bluuebunny/crossref_metadata_2025 huggingface dataset:
Discuss with Your Colleagues: It's All in the Area
Authors: Todd Lee J.
Published: November 2020

Examples and questions to help students understand the connection between the area under curves and samples, histograms, and various other indicators to which they have been exposed. The questions begin with the intuitive nature of empirical probabilities and progress from the frequencies to proportions, proportions to probabilities, and finally the ratio of areas in frequency charts to probabilities.
Technology Tips: November 1999
Authors: Barrett Gloria B.
Published: November 2020

Editors note: Contemporary statistics courses at both the secondary and collegiate levels expect students to interact with data and to discover and confirm ideas with simulations. Traditionally this objective has been accomplished with computer programs. However, students can perform many of these simulations on the home screen of a graphing calculator. Such an intimate interaction with the process of simulation may help develop a better understanding of the underlying concepts. Gloria Barrett, a teacher at the North Carolina School of Science and Mathematics and author of Statistics with the TI-83 (Meridian 1997), has convinced many of us of the power of this approach.
Japanese High School Entrance Examinations
Authors: Watanabe Tad
Published: November 2020

The results from the Third International Mathematics and Science Study (TIMSS) have renewed American interest in Japanese mathematics education (U.S. Department of Education 1996, 1997, 1998). Although many people are impressed with the high level of achievement of Japanese mathematics students, mathematics educators, both in and outside Japan, also realize the shortcomings of the Japanese educational system. For example, Nagasaki (1998) points out that a gender gap exists in both mathematics achievement and attitudes toward mathematics. He also points out that many Japanese children do not see the relevance of mathematics to their daily lives.
Connecting Research to Teaching: Moving beyond Teachers' Intuitive Beliefs about Algebra Learning
Authors: Nathan Mitchell J. , Koedinger Kenneth R.
Published: November 2020

Think about this question: Why are algebra story problems considered to be the most difficult tasks facing algebra students? Teachers generally regard them as difficult (Nathan and Koedinger forthcoming), textbooks typically place these problems at the ends of chapters (Nathan and Long 1999), students find them least favorable, and even comic-strip folklore presents story problems as the bane of formal education. (See fig. 1). Are these perceptions held by teachers and textbook authors justified? This article examines teachers' judgments of the difficulty of algebra problems. Our findings may surprise readers, and we hope that they will motivate readers to reexamine some long-standing assumptions about mathematics learning and instruction.
The Coefficient of Determination: Understanding r squared and R squared
Authors: Barrett Gloria B.
Published: November 2020

Fitting least-squares lines to bivariate data is a topic traditionally discussed in introductory statistics courses, often in a unit of study that includes correlation. Recently, because calculators that graph bivariate data sets and compute regression equations have become widely available, this topic has also been included in many algebra and precalculus courses. After students learn about linear models, additional discussions may concern transforming data to achieve linearity so that students can find models of other functional forms. They can also use calculators or computers to fit curves without going through the reexpression process; this procedure would be more common in algebra or precalculus than in statistics classes.
Discovering an Optimal Property of the Mean
Authors: Schwertman Neil C.
Published: November 2020

This activity introduces a mathematical idea that has several applications, which include determining the optimal location for a radio station's transmitter and determining the best locations for a fire station. Radio stations must determine the location of their transmitters so that they can broadcast the strongest possible signal and offer the broadest coverage to their listeners. The circular area covered by a transmitter at the center increases in proportion to the square of the radius. As it covers a larger area, the strength of a transmitted signal decreases proportionally to the area, or equivalently, as the square of the distance from the transmitter. To furnish the strongest overall signal to its listeners, a radio station should locate the transmitter at a site that minimizes the total of the squared distances to the communities that it serves.
Projects: Young Scholars Program
Authors: Forrest Denise
Published: November 2020

The Young Scholars Program in Ohio is a cooperative effort of nine urban school districts and several major institutions of higher education in Ohio. The program identifies minority youths with college potential who might drop out of school. The students are selected in the sixth grade by a committee of educators from their local school districts. Selection criteria include a written essay, grades, and a standardized test. Students are advised and mentored in various ways during their high school and college years. They must maintain a 3.0 gradepoint average in high school and take collegepreparatory classes to remain eligible for the program. Middle school students take mathematics classes on Saturday mornings, and high school students attend two-hour tutoring sessions twice weekly after school. The program's developers believe that it is very successful. Students who have been through the program can succeed in nearly all the academic disciplines at the college level. A notable exception to this rule is mathematics.
Group Symmetries Connect Art and History with Mathematics
Authors: Natsoulas Anthula
Published: November 2020

Throughout history, different cultures have produced designs to be used as ornamentation, as part of ceremonies, and as religious symbols. Many of these designs are mathematical in nature, and their bases are often the transformations of reflection and rotation in the plane. The images form groupings that appear to have an underlying unity. Thus, history and art merge to create a medium through which students can study the concrete operations of reflection and rotation in the plane, as well as the more abstract concept of symmetry groups. The resulting patterns give students a sense of the potential for creativity inherent in mathematics.
College Entrance Mathematics in the Year 2000-What Came True?
Authors: Maurer Stephen B.
Published: November 2020

In 1984, at the request of the editorial panel of the Mathematics teacher, I wrote an article of predictions: “college entrance mathematics in the year 2000” (maurer 1984). This follow-up article restates my predictions, then compares them with what actually happened.
The Mathematics of Levi ben Gershon
Authors: Simonson Shai
Published: November 2020

Much of the history of mathematics focuses on the Greeks (500 B.C.E.–400 C.E.), the founders of modern Western mathematical rigor. For a long time, the common notion was that little happened from the end of the Greek period until the solution of the general cubic equation in the Renaissance. However, mathematics in the period between the Greeks and the Renaissance, 400 C.E.– 1500 C.E., showed originality in both presentation and content.
The Role of History in a Mathematics Class
Authors: Marshall Gerald L. , Rich Beverly S.
Published: November 2020

Over the last five years, interest in the role of history in teaching mathematics has grown markedly. A National Science Foundation–supported Mathematical Association of America Institute on the History of Mathematics and Its Use in Teaching was founded in summer 1995 to explore how the history of mathematics can be used in the classroom. It has produced modules for use in high school and college mathematics teaching. A study on this topic has been authorized by the International Commission on Mathematics Instruction and was the centerpiece of the most recent International Congress in Mathematics Education (ICME), which was held in Japan in the year 2000.
Sound Off! Mathematics Curricula and Guinea Pigs
Authors: Reys Robert
Published: November 2020

Today Standards-based mathematics curricula are often discussed. In this commentary, Standardsbased refers to mathematics curriculum materials that are designed to address the content and pedagogy guidelines outlined in the NCTM's Standards documents (NCTM 1989, 1991, 2000). Standardsbased mathematics curricula projects funded by the National Science Foundation (NSF) are available for use in schools and have sparked serious disagreements about what and how mathematics should be taught in schools. Charges and countercharges have been made (Becker and Jacob 2000; Kilpatrick 1997; Wu 1997). Ads in prominent media have served as outlets to polarize those with opposing viewpoints. They have also brought an unprecedented level of national attention to mathematics education. Strategies for improving the mathematics learning of our children have long been debated and researched within the mathematics education community. However, the recent organized use of national media outlets to broadcast disagreements has created an unhealthy and unproductive atmosphere.
Discuss with Your Colleagues: Learning Mathematical Symbolism: Challenges and Instructional Strategies
Authors: Rubenstein Rheta N. , Thompson Denisse R.
Published: November 2020

Attempts to sensitize teachers to challenges students have with mathematical symbols, and suggests instructional strategies that can reduce student difficulties.
Centroid of a Polygon-Three Views
Authors: Shilgalis Thomas W. , Benson Carol T.
Published: November 2020

A mathematical look at the center–of–mass concept.
Sharing Teaching Ideas: A Calendar Problem, Revisited
Authors: Gonzalez-Velasco Donna M.
Published: November 2020

In my honors geometry classes, we do a lot of smallgroup problem solving. The problems range in difficulty and can take as little as five minutes or as long as several class periods to solve. This experience is the first occasion that many students have had to persist more than a few minutes to solve a mathematics problem. Over the course of the year, they experience frustration and discover that patience and persistence are important in problem solving.
Dividing Any Angle into Any Number of Equal Parts
Authors: Hutcheson Thomas W.
Published: November 2020

Two problems, easily solved in three dimensions, that are often approached in two.
The Most Magical of All Magic Squares
Authors: Swetz Frank J.
Published: November 2020

The history of magic squares.
Connecting Data and Geometry
Authors: Erickson Timothy E.
Published: November 2020

Geometrical ideas and representations often help us understand other areas of mathematics. For example, we might use area models for multiplying polynomials, for summing series, or for conditional probability calculations. But we can also use other areas of mathematics to help us understand a geometrical situation. This article describes an activity that was adapted from Erickson (2000, pp. 111–13), in which students use data analysis and mathematical modeling to obtain insight into a geometrical question. This activity explores a simple case of what is called the isoperimetric inequality. A Web search for that term can furnish additional information.
Black Dots: Newton's Method and a Simple One-Dimensional Fractal
Authors: Fisher Tony J.
Published: November 2020

Students in a standard calculus course learn Newton's method for finding the root of a differentiable function. Although they may often see a diagram that visually demonstrates how this method works, it often soon becomes yet another algorithm to memorize or to program into a calculator. In addition, students are sometimes told that using Newton's method on simple complex-valued functions can lead to beautiful fractal patterns. However, the connection between the sequence of steps that they have learned and the corresponding fractal images is fuzzy at best. This article describes a calculator exercise that can help students develop a better visual and numeric feel for Newton's method and discover how Newton's method can lead to a simple, one-dimensional fractal.
Activities for Students: A Graphical Approach to Understanding the Fundamental Theorem of Algebra
Authors: Goel Sudhir Kumar , Reid Denise T.
Published: November 2020

The fundamental theorem of algebra states, Every polynomial equation of degree n ≥ 1 with complex coefficients has at least one complex root. This fact implies that these equations have exactly n roots, counting multiple roots, in the set of complex numbers.
Counting Pizzas: A Discovery Lesson Using Combinatorics
Authors: Nord Gail , Malm Eric J. , Nord John
Published: November 2020

How many distinct pizzas can be made in a restaurant that has n bins of toppings if the pizza maker reaches into the bins at random and puts k toppings on the pizza?
Media Clips: December 2002
Authors: Harlos Carol Ann
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 20191-1502.
Activities for Students: Developing Spatial Sense: Comparing Appearance with Reality
Authors: Kelly Gwen , Ewers Tim , Proctor Lanna
Published: November 2020

Drawing changed radically when someone discovered that three-dimensional objects could be made to look real on a two-dimensional canvas. The ability to work with perspective is critical in many fields, including engineering, architecture, and construction. Beyond that, most of us use perspective skills almost daily, for example, when interpreting pictures of three-dimensional objects.
Writing About the Problem Solving Process to Improve Problem Solving Performance
Authors: Williams Kenneth M.
Published: November 2020

Problem solving is generally recognized as one of the most important components of mathematics. In Principles and Standards for School Mathematics, the National Council of Teachers of Mathematics emphasized that instructional programs should enable all students in all grades to “build new mathematical knowledge through problem solving, solve problems that arise in mathematics and in other contexts, apply and adapt a variety of appropriate strategies to solve problems, and monitor and reflect on the process of mathematical problem solving” (NCTM 2000, p. 52). But how do students become competent and confident mathematical problem solvers?
Projects: Math and Science Solutions (MASS) Project
Authors: Naslund Beverly
Published: November 2020

The Math and Science Solutions (MASS) project is an Eisenhower Professional Development Grant that connects schools in diverse settings throughout Illinois using a variety of learning technologies. The goal of the MASS project is to improve instruction and student learning in middle school and high school mathematics and science classrooms by helping to increase teachers' content knowledge, technology integration skills, and engaged learning knowledge, in addition to creating a forum for collaboration on classroom and curricular ideas. A learning network has been formed by connecting mathematics and science teachers and administrators with faculty and preservice teachers from Western Illinois University, in Macomb, Illinois.
Mathematical Insight: Changing Perspective
Authors: Sloyer Cliff W.
Published: November 2020

Many great ideas in mathematics have resulted from examining problems and solutions from different perspectives or directions, but one-dimensional thinking in solving a problem is all too familiar. A number of approaches to teaching and learning mathematics, especially those involving algorithms of some form, lend themselves to a reinforcement of one direction in solving problems. Mathematical insight gained by changing perspective is occasionally suggested in specific problems, but the “big picture” of mathematical exploration, conjecture, and proof requires a deeper commitment to this process. This article focuses on problems that can benefit from a change in perspective by looking forward and backward. In particular, the transition from basic mathematical manipulations to higher-order levels of reasoning requires an awareness of such processes.
Birthday Lies
Authors: Hathaway Dale K.
Published: November 2020

The “Birthday Problem,” as posed by Richard von Mises in 1939, is well-known and is often demonstrated in classrooms. The demonstration often involves an informal wager. The teacher may make a claim similar to the following: “I bet that if we check the birthdays of the members of this class, we will find that at least two students have birthdays in the same month and on the same day.” Students love to win any informal wager with a teacher, since one-upmanship can be a powerful motivator. However, how badly do they want to win? What if some students in the class are less than truthful about the date of their birthdays so that they can try to cause the teacher to lose the bet? At the very least, these birthday lies would probably prolong the length of time needed for a birthday match to occur. But how many birthdays do we need to check to achieve the probability of a match that is better than 50 percent? Spencer (1977) dealt with dishonest students in a variation of the birthday problem. Analyzing this situation provides another generalization of the ever-popular classic birthday problem.
Problem Solving Can Generate New Approaches to Mathematics: The Case of Probability
Authors: Kahan Jeremy A. , Wyberg Terry R.
Published: November 2020

In this article, we discuss the World Series problem, which we have used many times in our work with preservice and in-service teachers of secondary mathematics. This problem yields to multiple approaches. Furthermore, the history of the underlying problem illustrates how mathematics sometimes develops in response to the need to solve a problem, a process that we want to let our students share. In fact, we believe that this problem is a strong illustration of Stanic and Kilpatrick's (1989) metaphor that the problem is a vehicle. Those who work on (ride) the problem arrive at significant mathematics (a destination). We begin with a presentation of the vehicle, then explore several routes that students might take and notice how these routes all lead to the same destination.
On Inscribed and Escribed Circles of Right Triangles, Circumscribed Triangles, and the Four-Square, Three-Square Problem
Authors: Hansen David W.
Published: November 2020

For more than twenty years, I have been studying the fascinating relationships between right triangles and their inscribed and escribed circles. An inscribed circle is one that is tangent to all three sides of a triangle and whose center lies inside the triangle. An escribed circle is one that is tangent to one of the sides of the triangle and to the extensions of the other two sides and whose center lies outside the triangle. Every triangle has one inscribed and three escribed circles, as shown in figure 1.
Delving Deeper: Integral Factorization of Quadratic Trinomials
Authors: Kandall Geoffrey A.
Published: November 2020

You are teaching a basic algebra course and arrive at the section that deals with the factorization of quadratic trinomials. The textbook devotes fifteen pages to this topic, which it treats by trial-and-error methods. It mentions reverse FOIL, factoring by grouping, and the ac-method. Even after all this hassle, students are only expected to be able to factor trinomials no harder than, say, 12x2 − 31x − 30. The trinomial 120x2 − 371x + 98 probably does not appear as an exercise; 323x2 + 2986x − 8029 certainly does not
Sound Off!: A Highly Qualified Mathematics Teacher in Every Classroom by 2005–2006: Blueprint or Fantasy?
Authors: Reys Robert E.
Published: November 2020

The no child left behind act (P.L. No.107-110, H.R. 1, 2001) mandates a highly qualified teacher in every classroom by the 2005–2006 school year. It sounds great, but what does it mean and can it be achieved? The No Child Left Behind Act (NCLB) calls for all teachers to hold a bachelor's degree, demonstrate competence in the subject matter that they teach, and have full state teacher certification; thus, their certification requirements cannot be waived nor can they have an “emergency, provisional, or temporary” certificate.
Promoting Learning through Inquiry
Authors: Chissick Naomi
Published: November 2020

The process of including open-ended tasks and inquiry-based learning in the mathematics syllabus of high school. Article discusses 3 levels of open ended investigations to enhance learning. Includes 3 open ended investigations.
Sharing Teaching Ideas: Why Is the square root of 25 Not ±5?
Authors: Roach David , Gibson David , Weber Keith
Published: November 2020

You might want to try this brief experiment in one of your mathematics courses.
Media Clips: January 2004
Authors: Vennebush G. Patrick
Published: November 2020

“Media Clips” aims to offer readers contemporary, authentic applications of quantitative reasoning that are based on print or electronic media. We encourage submissions, which may include questions as well as clips. We especially welcome examples of flawed quantitative reasoning. Clips may be in text or graphic format, and clip sources may be either print or electronic media. Submissions must include a complete citation of the source along with the full text of the cited material. If possible, submissions should also include a copy of the actual clip. Please send submissions to “Media Clips,” Mathematics Teacher, 1906 Association Drive, Reston, VA 20191-1502.
Technology Tips: January 2004
Authors: Hollebrands Karen Flanagan , Stohl Hollylynne
Published: November 2020

This month, we provide an example of a rich mathematical task that leads to many different connections. The task was posed to a class of high school seniors who were using a dynamic program for geometry called Cabri Geometry II. This tip includes directions for creating this problem with technology and suggestions for exploring it. The Cabri II software is available for Macintosh and PC computers from www.cabrilog.com/en or education.ti.com. It is also available for several different Texas Instruments calculators (TI-83 Plus, TI-83 Plus Silver, Voyage 200, TI-89, and TI-92 Plus). The program is similar to The Geometer's Sketchpad, and users who are familiar with The Geometer's Sketchpad should be able to easily adapt this task to use with it.
Recruiting Mathematics Teachers: Strategies to Consider
Authors: Reys Barbara J. , Reys Robert E.
Published: November 2020

The federal No Child Left Behind Act (Public Law 107-110, HR 1, 2001) calls for all teachers in schools receiving federal funds to be “highly qualified.” That is, they must hold a bachelor's degree, demonstrate competence in the subject matter that they teach, and have full state teacher certification—their certification requirements cannot be waived nor can they have an “emergency, provisional, or temporary” certificate. These requirements are mandatory by the 2005–2006 school year. However, a serious shortage of mathematics teachers continues to exist in middle and secondary schools. For example, the Missouri Department of Elementary and Secondary Education reported that during the past fifteen years, an average of fewer than 200 mathematics teaching certificates (middle school and secondary) were issued annually by thirty-four different colleges and universities in the state. This average is far short of the more than 400 job listings that Missouri school districts annually post for middle school and secondary mathematics teachers.
What Do You See? A Case for Examining Students' Work
Authors: Kersaint Gladis , Chappell Michaele F.
Published: November 2020

Various interpretations of an area and volume problem, A rod-staircase problem to showng that different student interpretations lead to different solutions. Includes student work with different solutions.
Activities for Students: Demonstrating Boolean Logic Using Simple Electrical Circuits
Authors: McElhaney Kevin W.
Published: November 2020

The honors geometry classes at parkway North High School in Creve Coeur, Missouri, typically do a unit on Boolean logic operators as a precursor to the formal thinking required for geometric proofs. To provide a more direct and tangible connection of logic with a real-world application, I developed this enrichment activity. The activity uses simple electric circuits to model the basic logic operators AND and OR, thus allowing students to physically visualize the true-or-false patterns of compound logic statements, make conceptual connections between logical operators and the physics of electrical circuits, and gain an appreciation for the design of common electronic products.
Media Clips: February 2004
Authors: Vennebush G. Patrick
Published: November 2020

“Media Clips” aims to offer readers contemporary, authentic applications of quantitative reasoning that are based on print or electronic media. We encourage submissions, which may include questions as well as clips. We especially welcome examples of flawed quantitative reasoning. Clips may be in text or graphic format, and clip sources may be either print or electronic media. Submissions must include a complete citation of the source along with the full text of the cited material. If possible, submissions should also include a copy of the actual clip. Please send submissions to “Media Clips,” Mathematics Teacher, 1906 Association Drive, Reston, VA 20191-1502.
Ron's Theorem and Beyond: A True Mathematician and GSP in Action
Authors: Martínez-Cruz Armando M. , McAlister Ron , Gannon Gerald E.
Published: November 2020

Since geometry has visual appeal, it has long been considered the ideal subject for teaching proof. But for many, the concept of proof seems tedious and contrived. This article describes a situation when proof was quite meaningful and was the logical outcome of a student's investigation. Ron McAlister, a middle school teacher, became fascinated with geometric patterns produced with The Geometer's Sketchpad (Jackiw 1995), explored an idea, conjectured a result, and proved it. His activities constitute the essence of this article.
Tessellating the Sphere with Regular Polygons
Authors: Soto-Johnson Hortensia , Bechthold Dawn
Published: November 2020

Spherical geometry can be used to entice students into a deeper understanding of Euclidean geometry. Determining which regular spherical polygons tessellate the sphere is another motivating topic that is accessible to high school students. The most recognizable tessellations of the sphere are found on balls, such as soccer balls, volleyballs, and golf balls. Even Spaceship Earth at Epcot Center in Disney World involves tessellation.
Activities for Students: Using Geometry Software to Revisit the Ellipse
Authors: Jung Inchul , Kim Yunghwan
Published: November 2020

Because geometry focuses on figures instead of numbers, geometrical features can be explored by drawing on the ground with only a stick or a finger. Perhaps this ease of exploration explains why geometry has such a long history in mathematics and why, from a historical perspective, it can be considered the foundation of mathematics. Geometry involves a lot of graphics and manipulatives that students can touch and use to explore. Since those graphics serve as an important link between mathematical models and real-world phenomena, students sometimes are more comfortable with geometry than with more abstract mathematics
Calendar Problems: March 2004
Authors: Benson Steve
Published: November 2020

All the problems on this month's calendar originally appeared in, or were adapted from, problems that came from New York Interscholastic Mathematics League contests in the 1960s and 1970s.
Discover Mathematical Knowledge through Recreational Mathematics Problems
Authors: Sodhi Amar
Published: November 2020

A high-quality recreational mathematics problem should be an intriguing problem that is easy to understand but challenging to solve. If the problem leads each would-be solver to a discovery of myriad ideas in mathematics, then so much the better. An example of such a problem was posed by the Reverend T. P. Kirkman in the Lady's and Gentleman's Diary for 1851 (Kirkman 1850).
Technology Tips: May 2004
Authors: Hollebrands Karen Flanagan , Stohl Hollylynne
Published: November 2020

IN THIS MONTH'S “TECHNOLOGY TIPS,” MARY ANN Connors introduces readers to the use of scripts on the TI-89, TI-92, TI-92 Plus, or Voyage 200 calculators. She demonstrates how teachers can create scripts to be used either during a whole-class calculator demonstration or by students while they engage in a mathematical exploration.
Mathematical Lens: August 2004
Authors: Lancaster Ron
Published: November 2020

Welcome to “Mathematical Lens,” a new department in the Mathematics Teacher. “Mathematical Lens” gives the readers of the Mathematics Teacher an opportunity to link their photographs with mathematical ideas. The idea for this column began many years ago as a way to give students a visual connection with mathematical topics that they have studied. This department is looking for photographs of your mathematical sightings plus mathematical questions related to the images that you capture. The mathematics can involve an application or it can be whimsical, playful, and creative. You might find that students may even create additional questions related to the scenes depicted in the photographs. So go for a walk with your camera, take some photographs of mathematics, and share your imaginative images and questions with the “Mathematical Lens.”
Delving Deeper: The Chebyshev Polynomials: Patterns and Derivation
Authors: Sinwell Benjamin
Published: November 2020

Pafnuty Lvovich Chebyshev, a Russian mathematician, is famous for his work in the area of number theory and for his work on a sequence of polynomials that now bears his name. These Chebyshev polynomials have applications in the fields of polynomial approximation, numerical analysis, graph theory, Fourier series, and many other areas. They can be derived directly from the multiple-angle formulas for sine and cosine. They are relevant in high school and in the broader mathematical community. For this reason, the Chebyshev polynomials were chosen as one of the topics for study at the 2003 High School Teachers Program at the Park City Mathematics Institute (PCMI). The following is a derivation of the Chebyshev polynomials and a mathematical exploration of the patterns that they produce.
Note: Figures Not Drawn to Scale
Authors: Bremigan Elizabeth George
Published: November 2020

Mathematical problems that include a visual representation with the note “Figure not drawn to scale” are analyzed. Strategies that students can use in solving such problems are discussed. Also presented are sample tasks that include a figure not drawn to scale for teachers to use in their own classrooms. Ideas for teaching students to interpret figures not drawn to scale.
Activities for Students: Managing Returns in a Catalog Distribution Center
Authors: Gates Joyce , Stuart Julie Ann , Bonawi-tan Winston , Loehr Sarah
Published: November 2020

Iwas amazed by the complexity of the returnshandling problem at a catalog distribution center that I studied with a Purdue University research team. Initially, the problem appeared to be purely accounting in nature. I proposed storing returned items if the cost of processing a return back to inventory was lower than the potential profit generated by selling it. I discovered that many other factors, such as the speed with which the catalog distribution center fills an order, must be considered in the decision-making process. The research team developed an algorithm that considers several different factors, in addition to cost, to help catalog distribution centers process their returns more efficiently (Stuart et al. 2003).
Using Open-Ended Problems for Assessments
Authors: Goetz Albert
Published: November 2020

In my high school precalculus classes, students spend a great deal of time working in cooperative groups. During the course of the year, opportunities to assess students on group assignments will frequently present themselves. Many of the usual labs and worksheets require collaboration with one or more partners. Particularly in the honors class (with most of the class on track for the BC calculus level of the Advanced Placement course) I can make the group worksheets and labs a little more challenging, with the expectation that three or four minds are better than one. This approach works nicely in units on probability or counting, where it is easy to find problems that are both nonroutine and will stretch the students' minds a bit. When class time is devoted to these exercises, I can circulate and make sure that groups are moving in the right direction.
Delving Deeper: Things Fall Apart: A Recurrence of Tiling
Authors: Rogers Douglas G.
Published: November 2020

This article by Douglas Rogers is one of many we have received that investigate recurrence relations—sequences in which each term is determined by one or more previous terms. Readers of “Delving Deeper” will recognize this theme from previous articles (see Askey 2004, 2005; Huang 2005; Sinwell 2004), and Rogers's article provides yet another approach to the problem of finding closed forms for recursively defined sequences. The article caught our attention for two other reasons.
Helping Students Connect Functions and Their Representations
Authors: Moore-Russo Deborah , Golzy John B.
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.
Connecting Research to Teaching: Goals for Achieving Diversity in Mathematics Classrooms
Authors: Herzig Abbe H.
Published: November 2020

Commenting on the film Good Will Hunting, a mathematician noted, “We would do well to remember, in our efforts to include members of underrepresented groups in mathematics, that there can be as much resistance to our efforts from the students we work with as from the system we work in– (Saul 1998, p. 501). In other words, try as we may to include people in mathematics—in our version of mathematics—they might not be interested. Alternatively, some individuals may reject mathematics not out of a sense of choice but because they feel that mathematics has rejected them. Students' reactions to mathematics are affected both by their interests, abilities, and goals and by the particular ways that mathematics is conceived and taught within the mathematics classroom. It is possible that the people who succeed in mathematics are those who are able or willing to adapt themselves to the existing structure of mathematics education in schools. Individuals whose talents, values, skills, or interests make it difficult or undesirable for them to adapt to that structure may not be able to negotiate successfully the educational systems in a way that allows them to succeed in mathematics. In this argument I turn attention away from features of students—for example, their preparation or their ability—and toward our assumptions about mathematics education itself, presenting a unique challenge for us to build a context for mathematics education that is truly accessible and inviting to a broad range of students.
Activities for Students: Biology as a Source for Algebra Equations: The Heart
Authors: Horak Virginia M.
Published: November 2020

At the high school level, physics is often the area of science looked to when teachers seek applications and connections with mathematics. However, biology also can provide opportunities to represent relationships and solve problems using simple algebra. This article developed from work in an inquiry-oriented, interdisciplinary high school course that integrated firstyear algebra with an introductory environmental biology/anatomy and physiology course. Lessons and activities were derived by identifying areas where mathematics and biology content intersect (Horak 1997, 2000; Horak and Horak 1999). Team-teaching the courses in a two-hour block allowed for seamless study of the two areas, which deepened students' understanding and appreciation of both subjects.
Bugs, Planes, and Ferris Wheels: A Problem-Centered Curriculum
Authors: Campbell William E. , Kemp Joyce C. , Zia Joan H.
Published: November 2020

Mathematical power includes the ability to explore, conjecture, and reason logically; to solve nonroutine problems; to communicate about and through mathematics; and to connect ideas within mathematics and between mathematics and other intellectual activity.
Delving Deeper: Trisecting the Circle: A Case for Euclidean Geometry
Authors: Posamentier Alfred S.
Published: November 2020

As an undergraduate mathematics major, a prospective teacher usually takes at least one geometry course. Typically, these courses focus on non–Euclidean geometry (sometimes presented as Modern Geometry), or vectors, transformations, or topology. Instead, we at the City College of New York offer a course on more advanced Euclidean geometry in which prospective teachers investigate a plethora of geometric theorems (or relationships) that enrich their understanding of Euclidean geometry and, consequently, their teaching of it.
Approximating Pi with the Golden Ratio
Authors: Linn Stacy L. , Neal David K.
Published: November 2020

Around 225 BC, Archimedes wrote the treatise Measurement of a Circle, which contained the first derivation of a formula for the area of a circle and the first formal approximation of the constant we now call π. At that time, it was already known that the ratio of the areas of two circles equals the ratio of the squares of their diameters. This result had appeared in Euclid's Elements (Euclid 1956) as Proposition 12.2.
Advice for Solving Equations
Authors: Stueben Michael A. , Torbert Shane M.
Published: November 2020

This article describes three different approaches to finding the roots of an algebraic equation and concludes with a set of heuristics for solving equations in general. Our principal example is an earlyround elimination problem from the 1939 Moscow Mathematical Olympiad for students in grades 11 and 12 (Shklarsky 1993, problem 168). As we tell our students, you will get much more out of a problem if you work on it for 15 to 20 minutes and fail than if you turn to the solution after only 3 minutes.
From Classroom Discussions to Group Discourse
Authors: Manouchehri Azita , John Dennis St.
Published: November 2020

The vision to transform mathematics classrooms into learning communities in which students engage in mathematical discourse is a remarkable hallmark of the current movement, led by the National Council of Teachers of Mathematics, to reform mathematics education (NCTM 1991, 2000). According to NCTM, “the discourse of a classroom—the ways of representing, thinking, talking, agreeing and disagreeing—is central to what students learn about mathematics as a domain of human inquiry with characteristic ways of knowing” (NCTM 1991, p. 34). Indeed, both the Principles and Standards for School Mathematics (2000) and Professional Standards for Teaching Mathematics (1991) recommend that teachers of mathematics provide opportunities for children of all ages to participate in mathematical discourse.
Delving Deeper: Powers, Representations, and Counting
Authors: Kobayashi Yukio , Metz James
Published: November 2020

This installment of “Delving Deeper” contains two short articles that take different slants on themes that have run through many of the articles we have published: counting problems, Pascal's triangle, geometric series, and recurrence relations. It seems there is no end to the number of ways people can think about these topics.
ON THE SNAPPING OF A THIN SPHERICAL CAP
Authors: HU HAI-CHANG  ,   中国科学院数学研究所
Published: November 2020

In this paper, the snapping of a thin spherical cap under edge moment is considered. The snapping of the same cap under line load distributed along a circle as shown in Fig. 1 has been disscussed by Chien Wei-zang in two unpublished papers. His results are briefly summarized in this paper.
INTERNAL FRICTION PEAK ASSOCIATED WITH THE PRESENCE OF HYDROGEN IN STEEL
Authors: KE TING-SUI  , YUNG PAO-TSUI  ,   中国科学院金属研究所
Published: November 2020

Hair-line cracks or flakes in steel are known to be associated with the presence of hydrogen. A prevailing theory states that the crack formation is due to disruptive hydrogen pressure being built up in defects in steel. However, little was known regarding to the nature of these defects and the detailed mechanism of the hair-line crack formation. An attempt was made in the present research to investigate this problem by means of internal friction measurements. The preliminary experiments show: (1) The presence of hydrogen in certain steels may give rise to the appearance of an internal friction peak when internal friction is plotted against the temperature of measurement; (2) a slight amount of cold-work or soaking at a given temperature has a considerable effect on the optimum temperature of the observed internal friction peak, which indicates that the internal friction peak may be connected with the defects or stress condition in steel; (3) the internal friction peak shows an anomalous amplitude effect, i.e., at a given temperature of measurement an internal friction peak is obtained when the internal friction is plotted against the stress amplitude. Such an anomalous behavior is similar to that previously observed by. one of the authors (Ke) in the case of aluminimum containing 0.5% Cu. It is therefore considered that the internal friction peaks observed may be related to the interaction of hydrogen with the dislocations in steel.
ON THE LARGE DEFLECTION OF CIRCULAR PLATE
Authors: CHIEN WEI-ZANG  , YEH KAI-YUAN  ,   (1)北京大学力学专业; (2)清华大学及中国科学院数学研究所
Published: November 2020

Equations for the large deflection of thin plate established by Th. von Karman has been well known for many years. But so far there have been only a few iproblems studied with numerical certainty. S. Way was the first to apply these equations to solve the problem of a clamped plate under uniform pressure by the method of power series. After this, S. Levy found the solution of the simply supported rectangular plate under uniform load by the method of double trigonometric series. Both methods are too labourious to be applicable to other more important cases. Lately, Chien Wei-zang treated Way's problem again by means of the perturbation method and obtained excellent results. By the method as given by Chien Wei-zang, Yeh Kai-yuan worked out the problem of circular plate with a central hole under central concentrated load.In this paper, more results are given for various circular plates under various edge conditions. These include uniformly loaded circular plate under various edge conditions (section 2) and central concentrated loaded circular plate under various edge conditions (section 3). Such edge conditions are: (1) simply supported, (2) simply hinged, (3) rigidly clamped, (4) clamped but free to slip, (5) edge clamped but with possible slipping in horizontal direction, (6) edge simply supported but elastically fastened, and (7) edge clamped in elastic wall.All these results are presented in such a form that direct application in design problem is possible. In particular cases, under edge conditions (1) to (4), as σ=0.3, design formulae and curves for central deflection, radial tensile stress and radial bending stress are presented.
THE IMPACT PROPERTIES OF NODULAR CAST IRON AT VARIOUS TEMPERATURES
Authors: CHANG TSO-MEI  , SHü U-YUNG  ,   中国科学院金属研究所
Published: November 2020

Impact tests have been carried out on nodular cast iron in -the cast and forged conditions from -85℃ to 1070℃. The results indicate that nodular cast iron of predominantly ferritic matrix possesses two brittle temperature ranges (below 20℃ and between 550 and 700℃), and one comparatively more ductile range (between 850℃ and 1050℃).The fractures and the bending of the specimens at fracture have also been examined, and these have been correlated to the changes in structure of the matrix in the discussion.Finally, a brief account has been made of the possible applications of the results in the present research in evaluating the applicability of nodular cast iron at various temperatures; and, in particular, in the choice of proper hot working temperatures of this new material. The best hot working temperatures as indicated by the present results have been confirmed by actual forging and upsetting tests under the pneumatic and drop hammers respectively.
APPLICATION OF THE APPROXIMATE EVALUATION OF BESSEL FUNCTIONS TO FREQUENCY MODULATION SYSTEM
Authors: LIN WEI-GUAN  ,   华南工学院电讯工程系
Published: November 2020

In frequency modulation system the change of the percentage modulation affects the bandwidth of the transmitted signal. Outside the critical sideband the amplitude of all the higher order sidebands is a positive decreasing function of the order of the sidebands. In this paper we give an approximate determination of amplitude of this critical sideband as given by equation (8) or fig. (2); given also are the approximate magnitude of any side band above the critical and the ratio of the amplitude of any sideband above the critical to that of the critical, equation (11). By means of these two equations we can design the bandwidth of the transmission system by retaining a certain number of significant sidebands above the critical.
A PRELIMINARY REPORT ON MEASUREMENTS OF SOUND ABSORPTION COEFFICIENTS OF CHINESE ACOUSTICAL MATERIALS
Authors: WEI YUNG-CHIO  , YU CHUNG-CHI  ,   (1)东北人民大学物理系; (2)南京大学物理系
Published: November 2020

Normal sound absorption coefficients of some Chinese acoustical materials as measured by the tube method were tabulated and problem of conversion to random coefficients was discussed. Our results: most of the prefabricated units showed low absorption efficiency especially at low frequencies.In order to clarify the feasibility of these acoustical materials and to throw some light on the development of our future acoustical material industry, test data and relevant theoretical interpretations were presented for the effects of paint, thickness, flow resistance, and perforations etc. on absorption characteristics, and also for the functions of panel and perforated-panel absorbers. Emphasis was made for the last type for its versatile acoustical applications and insufficient use in this country. A conclusion was drawn with several suggestions both for the improvement of the existing products and for the manufacture of new ones.
DIE TEMPERATURABHANGIGKEIT DES PHOTOEFFEKTES DER Sb-Cs PHOTOKATHODEN IM TEMPERATURBEREICH-170℃—+ 20℃
Authors: ZS. NARAY  ,   中央物理研究所宇宙线研究室,布达佩斯
Published: November 2020

Es wurde die Temperaturabh?ngigkeit des Photoeffektes der Sb-Cs Photoka-thoden im Temperaturbereich -170℃-+20℃ untersucht. In der Nahe derGrenzwellenlange wurde ein monotoner Abfall der Empfindlichkeit mit abnehmen-den Temperaturen gefunden. Bei kurzeren Wellenlangen ist die Empfindlichkeit in einem weiten Temperaturbereieh unabhangig von der Temperatur und erst bei den tiefsten Temperaturen ergab sich ein steiler Abfall der Empfindlichkeit.