bluuebunny/crossref_metadata_2025 huggingface dataset:
New Projects
Authors: Hobbs Margie , Lott Johnny W. , Piccolino Anthony , Wheeler Margariete Montague , Whitman Nancy C.
Published: November 2020

The claim is often made that one can learn BASIC in a few hours or that a two-week programming supplement to an existing course can meet all the needs of the high school student with respect to computer programming. Although computer “literacy” can be part of any mathematics program through simple restructuring or such modest curricular additions as those cited above, many high school students have interests and capabilities that demand more depth than this. In fact, as society becomes more knowledgeable about the role and function of the computer and as freshmen college students find increasing use of the computer in their introductory courses, high schools are faced with a mandate to provide courses that offer a depth of knowledge in programming that exceeds the traditional high school computer courses. In its seven years of operation Briarcrest High School, an independent school in Memphis, Tennessee, has developed a computer curriculum that provides instruction beyond the introductory level.
A Method of Graphing f(x) = A sin (Bx + C) + D
Authors: John Hornsby E.
Published: November 2020

I have found after teaching trigonometry for many years that one of the most troublesome topics in the course is the graphing of trigonometric functions.
Implementing The Standards: Statistics and Probability
Authors: Burrill Gail
Published: November 2020

Statistics—the collection, organization, and interpretation of data; the art and science of an-alyzing information—was until the late 1960s the domain of those gifted in mathematics or those few who needed limited knowledge to make inferences within their chosen field. The school curriculum furnished little background for the (“science of numbers.” Statistics was a vast array of symbols, formulas, and rules that seemed to have little relationship to reality. During the 1960s, a combination of circumstances indicated a need to change the role of statistics in society: the development of computers with the capacity to create, store, and analyze large quantities of data; the formation of new, simple, and effective data-analysis techniques; and the occurrence of rapid changes in personal and working environments of society.
Soundoff: Mathematics: A Wonderful Kind of Play
Authors: Merow Craig B.
Published: November 2020

When Gerald L. Alexanderson interviewed Constance Reid for the book Mathematical People (Albers and Alexanderson 1985), he asked her what it was about mathematics and mathematicians that originally excited her.
Coordinate Geometry: A Powerful Tool for Solving Problems
Authors: Taback Stanley F.
Published: November 2020

In calling for reform in the teaching and learning of mathematics, the Curriculum and Evaluation Standards for School Mathematics (Standards) developed by NCTM (1989) envisions mathematics study in which students reason and communicate about mathematical ideas that emerge from problem situations. A fundamental premise of the Standards, in fact, is the belief that “mathematical problem solving … is nearly synonymous with doing mathematics” (p. 137). And the ability to solve problems, we are told, is facilitated when students have opportunities to explore “connections” among different branches of mathematics.
Combinatorial Scheduling
Authors: Dieffenbach Robert M.
Published: November 2020

Small projects like putting up a swing set and big projects like building a house all consist of interrelated tasks. Some of these tasks must be done in sequence, but often several tasks can be going on at once.
Sharing Teaching Ideas: Discovering a new Algorithm for area
Authors: Margulies Susan
Published: November 2020

During a recent discussion in my Advanced Placement calculus class regarding the area bounded by two curves, one of my students, Jon Brod, noticed that certain results occurred with great frequency.
Areas and Perimeters of Geoboard Polygons
Authors: Smith Lyle R.
Published: November 2020

Some problems in mathematics can be used to help students refine skills in computation and in application of generalizations. Other problems are useful to students in developing processes essential to inquiry and discovery. Problems that can help students to refine skills as well as to inquire and discover have particular appeal. This article focuses on such problems.
Soundoff: Let's Abolish General Math!
Authors: Seddon Thomas E.
Published: November 2020

Recently I taught “general math” in eleventh grade for the first time in almost twenty years of teaching. I was as unprepared as anyone for the generalmath challenge.
Implementing the Standards: Cooperative Learning
Authors: Artzt Alice F. , Newman Claire M.
Published: November 2020

Cooperative learning is a teaching strategy that holds promise for improving the mathematical skills and attitudes of students. With increased frequency teachers have been reporting their positive experiences with cooperative-learning techniques in the mathematics classroom (e.g., Artzt [1979]; Behounek et al. [1988]; Gilbert-Macmillan and Leitz [1986]; Rosenbaum et al. [1989]; and Slavin [1987]). Support for cooperative learning now comes from the NCTM's Curriculum and Evaluation Standards for School Mathematics (Standards) (1989): “Small groups provide a forum in which students ask questions, discuss ideas, make mistakes, learn to listen to others' ideas, offer constructive criticism, and summarize their discoveries in writing” (p. 79). Group assignments enable learners to work together, helping each other integrate new knowledge with prior knowledge and discover their own meanings as they explore, discuss, explain, relate, and question new ideas and problems that ar ise in the group.
A Mathematical Magic Show
Authors: Dehan Harriet Stone
Published: November 2020

What inspires people to become artists, athletes, scientists, and possibly even mathematicians? What ignites the spark of curiosity that drives young minds to explore science and mathematics so that they can try to unravel the mysteries of the world around them? How many of us have been profoundly affected by a passage in a book, the colors in a pain or an encounter with the elegant beauty and astounding power of mathematics? These questions suggest an approach that might be helpful to young students, many of whom are simply not motivated by the applications of mathematics to everyday life. An overview of mathematics, presented in a dynamic, compelling way, can give students new reasons to persevere in their studies. Although most people are aware of the utility of mathematics and its use in the sciences, relatively few know of its role in the development of Western culture, its influence on philosophy and the arts, and that it is a continually expanding field in which new and exciting things are being done.
Making Your Own Rules
Authors: Brutlag Dan
Published: November 2020

“Making conjectures, gathering evidence, and building an argument to support such notions is fundamental to doing mathematics” (NCTM 1989, 7). This statement represents one of the five curricular goals found in the introduction to NCTM's Curriculum and Evaluation Standards for School Mathematics. However, the work that students typically do in mathematics classes consists of proving someone else's conjectures, gathering evidence according to someone else's directions, and imitating someone else's arguments. This article presents a method I have used to help students start their own original investigations and create their own conjectures.
Sharing Teaching Ideas: The Gymnasium—A Dynamic Classroom For Teaching Slope
Authors: Grocki John M.
Published: November 2020

During our after-school gymnastics program, a student-gymnast commented that the inclined mat that we use for spotting had a slope. Having discussed slope in prealgebra class that morning, I asked the gymnasts to estimate the slope using a folding mat (fig. 1). We had discussed many times that estimates must be made from the materials at hand. I saw the moment and seized it for a great lesson. The following day I brought the students from my seventh-grade prealgebra class to the gymnasium for alesson whose objective was to help the students estimate and calculate slope. I assigned students to find slopes for the beams, the bars, and the braces on the gymnastics equipment (fig. 2). The students gathered the materials needed to measure and calculate the results: tape measures, calculators, pencils, and papers.
The Difference Equation Xn = axn−l + b
Authors: Spence Lawrence E.
Published: November 2020

The study of arithmetic and geometric progressions is part of the traditional twelfth-grade mathematics curriculum. This article discusses a generalization of these concepts that has many important applications. Furthermore, our principal formula can be obtained easily from familiar facts about arithmetic and geometric progressions.
Sharing Teaching Ideas
Authors: Greenwood James , Cunningham Robert F.
Published: November 2020

I had the pleasure of attending a presentation by Richard J. Light at the University of Hartford's College of Basic Studies 1992 Spling Colloquium. He teaches at Harvard University and is director of the Harvard Assessment Seminars. In one anecdote, he described the success of a history teacher who had photocopied students' work and distributed it in class. I adapted the idea to one of my mathematics classes with exciting results.
Are Most Fractions Reduced?
Authors: Shilgalis Thomas W.
Published: November 2020

Mathematics teachers delight when they encounter a problem that is interesting in its own right but that has the added feature of acting as a springboard for fruitful discussion in a broader context. My classes in probability and statistics for preservice middle school teachers have enjoyed both the question posed in the title and its analysis.
Sharing Teaching Ideas: Career Posters
Authors: Tibbs Peggy , Jordan Janette
Published: November 2020

After teaching high school mathematics for many years I found the perfect way to respond to the students' question, “How are we ever going to use this in the real world?” Two or three weeks into the school year I ask each student to make a career poster. The student must interview someone who uses mathematics in his or her job and write down an actual problem that person would have to solve as well as a paragraph explaining the problem. Most students think that they don't know anyone who uses mathematics at work, including parents, relatives, or neighbors. Usually they come back the next day to report, to their surprise, that their parents use mathematics! This discovery is a revelation to them.
The Converse of the Pythagorean Theorem
Authors: Rosenthal Jerome
Published: November 2020

An unusual monograph titled The Pythagorean Proposition by E. S. Loomis ([1927], 1968) is a collection of 256 proofs of this well-known theorem. Of fmther interest is the review written in the Mathematics Teacher (Christofferson 1928). A short excerpt from the review follows: “[I]t is impossible in a short review to do justice to this splendid bit of research work done for the love of mathematics.”
Sharing Teaching Ideas: Comparing three cursors
Authors: Reinford Daniel J.
Published: November 2020

On the Tl-82 graphing calculator, three cursors can be used to explore graphs. The free-moving cursor is shaped like a plus sign and has a blinking square at its center. The trace cursor is a square with five blinking squares—one at its center and one at each corner. The result cursor is shaped like a multiplication sign and also has a blinking square at its center.
Implementing the Assessment Standards for School Mathematics Quality Mathematics: How Can We Tell?
Authors: Clarke David , Wilson Linda
Published: November 2020

The motivation for this article comes from a conversation that occurred in my home recently.
Conjectures in Geometry and The Geometer's Sketchpad
Authors: Giamati Claudia
Published: November 2020

The Geometer's Sketchpad lets the user explore simple, as well as highly complex, theorems and relations in geometry. Although the user interface takes time to learn, students can use it to test a wide variety of conjectures once they have mastered this construction tool. The Geometer's Sketchpad has the ability to record students' constructions as A scripts. The most useful aspect of scripting one's constructions is that students can test whether their constructions work in general or whether they have discovered a special case.
Climbing Around on the Tree of Mathematics
Authors: Kennedy Dan
Published: November 2020

When I was in graduate school umpteen years ago, I was occasionally visited by anxiety attacks centered on such thoughts as “What am I doing here?” Specifically, I wondered how someone as obviously inconsequential as myself could hope to contribute any original thought to the vast lexicon of original thoughts known collectively as Mathematics. Without that obvious prerequisite, what would I use as a dissertation?
Technology Tips: Making Connections Using Embedded Software
Authors: Groden Claire , Pattison-Gordon Laurie
Published: November 2020

The NCTM's Curriculum and Evaluation Standards or School Mathematics (1989) calls for increased ttention to “connecting mathematics to other subjects and to the world outside the classroom.” Often, these connections are made with interdisciplinary projects and through the study of mathematics embedded in a real-world situation. We can also make connections by using software created for practical, real-world applications in the mathematics classroom
Revitalizing First-Year Algebra through Problem Solving with HALP
Authors: Dougherty Barbara J. , Matsumoto Annette N.
Published: November 2020

“We're ready!” students shout. Each group has just spent about eight minutes preparing a presentation about a homework problem. Agatha is randomly selected from her group to present their ideas. She puts up the transparency. See figure 1.
CORD Applied Mathematics: Hands-On Learning in Context
Authors: Pedrotti Leon S. , Chamberlain John D.
Published: November 2020

My and José are hurrying to their morning mathematic class. They are excited! Today they are scheduled to do a mathematics-laboratory assignment. Twice a week, their classroom turns into a laboratory where they use real measuting equipment—such as a vernier caliper, a carpenter's square, or a stopwatch. They collect and analyze data. They ee just how the mathematics they learn in the classroom helps them solve real-world problems. They really like these assignments.
Finding Quadratic Equations for Real-Life Situations
Authors: Korithoski Theodor
Published: November 2020

What simple techniques are available to integrate technology and real-life mathematics into an algebra class? What useful tools can assist teachers in this task?
Using a Feedback Form to Communicate with Students
Authors: Potter Margaret A.
Published: November 2020

The NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) has caused a stir of creativity, some of which has been used to devise ways to align instruction and teacher-student or student-teacher communication. One of the more visionary methods for assessment that teachers are currently using is journal writing. Journals can give both teachers and students great insight into a student's progress throughout a grading period.
Activities: Medical Applications of Systems of Linear Equations
Authors: Jabon David , Nord Gail , Wilson Bryce W. , Colfman Penny , Nord John
Published: November 2020

This activity is aimed at enriching the algebra strand by bringing to the classroom a medical application of systems of linear equations. These problems go beyond the traditional interpretation of systems of equations as intersections of lines and planes and furnish a nice mix of algebraic and geometric concepts. The activity is designed for small-group work.
From the Editorial Panel: A World of Ideas at Your Fingertips
Authors: Koss Roberta K.
Published: November 2020

A teacher's workday is filled with so many tasks—planning and presenting lessons, giving help to individual students, contacting parents, acting as advisors for extracurricular activities, serving on committees, assessing students' work—that adding another responsibility seems impossible. However, professional development is a necessary task that affects all aspects of a teacher's work. The Professional Standards for Teaching Mathematics (NCTM 1991) calls for teachers to take an active role in their own professional development and lists “reading and discussing ideas presented in professional publications“ (p. 16g) as an activity that will enhance professional growth. The necessity of reading professional journals can be a blessing in disguise because teachers can gain myriad ideas to help with their work. I shall share a few of the ways in which I plan to use the 1996–1997 Mathematics Teacher to help me plan my lessons, prepare student activities, and grow professionally.
Projects: The Connected Geometry Project
Authors: Stenglein Sharon
Published: November 2020

Connected Geometry is a secondary school curriculum development project funded by the National Science Foundation and housed at Education Development Center in Newton, Massachusetts.
Coin Tossing
Authors: Haruta Mako E. , Flaherty Mark , McGivney Jean , McGivney Raymond J.
Published: November 2020

The idea for this article came from a problem that was published before the widespread availability of graphing calculators (North Carolina School for Science and Mathematics 1988). In that publication, Geometric Probability, an interesting analytic solution to the problem about comparing the areas of squares was given and is described later in Algebraic-Geometric Solution. We have adapted the problem and have used successfully another, equally interesting method of solution in numerous classes from seventh-grade prealgebra through precalculus, as well as with several groups of teachers. This article presents our solution and, in addition, an example of how a rational function, a type not commonly found in applications of mathematics at this level, can model a solution of a related problem. The problem that we use with our students follows:
Sharing Teaching Ideas: How Big Was That? Parade Prompts Turkey Talk
Authors: Reynolds Marie J.
Published: November 2020

Who would ever think that mathematics is alive and well in the minds and pens of the writers of the Macy's Thanksgiving Day Parade commentary? While the turkey was in the oven and its delicious aroma was wafting throughout the house, I settled down to watch some of the floats and bands appearing in the Macy's parade. I snapped my fingers to the catchy Broadway show tunes, tapped my feet to the rhythm of the bass drums in the marching bands, oohed and aahed as the brilliantly decorated floats passed on my television screen, and marveled at the size of the massive character balloons. Soon I recognized a pattern in the commentator's description of each balloon and began to scribble them down: “as large as two doors,” “bigger than a bedroom,” and “barely fits under the Brooklyn Bridge.” Estimation and problem solving—mathematics on Thanksgiving Day!
Connecting Research to Teaching: Block Scheduling and High School Mathematics Instruction
Authors: Kramer Steven L.
Published: November 2020

Block scheduling is not a new phenomenon. It has been widely used in British Columbia, Ontario, and Alberta since the 1970s. In the United States, block schedules have become increasingly popular throughout the 1990s, and currently they are spreading to high schools in many regions.
Mathematics Applied to Domestic Arts
Authors: Ball Katharine F.
Published: November 2020

A diligent search in mathematics textbooks that offer even the slightest promise of the application of mathematics to domestic arts, reveals how little attempt has been made to discover the part that mathematics plays in what has come to be considered peculiarly “woman’s sphere.” This is not because the domestic arts do not offer an opportunity for the application of mathematics, but because of general social and educational conditions that have affected the kind of training given to girls.
Mathematics Clubs
Authors: Webster Louisa M.
Published: November 2020

Believing that courses in mathematics are second to none in value as a mental discipline, it seems meet that the teacher’s best efforts may be profitably spent in devising plans which will stimulate a desire for research. I speak from personal experience when I say that one of our most difficult problems lies in making provision for the many points of a crowded curriculum which must be treated lightly, assigned for outside work or entirely omitted.
A Popcorn Project for All Students
Authors: Taylor Lydotta M. , King Joann L.
Published: November 2020

The NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) encourages teachers to include activities that help students “construct and draw inferences from charts, tables, and graphs that summarize data from real-world situations” (p. 167) and “express mathematical ideas orally and in writing” (p. 140). The following activities combine data gathering and analysis with cooperative learning, mathematical connections, reasoning, problem solving, and communication.
Proof: The Power of Persuasion
Authors: Leonard Bill
Published: November 2020

One dictionary's definition of prove is “[t]o convince someone by offering arguments.” So many times in mathematics classes when we thought we did not understand a proof, perhaps it had failed in its main purpose and thus for us it was no proof.
Solutions to Calendar
Authors: McLoughlin John Grant
Published: November 2020

Problems 1–5 were contributed by Michael A. Steuben, 4651 Brentleigh Court, Annandale, VA 22003. Problems 6–11 were prepared by Peter Booth of the Mathematics and Statistics Department of Memorial University of Newfoundland, StJohn's, NF A1C 5S7. Problems 15–12 (working backward) were offered by William H. Kraus, Wittenberg University, Springfield, OH 45501. Problems 16–18 represent the contribution of James E. Beamer and Bikkar S. Randhawa of the University of Saskatchewan. Saskatoon, SK S7N OWO, and Cheuk Ng of Athabaska University, Athebaska. Alberta. Problems 19, 20, and 22 were provided by Barry Scully, York Region Board of Education, Aurora, ON lAG 3H2. Problems 21 and 23–26 were prepared by students in Betty J. Thomson's History of Math class at the Community College of Rhode Island, Warwick, RI 02886. The students were Marg McLellan, Laurie Nayman, Christine Nye, Diane Pardini, Andre Sabo, and Rick Wilson. Problem 27 was taken from 101 Puzzle Problems by Nathaniel B. Bates and Sanderson M. Smith (Concord, Mass.: Bates Publishing Co., 1980). Problems 28–31 were originally prepared for the Hamilton Junior Mathematics Contest by Eileen Shannon, Westmount Secondary School, Hamilton, Ontario, who generously provided them for the Calendar.
Data Analysis and the Hardrock 100
Authors: Frantz Marny , Lazarnick Sylvia
Published: November 2020

We are always in search of stimulating, relevant, and new data sets to use with our students, and we have found something exciting—ultraracing! Ultraraces are footraces that are longer than a marathon; participants in the United States take part in 50-kilometer, 50-mile, 100-kilometer, and 100-mile tests of running skill.
Issues and Aids for Teaching Mathematics to the Blind
Authors: Dick Thomas , Kubiak Evelyn
Published: November 2020

Efforts at inclusion and mainstreaming are pervasive in classrooms across North America. One of the many special populations formerly segregated but currently mainstreamed includes blind or partially sighted students. In this article we discuss some of the challenges faced by blind and visually impaired students and some of the tools available to help these students in their efforts to learn mathematics.
Reader Reflections
Authors: Paul McGuire H.
Published: November 2020

Natural-born roots: I hereby bequeath to the world a formula that I have taught for thirty-five years but have never seen in any textbook. I published it in The Oklahoma University Mathematics Letter in 1963.
Activities: Activities for the Logistic Growth Model; or, Invasion of the Killer Moths
Authors: Iovinelli Robert (Bob)
Published: November 2020

Teacher's Guide: When students begin to study exponential growth and they see a model for the rapid growth of, say, an insect population, they may wonder, “Why has this little bug not taken over the world if it can grow so fast? It is a reasonable question that allows the introduction of a function that models the situation better than the straightforward exponential function. The growth of the population of a species, when first introduced into an environment, can be subdivided into several different stages only one of which is exponential.
Active Learning in Mathematics: Desktop Teaching
Authors: Draper Roni Jo
Published: November 2020

Imagine a mathematics fair. Imagine young mathematicians, excited to share their ideas, standing as fair barkers and pleading for people to come and learn mathematics. Imagine fair goers discussing mathematics, learning mathematics, and winning prizes: chocolate, pie, and cookie treats.
Changing Assessment Practices in an Algebra Class, or Will this Be on the Test?
Authors: Murphy Tricia
Published: November 2020

After several years of teaching high school mathematics, I was pleased to be asked to teach college algebra at our high school as part of a program designed to prepare students to “test out” of college algebra at area colleges and universities. I had taught this course at our local university, and I looked forward to teaching it again.
Sharing Teaching Ideas: Course Diary: A Valuable Information Source
Authors: Gronseth Philip W.
Published: November 2020

In one week, classes begin. Your principal has assigned you a different course to teach. The course has been taught for a number of years with the same textbook and by the same teacher; however, you never taught it. You are handed a textbook, teacher's guide, and test masters and are told to do your best. How do you feel? Are you confident that you can do your best? Do you have the necessary resources? I believe that one additional resource will help tremendously as you prepare for the year. A course diary will answer questions about how a course is taught in your particular school; those questions are not addressed in a teacher's guide.
Sharing Teaching Ideas: Why Not Three Dimensions?
Authors: Legé Steve , Fidler Mark
Published: November 2020

When I reviewed our high school curriculum, I realized that my students did not have enough practice with situations that required analysis in three dimensions. Few situations that applied the threedimensional ideas studied in geometry were found in second-year algebra and precalculus, and calculus students faced with rotations that produced solids had very little experience to help them with the visualizations required. As a response, I developed several projects that I integrated into my courses at all three levels. What follows are descriptions of a three-year sequence of projects that help to develop students' spatial skills. The photographs are all examples of student-produced work.
Secondary Students' Performance on Data and Chance in the 1996 NAEP
Authors: Michael Shaughnessy J. , Zawojewski Judith S.
Published: November 2020

Are you increasing your emphasis on probability and statistics with students? Are more of your students studying statistics or probability during secondary school? Are your students improving in their performance in data and chance? If you answered yes to any of these three questions, you are not alone, according to the most recent National Assessment of Educational Progress (NAEP). The NAEP is administered approximately every four years to students attending a representative sample of schools across the United States. It includes a student test of mathematical concepts and processes, a student survey, and a survey for the teachers of the participating classes in grades 4, 8, and 12. The student test and the surveys are revised to respond to changing needs and trends in society, as well as to new research and curriculum development in mathematics education.
German Tanks: A Problem in Estimation
Authors: Flaspoher David C. , Dinkheller Ann L.
Published: November 2020

Estimation is covered extensively in elementary statistics courses. The example discussed in this article describes a real-world situation and a simulation of that problem in which the selection of a suitable estimate is less apparent. This problem can be used at various levels. In a more elementary setting, the problem is useful to describe the concept of estimation. An advanced class can use the problem to discuss unbiasedness, minimum variance, and best estimates. At any level, the problem furnishes an excellent opportunity to make connections to the social studies curriculum and demonstrates an application of statistical techniques.
Using Simulation on the Internet to Teach Statistics
Authors: Ng Vee Ming , Wong Khoon Yoong
Published: November 2020

Reflections about using simulation on the Internet to teach statistics. Because Internet resources keep appearing at a phenomenal rate, giving a comprehensive listing of all such resources within an article is impossible. The authors focus on a few ideas that are helpful for teaching and learning.
Fearless Learning
Authors: Izen Stanley P.
Published: November 2020

Recently, more and more, I have been reminded of the importance of failure as a necessary ingredient in success. One reminder was the Michael Jordan commercial in which he says, “I've missed more than 9000 shots in my career. / I've lost almost 300 games. / 26 times I've been trusted to take the game-winning shot / and missed. / I've failed over and over and over again in my life. / And that is why / I succeed” (Zorn 1997). A New York Times article a couple of years ago included the following quote by a professor at New York University referring to a student of his, a young woman who had just become the first woman to win the prestigious Putnam Mathematical Competition, the “Olympics” of math competitions: “ ‘What really got me was her fearlessness,’ he said. ‘To be good at mathematics, you have to go right at it and start playing around with it, and she had that from the start’ ” (Arenson 1997).
Astronomical Math
Authors: Ryden Robert
Published: November 2020

High school mathematics teachers are always looking for applications that are real and yet accessible to high school students. Astronomy has been little used in that respect, even though high school students can understand many of the problems of classical astronomy. Examples of such problems include the following: How did classical astronomers calculate the diameters and masses of Earth, the Moon, the Sun, and the planets? How did they calculate the distances to the Sun and Moon? How did they calculate the distances to the planets and their orbital periods? Many students are surprised to learn that most of these questions were first answered, often quite accurately, using mathematics that they can understand.
Sharing Teaching Ideas: A Book of Creative Geometry
Authors: Kane Jill A.
Published: November 2020

Writing is an important part of the curriculum at Indian Hills High School. Students are encouraged to become good and effective writers by “writing across the curriculum.” Teachers are encouraged to have students write in mathematics and science classes, as well as in English classes. Writing in journals and creative-writing assignments allow students to sort out ideas, reinforce their understanding of concepts, or discover weaknesses in their understanding of concepts.
(A + B + C) cubed
Authors: Forringer Richard S.
Published: November 2020

IN MANY FIRST-YEAR-ALGEBRA COURSES, a standard method for teaching or reinforcing the concept that (A + B) 2 = A2 + 2AB + B2 is to ask the question “What is the area of a square whose side measures A + B?” Defining A and B to be clearly different lengths and showing that the four rectangles do in fact fit together to form a square whose area is A2 + 2AB + B2 are enough to prevent many first-year-algebra students from making the dreaded mistake of writing (A + B) 2 = A2 + B2. See figure 1.
Activities for Students: Will the Best Candidate Win?
Authors: Magnus Teresa
Published: November 2020

Students think that they are familiar with the concept of voting. After all, they have heard about governmental elections, Academy Award voting, and the ranking of football teams. They have probably participated in club and school elections. Yet if you ask students about voting methods, most can describe only one voting method, namely, plurality. Not only have they never questioned its fairness, but considering other methods is new for them.
A Geometry Solution from Multiple Perspectives
Authors: Nissen Phillip
Published: November 2020

It is perhaps informative for students to see that no one approach to geometry is the best. Students should be encouraged to try a variety of approaches when attempting a solution to any problem, experimenting and discussing which method is helping them find an answer. For some problems, students could develop a hybrid method, which would allow them to experience the creative aspects of mathematics.
Learning Mathematics Vocabulary: Potential Pitfalls and Instructional Strategies
Authors: Thompson Denisse R. , Rubenstein Rheta N.
Published: November 2020

Do your students speak mathematics, or do they think that the mathematics classroom is another country where they must use a foreign language? Are they sometimes confused or overwhelmed by new vocabulary? Do they misuse words, forget key terms, or ignore important distinctions between words? Do they ask, “Where did anyone ever get a strange word like asymptote?” or “I forget, is twelve a factor or a multiple of twenty-four?”
Sound Off!: Who? How? What? A Strategy for Using History to Teach Mathematics
Authors: Wilson Patricia S. , Chauvot Jennifer B.
Published: November 2020

In classrooms, we often treat mathematics as if we were learning on an island. We travel to that island once a day for mathematics and delve into a study that is pure, clean, and logically solid and has clear lines and no dirty corners. Students think that mathematics is closed, dead, emotionless, all discovered…. By including [the history of mathematics], we can rescue students from the island of mathematics and relocate them on the mainland of life that contains mathematics that is open, alive, full of emotion, and always interesting.
Benoit Mandelbrot: The Euclid of Fractal Geometry
Authors: Camp Dane R.
Published: November 2020

This article cites Mandelbrot as an exemplar of one who learned the language of the universe -- mathematics -- biding his time until he could employ his knowledge both as a means of creative expression and as a tool for comprehending the intricacies of the world around us.
An Extension: Gather Your Students around the Gathering Circles
Authors: House Peggy
Published: November 2020

You and your students can investigate arrangements of tightly packed circles by using drinking straws and string or rubber bands. Wrap the string or band loosely around a handful of straws, and pull it taut. Observe the way that the straws align on the end when the string is pulled as tightly as possible. See figure 1. Looking at the cross section of the bundle of straws, notice that the centers of the circles and the points of tangency form the lattice points of an isometric grid, as shown in figure 2. You can use isometric paper to record the “packed” arrangements of various numbers of circles (fig. 3).
Selected Resources for Encouraging Females in Mathematics
Authors: Wiest Lynda R.
Published: November 2020

Girls seem to do fine in mathematics until the middle grades, when a number of factors begin to influence their mathematics achievement and attitude negatively (e.g., NCES 1997). As Yusuf (1995) points out, “Gender differences in mathematics performance are predominantly due to the accumulated effects of sex-role stereotypes in family, school, and society” (p. 187). Unfortunately, research shows that schooling exacerbates rather than improves this situation, even in the classrooms of well-meaning teachers of either gender or any ethnic background (e.g., Sadker and Sadker [1995]).
Promoting Conceptual Understanding of Matrices
Authors: Worrall Laura J. , Quinn Robert J.
Published: November 2020

This lesson presents an approach to teaching matrices that includes an emphasis on conceptual development. Too often, students view matrices as nothing more than abstract rows and columns with which they demonstrate their arithmetic skills. They need to go beyond these manipulations and acquire the knowledge needed to connect matrices with the real world. This lesson encourages students to understand what they are doing rather than follow steps leading to an answer that they do not understand.
Connecting Independence and the Chi-Square Statistic
Authors: White Wes
Published: November 2020

The chi-square statistic is used to test goodness of fit, homogeneity of proportions, and independence. Although the first two uses were at least somewhat new to my students, the concept of independence was not. This article begins by reviewing the use of the chi-square statistic for goodness of fit and then looks at how I have helped my students relate their earlier work with independence to the use of the chi-square statistic.
Beyond the Golden Ratio: A Calculator-Based Investigation
Authors: Glidden Peter L.
Published: November 2020

NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) calls for increased emphasis on problem solving, mathematical reasoning, mathematical communication, and mathematical connections. This call is reaffirmed in Principles and Standards for School Mathematics (NCTM 2000). A preferred way of achieving these goals is by having students perform mathematical investigations in which they explore mathematics, search for patterns, and use technology when appropriate. In short, students should be given opportunities to learn mathematics by doing mathematics. Of course, if students are to learn mathematics through investigations, teachers must have a ready supply of such investigations available for classroom use.
Pythagorean Triples from Harmonic Sequences
Authors: DiDomenico Angelo S. , Tanner Randy J.
Published: November 2020

Pythagorean triples have intrigued generations of mathematics explorers, including students, since ancient times. One of their most charming features is their connection with various other areas of mathematics. In the Mathematics Teacher, for example, authors have shown that Pythagorean triples can be generated from the Fibonacci numbers (Bertucci 1991), from geometric sequences (Carbeau 1993), and from both the addition and multiplication tables of whole numbers (DiDomenico 1993, 1995). These findings are indeed fascinating; when shared with students, they spark interest and curiosity and lead to a truly enriching mathematical experience. Students, in fact, independently found that Pythagorean triples could be generated from Fibonacci numbers and geometric sequences. This article reveals another surprising connection that shows how all primitive Pythagorean triples can be generated from harmonic sequences.
Kissing Pennies and Eating Pi
Authors: Buhl David A.
Published: November 2020

A classroom inquiry into expressing pi as the limit of a sequence of different ratios using relationships among coins and the radius of an inscribed circle.
Sharing Teaching Ideas: The Equation of a Triangle
Authors: Amit Miriam , Fried Michael N. , Satianov Pavel
Published: November 2020

In studying algebra, a segment of the number line is inevitably related to a system of inequalities in one variable. Similarly, a convex region of the plane is inevitably related to a system of inequalities in two variables. Indeed, we drew this unsurprising conclusion from a questionnaire that we gave to a group of high school teachers and a large group of advanced twelfth graders. In the questionnaire, we simply asked what kind of mathematical expression is needed to determine the points in a line segment and in a triangular region.
Mental Mathematics beyond the Middle School: Why? What? How?
Authors: Rubenstein Rheta N.
Published: November 2020

Mental mathematics has value beyond middle school. It builds students' confidence, promotes flexible thinking, and supports learning other mathematics. The article shares sample objectives and ideas for teaching and assessing mental mathematics objectives.
Using Learning Logs in Mathematics: Writing to Learn
Authors: McIntosh Margaret E. , Draper Roni Jo
Published: November 2020

Explanations and several examples of learning logs and discussion about the effective use of learning logs in a mathematics classroom so that students have opportunities to communicate mathematically.
Just Say Charge It!
Authors: Yarema Connie H. , Sampson John H.
Published: November 2020

The use of a recursive sequence with a graphing calculator to involve students in analyzing credit card debt.
The Spot Problem: Connecting Points, Connecting Mathematics
Authors: Kahan Jeremy A. , Wyberg Terry R.
Published: November 2020

In this article, we discuss the spot problem, which we have used many times in our work with preservice and in-service teachers of secondary mathematics. This problem appeals to us because of its multiple connections. First, it illustrates that patterns and proof are both necessary in mathematics and are related to each other. Moreover, in working on this problem, students connect geometry, algebra, graph theory, and combinatorics. Finally, the problem connects with another well-known problem, the pizza problem, which we describe subsequently.
Sharing Teaching Ideas: Reviewing Made Fun!
Authors: Barlow Angela T.
Published: November 2020

The purpose of this article is to share a review game that has successfully prepared students for tests and quizzes. Part of the success of this game lies in its ability to involve everyone in working problems. Adding to this success is the fact that most students enjoy playing the game. The game has been played by mathematics classes ranging from the middle school level to high school. A description of the setup and the rules for the game follow.
Sharing Teaching Ideas: Motivating Students to Achieve Higher-Order Thinking Skills through Problem Solving
Authors: Giannetto Mary L. , Vincent Lynda
Published: November 2020

Many high school teachers have concerns about their students' ability to apply mathematical skills to other disciplines and situations. Some teachers believe that students should not “learn math in a vacuum.” One way to enhance students' understanding of the concepts learned in mathematics class is to apply mathematical skills to other subject areas, especially science and technology.
Connecting Research to Teaching: Do Teachers Need to Incorporate the History of Mathematics in Their Teaching?
Authors: Liu Po-Hung
Published: November 2020

The merits of incorporating history into mathematics education have received considerable attention and have been discussed for decades. Still, before taking as dogma that history must be incorporated in mathematics, an obvious question is, Why should the history of mathematics have a place in school mathematics? Answering this question is difficult, since the answer is subject to one's personal definition of teaching and is also bound up with one's view of mathematics. Fauvel's (1991) list of fifteen reasons for including the history of mathematics in the mathematics curriculum includes cognitive, affective, and sociocultural aspects. My purpose in this article is not to provide complete and satisfactory answers but rather, on the basis of theoretical arguments and empirical evidence, to attempt to pinpoint worthwhile considerations to help high school teachers think about what history really can do for the curriculum and for their teaching.
Activities: Massive Graphs, Power Laws, and the World Wide Web
Authors: Charles Biehl L.
Published: November 2020

Mathematical modeling has seen many changes over the years. These changes range from the types of situations being modeled to the types of tools used for the modeling. An extremely powerful modeling tool for many situations is the vertex-edge graph (hereafter simply called a graph). In this type of graph, a set of points, called vertices (or nodes), represent objects, people, or other ideas. The nodes can be connected with (not necessarily straight) line segments, called edges (or arcs), to show a relationship between the nodes. Graphs are used to model everything from transportation networks to groups of friends.
The Original New Math: Storytelling versus History
Authors: Roberts David L. , Walmsley Angela L. E.
Published: November 2020

The 1950s and 1960s were a lively time for mathematics education in the United States, but that era has now become sufficiently remote that many teachers have only vague or secondhand knowledge of the issues and personalities involved. The phrase “new math,” which is often used to designate the reforms of that period, has never been very informative in itself, and it continues to generate misinformed commentary. Some of this commentary has been promulgated by critics of programs and textbooks associated with the NCTM's Standards, which these critics have dubbed the “new, new math” (Gardner 1998). More confusingly still, the newer reforms themselves are sometimes called the new math; the term is used in this manner in a 1997 Newsweek article (Kantrowitz and Murr 1997) and in a 2000 opinion piece by former Secretary of Education William Bennett (Bennett 2000). The phrase may eventually detach itself entirely from the 1950s and 1960s. This article aims to help teachers better understand the original new math by examining several stories that have been told and that continue to be told about it. These stories contain outright fallacies, half-truths, and not-quitethe-whole truths, as well as indisputable facts. Understanding the nuances among these cases should help us as we continue the never-ending process of providing a better mathematics education to students.
Sharing Teaching Ideas: The Area under a Curve: Conjecturing the Fundamental Theorem of Calculus
Authors: Suzuki Jeff
Published: November 2020

The NCTM standards for courses designed to prepare students for the “new calculus” include using “all appropriate calculator and computer technologies,” as well as promoting “the use of experimentation and conjecturing” (NCTM 2000, p. 289). Both these goals can be met in the following experiment to find the area under a curve. The experiment is a modification of a classroom exercise that I conducted in an introductory college calculus course for nonmajors.
Delving Deeper: Regression Lines through Conic Sections
Authors: Cuoco Al , Goldenberg E. Paul
Published: November 2020

From the first time that we used such inter-active geometry programs as The Geometer's Sketchpad and Cabri Geometry, we were intrigued by their potential to help students develop ways of thinking that underlie calculus and analysis (Cuoco, Goldenberg, and Mark 1995; Cuoco and Goldenberg 1997). One theme in this approach has been to look at optimization problems as geometrically defined functions, for example, the sum of the distances from a point to three fixed points. In an interactive geometry world such as Sketchpad or Cabri, the user does not need to impose coordinates on the plane and specify the distances algebraically at the outset. He or she may define the function directly as a geometric relationship; manipulate its variable (the movable point); and observe, numerically or geometrically, the value (sum of distances) that results. The algebraic step is also valuable; both geometric and algebraic interpretations lead to important insights. But the geometric step is often a particularly productive starting place for generating the ideas that one may want to revisit algebraically.