issues. Hoyles and Noss have devised a simple way, “messing up,” to make this distinction clearer for students.
Demystifying Polar Graphing
Authors: Germain-McCarthy Yvelyne
Published: November 2020

If we are to follow NCTM's Curriculum and Evaluation Standards (1989), teachers should connect various topics of mathematics so that students see the subject as a unified body of knowledge. This article discusses a method of graphing polar equations using information from the Cartesian graphs of trigonometric functions. This method will help students see relationships between the two types of graphs.
Conditional Probability: Its Place in the Mathematics Curriculum
Authors: Watson Jane M.
Published: November 2020

The increased attention given to probability and statistics in the NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) and other national curriculum statements around the world (e.g., Department of Education and Science [DES] 1991; Australian Education Council [AEC] 1991) has been applauded by most mathematics educators and many teachers. When they leave school, students need to make decisions when presented with claims about data. The fact that these challenges will arise is illustrated every day in reports in newspapers, magazines, and television news broadcasts.
Building Mathematical Models of Simple Harmonic and Damped Motion
Authors: Edwards Thomas
Published: November 2020

Given the recent public mania over bungee jumping, stimulating students' interest in a model of that situation should be an easy “leap.” Students should investigate the connections among various mathematical representations and their relationships to applications in the real world, asserts the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989). Mathematical modeling of real-world problems can make such connections more natural for students, the standards document further indicates. Moreover, explorations of periodic real-world phenomena by all students, as well as the modeling of such phenomena by college-intending students, is called for by Standard 9: Trigonometry.
Concept Worksheet: An Important Tool for Learning
Authors: Toumasis Charalampos
Published: November 2020

A Student's ability to learn mathematics is directly related to his or her understanding of mathematical concepts and principles (Cooney 1977). Experienced teachers know that concepts are the basic building blocks of thinking, particularly higher level thinking in mathematics. This article explores the concept worksheet, which can help students identify and analyze important mathematical concepts and is a simple assignment that can be completed by students overnight. The following four types of exercises are part of the worksheet.
Increasing Mathematics Confidence by Using Worked Examples
Authors: Carroll William M.
Published: November 2020

An important way to help our students learn and strengthen their knowledge of mathematics is by studying already-worked examples of problems. A considerable amount of evidence suggests that analyzing worked-out problems can be as conducive to learning as actually solving practice problems and that it may be more effective in helping students to recognize underlying similarities between problems (Sweller 1989; Zhu and Simon 1987; Sweller and Cooper 1985). Well-constructed worked examples do more than merely teach rote procedures; they illustrate mathematical principles and classes of problem situations
Graphing, Bit by Bit
Authors: Cave Richard C.
Published: November 2020

One topic area where technology can have an immediate impact in the mathematics classroom is graphing. Because of a variety of readily available software packages, graphs have become a universal method of representing mathematical relationships. From the charts in USA Today to the many diagrams of Ross Perot, the general public sees more graphs than ever before, and they need to have a better understanding of what graphs really represent. In the past, a graph could only be created by hand; therefore, most curricula emphasized the actual graphing of equations. With the help of technology, teachers can now concentrate on teaching students how to investigate what the graphs represent as well as how to interpret the graphs.
Connecting Research to Teaching: Mathematics as Reasoning
Authors: Galbraith Peter
Published: November 2020

The Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) defines a role for reasoning in school mathematics that is far different from the norm of recent practice. Until recently, the study of mathematical reasoning was largely confined to high school geometry. Further, as Schoenfeld (1988) pointed out, the approach used in geometry was often so rigid that it conveyed the impression that the style of the response—for example, the two-column-proof format—was more important than its mathematical quality. The Standards document notes that reasoning is to have a role in all of mathematics from the earliest grades on up and that the form of justification need not follow a pre scribed format. Indeed, students are encouraged to explain their reasoning in their own words. Teachers are asked to present opportunities for students to refine their own thoughts and language by sharing ideas with their peers and the teacher.
Implementing the “Assessment Standards for School Mathematics”: An Open-and-Shut Case? Openness in the Assessment Process
Authors: Lambdin Diana V.
Published: November 2020

Open assessment involves shared responsibilities by students, teachers, and the public. It contributes to a collective understanding of high performance criteria for mathematics.… Everyone is best served by an assessment process that is public, participatory, and dynamic.
Precalculus Explorations of Function Composition with a Graphing Calculator
Authors: Lum Lewis
Published: November 2020

In a 1993 Mathematics Teacher article, Will Hansen described a geometric visualization of iterative function composition and used it, with the aid of a graphing calculator, in student-centered explorations of convergence. Near the end of the article, he hinted at composing two different functions an idea that has been successfully employed in investigations of composition functions in general (Lum 1993).
Sharing Teaching Ideas: Family Algebra Night
Authors: Perlstein Ruth
Published: November 2020

At the end of October, all the first-year-algebra teachers at our school invited students experiencing difficulty to an evening workshop with their parents. Our team of algebra teachers and a counselor developed the program to address the concerns and frustrations of students having problems in this course. Since this program focuses on sharing with families strategies for student success in mathematics, it can be applied to various situations and courses.
Sharing Teaching Ideas: Projects in Precalculus
Authors: Basinait Linda M.
Published: November 2020

For the past few years, I toyed with the idea of having my precalculus students do a project as the last unit in the course. But each year I never got around to writing the guidelines or suggesting the topics. Last fall, I announced to my two precalculus classes that in the spring they would be completing a project based on something they learned during the year. The same announcement was made at Parents' Night later in the month. Perhaps I believed that if I kept announcing a project, I would have to follow through with it.
Exploring Parametric Transformations of Functions
Authors: Embse Charles Vonder
Published: November 2020

NCTM's Curriculum and Evaluation Standards for School Mathematics (1989, 154) identifies the study of functions as a central idea in the secondary school curriculum. In particular, the Standards document recommends that [i]n grades 9–12, the mathematics curriculum should include the continued study of functions so that all students can … analyze the effects of parameter changes on the graphs of functions; and so that, in addition, college-intending students can understand operations on, and the general properties and behavior of, classes of functions.
Technology Tips: Spreadsheets: Helpful for Understanding Mathematical Structures
Authors: Neuwirth Erich
Published: November 2020

Since the origins of spreadsheets are connected to accounting, many people consider them a utilitarian but uninteresting tool, useful for performing rote calculations only. But spreadsheets can also be tools for understanding mathematics better.
Perimeters, Patterns, and Pi
Authors: Barnes Sue
Published: November 2020

Students taking geometry in the ninth or tenth grades do not always have their “but why?” mathematical questions answered from previous grades.
Cutting a Chain—a Problem Made to Be Generalized
Authors: Gannon Gerald , Martelli Mario
Published: November 2020

Long before the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) called for an emphasis on problem solving and on the subsequent generalization of the problem whenever possible, good teachers were on the lookout for problems with these characteristics: (1) the problem was entertaining and reasonable for a wide range of students to explore and (2) forming and verifying a generalization would be within the reach of most students. Many teachers found that criterion (2) was usually harder to satisfy.
Implementing the Professional Standards for Teaching Mathematics: Analyzing Teaching and Learning: The Art of Listening
Authors: Arvold Bridget , Turner Pamela , Cooney Thomas J.
Published: November 2020

The visions of teaching set forth in the Professional Standards for Teaching Mathematics (NCTM 1991) do not, in themselves, ensure that we can successfully teach all students mathematics. To reach all students, we must understand how students think and then develop instruction compatible with their thinking. To begin to understand, we must observe, listen, and gather a variety of evidence of what and how students are learning. Although we might view a mathematical concept or algorithm as simple, it is “a mysterious, almost inexplicable phenomenon from the point of view of the outsider” (Davis and Hersh 1981, 43). Examining students' thinking through their interactions with mathematical tasks can help unlock a bit of the mystery. This process of analysis is the amalgamating item in the “Standards for Teaching Mathematics” section of the Professional Standards for Teaching Mathematics and emphasizes the need to listen carefully to our students.
Do Baseball Positions Correspond with a Player's Race?
Authors: Hildreth David J.
Published: November 2020

Recently, applications of mathematics have received more attention in school mathematics. The NCTM's Curriculum and Evaluation Standards (1989), the Mathematics Teacher, and many textbooks reflect this much-needed emphasis. Many of these applications, some old and many new, reflect the need to apply mathematics to science and technology.
Projects
Authors: Gay Susan , Stenglein Sharon
Published: November 2020

Graphing calculators are still not widely used in some classrooms, and classroom sets of graphing calculators are still far from common in many schools. As everyone who has taught a class of students working with graphing calculators can testify, this powerful technology is a significant aid to students' learning. One alternative to the classroom set has been the overhead graphing calculator used for demonstrations. With support for training and donations of overhead graphing calculators. the Oklahoma Graphing Calculator Project, over several phases, has made these calculators available in the classrooms in Oklahoma schools.
Winning at Quarto!
Authors: Kulig Christopher J. , Carm O.
Published: November 2020

I was walking through Union Station shopping mall in downtown Washmgton, D.C., taking a break from my theological studies, when I stumbled aeross a new game in the Brookstone store. The game, Quarto! by Gi Gamic, immediately grabbed the attention of the mathematician in me. The game is played on a four-by-four grid with sixteen game pieces. Each piece has four different characteristics: tall or short, light or dark, round or square, and solid or hollow. The game is played by two players, who take turns placing pieces on the game board. The goal is to be the first player to create a row, column, or main diagonal with four pieces that share a single characteristic, that is, with four short pieces, four round pieces, and so on. The interesting rub to the game is that each player places a piece that is received from his or her opponent!
Representing Probabilities with Pipe Diagrams
Authors: Konold Clifford
Published: November 2020

Probability is a notoriously difficult concept. Even after much instruction, many students remain confused both about methods used to calculate a probability and about its meaning (Konold 1991). In this article, I shall describe a modified version of the tree diagram that many of my students at both the high school and college levels have found helpful in making probabilities more meaningful. I refer to these representations as pipe diagrams. Although most readers are probably familiar with tree diagrams, I review a few of their basic features before introducing pipe diagrams and discussing their educational advantages.
Starting a Euclid Club
Authors: Brodkey Jeremiah Joseph
Published: November 2020

A wonderful summer educational experience at Saint John's College in Santa Fe, New Mexico, directly led to my initiating a most unusual Euclid Club at our high school.
Reflections: A Lesson in Related Rates
Authors: Nichols Joe D.
Published: November 2020

My objective in teaching an Advanced Placement calculus course is not only to help students pass the AP examination each spring but also to help them develop insights into advanced problem solving and real-world applications. I continually search for examples in my students' daily environment that can help them make a tangible connection from the classroom textbook to the real world. In this article, I discuss a general application of a basic concept with which all first-year-calculus students must contend: the problem of related rates.
Architecture and Mathematics
Authors: Reif Dan
Published: November 2020

It's got to have a center staircase!” “I've always wanted a kitchen with an island!” “Where is the best place for a skylight?” Enthusiastic comments such as these were typical of those I overheard while conducting a two-week workshop in which students used three-dimensional-model-building kits to plan and construct detailed scale models of their own home designs and to develop and sharpen mathematics skills related to architecture.
Media Clips
Authors: Lancaster Ron , Marion Charlie
Published: November 2020

We welcome and encourage submissions to this department, which will comprise resources that teachers can use immediately or file away for future use. The assignments generated by these entries may range from starters for an entire class to full-blown projects for optional individual pursuit. Each lesson, though, should help convince our students that mathematics is useful, current, and newsworthy. Please send submissions to Media Clips, Mathematics Teacher, 1906 Association Drive, Reston, VA 22091-1593.
Student Self-Assessment and Self-Evaluation
Authors: Stallings Virginia , Tascione Carol
Published: November 2020

Current recommendations for alternative assessment call for more student involvement in the evaluation process. Although very little research on student self-evaluation has been carried out beyond determining how well students' perceptions of their achievement align with their teacher's perceptions, the Assessment Standards for School Mathematics (NCTM 1995) supports activities that involve students in evaluating their own progress. According to the Assessment Standards, student self-assessment can be used to improve students' confidence in their ability to do mathematics and allow them to become more independent in their learning of mathematics. We have employed student self-assessment and self-evaluation in our high school and college mathematics classes and have found that the processes engage students in evaluating their progress, aid in developing their communication skills, and increase their mathematics vocabulary. Most important, students reflect on their understanding of mathematics and on their ability to learn mathematics. Although our procedures differ somewhat, as we describe subsequently, our goal of involving students in the evaluation process is the same.
Geometry, Iteration, and Finance
Authors: Landy Godbold A.
Published: November 2020

The mathematics offinance has long been a staple of the algebra-precalculus-calculus curriculum continuum. Whether in the study of geometnc sequences and series, logarithms and exponential functions, or limits and derivatives, topics from finance have presented interesting settings in which to apply the mathematics being studied. However, in all but the most advanced classes, such applications usually begin with the familiar assumption associated with exponential behavior—no deposits or withdrawals are permitted to or from any account. This article develops a view of linear functions that permits insight into the processes behind realistic financial applications.
Mathematical Structures: Answering the “Why” Questions
Authors: Bush Doug , Jones William S.
Published: November 2020

Understanding the answers to “why” questions is an important part of secondary school mathematics. Over the past few years, we have taught naturally curious high school and college students who have asked these questions as they learned mathematics.
Activities: Cryptography: Cracking Codes
Authors: Myerscough Don , Ploger Don , McCarthy Lynn , Hopper Hallie , Fegers Vicki G.
Published: November 2020

Cryptography is the art and science of encoding and decoding messages. It plays an important role in business as a means to protect corporate secrets, in banking to secure accounts, in the military to protect information, and in personal security devices to protect individuals. (See Kahn [1967] for a review of cryptography.) In this activity we introduce cryptography into the high school classroom as a context for doing mathematics.
Editorial
Authors: Smith Eugene R. , Gaylord Harry D. , Chambers Geo. Gailey , Breckenridge William E.
Published: November 2020

The use of the slide rule in mathematics classes is rapidly increasing. One school of a technical nature reports that every student in the first term is taught the use of the rule.
The Content of a Mathematical Course for the Junior High School
Authors: Gentleman F. W.
Published: November 2020

In view of the fact that junior high schools are actually being established in different parts of New England, it becomes the duty of this association to consider what shall be the nature of the work in mathematics for the course.
Implementing the “Assessment Standards for School Mathematics”: Some Practical Possibilities for Alternative Assessment
Authors: Schloemer Cathy G.
Published: November 2020

Imagine that you are a teacher in a secondary mathematics classroom, working hard to implement the precepts of the Standards documents {National Council of Teachers of Mathematics 1989, 1991, 1995). In particular, you are focusing on getting your students to (1) make connections between mathematics and the real world, (2) reason and communicate mathematically, and (3) value mathematics. A colleague with whom you often share classroom concerns and successes stops by during a rare quiet moment and, in the course of conversation about your most recent classroom endeavors, asks, “How is it going?” How might you reply?
How One Physics Teacher Changed his Algebraic Thinking
Authors: Nzcol Marsha P.
Published: November 2020

By his own admission, Mike Smith, Worthington Christian High School's physics teacher, has dramatically changed his concept of mathematics. Although I was the researcher in this study, the story that follows is Mike's story. Because Mike is a professional colleague, I want him to be recognized and credited for his contributions to educational reform (see Shulman [1990]). I prefer to tell his story with full recognition of who he is and where be works.
Ideas about Symbolism That Students Bring to Algebra
Authors: Stacey Kaye , MacGregor Mollie
Published: November 2020

eginning to learn algebra should be easy. To get started, students need to know that letters of the alphabet are used to stand for numbers and that “answers” often have operation signs in them. If we add 5 to x, we get x + 5; if we take away 5 from x, we get x − 5. So far so good. Algebra may be puzzling, but it is easy; we do not even have to “work ou” the answers. Then comes a stumbling block. The teacher says that 5x means 5 multiplied by x, whereas students might think that it signifies addition, like 5 1/2, or place value, like 53.
Multiple Representations—Using Different Perspectives to Form a Clearer Picture
Authors: Piez Cynthia M. , Voxman Mary H.
Published: November 2020

The concept of function is central to the study of precalculus and calculus. As educators explore ways to improve and invigorate the teaching of mathematics at the secondary and undergraduate levels, and as technology becomes more common in the classroom, the use of multiple representations to study functions and other concepts is gaining more attention in the curriculum. Because each representation emphasizes and suppresses various aspects of a concept, we believe that students gain a more thorough understanding of a function if it is explored using numerical, graphical, and analytical methods.
Reader Reflections
Authors: Petrella Gerald
Published: November 2020

What is your opinion of giving students the answers to numerical homework exercises so that they can see immecliately whether they did them right? Many textbooks have answers to the oddnumbered exercises in the back so that students can confirm that they did the exercises correctly. If they did get them right, they get unmediate positive feedback. That is good. If they get an exerctse wrong, they get immediate feedback that tells them to try again, lookmg for errors in their calculations or method. That is also good. Either way, the immedtacy of the feedback is pedagogically much better than finding out a day or more later that some work they did was wrong. Often students are not required to redo incorrect work, and they may already have internalized whatever erroneous process they used the first time. Wtthholding correct answers for a day cannot be good for learning.
Projects
Authors: Stenglein Sharon
Published: November 2020

ln this National Science Foundation (NFS) Teacher Enhancement Project, fifty Minnesota middle school and high school mathematics teachers are collaborating with three Saint Olaf CoUege mathematics professors to integrate inquiry-based geometry and visualization across their secondary mathematics curricula.
Sharing Teaching Ideas: A Novel Way to Practice Slope
Authors: Kennedy Jane B.
Published: November 2020

My students enjoy using a tic-tac-toe format to practice finding the slope and identifying parallel and perpendicular lines from various equation formats. After students learn different definitions and methods for identifying slope, I use this approach in a homework sheet or as a timed quiz. The tic-tac-toe format seems to make the drill more palatable; in a timed quiz, it quickly identifies students who have mastered rapid-identification techniques. I have also used this format successfully as a review in both precalculus and calculus classes before students work with applications of analytic geometry.
Media Clips
Authors: Wood Dorothy
Published: November 2020

We welcome and encourage submissions to this department, which will comprise resources that teachers can use immediately or file away for future use. The assignments generated by these entries may range from starters for an entire class to full-blown projects for optional individual pursuit. Each lesson, though, should help convince our students that mathematics is useful, current, and newsworthy. Please send submissions to Media Clips, Mathematics Teacher, 1906 Association Drive, Reston, VA 20191-1593.
Implementing the Assessment Standards for School Mathematics: Teachers and Students Learning Together about Assessing Problem Solving
Authors: Petit Marge , Zawojewski Judith S.
Published: November 2020

Enhancing problem-solving capabilities for all students is a goal that can be reached only when curriculum, instruction, and assessment are aligned toward the same goals, along with student and teacher understanding of the expectations associated with reaching these goals (NCTM 1995). The assessment of problem solving in particular is a difficult aspect, even when teachers have developed adequate resources and teaching methods to facilitate problem-solving activities. Pertinent questions regarding problem-solving assessment include the following: How can students' work be evaluated fairly when many reasonable solution paths exist? How can teachers assess students' responses to open-ended questions in a way that yields documentation useful for preparing progress reports? How can teachers help students understand what is expected in their written responses?
Problem Solving Does Not Have to Be a Problem
Authors: Barb Cynthia , Quinn Anne Larson
Published: November 2020

What role does problem solving play in the mathematics classroom today? What methods should be used to help students become better problem solvers?
The Algebra Launching PAD
Authors: Brunner Ann , Sheehan Sharon
Published: November 2020

How can technology bring substantive algebra content to students traditionally excluded from algebra? Such “low achieving” students do not lack ability, motivation, or dedication when presented with appropriate and challenging material. Their talents can be enhanced with new content and instruction in all mathematics courses through the exploitation of computer and calculator technologies. A decade ago, we were challenged to use technology to “move our subject away from the routine and toward the content and processes that will give the class of 2000 originality, insight, judgment, initiative, and understanding” (McConnell 1988). Our use of technology transforms our sequence of courses for lowachieving students so that the “modified” algebra they get is no longer weakened and diluted but rather is modified for power, applications, and depth (PAD).
Extraneous Solutions and Graphing Calculators
Authors: Goolsby Ronnie C. , Polaski Thomas W.
Published: November 2020

The widespread use of graphing calculators in both secondary and postsecondary mathematics allows students to investigate the standard topics of classic algebra in new and meaningful ways. The reasons behind certain algebraic techniques are sometimes difficult for average students to fathom; technology can help them visualize a result and thus understand and use the algebraic technique. Graphing can also be a source of qualitative information about the nature of the solutions to algebraic equations, and this information can help students check their solutions. As an example of using graphing to understand algebra, we investigate the solution of equations of the form ax + b = cx + d
A Star to Guide Us
Authors: Norwood Rick
Published: November 2020

Perhaps it is time to retire both the times sign and the multiplication dot and to use the star for multiplication from the very beginning.
Geometry Problems Promoting Reasoning and Understanding
Authors: Manaster Alfred B. , Schlesinger Beth M.
Published: November 2020

The four related problems presented in this article provide examples of ways to include justifications of interesting mathematics in courses taught before a geometry course. This understanding requires that the students follow chains of reasoning that furnish convincing justifications of the correctness of the general results. The reasoning involves both algebra and geometry, but all the problems can be done before the student takes a formal geometry course.
Discuss with Your Colleagues: Now Here Is That Authority on Mathematics Reform, Dr. Constructivist!
Authors: Mikusa Michael G. , Lewellen Hester
Published: November 2020

Dr. Constructivist is a character who was “born” while we were trying to develop ways of teaching what constructivism is and ways to address the obstacles that teachers might have to overcome in adapting a mathematics-reform and constructivist approach in the classroom. We were both working with in-service and preservice teachers at all levels and found that the ideas of constructivism were still new to them.
Preparing Students for Modern Work: Lessons from Automobile Manufacturing
Authors: Smith John P.
Published: November 2020

Take the mathematics that you find back to school. With a bit of adaptation, mathematical work can generate real–life problems and projects for your students.
Algebra for All and Discuss with Your Colleagues: Algebra as Generalized Arithmetic: Starting With the Known for a Change
Authors: Philipp Randolph A. , Schappelle Bonnie P.
Published: November 2020

Examples in which we relate the syntactic—including symbol manipulation—and semantic—including meaningful use of symbols—aspects of algebra and examine algebra as generalized arithmetic.
Discouraging Cheating in Your Classroom
Authors: Murdock Tamera B.
Published: November 2020

Cheating. How prevalent is cheating in your classroom? And more important, what can you do to discourage it?
Smokey the Bear Takes Algebra
Authors: Goetz Albert
Published: November 2020

IN THE LATE SPRING AND EARLY SUMMER OF 1998, fires in Florida made headlines across the country when they raged uncontrolled over a vast area, threatening people, livestock, and property, not to mention the timber, valued at millions of dollars, that was destroyed. Controlling fires has become a significant environmental issue not just in the United States but in Central and South America as well. In particular, Brazil suffered tremendous damage during the 1998 “fire
Exploring Hyperbolic Geometry with The Geometer's Sketchpad
Authors: Dwyer Marlene C. , Pfiefer Richard E.
Published: November 2020

This article explores some principles of, and shows several constructions in, hyperbolic geometry using script tools originally created by Mike Alexander and modified by Bill Finzer and Nick Jackiw for The Geometer's Sketchpad (Jackiw 1995). These tools are compatible with The Geometer's Sketchpad 3.0 and 3.05 for both Macintosh and Windows users.
The Role of Technology in Introductory Statistics Classes
Authors: Bratton George N.
Published: November 2020

Giving today's students statistically capable technology, that is, computer software or a statistics-enhanced graphing calculator, can allow courses to reflect statistical practice. In such courses, students work individually or in small groups, plan and perform data collection, and rely on technology to perform the mathematics of data analysis. Such an approach allows emphasis on the conceptual basis of probability and data analysis.