bluuebunny/crossref_metadata_2025 huggingface dataset:
Soundoff: Coaching versus Competence
Authors: Meike Gerald
Published: November 2020

Recently educators have been beset by complaints from parents and businesses—and the news media has run related stories—that educational standards are falling. In partial response to these complaints and stories, some states now demand, as a condition for a high school diploma, a passing score on a standardized elementary mathematical competency test. The introduction of these tests could represent a positive step in the present reform movement in mathematics education—but only if the tests are integrated into the overall framework of that movement. A recent incident in my local school system suggests that a competency test can actually undercut the reform movement if the test is allowed to impose its own reference frame. The discussion that follows derives from that specific incident, but I believe it applies, in principle, to the handling of any standardized test imposed solely to establish a minimum competency requirement. Broader spectrum, multilevel instruments, such as the New York State Regents Examination, raise more complex issues that cannot be addressed here.
Merging Mathematics and English: One Approach to Bridging the Disciplines
Authors: Hansbarger J. Clark , Stewart Eric Lane
Published: November 2020

Ideally, the differences between the disciplines should not divide, so much as complement. Mathematics should help explain the physics that created the geography, which in turn shaped the cultures, which developed the arts and sciences, which then explained the world, and so on and so on. Such interdisciplinary connectedness is, of course, the goal of a well-rounded education.
The Advanced Placement Course in Statistics: Increasing Students' Options
Authors: Piccolino Anthony V.
Published: November 2020

The NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) stresses the need to empower all students and to encourage students to study mathematics throughout their four years in high school. For the past forty years, the Advanced Placement program of the College Board has offered students the opportunity to pursue collegiate-level course while in high school. The offering in mathematics consists of a full-year course in either Calculus AB or Calculus BC. Each Advanced Placement course culminates with an examination, typically three hours in duration, for which students may receive college credit or advanced standing when entering college.
Morgan's Theorem
Authors: Watanabe Tad , Hanson Robert , Nowosielski Frank D.
Published: November 2020

A ninth-grade student was the featured speaker at a special mathematics colloquium at Towson State University's Mathematics Department and the Center for Mathematics and Science Education in May 1994. A colloquium in the mathematics department was nothing unusual, but the age of the speaker distinguished this one. In this article, we describe what made the colloquium so intriguing and the subsequent activities it generated.
Soundoff!–Teaching Applications: Will the Pendulum of Reform Swing Too Far?
Authors: Glidden Peter L.
Published: November 2020

Before getting to the main point of this article, I need to make a confession: I am a product of the “new math.ȝ I had the “new math” from kindergarten all the way through high school. The “new math” influenced my understanding of what mathematics is and what doing mathematics means. Most of the time I am not conscious of this influence, but a recent experience made me nostalgic for two, interrelated ideas of the “new math.” I believe that these ideas are just as valid today as they were thirty years ago. More important, as the pendulum of school mathematics reform swings toward the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) and away from any residual legacy of the “new math,” we need to retain these two ideas.
Connecting Research to Teaching: Learning and Teaching Indirect Proof
Authors: Thompson Denisse R.
Published: November 2020

Proof! It is the heart of mathematics as individuals explore, make conjectures, and try to convince themselves and others about the truth or falsity of their conjecture. In fact, proving is one of the main aspects of mathematical behavior and “most clearly distinguishes mathematical behavior from scientific behavior in other disciplines” (Dreyfus et al. 1990, 126). By its nature, proof should promote understanding and thus should be an important part of the curriculum (Hanna 1995). Yet students and teachers often find the study of proof difficult, and a debate within mathematics education is currently underway about the extent to which formal proof should play a role in geometry, the content domain in which reasoning is typically studied at an intensive level (Battista and Clements 1995).
Technology Tips: Friendly Windows for Graphing Calculators
Authors: Embse Charles Vonder , Engebretsen Arne
Published: November 2020

With the proliferation of graphing calculators in many secondary and college classrooms, teachers are confronted with questions about how to make the most of these tools for teaching and learning mathematics. One important process to understand is how to set the size of the viewing window so that it is “user friendly” for all levels of students. This type of window is called the friendly window because (1) it shows a visually correct graph and (2) it helps students make the connection between the graph and the numerical representation given by the Trace cursor by highlighting points on the graph that have friendly coordinates.
The Final Tally
Authors: Millard Ron
Published: November 2020

The Results Are In! Clinton Wins?

The election has been completed and the results recorded, but do we really have a majority winner? Newspaper headlines announce, “The people have spoken. It's time for a change!” but another side claims, “I represent the 57 percent and look after their needs.” What do we have here? How can both sides claim to represent the wishes of the majority of the population?
Our Critics and their Viewpoints
Authors: Minnick J. H.
Published: November 2020

The purpose of this paper is to present the viewpoints of those who criticize present practices in the teaching of mathematics. It has been impossible to make an exhaustive study of the criticism against mathematics.
Telephones and Algebra
Authors: Appelbaum Elizabeth Berman
Published: November 2020

Do students yawn at problems about upstream and downstream rowboats? Do Boyle's law and Ohm's law leave them cold? Applications of mathematics are important, yet too many applications in textbooks are esoteric, complicated, obsolete, or academic. Cellular telephones and their rates are a trendy application that may interest students.
Technology Tips: Using Technology to Introduce Radian Measure
Authors: Klein Raymond J. , Hamilton Ilene
Published: November 2020

Each year, at the beginning of the trigonometry unit, many mathematics teachers are faced with the challenge of effectively introducing the concept of radian measure. One approach is to use manipulatives to explain how many radii fit around a circle. Students can cut a pipe cleaner so that it has the same length as the radius of an arbitrary circle and then find how many of these same-length sections fit around the circle. This idea led us to the following classroom-related activity that accomplishes, in a similar fashion, the same result. The technology used for this activity is the Cabri Geometry II software (Bellemain and Laborde 1995) for the computer or the built-in geometry software based on Cabri Geometry II that is found in the TI-92 advanced graphing calculator.
Applications: Computing Airplane Stopping Distance: Applications of Derivatives
Authors: Louis Tylee J.
Published: November 2020

For beginning calculus courses, it is often helpful to have real-world applications that use derivatives. This article presents two such applications related to computing the distance required to stop an airplane. This problem is important because this distance must be within the constraints of the runways where an airplane might land.
Technology Tips: Exploring a Parabolic Paradox with the Graphing Calculator
Authors: Shilgalis Tom
Published: November 2020

The influence of technology on mathematics education is potentially deep and far-reaching, enabling students and teachers to discover and to demonstrate generalizations and properties with an impact unthinkable just a few years ago. This alticle is about one example; it might serve to generate others. Here i a warning, however. Some developmental material in the next several paragraphs may be difficult, not to say boring, for some readers to wade through. You may choose, without serious loss, to jump forward to the “techie” part starting at the “Wake Up!” heading and return to the background preliminaries later.
Integrating Mathematical Strands
Authors: Janovsky Andrew V.
Published: November 2020

Many readers are familiar with the traveling–salesperson problem: A salesperson must start at a home city, visit each city on a map exactly once, and then return home. This path is called a Hamiltonian circuit. Assuming that roads connect all pairs of cities, what is the shortest possible route that he or she could follow?
Even Perfect Numbers: (Update)2
Authors: Bezuszka Stanley J. , Kenney Margaret J.
Published: November 2020

Perfect Numbers may appear perfectly useless; however, they do have redeeming features. Specifically, the pursuit of perfect numbers leads us to examine the history of mathematics very closely to locate information about the progression of mathematicians who have discovered and worked with them. Students assigned to produce a report on perfect numbers and their properties will uncover some fascinating episodes. Further, perfect numbers are part of the frontier of the technological age. They are woven into the mystique of the supercomputer. Anyone with an interest in computing can try to determine how the computer helps in the search for perfect numbers and what algorithms are used for this purpose.
A New Angle on Stars
Authors: Peterson Blake E.
Published: November 2020

At some point in any geometry class, students examine the sums of the measures of interior and exterior angles of simple polygons. These concepts can be further reinforced by examining the sums of the interior angles of other polygonal shapes. What began as an investigation of a question about a fivepointed star with one of my classes became an extended problem-solving and problem-posing experience that led us to computer explorations on The Geometer's Sketchpad (Jackiw 1991) and conjectures of broad generalizations through inductive reasoning.
Aligning Assessment with Classroom Practices: A Promising Testing Format
Authors: Crowley Mary L.
Published: November 2020

The 1990s have emerged as a decade in which the mathematics community is redefining what we teach, how we teach, and, consequently, what and how we assess. What we are teaching has a stronger conceptual orientation than in the past. How we are teaching involves ouT students more actively than ever before. And what and how we are assessing … has caused us some problems. All too often, after creating an environment wherein students have, for example, used calculators and group work to investigate challenging and meaningful mathematical situations, we assess their learning through a standard in-class test. Individuals work independently on questions that, given the short time allowed to complete them, are by necessity less demanding than the previous class work. The assessment is not consistent with the curricular or pedagogical reforms.
A New Spin on Volumes of Solids of Revolution
Authors: Schimmel Judith
Published: November 2020

Finding or creating alternative assessment projects for calculus courses is often difficult. Teachers feel constrained by time and the desire to present students with opportunities to explore concepts that expand their understanding. Students, however, welcome the opportunity to expe1iment and test their ideas, particularly when it involves technology. I have found that the foiJowing project appeals to both student and teacher.
Constructing Radical Equations with Two Roots—a Student-Generated Algorithm
Authors: Schwartz Stu , Moulton Charles E. , O'Hara John
Published: November 2020

As a high school teacher for twenty-five years, I find it difficult to believe that something new can be discovered in the world of algebra. What can be more gratifying than students going far beyond the generation of a single solution and finding an elegant method that will always produce desired results?
Activities: The Soap-Bubble-Geometry Contest
Authors: Morgan Frank , Melnick Edward R. , Nicholson Ramona
Published: November 2020

For many students, playing with soap bubbles is a ritual of childhood. Others observe soap bubbles in action while washing the dishes or the dog. Working with soap bubbles in mathematics class allows students to recognize that mathematics can make common experiences more fascinating. The following soap-bubble-geometry contest allows students to mesh observation and mathematical reasoning and to discover that mathematics is much more than just number crunching. Apparently simple questions expose deep geometric concepts. Students find to their amazement that some simple questions have been answered only recently, by students, and that others remain unanswered today.
The Postage–Stamp Problem, Number Theory, and the Programmable Calculator
Authors: Shultz Harris S.
Published: November 2020

The “postage-stamp problem” is a classic question in the area of Diophantine equations. Students can explore this rich and accessible problem to deepen their understanding of linear equations.
Technology Tips: January 1999
Authors: McMullin Lin
Published: November 2020

One of my favorite rides at the amusement park is the Scrambler. The Scrambler has three long arms that revolve horizontally around a central post. At the end of each arm hangs a group of seats on which the “victims” sit. These seats revolve in a circle at the end of the arm (see fig. 1). The rider travels in a circle, which in turn is moving around a larger circle. I have always been intrigued by what the path the rider traverses really is—what it would look like from above. I have also wondered how the ride could be made more exciting.
Multiple Representations for Pattern Exploration with the Graphing Calculator and Manipulatives
Authors: Lapp Douglas A.
Published: November 2020

This investigation introduces the students to the process of conjecture followed by proof, and the combination of manipulatives and technology allows this process to be approached earlier than traditionally thought appropriate.
Forever May Only Be a Few Seconds
Authors: O'Connor James J.
Published: November 2020

One day, I brought a ball to mathematics class and dropped it. After developing the equations for the finite distance that the ball would travel, I asked the class whether the ball that I had dropped earlier was still bouncing. I was amazed at the enthusiastic discussion that followed.
The Mathematics of the Spirograph
Authors: Ippolito Dennis
Published: November 2020

The Spirograph and how it can be used.
Exploring Fractals in the Classroom
Authors: Naylor Michael
Published: November 2020

This article introduces students to fractals, allows them to study the properties of some famous fractals, and encourages them to create their own fractal artwork. Students will learn about iterative processes, exponential functions, limits, writing general formulas, creating and following algorithms, and computing areas and perimeters of increasingly complex figures.
Test Reliability
Authors: Clark Kenneth
Published: November 2020

This article explains and demonstrates a procedure that is commonly used to determine the reliability of a test in such a way that a person who has modest arithmetical skills can carry out the same analysis on a classroom test or examination. The article also presents issues that arise from this approach to assessing test quality.
Cooperative Teaching Opportunities for Introductory Statistics Teachers
Authors: Rumsey Deborah J.
Published: November 2020

This article presents ideas based on our experiences; they can apply to a local or regional community of high school teachers or a department of college teachers.
Data Analysis and Baseball
Authors: Talsma Gary
Published: November 2020

An investigation that leads to one of the significant contributions that sabermetrics has made to our understanding of baseball. Along the way, we illustrate the application of several principles of data analysis (Moore 1995, 95) in a context that is familiar to, and motivating for, many of our students.
Maclaurin Taylor Series for Transcendental Functions: A Graphing-Calculator View of Convergence
Authors: Stick Marvin E.
Published: November 2020

Most calculus students can perform the manipulation necessary for a polynomial approximation of a transcendental function. However, many do not understand the underlying concept. Graphing-calculator technology can be used to bridge this gap between the concept of an interval of convergence for a series and polynomial approximations. Calculus-reform textbooks usually treat this topic by displaying lower-order Maclaurin series approximations to selected transcendental functions to encourage discussions of intervals of convergence. Some textbooks display lower-order Taylor polynomials for ln x expanded about x = 1. This article presents a way to examine the topic in more depth.
Sharing Teaching Ideas: Human Coordinates and Floor Tiles
Authors: Johnson Carl
Published: November 2020

To the casual observer, it is just a dusty classroom floor. As for me, I see the square-foot tiles forming the coordinate system on which human geometric models will be constructed!
Technology Tips: January 2000
Authors: Coons Albert , McMullin Lin
Published: November 2020

Such presentation software as Microsoft Power-Point can be used to review content in many areas of mathematics. I recently created a slide-show presentation called “math Jeopardy” as a chapterreview activity for my students. I used a mobile computer connected to a large-screen television monitor to display the presentation in front of the classroom.
Photographs and Committees: Activities That Help Students Discover Permutations and Combinations
Authors: Szydlik Jennifer Earles
Published: November 2020

The vision of Mathematics Curriculum promoted by the NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) is based on two guiding principles: “First, activities should grow out of problem situations; and second, learning occurs through active as well as passive involvement with mathematics” (1989, 9). In particular, curriculum should be designed to support students in constructing their own mathematical ideas and connections. Students should solve problems, communicate ideas both orally and in writing, engage in mathematical reasoning, and search for mathematical connections.
Technology Tips: February 2000
Authors: Coons Albert , McMullin Lin
Published: November 2020

Formal deductive proof is often a difficult structure for students to master. Why not introduce proof within a context to which they can relate rather than with geometric statements? In that way, students can more easily focus attention on the structure of a deductive proof.
Liquid Assets: Increasing Students' Mathematical Capital
Authors: Winter Mary Jean , Carlson Ronald J.
Published: November 2020

One of the visions common to all the Standards is that students and teachers should use mathematics in a variety of meaningful settings. Certainly, direct instruction, acquisition of basic skills, and textbooks still have an important place in teaching mathematics, but students must be actively involved in mathematical processes while they practice skills, master concepts and structure, and extend what they have learned.
The Prisoner Problem-a Generalization
Authors: Gannon Gerald E. , Martelli Mario U.
Published: November 2020

Problem solving is generally recognized as one of the more important functions of mathematics, and producing “problem solvers” is one of the more important jobs of a mathematics teacher. In most problemsolving strategies, the final step is taking a look back after the problem has been solved to see whether the problem and the solution can be generalized. We believe that most classroom teachers would agree that this step is often the most difficult one in the problem-solving process. Hence, our purpose here is to suggest a possible generalization to a classic problem, one that is inherently interesting and that has a solution that is within the reach of most high school mathematics students.
Connecting Research to Teaching: Kinematics and Graphs: Students' Difficulties and CBLs
Authors: Hale Patricia
Published: November 2020

In a typical course that covers the concepts of functions and graphs, the teacher might use a graph to represent the acceleration of a moving object. After drawing a simple graph for an object with constant acceleration of –0.5 m/s2, she then asks the class, “If the object—a car or person perhaps—has an initial velocity of –2m/s, describe the velocity of the object.”
The Probability of Winning a Lotto Jackpot Twice
Authors: Noone Emeric T.
Published: November 2020

Studying state lotto games offers an opportunity for students to apply basic probability concepts in a practical setting. Students enjoy the discussion and become enthusiastic about investigating the probabilities of winning the jackpot. The probability of winning the jackpot in one play of a state lotto game is quite easy to compute. A more interesting problem is computing the probability of winning it twice. Some state lotteries have produced two-time winners of large jackpots. For example, an Ohio man won $4.3 million in August 1990, then won $12 million in March 1991. This article addresses the probability of winning twice.
Cost Allocation: An Application of Fair Division
Authors: Goetz Albert
Published: November 2020

Although the subject of cost allocation has been extensively discussed in the literature of political economics, it has been generally neglected in mathematical literature. However, cost allocation affords a practical extension of fair-division techniques–one that is readily accessible to secondary school students and that gives them a simple yet powerful application of mathematics to real-world problem solving. A study of the concepts and the mathematics involved in cost allocation is most appropriate in a discrete mathematics course or a modeling course, but a case can be made for including this topic in other courses, as well. This article presents a typical cost-allocation problem with possible solutions and includes suggestions for presenting similar problems in the classroom. The basics of the problem follow closely from Young (1994).
Gathering Circles: An Experience in Creativity and Variety
Authors: Nakagomi Koji
Published: November 2020

My idea was born from a combination of a student's notebook and the tile pattern on a floor.
Cubic Polynomials
Authors: Lipp Alan
Published: November 2020

A quadratic polynomial in two variables graphs as a conic section; conic sections are an important area of study in a traditional mathematics curriculum.
Media Clips: January 2001
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.
A History of Mathematical Dialogue in Textbooks and Classrooms
Authors: Mendez Edith Prentice
Published: November 2020

Mathematical communication is an important goal of recent educational reform. The NCTM's Principles and Standards for School Mathematics (2000), continuing an emphasis on mathematical discourse from the 1991 Professional Standards for Teaching Mathematics, has a Communication Standard at each grade level. This article examines textbooks and classrooms from antiquity through the nineteenth century in search of historical precedents for mathematical communication in the form of dialogue between teacher and student. Although we have no way of knowing how prevalent this mode of teaching has been, interest in dialogue as a tool for helping students learn mathematics has been ongoing.
Connecting Research to Teaching: Making Better Use of Computer Tools in Geometry
Authors: Glass Brad , Deckert Walter
Published: November 2020

Mathematics classrooms can be very busy places. An observer peering inside one of these classrooms will probably see hardworking students using various forms of technology. For example, students in a geometry class can use computer software to manipulate shapes on the screen while exploring patterns, developing conjectures, and testing the validity of the conjectures. The observer may see colorful figures moving across the computer screens while students rapidly write on their papers.
Geometric Approaches to Quadratic Equations from Other Times and Places
Authors: Allaire Patricia R. , Bradley Robert E.
Published: November 2020

Historical solutions to quadratics problems that do not rely on symbolic manipulation.
Don't Be Square-a Geometric Excursion
Authors: House Peggy A.
Published: November 2020

Extensions of a geometry problem that give students firsthand experience with what it means to think mathematically.
An Analysis of Bending Wood by Using Kerf Cuts
Authors: Barkow Denise , Gillman Rick , Hoffman Chris , Wingstrom Josh
Published: November 2020

An analysis of a carpenter's technique for creating wood arches utilizes elementary geometric and trigonometric information. An error analysis reveals the effectiveness of the technique.
Connecting with Geology: Where's the End of the Cave?
Authors: Vacher H. L. , Mylroie John E.
Published: November 2020

Such subsurface geological resources as mineral deposits and fossil fuels are detected and analyzed by drilling and by remote sensing. Caves, however, can be traversed by people. Unlike mountains, lakes, or rivers, which can be seen in their entirety from a distance and mapped from the air, caves cannot be characterized unless they are surveyed from within. Until a cave has been explored, its extent, branches, and end cannot be predicted. Cave surveyors are isolated from the surface world of benchmarks, air photographs, and global positioning satellites (GPSs). They must rely solely on a survey line run through the cave passages to determine position. Each segment of passage has length and direction. Each segment is a vector in the earth. A cave map is a representation of these vectors (Dasher 1994).
Activities for Students: A S(t)imulating Study of Map Projections: An Exploration Integrating Mathematics and Social Studies
Authors: Wilkins Jesse L. M. , Hicks David
Published: November 2020

As technological advances continue to help more people make connections with the entire world, students must understand how to use and interpret information shown in different maps of the world (Geography Education Standards Project 1994; Freese 1997). However, mental-mapping research suggests that students in the United States have major misconceptions about proportions, locations, and perspective when they work with maps (Dulli and Goodman 1994; Stoltman 1991).
A Truth Table on the Island of Truthtellers and Liars
Authors: Baltus Christopher
Published: November 2020

My experience teaching truth tables left me looking for problems that exercise students' critical reasoning. This article grew from efforts to find such problems.
Fostering Mathematical Curiosity
Authors: Knuth Eric J.
Published: November 2020

Problem solving is an important component of learning mathematics, and that topic continues to receive significant attention in recommendations for school mathematics (NCTM 2000). Yet despite Brown and Walter's (1990) contention that problem posing is an integral part of problem solving, “problem posing is almost always overlooked in discussions of the importance of problem solving in the curriculum” (Silver, Kilpatrick, and Schlesinger 1995, p. 15).
Media Clips: February 2002
Authors: Collier C. Patrick
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.
Connecting Research to Teaching: Cooperative Learning and Its Effects in a High School Geometry Classroom
Authors: Walmsley Angela L. E. , Muniz Joe
Published: November 2020

During our experiences as high school mathematics teachers, we have discovered these three things. First, many high school students do not like to take mathematics courses. These students sometimes find that mathematics is boring and believe that it will be of no use to them after they graduate from high school. Stuart (2000) states that many people think of mathematics as something that causes stress and is unpleasant. Such students have high anxiety about learning mathematics and trying to succeed. Second, students have difficulty expressing their thoughts on paper or in front of their mathematics class. This phenomenon may occur because many traditional mathematics classrooms foster a competitive atmosphere among students (Johnson and Johnson 1989). Third, the students are not accustomed to taking an active role in learning mathematics. In light of these discoveries, we wanted to find a method of teaching high school mathematics classes that would help our students understand and enjoy the mathematics. In particular, we asked the following question: Would our students understand and enjoy mathematics more if we tried a cooperative learning approach rather than the traditionally taught teacher-centered method?
Sharing Teaching Ideas: Problem Solving the Problems of Society
Authors: Cushner Jason
Published: November 2020

By learning problem solving in mathematics, students should acquire ways of thinking, habits of persistence and curiosity, and confidence in unfamiliar situations that will serve them well outside the mathematics classroom. In everyday life and in the workplace, being a good problem solver can lead to great advantages.
Delving Deeper: Gauss, Pythagoras, and Heron
Authors: Kerins Bowen
Published: November 2020

The Park City Mathematics Institute (PCMI) IS a three-week-long intensive mathematics experience for mathematicians of all levels. Each summer, it brings together high school and university mathematics educators, as well as researchers, undergraduates, and graduate students. In addition to a series of cross-program activities, each group has a specific program.
From Exploration to Generalization: An Introduction to Necessary and Sufficient Conditions
Authors: Bonsangue Martin V. , Gannon Gerald E.
Published: November 2020

In important concept in mathematics that is often difficult for students to grasp is the idea of necessary and sufficient conditions of a statement. This article presents a problem that has helped students understand this notion and has been enjoyable because of the nature of its solution. Moreover, the problem allows students to make a bridge from numerical examples to a general solution, thus helping them develop a deeper level of algebraic thought (see Shiflett and Shultz [2002]).
Using Bloom's Taxonomy as a Framework for Classroom Assessment
Authors: Kastberg Signe E.
Published: November 2020

AS A MATHEMATICS TEACHER, I WANT MY CLASSroom tests to reflect what my students have had an opportunity to learn so that I can assess both their learning and my teaching. I find, however, that often I create tests haphazardly. As a result, the tests that I give accomplish only part of what I had intended them to do. In an attempt to discover ways to be more systematic in my test preparation, I read Assessment Standards for School Mathematics (NCTM 1995). That document contains a variety of helpful advice, including a description of an assessment framework. An assessment framework sounded like just what I needed to turn my classroom assessment practices from haphazard to systematic.
Teaching with Technology: Using FreeCell to Teach Mathematics
Authors: Baker Paul L.
Published: November 2020

Various ways that the computer game FreeCell can be used to teach mathematical concepts.
Transition to a Problem-Solving Curriculum
Authors: Schettino Carmel
Published: November 2020

“How are we supposed to learn anything if you don't tell us what we need to know?” My new students sat there staring at me, possibly considering whether I was insane. They were about to embark on an interesting and challenging adventure with me as I attempted to expose them to a problemsolving curriculum (PSC) for the first time. “What do you mean these problems aren't from the book?” and “You can't think I can do any math problems without having a teacher explain them to me first” were statements that I often heard during those first few weeks. I was asking students who were accustomed to a traditionally run mathematics classroom to think creatively and independently; and some of them were being asked to do so for the first time. My goal is to share this experience, which has been enlightening, educational, and at times sobering. However, I have been encouraged and pleased that, for my students, a transition to a PSC is under way.
Constructing a Personal Understanding of Mathematics: Making the Pieces Fit
Authors: Ellis Mark W.
Published: November 2020

Reflecting on his own development as a teacher of mathematics, the author provides a glimpse into how a shift in his thinking about and understanding of mathematics powerfully impacted the nature of the learning environment he tried to create within his classroom. It is argued that students and teachers alike need opportunities to develop personal, meaningful understanding of mathematics if the vision of the NCTM standards is to become a reality.
We Have a Problem Here: 5 + 20 = 45?
Authors: Wagner David R.
Published: November 2020

In his discussion about the source of good mathematics problems, Kilpatrick asserts, “one person cannot give a problem to another person” (1987, p. 124). Outside the context of mathematics, two conditions are necessary for me to believe that I have a problem—having an unfulfilled need and finding no obvious way of fulfilling that need. I cannot be given a problem. I cannot even have a problem in the sense that I have control over it. Rather, my problems hold me—they make me captive. I suggest that the richness of classroom mathematical activity might be measured by the extent to which students are being captivated by mathematical problems.
Activities for Students: Let's Play Plinko: A Lesson in Simulations and Experimental Probabilities
Authors: Lanier Susie , Barrs Sharon
Published: November 2020

Fans of the television game show The Price is Right call Plinko their favorite pricing game. Although traveling to the CBS studios in Los Angeles, California, to be a contestant on The Price Is Right is only a dream for most students, the exciting game of Plinko can be used to teach students such mathematics as simulations and experimental probabilities.
Challenges Associated with Developing Discursive Classrooms in High-Poverty, Rural Schools
Authors: Kitchen Richard S.
Published: November 2020

That teacher was discussing the challenges associated with initiating mathematical discourse with his Navajo students. Although he is interested in developing a classroom in which students regularly share their mathematical thinking with one another, such a discursive classroom may in fact be incongruent with the students' culture. This example demonstrates one of many issues that impede secondary-level mathematics teachers in their efforts to negotiate toward a classroom in which students' ideas are valued and frequently solicited.
Using Conjectures to Teach Students the Role of Proof
Authors: Cox Rhonda L.
Published: November 2020

As a geometry teacher for twelve years, I have found that one of the most difficult parts of my job has been teaching students to write proofs. No matter how many different ways I tried to get the concept of proof across to my students, only a select few of them really ever got it. Even more unfortunate, most students greatly disliked the process and did not understand the need to learn how to write a proof (Senk 1989; Clements and Battista 1992). To the students, proof writing was an exercise in futility that they completed for the teacher (Alibert 1988). After years of frustration, I decided to design a unit that would not only improve students' ability to write proofs but would also show them the role of proof writing in mathematics and actually involve them in the activity of being a mathematician. This goal is consistent with Herbst (2002, p. 200), who calls for teachers to find “alternative ways of engaging students in proving.”
Laptops, Technology and Algebra 1: A Case Study of an Experiment
Authors: Levine Lawrence E. , Wasmuth Victorina
Published: November 2020

During the spring 2002 semester, an informal experiment at McNair Academic High School in Jersey City, New Jersey, dealt with the use of laptop computers as a teaching and learning tool in an algebra 1 class. One class of students used laptops as an integral part of their study of algebra, whereas a second class studied the subject in a traditional manner. This article reports on the experiment. It details how the study was conducted, the classroom approach taken by the teacher, problems encountered, the students' reaction, and so on.
Connecting Research to Teaching: High School Students' Intuitive Understandings of Geometric Transformations
Authors: Hollebrands Karen F.
Published: November 2020

Before designing, selecting, or implementing a lesson, understanding the knowledge that your students already have (or do not have) is helpful, regardless of the topic that you are teaching. When I began to teach geometric transformations to a class of tenth-grade honors geometry students, I attempted to assess their knowledge. What I learned about these students' initial understandings of geometric transformations was surprising, as well as extremely useful for planning instruction. Knowing in advance the difficulties that students may experience when learning new mathematical concepts and skills can help prepare teachers for the classroom.
Delving Deeper: In-Depth Mathematical Analysis of Ordinary High School Problems
Authors: Stanley Dick , Walukiewicz Jolanta
Published: November 2020

IN THIS ARTICLE, WE GIVE AN EXAMPLE OF WHAT “delving deeper” might mean with respect to standard, rather ordinary high school problems. The purpose is to illustrate the mathematical depth that is potentially present, even in simple problems. We use what we call an extended analysis of a problem, which is an analysis from a mature mathematical perspective, with careful attention paid to mathematical reasoning and to using good mathematical habits of mind. (See Cuoco, Goldenberg, and Mark 1996.) Our intent is to foster powerful ways of thinking that are characteristic of mathematics and science.
Using Proportional Reasoning to Solve Geometric Problems
Authors: Pandiscio Eric A.
Published: November 2020

The proportional nature of geometric work, and the ways in which proportional reasoning can be employed as a generalized problem–solving tool. A geometric construction is presented for students to solve. The author gives reasoning via proof and proportions and provides extension activities.
Connecting Research to Teaching: Combinatorial Mathematics: Research into Practice
Authors: Sriraman Bharath , English Lyn D.
Published: November 2020

For many students, combinatorics is associated with negative experiences calculating permutations and combinations, often confusing one with the other. What exactly is combinatorics? Combinatorics can be defined as the art of counting, or more specifically, as “an area of mathematics in which we study families of sets (usually finite) with certain characteristic arrangements of their elements or subsets, and ask what combinations are possible and how many there are” (Rusin 2002).
Activities for Students: Unit Fractions and Their Basimal Representations: Exploring Patterns
Authors: Herman Marlena , Milou Eric , Schiffman Jay
Published: November 2020

Major foci of secondary mathematics include understanding numbers, ways of representing numbers, and relationships among numbers (NCTM 2000). This article considers different representations of rational numbers and leads students through activities that explore patterns in base ten, as well as in other bases. These activities encourage students to solve problems and investigate situations designed to foster flexible thinking about rational numbers. Preservice teachers in a college-level mathematics course carried out these activities. Their conjectures and ideas are incorporated throughout this article.
That Ubiquitous Sum: 1 + 2 + 3 + … + n
Authors: Bezuszka Stanley J. , Kenney Margaret J.
Published: November 2020

There is no doubt about it. Carl Friedrich Gauss (1777–1855) would be pleased. The question that was purportedly asked of him when he was a young schoolboy has become the centerpiece of many mathematics lessons. Indeed, it has created quite a stir, since many people want to know just how he calculated the sum of 1 + 2 + 3 + … + 100 so rapidly. Nowadays, it is commonplace for the sum 1 + 2 + 3 + … + n to be the focus of investigations in mathematics classes ranging from elementary through secondary levels. For example, young children work with instances of the sum pattern while they engage in counting chips in triangular models similar to the ones in figure 1.
The Power of L-systems in Fractal Construction and Theory
Authors: Perham Arnold E. , Perham Faustine L.
Published: November 2020

In searching the Web for an interactive applet that would help high school geometry students construct line fractals, we discovered a unique method called L–systems. Applets that integrate L–system theory with Turtle Geometry provide students with an easy–to–use graphics tool capable of creating a wide variety of fractals. In our experience, when students construct their own fractal drawings, rather than using prepared ones, they exhibit increased interest, insight, and enthusiasm. Mandelbrot and Frame (2002) make a strong case for fractals in the classroom, noting that the study of fractals heightens interest for non–mathematics/science–bound students.
Geometry of Transformations: Teacher and Unit Under Construction
Authors: Harkness Shelly Sheats
Published: November 2020

Have you ever abandoned the textbook to try new activities? What happened to your teaching practice and your beliefs about teaching and learning as a result? This article describes the story of how I created a new geometry unit related to transformations for my prealgebra and general mathematics classes and why I continually reflected upon and refined the unit each time that I used it. The process of change–for me and also for the unit–has been a gradual one. I changed my teaching by small steps rather than by leaps and bounds.
Partitioning the Interior of a Circle with Chords
Authors: Parker Dennis
Published: November 2020

Pattern recognition is one of the most important devices that we use in understanding mathematical concepts and principles, as well as in solving mathematics problems. In middle school and high school mathematics classes, students commonly study several specific cases of a problem situation and then attempt to generalize to the nth case. A problem about partitioning a circular region with chords, sometimes called Moser's circle problem, illustrates that one must be careful when using inductive reasoning to generalize.
Conic Sections: Draw It, Write It, Do It
Authors: Leapard Barbara B. , Caniglia Joanne C.
Published: November 2020

A challenging activity for integrating mathematics and art using conic sections. Students create a drawing that is formed by the graphs of linear equations and conic sections and record the equations with domain and range for each. The art work incorporates graphing calculators and pencil and pencil and paper graphs.
Interactive Geometry Software in the B.C. (Before Computers) Era
Authors: Whittaker Heather , Johnson Iris DeLoach
Published: November 2020

The use of 3×5 cards to explore geometric relationships through the first three van Hiele levels of geometric reasoning. Students engage in reasoning and proof as they explore concepts related to parallel lines and quadrilaterals.
Sharing Teaching Ideas: Say What You Mean and Mean What You Say
Authors: Csongor Julianna , Craig Carolyn
Published: November 2020

Communication, whether verbal or written, is an essential tool in our society. Yet we find students in mathematics classes resisting efforts aimed at improving their skills in this area. They consider the usual writing assignments boring. As requirements in the mathematics curriculum shifted to call for better communication abilities from students, many teachers discovered that the mathematics classroom is an ideal setting for the development of a more precise use of language that leads to better communication.
Sound Off! A Dialogue between Calculator and Algebra
Authors: Wade William R.
Published: November 2020

Calculator and Algebra were hanging out one sunny day, talking about mathematics. Algebra had been king for many years, but lately Calculator seemed to be getting the upper hand. More and more problems were made with Calculator in mind. Big numbers were not Algebra's strong suit, and Calculator knew it. He decided to press the advantage by dazzling Algebra with his latest discovery.
Teaching about Functions through Motion in Real Time
Authors: Fernández Maria L.
Published: November 2020

How do you think your students would respond to the question in figure 1? When I presented this problem to a group of first and second year algebra students near the end of the school year, the majority answered incorrectly, choosing graphs (a) or (d). Their selection revealed the common phenomenon of students' confusion about the link between the shape of a graph (circular) and the actual event (motion of the Ferris wheel). Calculator-Based Laboratories, or CBLs (used here for CBL and CBL 2), and motion detectors are tools that clarify and enhance students' understanding of functions. After using these tools, the majority of those same first and second year algebra students correctly chose graph (b).
Activities for Students: A Natural Bridge from Algebra and Geometry to Trigonometry
Authors: Calzada Maria E. , Scariano Stephen M.
Published: November 2020

Studying trigonometry for the first time, students can be overwhelmed by the totally new mathematical setting in which they find themselves. We recommend the following smooth transition into trigonometry, starting with familiar concepts from algebra and geometry. The teacher-guided, group activities for students we suggest here depend only on the concepts of area and the Pythagorean theorem. Using these ideas, students actively learn the foundational concept of invariance, which is the cornerstone of trigonometry. In this discovery process, the artificial barriers our students place between geometry, algebra, and trigonometry are broken down, and students begin to see the interconnectedness of these three strands of mathematical thought.