bluuebunny/crossref_metadata_2025 huggingface dataset:
Implementing the “Assessment Standards for School Mathematics”: Using Students' Portfolios to Assess Mathematical Understanding
Authors: Asturias Harold
Published: November 2020

In 1989, NCTM published the Curriculum and Evaluation Standards for School Mathematics, which presented the mathematics profession with a broad view of the important mathematics that should be taught in schools. Two years later, the Professional Standards for Teaching Mathematics gave teachers the opportunity to address the pedagogical issues inherent in teaching a broad-based, thinking curriculum as described in the curriculum standards. The next link, assessment, though part of the first document, required specific attention. Assessment Standards for School Mathematics, currently in progress, will present the criteria for judging the appropriateness and quality of assessment tools and systems.
Soundoff: Make Room for Dancing
Authors: Askins Larry E.
Published: November 2020

As mathematics teachers, we are eager for an optimistic view of what our classrooms can become during this decade and beyond. I believe that NCTM's Professional Standards for Teaching Mathematics (1991) and Curriculum and Evaluation Standards for School Mathematics (1989) present a clear vision for making mathematics education successful in the 1990s. However, the documents mean nothing if individual teachers fail to take deliberate steps toward realizing that vision.
Circular Graphs: Vehicles for Conic and Polar Connections
Authors: Germain-McCarthy Yvelyne
Published: November 2020

A unified treatment of conic sections and polar equations of conics can be found in most calculus books where the reciprocals of limafçons are shown to be conic sections. The treatment, however, is from an algebraic standpoint and does not refer to the inherent connection between polar graphs and the graphs of trigonometric functions and conics. Beginning with information gained from the graphs of circular functions of the form y = A + B sin x, students can be guided to graph conic sections on the polar plane without using a table of values. This approach helps students to appreciate the roles that both algebra and coordinate geometry play in weaving various sections of mathematics into a meaningful whole.
Sharing Teaching Ideas: A Unique Slope For A Parabola
Authors: Amick H. Louise
Published: November 2020

While exploring the sketching of parabolas in the form y = a(x − h)2 + k in a precalculus class, the discussion turned to the effect of a on the appearance of the curve. We observed that for | a | > 1, the parabola stretched away from the x-axis, making its sides steeper, whereas for | a | < 1, the parabola was seen to contract toward the x-axis, making its sides less steep. A student then asked, “Is a similar to m in the equation y = mx + b, so that it could be called the slope of the parabola?”
Discrete Mathematics And Historical Analysis: A Study Of Magellan
Authors: Perham Arnold E. , Perham Bernadette H.
Published: November 2020

Historical narratives often involve the lives of kings and queens, periods of exploration, or political movements. In these narratives, identifying the strategic elements and analyzing their interrelationships are interesting problems. Topics in discrete mathematics used to model conflict, like graph theory and game theory, could be helpful modeling tools of history.
Applications: An Example of Algebra in Lake Roosevelt
Authors: Nord Gail D. , Nord John
Published: November 2020

The NCTM's Standards documents (1989, 1991) encourage teachers to give students problems drawn from real-world contexts. This article demonstrates the mathematical nature of a marine environment-pollution problem. Examples discussed could be used to motivate students in a beginning-or intermediate-algebra classroom.
Implementing the Assessment Standards for School Mathematics: The Mathematics Test: A New Role for an Old Friend
Authors: Manon John Rahn
Published: November 2020

Most of us have, for many years, relied on timed teacher-designed mathematics tests to evaluate our students' progress in mathematics. We have often assessed the efficacy of our own instruction by the level of students' achievement on these same tests. Indeed, tests have come to bear the lion's share of the burden in determining whether we and our students have been successful in our respective roles of teaching and learning mathematics. Perhaps we have placed far too great an emphasis on our old friend, the mathematics test.
A Geometric Approach to the Discriminant
Authors: Daniel Hurwitz R.
Published: November 2020

An important connection between mathematical ideas to which the NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) repeatedly refers is using graphs to help understand algebraic concepts. For many students, the ability to “draw a picture,” either by hand or with the aid of a graphing utility, is valuable. For this reason, the second course of many integrated mathematics sequences places chapters on coordinate geometry, and perhaps even transformation geometry, before the chapter on quadratic equations (for example, see Keenan and Dressler [1981, ch. 12- 15] or Bumby and Klutch [1979, ch. 8-12].
Network Neighbors
Authors: Finzer William F.
Published: November 2020

Imagine that as you work on a problem, another person's attempts to solve the same problem are visible to you, changing in real time. You might both be in the same room, or across a continent. When you get stuck, you look at her work for ideas. Likewise, she tracks your progress. Your coll aboration may become explicit as you ask each other questions about how you did things, or it may remain in the background as a glance gives you a gentle hint about a possible direction for solution.
Nurturing Mathematical Talent in New Zealand
Authors: Holton Derek
Published: November 2020

During the last seven years, a program has been developing to assist high school students in New Zealand who have an aptitude for mathematics. It is the only New Zealand program in any subject that aims to help all talented students, no matter where they live or what school they attend.
Women in Consortium Calculus
Authors: Miller Nancy C.
Published: November 2020

For many women, traditional calculus courses have usually not been encouraging places.
Editorial: Welcome to Our Sampler: Emerging Programs for the First Two Years of High School
Authors: Rubenstein Rheta N.
Published: November 2020

In the late 1980s we challenged ourselves as a profession to meet the goals described in the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989). We envisioned programs that would “encourage and enable students to value mathematics, gain confidence in their own mathematical ability, become mathematical problem solvers, communicate mathematically, and reason mathematically” (NCTM 1989, 123). In particular, for secondary school students we sought to present a core curriculum consisting of a common body of mathematical ideas accessible to all students. These visions were further detailed in the specific content standards, in the Professional Standards for Teaching Mathematics (NCTM 1991), in an assortment of Addenda books, and, this past spring, in the Assessment Standards for School Mathematics (NCTM 1995). For many people, however, among them teachers, parents, students, and school administrators, these Standards documents were merely visions, perhaps even pipe dreams.
Technology Tips: Spreadsheets—a Tool for the Mathematics Classroom
Authors: Hunt William J.
Published: November 2020

After teaching Newton’s method for approximating roots to a precalculus class as a paper-and-pencil and calculator investigation, I was faced with the problem of evaluating my students’ work. I knew that in many instances they would have made arithmetic errors and that often such errors would be hard to spot. A spreadsheet was an ideal tool for checking their work. At first, I would program the spreadsheet to model Newton’s method, enter a student’s first guess, and check the resulting approximations. I could then proudly circle an error, and the students would be impressed at my diligence in tracing through their arithmetic.
The Case of Trapezoidal Numbers
Authors: Feinberg-McBrian Carol
Published: November 2020

Several sets of numbers are associated with geometric shapes. Most teachers are familiar with the square numbers, the triangular numbers, the cubic numbers, and the rectangular numbers. (See fig. 1.) To this list of sets I propose adding the trapezoidal numbers.
Technology Tips: Teaching Statistics with Computer Networks
Authors: Sanders Mark
Published: November 2020

Because computers are so often used for statistical analysis in real-world situations, they should be highly integrated into statistics classrooms. Computers can be used not only to analyze data but, with the availability of computer networks, to collect data as well. Networks can also assist students in communicating with people who are far away, thus helping to interest students in the statistics. This article chronicles an example of using the Internet, the world's largest network, to teach high school statistics.
Exploring Quadratic Functions: From a to c
Authors: Edwards Thomas G.
Published: November 2020

A variety of instructional methods should be used in classrooms in order to cultivate students' abilities to investigate, to make sense of, and to construct meanings from new situations; to make and provide arguments for conjectures; and to use a flexible set of strategies to solve problems,” according to the NCTM's Curriculum and Evaluation Standards for School Mathematics (1989, 125). This article examines a way to recycle and enrich an old idea—exploring the effects of varying the coefficients in the general quadratic function—by using graphing calculators.
Stimulating Mathematical Interest with Dynamical Systems
Authors: Durkin Marilyn B.
Published: November 2020

Suppose that you have just been hired by the State Fish and Game Department. The management is considering opening a remote lake for fishing and wants to know if preliminary stocking should be done.
Implementing the Assessment Standards for School Mathematics: Informal Assessment: A Story from the Classroom
Authors: Vincent Mary Lynn
Published: November 2020

In fall 1994, I agreed to collaborate with Linda Wilson of the University of Delaware in a study of assessment in an algebra classroom. We were particularly interested in the types of strategies a teacher could use to document observations of students' work.
Solution or Bused
Authors: Robertson James W.
Published: November 2020

For my early morning bus duty, I had to determine whether all the buses had arrived in a timely manner. Each bus prominently displayed a number between 1 and 8, inclusive. The buses did not arrive in the same order each morning. I began to characterize the order of arrival as a “success” or a “failure.” For example, I considered the patterns 76854321 and 43251678 as successes, whereas 76821543 and 34251786 were failures. The challenge of deducing the criterion for success or failure could be presented to any class by adapting it to the appropriate level. A younger audience could be challenged by fewer buses. When using five buses, for example, the numbers 23145 and 43215 would be successes and 43125 and 14253 would be failures. Students could generate their own pattern and ask the designated bus monitor whether it is a failure or a success. If the students cannot discern the criterion, a pattern could be declared a failure at the digit at which it becomes one. For instance, in the foregoing five-bus failures, 431 and 14 would be declared failures.
How Many Cards? Let’s Ask the LCM
Authors: Zerger Monte J.
Published: November 2020

Most of us love a good game of cards or a puzzling card trick. Students are certainly no exception. Playing cards can also be used to pose stimulating mathematical problems. A deck of cards has an aura of color and excitement about it that can spark a class to study the mathematics involved. Many fascinating ways to blend card games and mathematics can be found in the sources listed in the bibliography.
Sharing Teaching Ideas: The Graphing Calculator and Division of Fractions
Authors: Pelech James , Parker Jacquelyn
Published: November 2020

How does a mathematics teacher develop a strategy to use more technology in the classroom? The answer is simple: work with an English teacher in a year-long in-service program! Jacquelyn Parker and I worked together in the Content Area Reading Program, which emphasizes different reading and teaching strategies. For the visual learner, Parker recommended that I use a graphic organizer. She also reminded me that as a teacher of algebra, geometry, and trigonometry, I had a built-in graphic organizer at my fingertips—the graphing calculator.
A Mean Solution to an Old Circle Standard
Authors: Samide Andrew J. , Warfield Amanda M.
Published: November 2020

Students have an uncanny knack for inventing solutions to problems that intrigue classmates who immediately want to know if they hold true in all cases.
The Write Way: A Look at Journal Writing in First-Year Algebra
Authors: Dougherty Barbara J.
Published: November 2020

With changes in mathematical content and instructional techniques in first-year algebra, we are forced to change the tasks we give students as a means of expressing their understandings. Homework papers consisting of endless drill with no thrill cannot give a good indication of what students are learning. Test questions that focus on applying an algorithm cannot assess how students connect mathematical ideas. Neither traditional measure gives adequate information about how students have constructed generalizations or about attitudinal changes. Thus, traditional tasks do not give a clear picture of what students understand or believe.
Activities: Experiments from Psychology and Neurology
Authors: Hadley William S.
Published: November 2020

One topic advocated by the new curriculum-reform movement, but often missing from textbooks, is data collection and analysis. As a classroom teacher, I have tried many different experiments in an attempt to produce good linear-data sets: measuring different body parts, burning candles, and so on. Unfortunately, students were often unable to make accurate enough measurements to obtain usable data, and many experiments required students to collect data over a long time or outside of class. As a result, I have often been disappointed.
Soundoff!–Who Is Dick Askey and Why Is He So Upset about the Standards?
Authors: Addington Susan , Roitman Judy
Published: November 2020

Dick Askey is a mathematician at the University of Wisconsin. He is passionately concerned about students' understanding of mathematics. He does not talk the language of the Standards (NCTM 1989, 1991, 1995). In fact, he gets very upset by the language of the Standards. But he, and others like him, have a lot worth saying, and we need to listen. We have picked him as a representative of thoughtful critics of reform, and, yes, it is possible to be both thoughtful and critical of reform.
Activities: Pentagrams and Spirals
Authors: Douglas Lew
Published: November 2020

The inspiration for this activity is the mathematics videotape Donald in Mathmagic Land, originally produced as a movie by the Walt Disney Company in 1959. Anticipating current music videos and some commercials, the images are tantalizing and rapid, encouraging some teachers to show it to students as early as third or fourth grade. The mathematical presentation, however, is deep and brilliant, so I had a natural desire to slow it down and make it truly accessible to students. I generally show the videotape to students just after we complete the exercises presented in this article, and several have told me that it was the third or fourth time they had seen it but the first time they understood it.
Projects
Authors: Stenglein Sharon
Published: November 2020

Getting Ready for Algebra is a unique staff-development program that models curriculum strategies and practices for pre-algebra and first-year-algebra teachers. It is intended to offer teachers the opportunity to observe their students learning from rich classroom activities and help them understand the underlying curriculum design so that they can adopt and adapt these strategies and become more effective at teaching all students of algebra.
Sharing Teaching Ideas: Dry-Erase Boards for Students' Responses to Classroom Exercises
Authors: Clare Scott M.
Published: November 2020

Encouraging all students to participate in critical thinking and generating mathematics discourse among students are important goals set by the NCTM's Professional Standards for Teaching Mathematics (1991). Johnson (1982, 8-16), in a chapter titled “The Art of Questioning,” encourages teachers to involve all students in answering questions and classroom exercises. He has his students work a problem with pencil and paper, and he then walks around to spot-check the work of each student.
Revisiting and Extending the Hog Game
Authors: Bohan James F. , Shultz John L.
Published: November 2020

The instructional value of any problem can be determined in many ways. One criterion might be the degree to which it permits multiple levels of solutions with varying degrees of sophistication and abstraction. Another consideration might be the degree to which it engages students in terms of their curiosity and level of involvement in the investigation of the problem and its eventual solution.
The Established Results of the New Psychology as it Bears upon the Teaching of Mathematics
Authors: Rogers Agnes L.
Published: November 2020

By the New Psychology we indicate the comparatively modern psychology based upon experiment and measurement. Its results have been obtained from investigations, which have been conducted under controlled conditions and which can therefore be repeated by other investigators and verified.
Middle and Western New York State
Authors: Decker Floyd F.
Published: November 2020

Through this district there is little criticism of mathematics teaching. The requirements in school and college have not been changed to any appreciable extent. The question as to the relative number of pupils taking high-school mathematics seemed hard to answer definitely. The following table, however, gives the number of pupils per thousand of enrollment who presented answer papers to the regents in the years shown.
Western New York
Authors: Sherk Wilfred H.
Published: November 2020

In gathering the material for this report of the results of the recent criticisms of mathematics teaching in western New York, a questionnaire was sent to one hundred and twenty high schools and to ten colleges.
New Jersey
Authors: Webb Harrison E.
Published: November 2020

Mr. Webb had not prepared any questionnaire but had made a good many informal inquiries. He felt that the chief critics of mathematics teaching in high school were those teachers or employers to whom students leaving high school were later to be responsible.
Recent Advances in the Teaching of Mathematics
Authors: Henderson R. H.
Published: November 2020

In attempting a discussion of our subject, we are confronted by three possible lines of attack: First, what advances, if any, in the subject matter that is presented in the ordinary courses in mathematics; second, what improvements are to be noted in the methods of presentation of mathematical subjects to the classes; and third, what advancement is worthy of note among teachers of mathematics as to their professional training and fitness to be recognized as leaders in their chosen profession. Any one of these lines of thought is capable of extended discussion which exceeds the limits of this paper. We shall, therefore, set forth under each some points which appeal to us as worthy of presentation on a subject of such vital interest to us all.
The Use of the Radical Symbol
Authors: Miller G. A.
Published: November 2020

Probably the most significant recent step towards the uniformisation of the notation of elementary and secondary mathematics is represented by the list of symbols, numbered from 1 to 139, proposed in Heft 17 of the Schriften des Deutschen Ausschusscs für den mathematischen und naturwissenschaftlichen Unterricht, 1913. In many cases remarks and explanations are added to the proposed symbols. Number 20 of these symbols relates to an nth root, and, as this may be of special interest, we shall reproduce it here, translated into English.
Editorial: Join the Referee Ranks
Authors: Barbeau Edward , Masunaga David
Published: November 2020

This journal is supervised by an Editorial Panel of nine members, each of whom serves for three years, with appointments staggered. This panel determines policy and ensurts that the contents of the Mathematics Teacher are sound as well as useful for you, the reader. Every article is submitted to three referees. of whom one is a practicing teacher. Depending on the content, the article may also go to a mathematician, an educational researcher. or an educator of teachers. Their reviews are forwarded to at least one member of the Editorial Panel, who will collate the opinions, use them to come to an independent judgment on the acceptability of the article, and formulate a response to the author. The expertise of careful and experienced referees is of enormous assistance.
Visualizing the Proof of the Mean-Value Theorem for Derivatives
Authors: Hurwitz Marsha
Published: November 2020

A student constructs a perpendicular to the base of an isosceles triangle to prove that the base angles are congruent. Another student looks at the function f(x) = ex − 1 − x to show that ex > 1 + x. A third student investigates the antiderivative of a function to show that the function has at least one zero in a given interval.
Technology Tips: Using a Graphing Utility as a Catalyst for Connections
Authors: Embse Charles Vonder , Engebretsen Arne
Published: November 2020

One of the most important ideas expressed in the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) is that of connections. The Standards document recommends that “the mathematics curriculum should include investigation of the connections and interplay among various mathematical topics and their applications … ”
Improving Classroom Tests as a Means of Improving Assessment
Authors: Thompson Denisse R. , Beckmann Charlene E. , Senk Sharon L.
Published: November 2020

Cunently much discussion is occurring within the mathematics-education community regarding assessment. In attempting to develop the mathematical power of students, the Assessment Standards for School Mathematics (NCTM 1995, 29) encourages teachers to make several changes in their assessment practices. Among these are the following:
From the Editorial Panel: Welcome to Our Focus Issue on Algebraic Thinking
Authors: Barbeau Edward , Brown Susan
Published: November 2020

Algebra is the core of the high school curriculum. Without it, advancement into most other areas of mathematics—analytic geometry, trigonometry, combinatorics, analysis—is impossible. The study of science beyond the descriptive stage requires algebra. Accorrungly, an increasing percent of the school population will need to study algebra.
Activities: Iterating Linear Functions—An Introduction to Dynamical Systems
Authors: Choate Jonathan
Published: November 2020

The arrival of computers has caused some major changes in both mathematics and mathematics education. One of the biggest shifts has been from an emphasis on symbolic methods to one on numerical methods. One field of mathematics, dynamical systems, requires considerable number crunching and is just coming into its own because we currently have the ability to perform extensive calculations easily. This article introduces students to this new field. The study of sequences created by using numerical iteration provides interesting new ways to approach many of the concepts central to the secondary mathematics curriculum, such as functions in general and linear and exponential functions in particular.
Implementing the Professional Standards for Teaching Mathematics: Supporting the Development of Mathematical Pedagogy
Authors: Brown Catherine A. , Smith Margaret S.
Published: November 2020

In the Professional Standards for Teaching Mathematics (NCTM 1991), the National Council of Teachers of Mathematics has explicated what a teacher needs to know and be able to do to teach mathematics in the spirit of reform. Teachers' current knowledge of mathematics and mathematics pedagogy may not be adequate to meet the new instructional goals. Toward this end, the Professional Teaching Standards document includes six standards that are intended to guide the preparation, support, and career development of teachers. This article focuses on one of these standards—Standard 4: Knowing Mathematical Pedagogy—which is integral to the effective teaching of mathematics.
Solving Equations in a Technological Environment
Authors: Yerushalmy Michal , Gilead Shoshana
Published: November 2020

What is it that a rich society owes to its citizens? Not, I think, warehousing under the pretext that they are in “ninth-grade algebra or learning Macbeth,” but something far deeper, more valuable, more personal, more meaningful. When a student is bored, I do not think it is necessarily a failure of the student, nor of the teacher, but more often a failure of the curriculum. (Davis 1994, 318)
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 fullblown 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.
A Mathematics Trail at Exeter Academy
Authors: Lancaster Ron , Delisi Vince
Published: November 2020

At a recent Phillips Exeter Academy Mathematics and Technology Conference, we decided to create and conduct a mathematics trail on the campus and in the town of Exeter, New Hampshire. The idea was to put together a series of problems and puzzles that would give teachers an opportunity to do some mathematics in an outdoor setting. An attempt was also made to look for novel questions that would permit teachers and their classes to look at things from a different perspective. Although our trail was highly successful, the most exciting aspect was watching other participants create questions of their own. A high point for both authors during the walk was meeting Debbie Merrill, a mathematics teacher at Exeter Junior High School. Merrill has created and used for many years a trail of her own, along which, among other things, she has her students search for golden rectangles and solve a number of geometrical and algebraic problems related to the town of Exeter.
Reciprocal Mappings: The Neglected Transformations
Authors: Metz James R. , Zilliox Joseph T.
Published: November 2020

The ability to make connections between graphs and algebraic representations is necessary for a strong conceptual foundation for understanding functions. According to the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989, 146),
Explorations in Geometry: The “Art” of Mathematics
Authors: Libow Herb
Published: November 2020

We often pose a mathematical situation, concept, or theorem and do not proceed to explore it fully. We do not experience the thrill of chasing our intuitions, the excitement of meeting the unexpected, the uplift of clarifying ideas, the feeling of enlightenment and pride upon discovering something new to us, and the rush of succinctly capturing the essence of complexity. In short, we miss the artistic experience in one of our great arts—mathematics.
A Geometrical Representation of Primitive Pythagorean Triples
Authors: Bonsangue Martin Vern
Published: November 2020

The geoboard has long been a useful tool for mathematics students of all ages to explore geometric patterns and relationships. Among the many interesting activities that can be done on geoboards is exploring properties of polygons, such as the relationship between a polygon's number of sides and the sum of its interior angles. Such software programs as The Geometer's Sketchpad (Jackiw 1990) can also be helpful in exploring these and even more sophisticated geometric relationships. These tools can be helpful not only in discovering known relationships but also in making new conjectures on the basis of observations and experimentations (NCTM 1991).
From Fibonacci Numbers to Geometric Sequences and the Binet Formula by Way of the Golden Ratio!
Authors: DiDomenico Angelo S.
Published: November 2020

Involving students in open exploration and a search for patterns and relations is a very exciting and rewarding experience. These activities promote stimulating discussions, the application of known skills and relations to unfamiliar settings, and the development of what the NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) has called “mathematical power”—the ability to probe and connect and to reason both inductively and deductively. Involvement of this kind can also lead to findings that capture the imagination and that foster a lasting interest in mathematics and an appreciation for its beauty.
Showing That a − b = ‒[b − a]
Authors: Bell Garry
Published: November 2020

Arecent discussion with a Year-9 class of fifteen-year-olds led to some interesting investigations, some novel explanations, and some acquisition of algebra knowledge. The algebra involved that difficult relation
Parental Involvement in the Reform of Mathematics Education
Authors: Peressini Dominic
Published: November 2020

High Schools across the nation have been, and are, engaged in efforts to implement the recommendations in the three Standards documents of the National Council of Teachers of Mathematics (1989, 1991, 1995). These efforts continue in the context of a larger educational-reform movement that spans all content areas (Fullan 1991; U.S. DOE 1994a). Many of the more recent reform documents, which point the direction for this movement, call for increaes in parental involvement and the promotion of partnerships between schools and communities (National Parent-Teacher Association 1994; U.S. DOE 1994a, 1994b). These aims are apparent in the U.S. DOE's national goals for the year 2000 (U.S. DOE 1994a):
Realistic Problem Formulation and Problem Solving
Authors: Ralston Anthony , Willoughby Stephen S.
Published: November 2020

Moreover, we seldom introduce students to problems that lend themselves to long-term, in-depth analysis at different levels of intellectual accomplishment. In this article, we discuss such a problem, one that allows significant mathematical endeavor beginning with second graders and extending all the way through college undergraduates.
Reasoning with Algebra
Authors: Friedlander Alex , Hershkowitz Rina
Published: November 2020

The question “What should algebra be?” has been discussed extensively in recent times. Many recommendations have been made to transform algebra from a skill-drilling sequence of exercises into a meaningful topic that can be approached through problem solving and connected to children's previous experiences in mathematics, science, and real-life situations (see, e.g., NCTM [1989]; “Board Approves Statement on Algebra” [1994]). According to these recommendations, the stage of beginning algebra should include an understanding of such algebraic concepts as variable expression, function, and equation, as well as the ability to construct and analyze multiple representations of number pattern and situations. The ability to model and generalize situations and patterns is also mentioned frequently. However, using algebra as a tool for reasoning, explaining, and justifying is seldom addressed or emphasized, in spite of its being a direct and natural continuation of the recommended processes of generalizing. This situation may be one of the unintentional legacies of the traditional approach. which viewed mathematical reasoning and proof a belonging almost exclusively to Euclidean geometry.
Activities: The Mathematics of the Global Positioning System
Authors: Nord Gail D. , Jabon David , Nord John
Published: November 2020

Teacher's Guide: The Global Positioning System (GPS) is a constellation of twenty-four satellites, orbiting approximately 20 200 km above sea level, that enable receivers to compute their position anywhere on the earth with remarkable accuracy. The mathematical theory and computation involved in the GPS are within the scope of the second-year-algebra curriculum. This activity illustrates an application of mathematics to modern navigation.
Sharing Teaching Ideas: Piecing Together Piecewise Functions
Authors: King Sybrina L.
Published: November 2020

“Why didn't more students understand this topic?” and “How can I better present this topic next year?” were two questions that I asked after attempting to teach the concept of piecewise functions for the first time. The answer to these questions came in the form of patty paper. Patty papers are the waxed papers that many restaurants use to separate hamburger patties. They are relatively inexpensive and can be used as tracing paper.
Reader Reflections
Authors: Roche James W.
Published: November 2020

Have students escort out the old year—1997—by generating all the numbers 1–100 using only 1, 9, 9, and 7, in that order. Here are a few examples:
Sharing Teaching Ideas: A Prop Is Worth Ten Thousand Words
Authors: Critchlow Carol M.
Published: November 2020

I find that calculus students have great trouble visualizing solids of revolution and their cross sections. To help them, I resort to a wide variety of props.
Nonperiodic Tilings: The Irrational Numbers of the Tiling World
Authors: Naylor Michael
Published: November 2020

Most tessellations that we see in art and nature can be reduced to one repeating unit. The honeycomb, for example, is made from one hexagon repeated in a regular pattern. A popular floor-tiling pattern might alternate octagons and squares; some manufacturers take advantage of this regularity and sell bricks in the shape of one octagon and one square connected along an edge.
Sharing Teaching Ideas: Quote the Student, Evermore
Authors: Jennings Edward M.
Published: November 2020

A handwritten note from one of my students contained the following inscription: “Mathematics is a rock of certainty, a true utopia, that resists all irrational things.” With this statement, the student, Jonas Verdeflor, sparked a thought in my mind that grew and developed into a wonderful series of classroom discussions. Not only did Jonas's thought cause me to recall inspiring quotations from famous mathematicians, it also gave me the idea of having students create their own mathematical quotations. As a result, I asked the students in my junior-year-mathematics class in an all-boys school to bring in two mathematics-related quotations: one written by a famous mathematician and one original statement that they had written.
Exploding the Ellipse
Authors: Good Arnold
Published: November 2020

Readers are advised to proceed with caution. Those with a weak heart may wish to consult a physician first. What we are about to do is explode an ellipse. This risky business is not often undertaken by the professional mathematician, whose polytechnic endeavors are usually limited to encounters with administrators.
Fractals in High School: Exploring a New Geometry
Authors: Lornell Randi , Westerberg Judy
Published: November 2020

A brief description of fractals and their characteristics, some examples from our unit, and an argument about the reasons for including fractals in the mathematics curriculum.
The Vortex Tessellation
Authors: Drost John P.
Published: November 2020

A description of vortex tessellations. Some examples include doves and bluebirds.
Analyzing and Making Sense of Statistics in Newspapers
Authors: Kitchen Richard S.
Published: November 2020

A project developed for students in grades 9—12 that uses real–world statistical data presented in the print media. The activities take advantage of the variety of statistical representations found in newspapers.
Algebra for All: Using the Elegance of Arithmetic to Enhance the Power of Algebra
Authors: McConnell Michael , Bhattacharya Dip D.
Published: November 2020

In this article, we produce algebraic solutions, thereby reducing the problems to equations, and arithmetic solutions, which do not use variables. We look at how the arithmetic solution helps students better understand the problem before they approach it algebraically and how logic used in solving the problem arithmetically can be directly linked to the algebraic solution.
Creative Writing in Trigonometry
Authors: Barnes Julia A.
Published: November 2020

These assignments bring an alternative form of assessment into my trigonometry classroom, and they combine topics from English and mathematics – two subjects that students often consider to be totally unrelated.
The Role of Instructors in Creating Math Anxiety in Students from Kindergarten through College
Authors: Jackson Carol D. , Leffingwell R. Jon
Published: November 2020

I just don't like math.” How often have students uttered these anxiety-based words? The primary purpose of this research was to investigate the types of instructor behavior that created or exacerbated anxiety. In addition, the authors wanted to determine the grade levels (K—college) in which mathematics anxiety first occurred in these students. In this article, the term instructor includes anyone who teaches at any level, kindergarten through college.
Counting Triples, Triangles, and Acute Triangles
Authors: McClintock Ruth
Published: November 2020

Activities involving counting triples, triangles, and acute triangles enrich the curriculum with excursions into modular arithmetic, the greatest-integer function, and summation notation. In addition, more advanced students can apply difference-equation techniques to find closed forms and can use mathematical induction to prove the formulas. Students may be learning about these topics for the first time, or they may be reviewing familiar ideas in different problem-solving contexts. In either situation, personal arsenals of problem-attacking skills are strengthened.
What Is Normal, Anyway?
Authors: Calzada Maria E. , Scariano Stephen M.
Published: November 2020

The Lilliefors test for normality has not been given the prominence it deserves in statistics education, perhaps because of its intensive reliance on graphics and numerical computation. Now that students have access to graphing calculators, the Lilliefors test for normality must become as important a tool as histograms and normal–quantile plots in exploratory data analysis.
Grasping Graphing
Authors: Patterson Anne C.
Published: November 2020

How can you say you taught us to graph linear equations if we didn't learn it?” This stinging statement made by an algebra student left me speechless.
Sharing Teaching Ideas: Optimization: A Project in Mathematics and Communication
Authors: Lowther Martha H.
Published: November 2020

I am always looking for lessons with multiple purposes: problems that can be approached from a variety of perspectives, hands-on experiences for the students, and attention-grabbers that lead to good discussion. When I saw the idea of having students use the graphing calculator to teach a parent mathematics, in the North Carolina School of Science and Mathematics Newsletter (Doyle 1995), I knew that I had found a winner. The article referred to an idea shared by Kent Reigner of Conestoga Valley High School. I did not use the problem suggested in the newsletter, but I updated and extended Jennifer Mayer's variation of a traditional problem and was extremely pleased with the results.
Media Clips: December 1999
Authors: Westegaard Susanne K.
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 fullblown 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 20191-9988.
Triangular Numbers in Problem Solving
Authors: Szetela Walter
Published: November 2020

A surprising number of mathematics problems have solutions that involve triangular numbers. This article gives some of these problems, as well as many of their interesting extensions. It also suggests other extensions for further investigation. In addition to exhibiting the ubiquity of triangular numbers in problem solving, a major goal of this article is to illustrate the observation of number patterns from which conjectures and generalizations can be made. Most of these generalizations can be formally proved by mathematical induction, but such a proof will be given just once for illustration.
A Mathematical History Tour
Authors: Kelley Loretta
Published: November 2020

One can invent mathematics without knowing much, if any, of its history. But one cannot have a mature appreciation of mathematics without a substantial knowledge of its history.
Isoperimetric Quadrilaterals: Mathematical Reasoning with Technology
Authors: Foletta Gina M. , Leep David B.
Published: November 2020

This article evolved as an extension of a lesson created in 1995 as part of the Kentucky Partnership for Reform Initiatives in Science and Mathematics (PRISM). We intended to work with quadrilaterals in general on our original lesson, but we thought that such a lesson might be too difficult for many students. Hence, we designed the original lesson to engage secondary school students in an investigation of the nature of isoperimetric parallelograms, that is, parallellograms with the same perimeter. We wanted the lesson to be appropriate for students at different levels of mathematical understanding.
Sound Off!: What Can We Expect from Research?
Authors: Hiebert James
Published: November 2020

As reform recommendations collide with traditional practices, the questions on the minds of many interested teachers and parents are these: Should we really adopt this new textbook series? Should we change the way we teach? What mathematics should students really be learning? It would be nice if research could give clear and simple answers to these kinds of “should” questions, but it does not and it cannot.
Functioning in a World of Motion
Authors: Dixon Juli K. , Glickman Cynthia L. , Wright Terri L. , Nimer Michelle T.
Published: November 2020

AN INTRIGUING SET OF QUESTIONS was posed as part of a class discussion involving the concept of function, specifically, distance as a function of velocity and time: Can two people ride on the same roller-coaster train and actually move at two different speeds? What if one person is riding in the front of the train, and the other is riding in the back? Would the person in the front have a faster or slower ride around a loop or through a corkscrew than the person in the back?
An Out-of-Math Experience: Einstein, Relativity, and the Developmental Mathematics Student
Authors: Fiore Greg
Published: November 2020

A LBERT EINSTEIN (1879–1955) had a passion for sailing. The natural counterplay of wind and water, stress and strain, action and reaction thrilled the physicist, who never learned to swim and who refused to have a life jacket on board. When the breeze died and the sails drooped, unlike other sailors, he was delighted. Out came his notebook. He would begin to scribble away, jotting a new idea or making a calculation, until the breeze picked up once again.
Bingo Games: Turning Student Intuitions into Investigations in Probability and Number Sense
Authors: Bay Jennifer M. , Reys Roberts E. , Simms Ken , Taylor P. Mark
Published: November 2020

Bingo is a game enjoyed by all ages. Community centers, schools, churches, and service organizations often have regularly scheduled bingo nights. Although bingo is sometimes called a game of chance, it has been used to raise money for special causes; to provide an enjoyable evening out; and in the classroom, to teach vocabulary, listening skills, mathematics facts, geography, language, and so on. Its history begins with the eighteenthcentury Italians, who played a similar game called lotto; a variation of the game, known by that name, still exists. In the nineteenth century, this game became popular in the United States and England, with the British renaming the game housey-housey. People in the United States began playing the game but called it keno, which comes from the French word quine, meaning set of five (Wykes 1964). In these games, the numbers selected for making the cards, as well as the shape of the grid, have varied. In the United States, a bingo card is a five-by-five grid displaying numbers from 1 through 75, with a free space in the center, as shown in figure 1
Media Clips: March 2000
Authors: Westgaard Susanne K.
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 fullblown 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 20191-9988.
Traveling toward Proof
Authors: Craine Timothy V. , Rubenstein Rheta N.
Published: November 2020

We have found that the extended metaphor of Aristotle Airlines is extremely helpful in supporting students' construction of proofs. It fits well with both flow and two–column forms, and it acts as a stepping–stone to paragraph proofs, as well.
Messy Monk Mathematics: An NCTM Standards-Inspired Class
Authors: Copes Larry
Published: November 2020

A description of inquiry–based teaching and learning.
Algebra in the Service of Geometry: Can Euler's Line Be Parallel to a Side of a Triangle?
Authors: Diemente Damon
Published: November 2020

This investigation of Euler's line has become a regular and valued unit in my honors–geometry syllabus. It originated with an intelligent question from a curious student. Its geometric foundation comprises sophisticated Euclidean triangle geometry. Its solution requires plentiful but not excessively complicated algebra. It culminates in the discovery of a conic locus that can be verified by construction on a computer screen.
Sharing Teaching Ideas: Making Connections between Sequences and Mathematical Models
Authors: Horton Bob
Published: November 2020

Almost all secondary mathematics curricula include linear and exponential models. More and more programs also include arithmetic and geometric sequences, as well. These topics are linked, but students often do not make the connections.
Media Clips: September 2000
Authors: Westegaard Susanne K.
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 fullblown 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 20191-9988.
Activities for Students: Modeling Soft-Drink Packaging
Authors: Froelich Gary
Published: November 2020

Mathematical Modeling is the process of describing real-world phenomena in mathematical terms, obtaining mathematical results from the mathematical description, then interpreting and evaluating the mathematical results in the real-world situation. A simple diagram, shown in figure 1, depicts the process.