bluuebunny/crossref_metadata_2025 huggingface dataset:
More on Physical Models for Factoring Polynomials
Authors: Gibb Allan A.
Published: November 2020

In a recent issue of the Mathematics Teacher, James K. Bidwell (1972) details a physical model for factoring quadratic polynomials, which he describes as an extension of a model described by Jerome Bruner (1966). This, in turn, is closely related to Dienes’s (1971) materials.
Dots and Cubes
Authors: Eaton Bryan L.
Published: November 2020

What seems more logical to a mathematician than the extension of a two-dimensional game into three dimensions? Commercially, this has been done with TIC-TAC-TOE. DOTS and CUBES is the three-dimensional counterpart of a well known two-dimensional game, DOTS and SQUARES.
Master of Tessellations: M. C. Escher, 1898-1972
Authors: Ranucci Ernest R.
Published: November 2020

An account of the contributions to geometry of a well-known artist.
How to Draw Tessellations of the Escher Type
Authors: Teeters Joseph L.
Published: November 2020

A short, clear discussion showing how you or your students can create tessellation art.
Discoveries in Mathematics: How Are They Made?
Authors: Arpaia Pasquale J.
Published: November 2020

Methods for squaring some numbers with special properties are explored.
A Plan to Combine Individualized Instruction with the Lecture Method
Authors: Schoen Harold L.
Published: November 2020

Teachers will be interested in reading about an attempt to strive toward individualization without sacrificing the benefits of group instruction. Have you tried to modify your teaching? Share your experience with the readers of the Mathematics Teacher.
More about Triangles with the Same Area and the Same Perimeter
Authors: Lichtenberg Donovan R.
Published: November 2020

Here is a simple, yet interesting, idea for your students to pursue. Every geometry teacher can use this to good advantage.
The Bliss Principle— or what you don't know won't hurt you
Authors: Schwartzman Steven
Published: November 2020

Problems can be solved even when data appear to be missing.
Using the Computer to help Prove Theorems
Authors: Hay Louise
Published: November 2020

Using a computer to generate possible counterexamples helps in finding a proof of a geometry theorem.
Building Birdhouses with Vectors and Linear Homogeneous Equations
Authors: Olson Alton T.
Published: November 2020

Some second-year algebra helps solve some equations with whole number solutions.
The Monte Carlo Method: A Fresh Approach to Teaching Probabilistic Concepts
Authors: Travers Kenneth J. , Gray Kenneth G.
Published: November 2020

Now you can find out how long it would have taken you to get that complete set of prizes f rom your candy boxes or cards from your pack ages of gum
Encounter with Introductory Calculus
Authors: Flick T. Michael
Published: November 2020

There are some nice ways to relate two different methods for generating a cylinder.
Clubs: Wayne Valley Computer Club
Authors: Reimann Robert
Published: November 2020

The Wayne Valley Computer Club (WVCC) is organized by and for students of Wayne Valley High School who are interested in computers and computer programming. The club has been in existence for approximately ten years and presently has a membership of thirty students. It operates under the leadership of four elected officers and a faculty advisor. The club meets once a week and any interested student may join.
The Golden Mean and an Intriguing Congruence Problem
Authors: Pagni David L. , Gannon Gerald E.
Published: November 2020

Try finding two tnanglts that art not congruent but which have five pairs of congruent parts.
Sharing Teaching Ideas: A New Look For Some Old Algebra Concepts
Authors: Johnson Maureen , Ueland Pat
Published: November 2020

During the past two years, our school district has adopted a different approach for teaching algebra to the “average” students. This change was made on the basis of the premise that these students do not need more work in arithmetic but need more time in algebra. Rather than offer a one-year prealgebra course and a one-year algebra course, we decided to offer a two-year algebra course. Students choosing the two-year course spend approximately one year on the first half of the text (the same text as the one used in the one-year algebra course) and the second year covering the other half of the text.
Why Data Analysis?
Authors: Scheaffer Richard L.
Published: November 2020

Recent years have witnessed a strong movement away from what might be termed classical statistics to a more empirical, data-oriented approach to statistics, sometimes termed exploratory data analysis, or EDA. This movement has been active among professional statisticians for twenty or twenty-five years but has begun permeating the area of statistical education for nonstatisticians only in the past five to ten years. At this point, there seems to be little doubt that EDA approaches to applied statistics will gain support over classical approaches in the years to come. That is not to say that classical statistics will disappear. The two approaches begin with different assumptions and have different objectives, but both are important. These differences will be outlined in this article.
Ages of Oscar-winning Best Actors and Actresses
Authors: Brown Richard , Davis Gretchen
Published: November 2020

The Curriculum and Evaluation Standards for School Mathematics (National Council of Teachers of Mathematics, Commission on Standards for School Mathematics 1989) has urged a greater emphasis on statistics in the curriculum. It recommends that students generate information related to their interests and then use the information to formulate and support conjectures. This article illustrates an example of implementing these recommendations by considering the work of one student who wondered if a gender difference exists in the ages at which actors and actresses win Oscars.
Soundoff: How Can Nctm's Standards For Statistics Be Achieved?
Authors: Bratton George N.
Published: November 2020

NCTM's proposed curriculum standards contain a well-planned, comprehensive strand of statistical science. The necessity for this strand, as well as the probability strand, is not debatable. Topics and ideas from probability and statistics permeate most aspects of our lives. However, the successful implementation of this curriculum will require much from the teachers of mathematics, primarily the reeducation of many teachers.
Implementing the Standards: Teaching Discrete Mathematics in Grades 7–12
Authors: Hart Eric W. , Maltas James , Rich Beverly
Published: November 2020

The NCTM's Curriculum and Evaluation Standards for School Mathematics (Standards) (1989) explicitly recommends discrete mathematics for inclusion in the 9-12 curriculum, and many of the recommendations for the middle grades can be addressed by teaching discrete mathematics in grades 7 and 8. In this article we examine how discrete mathematics can be taught in grades 7-12. We shall first briefly discuss what is meant by discrete mathematics.
Sharing Teaching Ideas: Integral Cubics
Authors: Gray Allan B. , Moore Charles G.
Published: November 2020

The teacher of differential calculus needs to be able to draw from a store of problems that have neat answers, solutions that are integers. These problems furnish valid items for quizzes or short tests because fundamental concepts involved in the problem can be tested without the student's becoming ensnared in peripheral details.
Sharing Teaching Ideas: Writing Study Cards for Understanding
Authors: Whitesitt John
Published: November 2020

How often have you heard your students say, “I understand that but I just can't explain it”? Nonsense! If they really understood they could explain it, and furthermore, requiring them to explain it in complete sentences helps them to understand the informat ion. Here is a technique that really helps students to identify and learn important mathematical concepts.
Mathematical Models of Waiting Time
Authors: Gordon Sheldon P. , Gordon Florence S.
Published: November 2020

The old saying goes, “Time and tide wait for no man.” In today's society, an equally apt line might be, “Everyone waits for almost everything.” This widespread experience with waiting lends a marvelous opportunity to develop some very nice mathematics and apply it to problems with which our students can easily identify. In this art icle, we shall consider several mathematical models that can be used to study different waiting situations. The mathematics used involves just simple ideas from probability and mathematical expectation. Some related ideas are given in Mathers (1976). We shall also consider how computer simulations can be introduced to bring an added dimension to these topics. Most important, we shall see how such mathematical ideas furnish an ideal vehicle for involving students in actual individual research projects to collect real-life data, analyze it, and compare the results to predictions based on the mathematical model.
Implementing the “Professional Standards for Teaching Mathematics”: Teaching Middle School Students with Diverse Cultural Backgrounds
Authors: Callahan Walter , Farrell Margaret A.
Published: November 2020

In this second article of the 1993-94 series, a middle school teacher reflects on his work with Hispanic and Haitian students. He used what he has learned to recommend ways to help preservice teachers learn about students of other cultures.– Ed.
Nested Platonic Solids: A Class Project in Solid Geometry
Authors: Hopley Ronald B.
Published: November 2020

Several years ago at a regional NCTM conference in Phoenix, the author was fascinated by a set of card board Platonic solids that were nested inside each other. The Platonic solids are polyhedra whose faces are congruent regular polygonal regions, such that the number of edges that meet at each vertex is the same for all vertices; only five are possible. since the set is no longer commercially available, the author decided to make a nested set for classroom demonstrations and instruction for students to make their own.
Introducing the Derivative through the Iteration of Linear Functions
Authors: Riddle Lawrence H.
Published: November 2020

The traditional approach to introducing the study of the derivative is through the tangent line as the limiting behavior of secant lines. This geometric discussion leads to the definition of the derivative, after which the differentiation formulas are developed and tangent lines can finally be computed. Unfortunately, this approach seldom gives the students any idea of what tangent lines actually are and why they might be of interest. Today's computer-graphics capabilities, however, allow exciting new ways to introduce the derivative. Topics from dynamical systems serve as interesting applications that are accessible even to students in precalculus courses (see, e.g., Devaney [1990]). Coupled with modern graphing technology, the iteration of functions, especially linear functions, not only allows an instructor to illustrate ideas from an area of great current mathematical interest but also affords an excellent opportunity to introduce the derivative and tangent lines.
Creative Teaching Will Produce Creative Students
Authors: Krulik Stephen , Rudnick Jesse A.
Published: November 2020

Almost every day in every classroom, an opportunity arises for discussing a problem, solving it, and extending it to help students engage in creative reasoning. This point is constantly emphasized in the preservice methods course that our undergraduate seniors take to coincide with their practice teaching. This article recounts a sequence of activities that occurred in a senior high school geometry class that was conducted by a practice teacher.
Spiral through Recursion
Authors: Choppin Jeffrey M.
Published: November 2020

Finding and emphasizing connections between strands of mathematics is Standard 4 of the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989). The activity described in this article is designed to review and connect a number of algebra and geometry topics, as well as to involve students in formal calculations, visualization, pattern discovery, series, estimation, and recursion. This activity centers on Baravelle spirals created in regular polygons, as discussed by Newman (1989).
How Much Does Camouflage Help?
Authors: Brunner Regina Baron , Brunner Carl E.
Published: November 2020

In the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989), activities that connect mathematics with other content areas are encouraged. Far too often, students do not realize that mathematics exists outside the mathematics classroom. When they enter another class, they forget their mathematics as they cross the threshold. This article describes an activity used with eighthand ninth-grade students who had some experience with graphing data.
Name That Graph
Authors: Greenword James
Published: November 2020

If the reader recognizes immediately the function in figure 1, then what follows may be anticlimactic. But 1f anythmg seems peculiar, some interesting surprises may be in store. A check of a few textbooks reveals a neglect of function, whose lessons are overlooked. The family of functions that includes this mystery curve is rich with opportunities for open-ended investigations by students.
Soundoff: Issues of Implementation
Authors: Wertheimer Richard D.
Published: November 2020

I believe in the goals and methods described in the NCTM's Curriculum and Evaluation Standards for School Mathematics and Professional Teaching Standards (1989, 1991). I make this statement from the perspective of teacher, supervisor, consultant, graduate student, and parent. In these roles, I am actively involved in implementing both documents. In working with teachers, I focus on those who are excited and empowered by the concepts presented in these two volumes. These teachers demonstrate powerful techniques and strategies in their classrooms that reflect the spirit of the standards documents. However, in my experience, these teachers are in the minority. As I continue to work with teachers, I meet many who are not enthusiastic about the Standards. These teachers represent the mainstream of our profession; they are hardworking individuals who are reluctant to incorporate change that alters everything they have worked for both as students and as teachers.
Soundoff: Some Worries about Mathematics Education
Authors: Cuoco Al
Published: November 2020

People all over the country are working on the restructuring of mathematics education. We are in the midst of the most serious and promising educational reform effort of the twentieth century.
Teaching Graphing Concepts with Graphing Calculators
Authors: Mercer Joseph
Published: November 2020

Graphing calculators calculate and plot a large number of points very quickly. Students find it interesting to see the results when equations that produce unusual curves are entered, and the most bizarre ones readily get swapped from calculator to calculator in much the same way that video games get passed around. The nice thing about this exchange is the curiosity it stimulates in the minds of those who see the excitement. The following five lessons demonstrate how graphing calculators can be used not only to teach the slope-intercept concept of linear equations but also to establish more general principles about two-dimensional graphs. The lessons were refined with a lower-level section of junior-year algebra, then with a regular junior-year algebra course and a sophomore-year honors-algebra course. These students had all been exposed to slope-intercept equations of lines in previous algebra and geometry courses. The classes met for forty minutes a day
Plinko, Probability, and Pascal
Authors: Haws LaDawn
Published: November 2020

In class, I recently referred to the television game show The Price Is Right and was surprised when nearly all the students broke into a grin of appreciation. As I thought about the show more seriously, I realized that many of the games played on the show contain an element of probability or problem solving. Analysis of some of the games is very simple and quite appropriate for an introductory probability class; others are a bit more complicated. This article discusses one of the most interesting games, Plinko.
Technology Tips: Say It with Machines
Authors: Martignette-Boswell Carol , Cuoco Albert A.
Published: November 2020

How many times have teachers drawn a picture like figure 1 on the chalkboard for their students?
The Case of the Blue Wooden Flower
Authors: Hollowell Kathleen A.
Published: November 2020

The following lesson combines statistics, introductory algebra, and economics. Students have the opportunity to create their own mathematical functions and make inferences from those functions. They engage in problem solving, communication, reasoning, and connecting mathematics to real-world situations.
Implementing the “Professional Standards for Teaching Mathematics”: “The Farther Out You Go…”: Assessment in the Classroom
Authors: Driscoll Mark
Published: November 2020

I can thank a student named Billy for teaching me about the importance of integrating assessment with instruction. It was the early 1970s, and I was teaching in an alternative high school that I had helped found the year before, in a converted warehouse in central St. Louis. Our students were drawn from the city's school-dropout population, and many had not been in a mathematics classroom for years. Luckily, our classes were relatively small, which permitted me on this day to reflect on what Billy had done on a task. I had put five decimals, all between 0 and 1, on the chalkboard and asked the class to rank them in order of number size—a list something Like .06, .607, .6, .6707, .067. Billy anayed them in descending order of length, longest to shortest. I asked, “Which is the number with the smallest value, Billy?” He pointed without hesitation to .6707. “How come?” I asked. This time, Billy thought a bit and seemed to be looking at what he had done through the mists of school memory. “I don“t know. I sort of remember one of my teachers saying, ‘The farther out you go in a decimal, the smaller the number.’”
National Council of Teachers of Mathematics Selling and Buying Reform: If We Build It, Will They Come?
Authors: Price Jack
Published: November 2020

The original title of the talk on which this article is based was “Mathematics Reform: Field of Dreams or Waiting for Godot?” In the film, Ray Kinsella's “field of dreams” brings thousands of people. He built it and they, cam e. In Waiting for Godot, two people talk to each other throughout he entire play, waiting for someone who never comes. A logical comparison consis so these questions: Will we bring thousands of people to join us in building a better mathematics education for all students, or are we talking to each other while waiting for the change that never comes?
Media Clips
Authors: Lancaster Ron
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.
Rethinking the First Two Years of High School Mathematics with the UCSMP
Authors: Hirschhorn Daniel B. , Thompson Denisse R. , Usiskin Zalman , Senk Sharon L.
Published: November 2020

The University of Chicago School Mathematics Project (UCSMP) was begun in 1983 as an attempt to implement the recommendations of many reports to improve school mathematics. The national reports available at the time (e.g., NACOME [1975); NCTM [1980]; CBMS [19821; College Board [19831; NCEE [1983)) called for a curriculum of broader scope that would include statistics, probability, and discrete mathematics and that would give strong attention to applications, use the latest in technology, and emphasize problem solving. To accomplish the curricular revolution recommended by these reports, it was essential that new, appropriate materials be written. History had shown that neither materials written for the best students, such as those from the new-math era, nor materials written for the slower students, such as those popular in the backto-basics movement, were appropriate for the vast majority of students without major revisions (Usiskin 1985). Thus UCSMP started with the goal of developing mathematics for all grades K–12 that would be appropriate for the majority of students in the middle.
Analyzing Tensile and Compressive Forces in Planar Trusses
Authors: Russell Jeremiah V.
Published: November 2020

Astudent, Mark J. Bentley, interested in structures and design asked if he could investigate the effects that applied loads have on the various structural elements of a bridge. In working on this project, he The truss is practical and economical developed a basic understanding of statics. The project was a fruitful opportunity for both student and teacher to experience mathematical problem solving in an applied context.
Group Theory: It's a SNAP
Authors: Huetinck Linda
Published: November 2020

Martin Gardner has compared the concept of a mathematical group to the grin of the Cheshue Cat. “The body of the cat (algebra as traditionally taught) vanishes, leaving only an abstract grin. A grin implies something amusing. Perhaps we can make group theory less mysterious if we do not take it too seriously” (Gardner 1966). A game like “It's a SNAP” can be used to introduce mathematical groups and make them appear less mysterious. This mathematics manipulative allows students to play with the concepts of group theory and develop an understanding of modern algebra.
Becoming Flexible with Functions: Investigating United States Population Growth
Authors: Shealy Barry E.
Published: November 2020

Real-world contexts are appearing more often in international curricula, and the arguments for using modeling and applications are broadening (Blum and Niss 1991). The National Council of Teachers of Mathematics, in its Curriculum and Evaluation Standards for School Mathematics (1989), suggests that modeling is a great context for developing problem-solving and reasoning skills. These types of experiences promote communication and allow students to make connections among mathematical ideas and between mathematics and other disciplines. Modeling activities are also consistent with the concept of a core curriculum, offering contexts for a variety of types and depths of problems. It is not surprising that the Curriculum and Evaluation Standards points out that students should be able to “apply the process of mathematical modeling to real-world problem situations” (NCTM 1989, 137)
Data Driven Mathematics: A Curriculum Strand for High School Mathematics
Authors: Burrill Gail
Published: November 2020

How well can you estimate the ages of famous people?
Implementing the Assessment Standards for School Mathematics: The Demands of Alternative Assessment: What Teachers Say
Authors: Cooney Thomas J. , Bell Karen , Fisher-Cauble Diane , Sanchez Wendy B.
Published: November 2020

Margaret is an experienced high school mathematics teacher who is always searching for ways to “get the kids turned on to math.” As a result, she is very concerned about understanding better the diverse student populations she teaches, and she recognizes that her traditional teacher-centered methods of teaching and assessing often did not allow her to reach all her students. For the past three years, she has been involved in an in-service program on alternative assessment, and she has used many new techniques, such as portfolios and open-ended questions, to enrich the mathematical environment in her classroom. She recalls her earlier days of teaching, which she describes as lecture and teacher oriented. Then, her students seldom worked together. She assigned them mostly algorithmic exercises, which also dominated her tests and quizzes. Although her students were generally successful, something was missing for Margaret. She wanted students to share her excitement about mathematics, but this enthusiasm was rarely apparent. She “wasn't brave enough to go outside the traditional setting” until she became involved with other like-minded teachers who shared her desire to teach and assess mathematics differently.
Solutions to Calendar
Authors: McLoughlin John Grant
Published: November 2020

Problems 1-4 were submitted by Mike Arcidiacono, Frazer Boergadine, Gene Maier, Ted Nelson. Kathy Pfaendler, and Mike Shaughnessy of the Math Learning Center at Portland State University, Portland, OR 97207. Problem 5 was submitted by Betty J. Thomson, Community College of Rhode Island, Warwick, RI 02886. The problem was provided by Ruth Sperry, a student in Math 1470, History of Math. Problems 6, 7, 19-22 were sent by Corbin P. Smith. 8750 Hunter's Way, Apple Valley, MN 55124. Credit for 20 was given to Duane Hinders, Woodrow Wilson Summer Institute for Statistics. Problems 8, 9, 16-18 were supplied by Susan L. Besancon, 5100 South Ninety-second Street, Fort Smith, AR 72903. Problems 10 and 23 were adapted from The Penguin Dictionary of Curious and Interesting Numbers by David Wells (London: Penguin, 1987). Problems 11-15 were adapted from Let's Solve Some Math Problems by Derek Holton (Waterloo, Ont.: Canadian Mathematics Competition, 1993). Problems 25-27 were adapted from After Math: Puzzles and Brainteasers by Ed Barbeau (Toronto: Wall & Emerson, 1995). Problem 29 was contributed by Gene Zirkel, Nassau Community College, Garden City, NY 11530-6793.
Implementing the Professional Standards for Teaching Mathematics: Systemic Reform: A New Mantra for Professional Development
Authors: Long Madeleine J.
Published: November 2020

Like a huge pendulum that indiscriminately sweeps aside everything in its path, educational reform sometimes adopts new ideas and approaches without fully understanding their implications for teachers, for programs, and, most important, for students. Too often, educators jump on the bandwagon, forgetting the complexities of educational progress and engaging in either-or thought and decision making.
Implementing the Assessment Standards for School Mathematics: Assessing Students' Ability to Analyze Data: Reaching beyond Computation
Authors: Curcio Frances R. , Artzt Alice F.
Published: November 2020

Quantitative literacy, which includes the ability to read, interpret, and analyze statistical data in visual displays, is an essential skill needed for living and functioning in our highly technological society. In light of this need, recommendations have been made that statistics be included in the secondary mathematics curriculum (NCTM 1989). To help students value the mathematics they are learning, further recommendations suggest that statistical problems involve real data that students will find interesting and informative. As indcated in the Assessment Standards for School Mathematics (NCTM 1995), assessment tasks and instruments communicate to students which activities and learning outcomes are valued. It is therefore essential that assessment tasks engage students in data interpretation and analysis that lead to interesting and meaningful discoveries about real-life data (Burrill et al. 1992).
Technology Tips: Visual Representations of Mean and Standard Deviation
Authors: Embse Charles Vonder , Engebretsen Arne
Published: November 2020

Summary statistics used to describe a data set are some of the most commonly taught statistical concepts in the secondary curriculum. Mean, median, mode, range, and standard deviation are topics that can be found in nearly every program. Technology empowers us to access these concepts and easily to create visual displays that interpret and describe the data in ways that enhance students' understanding. Many graphing calculators allow students to display nonparametric statistical information using a box-and-whiskers plot or a modified box plot showing a visual representation of the median, upper and lower quartiles, and the range of the data. But how can students visually display the mean of the data or show what it means to be within one standard deviation of the mean? One way to create this type of visual display is with a bar graph and constant functions. Unfortunately, graphing calculators, and some computer programs, only display histograms and not bar graphs. The tips in this issue focus on using graphing calculators to draw bar graphs that can help students visualize and interpret the mean and standard deviation of a data set.
Mathematics for the Junior High School
Authors: Evans George W.
Published: November 2020

We have had a committee of the Association of Teachers of Mathematics in New England appointed to deal with the criticisms brought against the teaching of mathematics in secondary schools. This attack has been vigorous, and in many cases thoughtful.
Mathematics and Psychology
Authors: Grove C. C.
Published: November 2020

The “measurements” herein spoken of as indirect are so called because they are physical measurements upon the physical causes or effects of mental activity.
Sharing Teaching Ideas: The Birthday Problem Explained
Authors: Johnson John M.
Published: November 2020

Ask students to solve the following problem: How many people do you need in a group to ensure that the probability of at least two of them having the same birthday is greater than one-half?
A Look at Parabolas with a Graphing Calculator
Authors: Johnson Luella H.
Published: November 2020

The use of a graphing calculator can encourage explorations that take the explorer well beyond a mundane exercise to look more deeply into a concept. The exploration described in this article was prompted by a routine exercise undertaken in a graphing-calculator workshop with high school mathematics teachers. The availability of technology reduced the tedium of calculation and extended what started as a minor exercise into a full-fledged, open-ended, ongoing mathematical exploration. Such experiences are the rule, not the exception, when one is teaching with technology. They can be generated by a teacher or a student.
Recent Improvements to the Sieve of Eratosthenes
Authors: Quesada Antonio R.
Published: November 2020

The development and growth of electronic transmissions of information as well as the need for computertesting algorithms are partly responsible for the resurgence of activity in the area of prime numbers that we have witnessed in the last thirty years. In particular, the search for efficient algorithms to generate tables of prime numbers has produced excellent new results (Bengelloum 1986; Mairson 1977; Pritchard 1981). It has also produced noticeable improvements of such traditional algorithms as the sieve of Eratosthenes. This article presents recently developed generalizations to the sieve of Eratosthenes, showing the principles underlying these improvements, which increase its efficiency without changing too much of its simplicity.
Using Dynamic Geometry Software to Teach Graph Theory: Isomorphic, Bipartite, and Planar Graphs
Authors: Quinn Anne Larson
Published: November 2020

I have always used concrete marupulatives, such as marshmallows and toothpicks, to create models for my geometry and discrete-mathematics courses. These models have come in handy when discussing volume, introducing the 4-cube, or illustrating isomorphic or bipartite graphs. However, after discovering what a dynamic geometry–software package could do for geometry teaching, which has been well documented by research (e.g., Battista and Clements [1995]), I realized that this type of technology also had much to offer for teaching graph theory in my discrete-mathematics course. Although this article discusses The Geometer's Sketchpad 3 (Jackiw 1995), any software that can draw, label, and drag figures can be substituted for Sketchpad.
Implementing the Assessment Standards for School Mathematics: Assessing Students' Performance on an Extended Problem–Solving Task: A Story from a Japanese Classroom
Authors: Shimizu Yoshinori
Published: November 2020

Japanese mathematics educators at the secondary level struggle with some of the same assessment issues that plague American educators. Yet in both cultures, teachers today are experimenting with alternative forms of classroom assessment. They want to encourage their students to think more deeply about the mathematics they are learning, and they want to improve their own teaching by finding out more about what their students know and can do. This article describes the use of an extended problem-solving assessment task with a Japanese secondary class.
Technology Tips: Fishy Formulas
Authors: Bosch William , Sizoo Jennifer , Curtis Anita , Klein Shannon , Micale Cheryl , Lin E-Sin
Published: November 2020

With increased emphasis on implementing national and local standards in the classroom, educators are searching for authentic learning tasks and evaluation strategies. The NCTM's Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) encourages teachers to provide students with real-life problems and mathematical explorations. One such problem, presented in sports-fishing magazines, is developing a formula that predicts the weight (W) in pounds of a game fish given its length (L) in inches. Since length is easier to approximate, or actually measure, than weight, the length of the fish is considered the independent variable, and weight, the dependent variable. Sternberg (1996, 58–59) and Giordano and Weir (1985, 59–62) suggest the formula
Cooperative Learning in Mathematics Teacher Education
Authors: Artzt Alice F.
Published: November 2020

How a cooperative–learning activity was used in a college mathematics–teacher–education course to enable preservice and in–service middle and high school mathematics teachers to experience, learn about, and reflect on the intricacies, complexities, and values of effective cooperative–learning strategies.
Area of Spherical Triangles
Authors: Davis Elwyn H.
Published: November 2020

Interest in teaching spherical geometry—both as a subject and as a means of increasing interest in geometry—has been growing. In this article, I hope to further this interest by describing a set of classroom activities or steps that can be used to determine the area of a triangle on a sphere.
Who's Winning at Monopoly Junior?
Authors: Hooley Donald E.
Published: November 2020

A “real” problem for many students with younger siblings, in the spirit of the NCTM's Curriculum and Evaluation Standards for School Mathematics (1989). This article also illustrates the use of modeling and simulation, which Burrill (1997) notes that mathematicians in industry report using more than any other content area.
Sharing Teaching Ideas: Studying Proportions Using the Capture–Recapture Method
Authors: Bisbee Gregory D. , Conway David M.
Published: November 2020

Proportions play an important role in mathematics and science and are widely used in conversions, similar triangles, direct variation, indirect variation, interpolation, extrapolation, and many other applications. Teachers of mathematics and science understand and value proportions as a powerful tool for solving problems. Their students, however, are a bit less enthusiastic. After all, when would anyone ever use proportions? The capture-recapture method offers an exciting demonstration of such a use.
A Survey on the Use of Writing-to-Learn in Mathematics Classes
Authors: Silver Jennifer Williams
Published: November 2020

A recent survey of mathematics teachers in New York (Silver 1997) furnishes information that may trouble or encourage educators, depending on their views of how mathematics should be taught. The 117 respondents to this survey teach at all levels and are all members of the National Council of Teachers of Mathematics (NCTM).
Sharing Teaching Ideas: The Rules of the Game
Authors: Gernes Don
Published: November 2020

Concept development is both important and challenging; important because understanding a concept allows it to be used correctly and recognized in new situations, challenging because of the difficulty of enabling students to “see” clearly that which is confusing or vague.
Discovery Algebra: Graphing Linear Equations
Authors: Thomas David A. , Thomas Rex A.
Published: November 2020

One teacher's effort to change the classroom environment. It reveals a teacher, strongly motivated by dissatisfaction with existing practices, experimenting with a new approach to teaching. It shows that change can be made. The story is told through the teacher's perception of classroom events.
Broken-Line Functions with Unbroken Domains
Authors: Satianov Pavel , Fried Michael , Amit Miriam
Published: November 2020

Although we no longer doubt the legitimacy of split domains as a function, students still have difficulty perceiving a graph having a shape like that of Euler and d'Alembert's string as the graph of a single function rather than of several joined functions (Vinner 1989). Given the centrality of the function concept in the school curriculum (NCTM 1989; Amit and Koren 1993), one cannot underestimate the importance of exposing students to this type of function. A method of introducing students to this concept while furnishing the teacher with material that has its own intrinsic interest (Satianov 1995).
A Chinese Zodiac Mathematical Structure
Authors: Lamb John F.
Published: November 2020

Many Chinese restaurants use place mats that depict the Chinese zodiac—a twelveyear cycle of years named after animals. Legend has it that long ago, one Chinese New Year, the Buddha invited all the animals in his kingdom to come before him. Unfortunately, only twelve turned up. The first to arrive was the rat, followed by the ox, tiger, rabbit, dragon, snake, horse, sheep, monkey, rooster, dog, and finally the boar. In gratitude, the Buddha decided to name a year after each of the animals. The mats usually describe the traits and talents of each animal, list the animals that are compatible with it, and indicate its opposite. People enjoy passing the time while waiting for a meal by finding the year of their birth on the mat and reading about themselves. The reader soon begins to notice some patterns in those animals that are compatible and those that are not.
Making Sense of Slope
Authors: Crawford Ann R. , Scott William E.
Published: November 2020

First-Year Algebra Has Shifted From A Course That Focused Primarily On Formal Procedures For Simplifying Symbolic Expressions And Solving Equations To A Course That Emphasizes Applications In Which Students Encounter A Wide Variety Of Situations Structured By Patterns. By Observing Tables And Graphs, They Learn To Describe And Extend Patterns, Create Equations With Variables To Represent Patterns, And Make Predictions On The Basis Of These Patterns. The Study Of Variability And Change Is An Essential Algebraic Concept That Helps Students Make Sense Of Our World Through Mathematics. Hence, Understanding Slope As A Rate Of Change For Variability Has Become An Important Concept For All Students.
The Test of Time
Authors: Norwood Rick
Published: November 2020

Roger Bacon (1214–1294) wrote in his Opus Majus, “If in other sciences we should arrive at certainty without doubt and truth without error, it behooves us to place the foundations of knowledge in mathematics” (Bacon 1962, 116). We trust mathematics with our lives. Where would we be if the power grid blacked out, if all our computers crashed, or even if something simple like the sequencing of traffic lights failed? We trust mathematics. But what is it about mathematics that earns our trust?
Project Jacobean: A Mathematical Exploration of a Literary Era
Authors: Binongo JoséNilo G. , Smith M. W. A.
Published: November 2020

In this investigation, differential calculus appears in a field where few would expect to see it: the study of literature, in particular, the attribution of an anonymous work.
Activities for Students Using Hooke's Law to Explore Linear Functions
Authors: McGlone Chris , Nieberle Gary M.
Published: November 2020

One recommendation in the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) is that students should make connections. These connections bridge various disciplines of mathematics, but students also need to make connections with other disciplines, particularly the sciences. The following activity, which deals with Hooke's law, is simple to implement and furnishes applications of linear equations and graphing to a physical situation.
Technology Tips: May 2000
Authors: Coons Albert , McMullin Lin
Published: November 2020

The following tips give ways that teachers and students can use animation in their demonstrations. They involve both computer programs and graphing calculators.
Projects: Virginia Tech's Math Emporium: A Revolutionary Approach to Learning
Authors: Kasten Peggy
Published: November 2020

Prices on software, books, and materials are subject to change. Consult the suppliers for the current prices. The comments reflect the reviewers' opinions and do not imply endorsement by the National Council of Teachers of Mathematics.
Discuss with Your Colleagues: Why Representations?
Authors: Schultz James , Waters Michael
Published: November 2020

Agrowing number of factors seem to influence the teaching and learning of mathematics. These factors include developing conceptual understanding, preparing students for higher-level mathematics, applying mathematics to real-world settings, effectively using more technology, and adapting to different learning styles.
The Angles of a Star
Authors: Lipp Alan
Published: November 2020

Many students harbor the limiting belief that mathematics problems have only one right solution. A good problem can debunk this myth by provoking a variety of solutions. The angles-of-a-star problem, illustrated in figure 1, has produced much interesting thought and many elegant solutions from students and teachers alike. It is wonderfully easy to state: “Find the sum of the measures of the angles at the vertices of a five-pointed star.” This article shares some of that great diversity of solutions. Not all the methods presented are mathematical proofs, but each illustrates a powerful thinking strategy inspired by the challenge of the problem.
Activities for Students: Building Connections among Classes of Polynomial Functions
Authors: Buck Judy Curran
Published: November 2020

A major thrust of Principles and Standards for School Mathematics is that teachers should help students see mathematics as an integrated whole rather than as a series of isolated topics. “Mathematics makes more sense and is easier to remember and apply when students can connect new knowledge to existing knowledge in meaningful ways” (NCTM 2000, p. 20). The document maintains that new concepts should be introduced, whenever possible, as extensions of familiar mathematics. The activity that follows emphasizes the commonalities among classes of polynomial functions and the themes that are transferred from one class to another.
Activities for Students: Doing Mathematics with Bicycle Gear Ratios
Authors: Stump Sheryl L.
Published: November 2020

Bicycle gear ratios furnish an interesting context that allows students to explore connections among the concepts of ratio, slope, rate of change, speed, and function. In this activity, students use a bicycle to investigate the relationship between the number of pedal revolutions and the number of wheel revolutions. They also use the speed of pedal revolution and the gear ratio to predict the speed of the bicycle.
Mathematics in Tribal Philippines and Other Societies in the South Pacific
Authors: Lee-Chua Queena N.
Published: November 2020

Many history-of-mathematics textbooks begin with the four ancient centers of civilization: Egypt, with its pyramids and Rhind papyri; Babylon, with its cuneiform blocks and sexagesimal system; China, with its magic squares and arithmetic classics; and India, with its Sanskrit manuscripts and numeral system. The focus then shifts chronologically to Greek geometry, Arab algebra, Renaissance calculus, nineteenth-century specializations, and finally, the technologically aided wonders of our present age.
Sound Off! Algemetic
Authors: McMullin Lin
Published: November 2020

Item: A few years ago I ran across an ad for a wellknown symbol-manipulation software package. The left-hand page extolled the virtues of the software, and the right-hand page listed all the things that it could do. The list had three columns, running about two-thirds the length of the page. It went from simple addition through matrix arithmetic and multiple integration. I estimate that a typical student takes at least six years in high school and college to learn how to do what was listed on that two-thirds of a page.
Media Clips: March 2001
Authors: Westegaard Suzanne 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 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 20191-9988.
Discrete Dynamical Systems Meet the Classic Monkey-and-the-Bananas Problem
Authors: Gannon Gerald E. , Martelli Mario U.
Published: November 2020

Presents a solution to the three–sailors–and–the–bananas problem and attempts a generalization using an idea drawn from the branch of mathematics called “discrete dynamical systems.”
The Math Poem: Incorporating Mathematical Terms in Poetry
Authors: Keller Rod , Davidson Doris
Published: November 2020

A collaboration between an English and Math teacher to have students apply their knowledge of mathematics to another field while writing fresh, clever, and memorable poems. Includes student samples.
Consequences of the Varignon Parallelogram Theorem
Authors: Oliver Peter N.
Published: November 2020

Presents discovery activities for Elementary, Middle, and High school students based on the Varignon Parallelogram Theorem.
Sharing Teaching Ideas: Irrational Numbers on the Number Line: Perfectly Placed
Authors: Coffey Margaret E.
Published: November 2020

How do your students react to such numbers as 2 ± root3 or 1 + root3/2? If your students are like mine, they regard such numbers as unfortunate “answers” rather than numbers with a precise location on the number line. Some of my students do not easily distinguish between 2 + root3 and 2root3. I devised the following low-tech, hands-on activity to improve my students' understanding of these irrational numbers. Each student creates a number line from adding-machine tape and uses a square and a precisely folded triangle as the only measuring devices. The activity can be introduced in any algebra or geometry class after the Pythagorean theorem has been presented.
How Reform Secondary Mathematics Texts Stack Up against NCTM's Principles and Standards
Authors: Martin Tami S. , Hunt Cheryl A. , Lannin John , Leonard William , Marshall Gerald L. , Wares Arsalan
Published: November 2020

This analysis of the five NSF–funded secondary mathematics curricula describes their alignment with the Process Standards and Content Standards in Principles and Standards for School Mathematics. Distinctive features and examples are included.
Dynamic Visualizations of Calculus Ideas
Authors: Embse Charles Vonder
Published: November 2020

Three important calculus ideas –– tangent lines, the derivative definition, and the fundamental theorem of calculus –– are explored using Cabri Geometry II.
Projects: Making Mathematics: Developing Mathematical Research Skills
Authors: Foletta Gina M.
Published: November 2020

The Making Mathematics Project supports high school students—or small groups of students with a teacher—in their work on research projects in mathematics. This support is offered in the form of Web-based resources and online mentoring for secondary school students and their teachers.