bluuebunny/crossref_metadata_2025 huggingface dataset:
Albrecht Dürer's Renaislance Connections between Mathematics and Art
Authors: Walton Karen Doyle
Published: November 2020

The great German artist Albrecht Durer (1471-1528) is best known for his woodcuts, paintings, and copper engravings as well as the thousands of books and articles that have been written about him. However, the mathematics in Durer's work has been given inadequate recognition, except by the occasional mathematics or art historian who has appreciated the important connections Durer made between the two fields. The purpose of this article is to bring those connections to the readers of this journal.
Implementing the Professional Standards for Teaching Mathematics: Self-Evaluation as an Act of Teaching
Authors: Kouba Vicky L.
Published: November 2020

A common call heard today is that we want “reflective teachers” who have the skills and expetiences necessary to engage in a high-level examination of teaching. Indeed, the authors of the NCTM's Professional Standards for Teaching Mathematics (Professional Teaching Standards), published in 1991, recommend that teachers assume more of the responsibility for both self-evaluation and peer evaluation.
Sharing Teaching Ideas: A Guide to More Sensible Word Problems
Authors: Marks Daniel
Published: November 2020

No academic subject seems to terrorize students and their teachers more than algebra “word problems.” Students say they would do much better in algebra if it weren“t for the word problems, which is like saying, “I'd do better in history class if I didn't have to learn all that stuff about the past.”
Technology Tips: A Global Electronic Community
Authors: Manes Michelle A.
Published: November 2020

The whole world is going digital! As a teacher, I have used my home computer to prepare materials, to keep track of grades, and to type final reports. But my computer holds so much more potential. As an active member of what I like to call “the on-line culture,” I have planned entire vacations from my desk, including purchasing plane tickets and requesting vegetarian meals, making hotel reservations, and budgeting the cost; I have used electronic mail to keep in touch with my family and friends; and I have learned how to clean the bearings and rotate the wheels of my in-line skates thanks to Usenet newsgroups. These newsgroups are like worldwide electronic bulletin boards. People use their computers to post questions, answers, opinions, ideas, and tirades to be read by anyone anywhere. Groups exist for every professional and personal interest imaginable.
Match Geometric Figures with Trigonometric Identities
Authors: Ren Guanshen
Published: November 2020

In most high school and college algebra textbooks, basic trigonometric identities are often proved by algebraic operations with other identities or presented without proof. This article presents proofs of some trigonometric identities from a geometric point of view and shows that some geometric figures beautifully match some trigonometric identities. The identities are proved with the use of geometric figmes, a process that offers a visual way to relate trigonometric properties. Studying trigonometric identities from a geometric perspective helps students make connections among various branches of mathematics and raises students' interests and appreciations in geometry. This idea was introduced in the context of a mathematics club, and the responses from students have been very positive. Some students did their own research and presented a paper at the regional Pi Mu Epsilon student conference.
Technology Tips: Is Calculus Obsolete?
Authors: Dubinsky Ed
Published: November 2020

So why do we need calculus if we have all this technology? I mean, if you want me to find all the relative maximum values of a function, I'll just draw the graph, zoom in on the humps, and get as close an approximation as you want.
Technology Tips: Navigating the Web
Authors: Doerr Helen M. , Hecht Caroline G.
Published: November 2020

Technological advances in high-performance computing and communications over the past decade have brought unprecedented power to the desktops of teachers and students. Within the past three years, the use of the World Wide Web-known as WWW, W3, or simply “the Web”—has increased dramatically with the advent of such software as NCSA Mosaic, Netscape, and Lynx. This software has led to an explosion of resources on the international information superhighway. These Web resources bold many possibilities for both students and teachers in secondary mathematics education by allowing new opportunities for collaboration, communication, and exploration.
Implementing the Professional Standards for Teaching Mathematics: Highlighting the Humanistic Dimensions of Mathematics Activity through Classroom Discourse
Authors: D’Ambrosio Beatriz S.
Published: November 2020

Classroom discourse is often understood as the process of engaging the members of the classroom community—students and teachers—in talking with one another. In this discussion 1 use the term classroom discourse to mean the process of engaging the classroom community in real dialogue, wherein meaning is negotiated and assumptions are questioned. An underlying assumption throughout the discussion is that classroom discourse can help shape the views of the nature of mathematics that are held by students. The goal of this article is to raise questions and invite the readers to reflect on the issues raised rather than to answer any specific questions.
Applications Short-Term Sales Forecasting
Authors: Murdick Kent
Published: November 2020

A medical-supply company asked for help in solving a warehouse-inventory problem. They wanted a computer program to track the inventory of more than one hundred medical items, such as cases of bandages and syringes, and to predict the sales for the next business period. Thus, when the company needed to order a particular item, the quantity could be calculated automatically by the program. Specifically, the problem concerned the short-term prediction of future sales.
Everybody Talks about It!—Weather Investigations
Authors: Dion Gloria S. , Fetta Iris Brann
Published: November 2020

Everybody talks about the weather, but few people consider the mathematical explanations behind weather-related topics. The study of weather is an excellent source of applications for students at all levels because the topic is familiar and easily understood. In other words, it is interesting because it is real.
Analyzing Data from the Olympic Games for Trends and Inferences
Authors: Edgerton Richard T.
Published: November 2020

The NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) emphasizes classroom mathematics that engages students in meaningful activities through which they construct their own understanding of important concepts. Students' investigations are derived from problem situations that arise from real-world contexts. The Olympic Games furnish ample data for students to connect meaningful mathematics with real-world problems.
Sharing Teaching Ideas: Active Participation In the Classroom through Creative Problem Generation
Authors: Gonzales Nancy A. , Fernandez Albert , Knecht Carine
Published: November 2020

The authors collaborated on a project that was designed to change how students perceived their role in the mathematics classroom—from being passive recipients of mathematics problems to becoming active participants in the creation of mathematics problems. Accomplishing such a task called for the development of an activity that would involve all students, provide an opportunity for more mathematical communication in the classroom, and link creative thinking with mathematical content. This article focuses on one such activity.
National Council of Teachers of Mathematics: Building Bridges of Mathematical Understanding for All Children
Authors: Price Jack
Published: November 2020

The theme “Building Bridges of Understanding” is especially important to the mathematics-education community. It proclaims the next steps that we all must take to make implementing the NCTM's Standards a long-lasting, successful effort to enhance mathematics education. Over the past two years, I have seen exciting changes taking place in many classrooms:
Soundoff!–Mathematics Reform for K-16
Authors: Ross Kenneth A.
Published: November 2020

The past ten years have seen a remarkable amount of progress in improving mathematics education at all levels. The goal is to enable all students, including those from all racial and ethnic backgrounds and both sexes, to master and appreciate mathematics. The emphasis is on understanding mathematics instead of thoughtlessly grinding out answers. For various reasons. resistance is increasing to what is usually called “math reform,” which reflects some serious concerns that need to be addressed.
Why Were So Few Mathematicians Female?
Authors: Kelley Loretta
Published: November 2020

I have been asked the question in the title many times during my professional life in mathematics, often as a challenge: If men are not innately better at mathematics than women, why have male mathematicians so outnumbered females? My answer is twofold. First, many women have become mathematicians. Second, when I study their lives, I do not wonder why more women did not choose this career. Considering the little support that they have received and the many barriers that have been placed in their way, it is remarkable that so many women have accomplished so much in mathematics.
Problem Solving: A Method for All Seasons
Authors: Hess Alice
Published: November 2020

The ultimate goal of the project presented herein is to teach students methods of problem solving through an industrial-engineering environment. The project has been used with calculus classes for several years with minor adjustments and with considerable success. Students are awakened to the importance of sharing responsibility, understanding group dynamics, using resources efficiently, meeting deadlines, articulating mathematics in oral and written form, and, of course, applying the appropriate mathematics.
College Preparation: What is its Effect on What you Teach and How You Teach it?
Authors: English Harry
Published: November 2020

I have taken the liberty of treating this topic in a somewhat broader way than would appear at first sight to be intended, though it is far from my intention to avoid a direct answer to the obvious meaning. It is hoped however that nothing shall be construed to be other than an unprejudiced presentation of a few salient points for earnest consideration.
Report of the Geometry Committee
Authors: Chambers Geo. Gailey
Published: November 2020

At the Pittsburgh meeting of this association this committee presented a preliminary report which outlined a plan of procedure. That report was approved at fhat meeting and published in the December, 1913, issue of the MATHEMATICS TEACHER. In that report the committee said that one of the first tasks which it had undertaken was that of preparing a suitable non-geometrical test to determine the results of geometry teaching as to reasoning ability, and in the report the committee gave five questions indicating in a general way the nature of the questions that should constitute such tests.
Maryland
Authors: Smith Eugene Randolph
Published: November 2020

There has been little change in either school or college mathematics in Maryland. Some unimportant criticisms of the subject have been made and answered, but without effect on the general procedure. Individual schools are using the more modern' methods in arithmetic and are doing progressive work in the high-school subjects, but the state as a whole has not been greatly influenced by recent developments.
Activities: Packing Them In
Authors: Carter Claudia R.
Published: November 2020

Spheres and cylinders are usually packaged in rectangular boxes. Is this packaging the best choice? In this activity, students investigate some questions in the mathematics of packaging, and in the process they explore and use various concepts in geometry, including area and the Pythagorean theorem. The mathematics comes out of the discussion of the packaging of soft-drink cans into six-packs and focuses on cost-effectiveness in terms of the horizontal area used and on the need for the packages to be easy to juxtapose on a shelf. A side benefit of the activity is to help students realize how the knowledge of mathematics allows exact answers that lead to generalizations and formulas.
Investigating a Definite Integral—from Graphing Calculator to Rigorous Proof
Authors: Touval Ayana
Published: November 2020

A wonderful opportunity presented itself in my advanced calculus class when we tried to evaluate the following definite integral (Faires and Faires 1988, 419):
Projects
Authors: Stenglein Sharon
Published: November 2020

The Minnesota Best Practice Mathematics Network is a statewide group of teacher-leaders committed to and able to support standards-based change in mathematics education throughout the state. The concept of “best practice” comes from the idea of using what we know from research and experience about successfuJ teaching and learning of mathematics. Network members are selected with two essential qualifications:
Sharing Teaching Ideas: Descriptive-Paragraph Miniproject
Authors: Kern Cherlyn
Published: November 2020

Students often have difficulty assimilating the meaning of the myriad vocabulary terms used in geometry. To help students master the definitions while having fun, I developed a descriptive-paragraph miniproject (fig. 1). Students are encouraged to pay particular attention to those terms that describe two- or three-dimensional objects. They are given one week in which to write their paragraphs. Paragraphs written by former students may be used as samples. The level of difficulty expected depends on the skill levels of the students.
The Pythagorean Theorem: An Infinite Number of Proofs?
Authors: Hoehn Larry
Published: November 2020

Recently, while trying to compose for my geometry students a test item involving perpendicularity, I came up with the following question
Angled Sunshine, Seasons, and Solar Energy
Authors: Kim Hy
Published: November 2020

What makes winner cold and summer hot? Most students have learned that seasons change because the rotational axis of the earth is tilted from the perperticular to its orbital plane. Exploring the relationship between the tilt of the earth's axis with respect to the orbital plane and the relative amount of solar energy received by different latitudinal locations is a real-world situation that can be analyzed using mathematics. What follows is an integrated unit involving mathematics and science. This unit illustrates two strands of the NCTM's Curricutum and Evaluation Standards for School Mathematics (1989). using mathematics as a problem-solving activity and making mathematical connections.
The Perils of Conditional Statements and the Notion of Logical Equivalence
Authors: Gregg Jeff
Published: November 2020

Increased attention should be given to the expression of deductive arguments in high school geometry, according to the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989). In general, the standards for grades 9-12 emphasize helping students develop their logical-reasoning skills. One means by which high school geometry textbooks typically introduce the notion of deductive reasoning and the principles oflogic is by examining conditional statements and their converses, inverses, and contrapositives. This article describes an episode involving conditional statements and the notion oflogical equivalence that occurred in a tenth-grade college-preparatorygeometry class. The episode illustrates some of the confusion that can arise in connection with this topic for both students and teachers. The ensuing discussion
Applications: Trigonometry for the Energy-Conscious Architect
Authors: Stephens Gregory P.
Published: November 2020

My favorite trigonometry problem has to do with controlling the entry of sunlight through a window. Sunlight is radiant energy and a significant source of heat in a building. I ask students to be energy-conscious architects and design a window overhang that will block the midday summer sun but allow plenty of sunlight in the winter. Both goals are reachable because the apex of the sun's circuit in the sky is much higher in summer than in winter.
Technology Tips: Spreadsheets: Powerful Tools for Probability Simulations
Authors: McClintock Edwin , Jiang Zhonghong
Published: November 2020

The Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) recommends that high school students extend their K-8 experiences with simulations and experimental probability to continue to improve their intuition and build more formal concepts of theoretical probability based on these experiences. In keeping with this idea, we use spreadsheets frequently in studying and investigating probabilistic situations with both high school students and preservice mathematics teachers. Our approach includes simulations but goes beyond the simulation process as a way of learning important concepts and principles of probability. We have found that spreadsheets are a very powerful tool for simulating probabilistic situations, not only for simple problems, such as simulating coin Ripping and die tossing, but also for rather complex problem situations. In this article, we illustrate the use of spreadsheets as a simulation tool for solving a collection of probability problems. The spreadsheet program we use is Microsoft Excel (1995), but other spreadsheet programs, such as Quattro Pro (Borland 1994), can be used with the same degree of effectiveness.
Activities: Exploring Knots
Authors: Adams Colin , Furstenberg Eric , Jie Li , Jodi Schneider
Published: November 2020

Teacher's Guide: Knot theory is a relatively new field of mathematics, initiated only at the end of the last century. Although much progress has occurred, a multitude of fascinating unanswered questions remains. Whereas some of these questions are very complex, others are simple enough to be explored by high school students and their teachers.
Numbers, Please! The Telephone Directory and Probability
Authors: Brunner Regina Baron
Published: November 2020

Coll aborative-group assignments in my introductory probability and statistics class have helped students understand such concepts as Monte Carlo simulation. Research has shown that cooperative-learning experiences improve problem-solving ability and conceptual learning when students in small groups communicate about mathematics (Johnson_and Johnson 1990).
Evaluation of Empowerment in a High School Geometry Class
Authors: Ridgway Carolyn , Healy Christopher
Published: November 2020

Since the publication of the Curriculum and Eualuation Standards for School Mathematics in 1989, the National Council of Teachers of Mathematics has been encouraging teachers to give more responsibility and choice to students. Students become mathematically empowered as they solve problems together in a community oflearners, communicate with one another concerning mathematical ideas, and use reason and logic to defend their work. To teach in accordance with these standards has required teachers to sruft the ways in which they view and manage their classrooms (Frye 1991).
The Recurring Fibonacci Sequence: Using a Pose-and-Probe Rubric
Authors: Shaw Kenneth L. , Aspinwall Leslie
Published: November 2020

As amateur Fibonacci explorers, we have investigated a known problem involving the Fibonacci sequence and tried our hand at extending it. Not only have the explorations been fun, they have been mathematically rewarding. The aesthetics of seeing the Fibonacci sequence recur in the strangest places is exhilarating. We often think back to our school days and wonder why our teachers presented countless rigid and uncompromising procedures and formulas instead of making the learning of mathematics more dynamic. In this article, we share some explorations and then furnish several problems for further investigation.
Teaching Statistics Using Humorous Anecdotes
Authors: Friedman Hershey H. , Halpern Noemi , Salb David
Published: November 2020

A demonstration of how fictitious anecdotes, especially humorous ones, can be used to teach statistical concepts effectively and to dispel some common misconceptions in statistics.
Whelk-come to Mathematics
Authors: Keller Brian A. , Thompson Heather A.
Published: November 2020

Do you ever wonder why animals behave as they do? For instance, scientists have noticed that northwestern crows drop whelks, large marine snails, consistently from a height of 5 meters. Why 5 meters? The answer to this question may seem obvious; they wish to break open the shells to eat the soft meat inside. But why 5 meters? This situation raises many interesting questions that are open to mathematical investigations. For instance, how many times must a whelk be dropped to break open its shell? What factors influence the height from which the whelk is dropped? Is 5 meters the optimum height to minimize the work?
Titanic: A Statistical Exploration
Authors: Takis Sandra L.
Published: November 2020

The tremendously popular movie Titanic has rejuvenated interest in the Titanic and its passengers. Students are particularly captivated by the story and by the people involved. Consequently, when I was preparing to explore categorical data and the chisquare distribution with my class, I decided to use the available data about the Titanic's passengers to interest students in these topics. This article describes the activities that I incorporated into my statistics class and gives additional resources for collecting information about the Titanic.
Understanding Connections between Equations and Graphs
Authors: Knuth Eric J.
Published: November 2020

General consensus holds within the mathematicseducation research community that functions are among the most important unifying ideas in mathematics (Romberg, Carpenter, and Fennema 1993). In fact, “it can be argued that functions form the single most important idea in all of mathematics, at least in terms of understanding the subject as well as for using it” (Dubinsky 1993, 527). Further, the introduction of algebraic and graphical representations of functions can be seen as a crucial moment in mathematics learning and represents “one of the earliest points in mathematics at which a student uses one symbolic system to expand and understand another” (Leinhardt, Zaslavsky, and Stein 1990, 2).
Algebra for Students with Learning Disabilities
Authors: Feigenbaum Ruth
Published: November 2020

The learning environment created in the special algebra classes, combined with the teaching strategies developed, has produced positive results for students with learning disabilities.
Sharing Teaching Ideas: Relay Review
Authors: Gaglione Jeffrey T.
Published: November 2020

Being a mathematics teacher means more than simply teaching the subject itself. Students also need to see the relevance of mathematics to their everyday lives, and they should enjoy learning the subject. If a student comes into my classroom in September and leaves in June without seeing the relevance of mathematics and having fun in class, I do not think that I have done that student—or the subject—justice. Group work and projects are two ways to accomplish these goals in the classroom. Another way is to play review games. Students rank one of the games that we play, “relay review,” as their favorite year after year.
Analyzing Unstable Systems of Linear Equations
Authors: Shultz Harris S.
Published: November 2020

The Curriculum and Evaluation Standards for School Mathematics (NCTM 1989, 146) states, “Students who are able to apply and translate among different representations of the same problem situation or of the same mathematical concept will have at once a powerful, flexible set of tools for solving problems and a deeper appreciation of the consistency and beauty of mathematics”. The Standards also states, “The connections between algebra and geometry are among the most important in high school mathematics” (1989, 147). This article shows how the phenomenon of instability in the solution of a system of linear equations can be analyzed both algebraically and geometrically.
Strength of Wood Beams: An Engineering Application
Authors: Dengerud-Au Mary
Published: November 2020

NCTM's Curriculum and Evaluation Standards for School Mathematics (1989, p. 76) states that allowing students to experience problems “that have multiple solutions, each with different consequences, will better prepare them to solve problems they are likely to encounter in their daily lives.” Structural engineering furnishes an ideal context in which students can learn how different solutions to a problem can affect human lives. In designing a structure, engineers must consider economic constraints, aesthetics, and most important, safety. One engineer might use too much material in a given structure, thus making it more expensive than it should be. Another might save money by using less material but compromise the integrity of the structure. Still another engineer might find an acceptable balance between cost and safety but design a structure so aesthetically unpleasant that nobody wants to look at it.
Mathematics in the Age of Jane Austen: Essential Skills of 1800
Authors: Gray S. I. B.
Published: November 2020

THE ACADEMY AWARD–WINNING MOVIE Sense and Sensibility presented a wonderful vision of life in early nineteenth-century England. In the absence of television, radio, movies, and videos, families sought entertainment in a manner far different from today's. The Dashwood girls—Elinor, Marianne, and Margaret—filled their days with visiting, reading, practicing the pianoforte, needleworking, and letter writing, not to mention gossiping and matchmaking. Long days were highlighted by a wonderfully relaxed midday family meal, during which conversation was paramount. Above all, Jane Austen portrays a concern for the thoughts and feelings of one's immediate acquaintances and pride in one's village.
Activities for Students: Big Box—Office Bucks
Authors: Powers Robert A.
Published: November 2020

One of the challenges that teachers face when they teach data analysis to their students is getting them to see the “big picture.” Using multiple representations is one way to help students make sense out of a large amount of data. For example, the data can be recorded in a table, graphed in a rectangular-coordinate system, and analyzed for an equation of best fit. These techniques are standard practice in data analysis. However, furnishing an interesting context for teaching these methods is sometimes difficult. One way to motivate students to see the big picture of data analysis is to explore an example from the motion-picture business. Since the release of James Cameron's movie Titanic in December 1997, the film has become a part of the popular culture. It earned more than $1 billion worldwide at the box office, which makes it the highest-grossing film of all time. Inevitable articles and programs documented and commented on the success of this epic film. One article, in particular, in Newsweek was titled “Our Titanic Love Affair” (Ansen 1998). A simple statement of earnings trends appeared in a caption next to a graph showing the longevity of the weekend revenues generated by the movie. It read as follows: “Most hit movies enjoy big opening weekend sales, then revenues decline quickly.” Both the graph and the caption raise two interesting questions: What do the graphs and trends of “most hit movies” look like? And how are the weekend earnings of Titanic different?
Media Clips: February 2001
Authors: Westegaard Suzanne K.
Published: November 2020

We welcome and encourage submissions to this department, which will comprise resources that teachers can use immediately or file away for future use. The assignments generated by these entries may range from starters for an entire class to full-blown projects for optional individual pursuit. Each lesson, though, should help convince our students that mathematics is useful, current, and newsworthy. Please send submissions to “Media Clips,” Mathematics Teacher, 1906 Association Drive, Reston, VA 20191-9988.
Discuss with Your Colleagues: Equiareal Polygons: A Mathematical Conversation about a New Concept
Authors: Winicki-Landman Greisy
Published: November 2020

This article describes an activity that is connected with mathematical definitions and that illustrates the process of gradual refinement as a way to understand and construct knowledge. It presents a gradual construction of a specific geometry concept that was the result of the interaction among the participants in a mathematical discourse (Pimm 1987). This activity took place at the end of a professional development program for teachers of secondary school geometry. During this fourteen-week program, the participants were exposed to the NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) and to several activities adapted from two volumes in the Addenda Series: Geometry in the Middle Grades (Geddes 1992) and Geometry from Multiple Perspectives (Coxford 1991).
Sharing Teaching Ideas: Less Is Sometimes More
Authors: Socha Susan
Published: November 2020

In 1998, the keynote speaker for the Virginia Council of Teachers of Mathematics Conference was Gail Burrill. I had read many of her articles and even used some of the materials in her books, but that occasion was the first time that I had heard her speak. During the talk, she described an elementary school classroom that was working on data collection and analysis. The teacher had not provided a template or worksheet for the students to use to collect the data, so they had to come up with their own formats. The teacher acted as a coach, walking around the room as the children counted pumpkins, leaves, and other objects and making suggestions, such as “How do you know which objects your tally marks represent?” The students had to decide how they should represent the objects and then how to present the data in a usable format. Burrill's point was that we sometimes give students too much structure and not enough real-world experience. In a work environment, no one hands employees a set of instructions with a worksheet to collect data and questions to assist in interpreting those data. Workers must design their own experiment as part of the process.
Determining the Endpoint of a Segment
Authors: Mustafa Asif Khan
Published: November 2020

A new method for determining the coordinates of the endpoint of a line segment, given its slope, point of origin, and length.
Understanding Transformations of Periodic Functions through Art
Authors: Miller Syrilda
Published: November 2020

The use of interdisciplinary units can satisfy requirements found not only in the NCTM's Principles and Standards for School Mathematics (2000) but also in such state guidelines as New Jersey's Core Curriculum Content Standards. The secret to developing connected curricula is to use professional connections, which include many resources: professional organizations, teachers, and your own students.
Rugby and Mathematics: A Surprising Link among Geometry, the Conics, and Calculus
Authors: Jones Troy , Jackson Steven
Published: November 2020

As mathematics teachers, we are always on the lookout for motivational applications for the mathematics that we teach. Applications are especially useful if they help students make connections among the various topics of mathematics and can be used by students at several different levels. While thumbing through The Penguin Dictionary of Curious and Interesting Geometry, by David Wells (1991), we ran across an entry that led to a very interesting discovery
Flying through Graphs: An Introduction to Graph Theory
Authors: McDuffie Amy Roth
Published: November 2020

The mathematics education community has called for changes in the high school curriculum to increase the emphasis on meaningful problem solving and on topics in discrete mathematics (National Council of Teachers of Mathematics 1989, 1991, 2000). This recommendation resulted from changes in knowledge and revisions in problem-solving needs because of advances in such fields as information processing and computer technology. Including graph theory in the curriculum is one way to meet these goals. Graphs present an opportunity to model and analyze such problem situations as networks and circuits. This activity incorporates basic terminology, concepts, and solution methods of graph theory in the context of solving problems related to air travel.
Sharing Teaching Ideas: Roots in Music
Authors: Houser Don
Published: November 2020

Mathematics and music? In college, as a mathematics major with a music minor, I never realized that connections between the two–other than mundane applications of fractions to note values–might exist. Subsequent teaching experiences have furnished me with delightful and teachable connections that I enthusiastically share with my students. I have even observed that the musicians “perk up” when I do so.
Sharing Teaching Ideas: Using the Law of Cosines to Teach the Ambiguous Case of the Law of Sines
Authors: Harrison Elizabeth P.
Published: November 2020

The laws of sines and cosines easily lend themselves to links with other areas of algebra and geometry. The most-used link is probably that of congruent triangles, but additional links exist with imaginary numbers, the quadratic formula, parabolas, zeros of functions, and the triangle inequality.
More Meaning From the Geometric Mean
Authors: Dorner Bryan C.
Published: November 2020

We are all familiar with the average, or arithmetic mean, of two numbers. Less frequently used is the notion of the geometric mean. In “Geometric Meaning in the Geometric Mean Means More Meaningful Mathematics” in the March 2001 issue of the Mathematics Teacher, Matt E. Fluster shows how the geometric mean, s = _ab, of two positive numbers, a and b, can be used in a first-year algebra course to tie together geometric, algebraic, and computational investigations. In this article, I add a bit of history and an example suitable for more advanced courses. The example uses the geometric mean to compute square roots with Newton's method and does not require calculus. The history surrounding these concepts provides an opportunity to point out the universal nature of mathematics as a human activity. The basic ideas in this article were alive in the minds of ancient peoples who lived in what are now India, Pakistan, Iraq, and Egypt.
A Reasonable Restriction Set for Solving Radical Equations
Authors: J. Gurevich Vladimir
Published: November 2020

The traditional method for solving radical equations requires a final step of checking answers to identify and reject extraneous solutions. In this article, I introduce the reasonable restriction approach, a method that eliminates the extraneous solutions at the beginning of the solution process instead of at the end. This method offers students good “up-front” thinking rather than tail-end checking and adds to the understanding of extraneous roots and why they occur. It also offers opportunities to solve radical equations that are either impossible or very tedious to solve by the traditional checking method, as later examples show.
Calendar Problems: February 2004
Authors: Benson Steve
Published: November 2020

Problems 1–24 were all taken from problems originally appearing in The Ideas of Algebra, K–12, the 1988 NCTM Yearbook (Reston, Va.: National Council of Teachers of Mathematics, 1988). The problems were supplied by Terry Goodman, Central Missouri State University, Warrensburg, and Martin Cohen, University of Pittsburgh. Problems 25 and 26 came from the Math Forum Problems of the Week available at www.mathforum.org. Problems 27–29 originally appeared on the 2003 AMC12 examination, sponsored by American Mathematical Competitions (Web site www.unl.edu/amc).
Fostering Mathematical Inquiry with Explorations of Facial Symmetry
Authors: Edwards Michael Todd
Published: November 2020

AS A SECONDARY SCHOOL MATHEMATICS TEACHER, I have the opportunity to explore a wide range of mathematical topics with a vast array of students. Student variability–from student to student and from class to class–routinely challenges my instructional ingenuity. Although the mathematical concepts presented in such courses as first-year algebra or geometry are more basic than those presented in upper-level classes, entry-level courses are among the most challenging that I teach.
Media Clips: April 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.
Connecting Research to Teaching: Action Research: A Tool for Exploring Change
Authors: Evitts Thomas A.
Published: November 2020

Action research offers teachers the opportunity to investigate questions and concerns related to their teaching practice and to their students' learning. Often conducted intuitively and informally, action research constitutes a natural component of teaching (Fischer 1996). Nolan and Hoover (2004, p. 136) state, “Good teachers are constantly engaged in a process of asking questions about students and their learning and of gathering information … to answer those questions.” Through action research, teachers have “a chance to shape what happens in their classrooms and relate what they believe with what they practice” (Burnaford 1996, p. 58)
Examining Students' Reluctance to Use Graphs
Authors: Dyke Frances Van , White Alexander
Published: November 2020

What students do not understand about graphing and why they remain reluctant to use it for problem solving at the calculus level. The authors point out that graphing was not developed until the 17th century. Conclusions are that students lack understanding in the concept of function, the Cartesian connection, and the mathematical description of change.
Visualizing the Domain and Range of the Composition of Functions
Authors: Neger Nial , Frame Michael
Published: November 2020

Graphical methods for visualizing addition and subtraction of functions are familiar to every teacher. This article presents a simple graphical method for visualizing the composition of functions. This straightforward application enables users to find the domain and range of compositions. Finding the domain by standard approaches is challenging to the point that it often is ignored by composing functions with domains consisting only of real numbers. By contrast, finding the domain of the composition of functions with restricted domains is easy with this method. In addition, teachers and students can generate families of functions in which composition domains undergo all sorts of changes.
Placing a Deck of Cards in Order
Authors: Yates Mark
Published: November 2020

While playing cards one evening with my family, I became curious about whether I was wasting moves while reordering my cards. After each hand was dealt, I studied it for a minute until I thought that I had determined the smallest number of moves needed to arrange the cards by suit. After a while, I wondered whether I could approach this topic mathematically.
Another Way to Divide a Line Segment into n Equal Parts
Authors: Besteman Nathan , Ferdinands John
Published: November 2020

In summer 1995, two high school students, David Goldenheim and Dan Litchfield, discovered a way to divide a line segment into any number of equal parts. Their method differed from the standard method of Euclid. Together with their teacher Charles Dietrich, they wrote an article on their method, which appeared in the January 1997 issue of the Mathematics Teacher (Litchfield, Goldenheim, and Dietrich 1997). The discovery received considerable publicity in the popular media and was written up in the Wall Street Journal and the New York Times. The authors gave talks at several professional conferences and were invited to meet the secretary of education.
A Proposal to Address the Shortage of Highly Qualified Mathematics Teachers
Authors: Paul Clyde A.
Published: November 2020

Available research supports a belief that a highquality mathematics education for high school students cannot be accomplished unless a highly qualified mathematics teacher is in the classroom. In the January 2004 issue of the Mathematics Teacher, Robert Reys addressed the shortage of highly qualified teachers. In the February 2004 issue Robert and Barbara Reys provided a summary of the work they are leading in Missouri to recruit persons who have the potential to become highly qualified teachers of mathematics. I have assisted them in that work. The undertaking is important and, as they report, appears to be achieving some short-term success. I believe that the national shortage of high-quality mathematics teachers will not be significantly alleviated, however, until we focus our major effort on the retention side of the issue.
The Legacy of Marin Mersenne: The Search for Primal Order and the Mentoring of Young Minds
Authors: Wanko Jeffrey J.
Published: November 2020

The search for prime numbers has long held a great fascination for mathematicians and for mathematics enthusiasts. Whether as a mathematical recreation or as a serious study within number theory, this quest has resulted in some profound mathematical advances and in a few surprising results that held some unforeseen applications and connections to other areas. For example, the problems of finding perfect numbers and constructible regular polygons have both been simplified through the search for prime numbers (Bell 1937, Clawson 1996).
You've Heard of Cramer's Rule, Now Try Comer's: An Alternative Approach to Finding Determinants
Authors: Genovese Michelle J.
Published: November 2020

Ihave an article on my bulletin board that I cut out years ago about two summer school students from Connecticut's Greens Farms Academy. Usually, near the beginning of every year, a few of my students read about how the two students from Greens Farms solved a problem that “geometry students had wrestled with for centuries” (Wagner 1996). Often a student will suggest that he or she wants to discover something new, too.
Sharing Teaching Ideas: Multiplying Polynomials
Authors: O'Neil Michael C.
Published: November 2020

I find that the method demonstrated below works very well in my algebra classes, as well as for students in other mathematics classes. Several of these students who have been diagnosed with a learning disability receive individual instruction in a resource room setting and most likely are on an Individualized Educational Plan (IEP).
An Engaging Puzzle to Explore Algebraic Generalizations
Authors: Shockey Tod L. , Bradley David M.
Published: November 2020

In this article, we discuss a popular puzzle we have used numerous times with our preservice elementary and secondary mathematics teachers. Students find the puzzle engaging; its primary appeal to us lies in the mathematics behind it.
The Human Body's Built-In Range Finder: The Thumb Method of Indirect Distance Measurement
Authors: Wong Michael
Published: November 2020

By using body parts and sightlines to find the distance to an object, students can reinforce the study and application of similar triangles. This activity is a good combination of tactile learning, practical application, careful observation and modeling, and higher-order thinking. The teacher may subsequently pursue the material in greater depth or use it as a starting point for excursions into other fields of study.
Technology Tips: Sliders and Parameters Extend Interactive Software
Authors: (Rod) Rodrigues Mariano
Published: November 2020

I write this realizing it is my final “Technology Tip” column as editor. I wish to thank everyone who contributed a tip or “Surfing Note” this year. I would also like to thank Shannon Driskell, my co-editor, Rod Rodrigues and Ruth Casey, our liaisons from the editorial panel, and Albert Goetz, the journal editor, for their collaboration in preparing each column this year.
Key Ideas and Insights in the Context of Three High School Geometry Proofs
Authors: Raman Manya , Weber Keith
Published: November 2020

According to the NCTM Standards (2000), conjecturing and proving should be central activities throughout students' mathematical education. However, the question of how we can help students generate proofs, especially the formal proofs expected at the high school level, is still a difficult one. In this article, we argue that one cause of students' troubles with proof is that they are not accustomed to making the important, but difficult, connections between their intuitive sensemaking and the formal proofs they are supposed to produce. Thus as teachers, we should provide opportunities for students to make these connections.
Spatial readout of visual looming in the central brain of Drosophila
Authors: Morimoto Mai M , Nern Aljoscha , Zhao Arthur , Rogers Edward M , Wong Allan M , Isaacson Mathew D , Bock Davi D , Rubin Gerald M , Reiser Michael B
Published: November 2020

Visual systems can exploit spatial correlations in the visual scene by using retinotopy, the organizing principle by which neighboring cells encode neighboring spatial locations. However, retinotopy is often lost, such as when visual pathways are integrated with other sensory modalities. How is spatial information processed outside of strictly visual brain areas? Here, we focused on visual looming responsive LC6 cells in Drosophila, a population whose dendrites collectively cover the visual field, but whose axons form a single glomerulus—a structure without obvious retinotopic organization—in the central brain. We identified multiple cell types downstream of LC6 in the glomerulus and found that they more strongly respond to looming in different portions of the visual field, unexpectedly preserving spatial information. Through EM reconstruction of all LC6 synaptic inputs to the glomerulus, we found that LC6 and downstream cell types form circuits within the glomerulus that enable spatial readout of visual features and contralateral suppression—mechanisms that transform visual information for behavioral control.
Casein kinase 1G2 suppresses necroptosis-promoted testis aging by inhibiting receptor-interacting kinase 3
Authors: Li Dianrong , Ai Youwei , Guo Jia , Dong Baijun , Li Lin , Cai Gaihong , Chen She , Xu Dan , Wang Fengchao , Wang Xiaodong
Published: November 2020

Casein kinases are a large family of intracellular serine/threonine kinases that control a variety of cellular signaling functions. Here we report that a member of casein kinase 1 family, casein kinase 1G2, CSNK1G2, binds and inhibits the activation of receptor-interacting kinase 3, RIPK3, thereby attenuating RIPK3-mediated necroptosis. The binding of CSNK1G2 to RIPK3 is triggered by auto-phosphorylation at serine 211/threonine 215 sites in its C-terminal domain. CSNK1G2-knockout mice showed significantly enhanced necroptosis response and premature aging of their testis, a phenotype that was rescued by either double knockout of the Ripk3 gene or feeding the animal with a RIPK1 kinase inhibitor-containing diet. Moreover, CSNK1G2 is also co-expressed with RIPK3 in human testis, and the necroptosis activation marker phospho-MLKL was observed in the testis of old (>80) but not young men, indicating that the testis-aging program carried out by the RIPK3-mediated and CSNK1G2-attenuated necroptosis is evolutionarily conserved between mice and men.