# Mathematics Teacher

Publications
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.

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.

Autograph is a two- and three-dimensional dynamic statistics and graphing utility, developed in England, that has grown out of direct classroom experience. A simple select-and-right-click interface, together with tools such as Autograph's unique Slow Plot, Scribble Tool, and dynamic Constant Controller help make the classroom experience interactive. Autograph can be used to introduce most students to their first nonlinear function. To their surprise, they will discover that parabolas are everywhere. In this article, the author illustrates applications and solutions in finding the angle of elevation for games like shot-put and javelin toss using Autograph. (Contains 12 figures.)

Considered is the following question: If a consumer loan is prepaid before the contract time, how much of the total interest should be paid to the lending institution? Discusses the validity of the rule and demonstrates strategies for answering interest questions. (PK)

Graphing calculators have had a significant impact at many levels in high school and college mathematics in the past few years. We have had to reexamine what we teach and how we teach. Topics that were once the exclusive province of calculus, such as max-min problems, are currently studied by algebra students working with graphing calculators. We have added to our syllabi new techniques, such as zoom and trace; new concepts, such as complete graph and friendly viewing window; and new mathematics, such as chaos and fractals. Graphing calculators have helped us enhance, enrich, and enliven our courses, and they have empowered students, allowing them to discover and explore mathematics as active learners.

Discusses the importance of the use of rich problems to challenge gifted students. Highlights the characteristics of challenging problems and projects, and presents examples of these kinds of questions. (ASK)

Demonstrates that the functions known as the witch of Agnesi and the normal distribution are not identical by comparing the areas under the curves and the slopes of the lines tangent to the curves of the two functions. Suggests follow-up activities to the investigation. (MDH)

The National Council ofTeachers of Mathematics's Curriculum and Evaluation Standards for School Mathematics (1989) urges the implementation of technology in all areas of mathematics learning. In a high school algebra class, the graphing calculator can be a powerful aid in developing mathematical understandings. Equations involving absolute value are rich ground for graphical exploration. Students can apply their knowledge of graphs of lines and solution sets of simultaneous linear equations to enhance their conceptual understandings of these topics.

Algebra students often have difficulty in sorting out and correctly using the concepts involved in the solution of linear inequalities. The introduction of the concept of absolute value causes additional complications for many students. Two of the major problem areas are (1) the concept of absolute value as normally defined and (2) the recognition and appropriate use of the logical connectives and and or in the solution of problems that combine inequalities with absolute value. One way to alleviate students' confusion and promote their problem-solving ability in such a setting is to adopt an approach different from that found in current textbooks. This approach involves consideration of the function concept as the graph generated by associated ordered pairs of real numbers rather than functional values as obtained from the standard definition of absolute value.

Discusses students whose mathematics anxiety resulted from past abuse, verbal or physical, by a teacher or parent while doing mathematics. Presents two examples of math abuse, the resulting math anxiety, and how the issues were addressed. Contains 14 references. (ASK)

A mathematical problem is solved using the extension-reduction or build it up-tear it down tactic. This technique is implemented in reviving students' earlier knowledge to enable them to apply this knowledge to solving new problems.

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

Presented is an activity which examines an arithmetic concept from a geometric point of view. Objectives, prerequisite learning, and procedures are discussed. Four worksheets are provided with answers. (CW)

Presented are student activities that involve two standard problems from geometry and calculus--the volume of a box and the bank shot on a pool table. Problem solving is emphasized as a method of inquiry and application with descriptions of the results using graphical, numerical, and physical models. (JJK)

Presents a series of five activities that introduce division of fractions through real-world situations. Discusses problems related to resurfacing a highway, painting dividing stripes on a highway, covering one area A with another area B, looking for patterns, and maximizing the result of a division problem. Includes reproducible worksheets. (MDH)

The integration of mathematics and other content areas has long been advocated by mathematics educators and sought after by teachers of mathematics, from kindergarten through high school and beyond. Science is one area that provides context and content for many valuable and useful connections with mathematics. When attempts are made to integrate mathematics and science, it seems that all too often mathematics is thought of as a “computational tool” for the sciences. For example, students need to have these computational tools for data analysis, so they can work with the data they collect in their science experiments. This use is not bad. However, it does not help students see the full power, usefulness, and beauty of mathematics. Beyond computation, mathematics can be used to determine and model science relationships, explain complex applications and, of course, solve problems.

This article describes how a series of lessons might be used to allow students to discover the size of the Earth, the distance to the Moon, the size of the Moon, and the altitude of Mount Piton on the Moon. Measurement with a sextant, principles of geometry and trigonometry, and historically important scientists and mathematicians are discussed.

Activities involving counting triples, triangles, and acute triangles enrich the curriculum with excursions into modular arithmetic, the greatest-integer function, and summation notation. In addition, more advanced students can apply difference-equation techniques to find closed forms and can use mathematical induction to prove the formulas. Students may be learning about these topics for the first time, or they may be reviewing familiar ideas in different problem-solving contexts. In either situation, personal arsenals of problem-attacking skills are strengthened.

Recognizing the need for making adaptations for special students in regular classes, Project Train at Virginia Commonwealth University has developed a model for adapting the curriculum for mildly handicapped children (Wood 1985). The model is generic to all academic subjects and grades K-12. This article focuses on adapting the construction of teacher made mathematics tests for mildly handicapped children, that is, the educable mentally retarded, the emotionally handicapped, and the learning disabled, in the mainstream.

Discussed is the use of the book Flatland, written by Edwin Abbott, as a novel assigned for reading in secondary school geometry. The assignment is largely designed to reverse the low student interest in geometry. It is noted that overall the book was well received by students. (MP)

This article describes several enrichment activities that connect mathematics to science in an algebra 1 curriculum. It provides a basis and suggestions for teachers to include student-produced essays about the role of mathematics in the history of civilization. (Contains 6 figures and 1 table.)

The College Board's Advanced Placement (AP) offering in computer science is in its first year of operation. In the spring of 1983 the board (1984) published a course description to serve as a guide to those secondary schools that wish to offer AP Computer Science. This course description is also the basis of the first AP examination in computer science that is being administered in May 1984. A teacher's guide for AP Computer Science (College Board 1983) has also been prepared to assist secondary school teachers in planning and teaching the course.

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

Describes a five-number summary which is a display of the minimum value, lower quartile, median, upper quartile, and maximum value. Indicates how to draw box plots as graphical representations of a five-number summary. (YP)

Discusses the concepts of fractals, fractal dimension, and the generation of fractals in LOGO. Provides diagrams of several fractals and their LOGO programs. Lists eight references. (YP)

Computer programs are now available that perform most of the algebraic and numerical-manipulation procedures on which school mathematics now concentrates. The defining characteristic of these symbolic mathematical systems is that, unlike many of the popular computer languages, they can manipulate variables as well as numbers. They can perform rationalnumber arithmetic, solve equations, produce equivalent expressions of a variety of types, apply trigonometric identities, evaluate limits and sums, compute the algebraic form of derivatives and integrals, perform matrix manipulations, and produce accurate numerical answers-with hundreds of digits if appropriate. In short, they can perform with more dependable accuracy and greater speed most of the algebraic and numerical procedures on which students now spend most of their mathematical careers.

If students are to understand inferential statistics successfully, they must have a profound understanding of the nature of the sampling distribution. Specifically, they must comprehend the determination of the expected value and standard error of a sampling distribution as well as the meaning of the central limit theorem. Many students in a high school or college introductory statistics class (including those taking AP Statistics) do not have the mathematical background to understand the derivation of these concepts. Consequently, they tend to memorize the results. Such a rote approach leads to a less than satisfactory understanding of inference--and can be boring. To overcome these problems, the authors, working independently, found that using simulations and combining them with a discovery approach gives students a means to discover the concepts associated with sampling distributions. One author used statistical software to conduct the activity discussed in this article; the other used a graphing calculator. Both approaches are presented for use by teachers of introductory statistics courses. (Contains 2 tables and 14 figures.)

Recently we had the pleasure of observing high school mathematics classrooms in which teachers were demonstrating effective teaching strategies that they had learned at in-service sessions sponsored by the Eisenhower and National Science Foundation (NSF) programs. Unfortunately, we also noticed a distinct lack of African American students in the advanced mathematics classes. In most of the schools, Aftican American students made up 10 to 30 percent of the enrollment, but seldom was more than one African American student enrolled in the advanced mathematics classes.

Algebra teachers find themselves overwhelmed by the sheer volume of information that they have to teach their students. One way to address this issue is to have teachers teach for understanding by focusing on concepts. In this article, I share how language can help uncover the core ideas that define the concept of function and share activities that have been successfully used with students to introduce the core ideas and illustrate the usefulness of a function outside the classroom. (Contains 8 figures.)

Presented is the ancient Egyptian algorithm for the operations of multiplication and division of integers and fractions. Theorems involving unit fractions, proved by Fibonacci, justifying and extending the Egyptian or Ahmes' methods into the Hindu-Arabic numeric representational system are given. (MDH)

Suggestions are given concerning the use of the pocket calculator for computations that are too trivial to be processed by a computer but too time-consuming for manual calculation. Several examples from arithmetic, algebra, trigonometry, and calculus are included. (DT)

Brief mystery stories such as the three presented here can be used to help students develop critical reading and thinking skills, as well as to introduce the notion of indirect proof. (SD)

A computation learning lab was planned, implemented, and evaluated as a way of helping learners conquer computation problems. Features include: emphasis on a premotivation activity; the use, on a regular basis, of eight instructional strategies; and active parent support. The lab students showed dramatic gains. (MP)

Discusses 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. Argues that much of the language of mathematics relies heavily on visual reference. (DDR)

Introduces a room air-exchange activity designed to assess student understanding of the concept of volume. Lists materials for the activity and its procedures. Includes the lesson plan and a student worksheet. (KHR)

The problems involved in making reservations for airline flights is discussed in creating a mathematical model designed to maximize an airline's income. One issue not considered in the model is any public relations problem the airline may have. The model does take into account the issue of denied boarding compensation. (MP)

Data analysis plays a prominent role in various facets of modern life: Schools evaluate and revise programs on the basis of test scores; policymakers make decisions on the basis of information gleaned from polling data; supermarkets stock shelves on the basis of data collected at checkout lanes. Data analysis provides teachers with new tools and new methods for exploring mathematical concepts. In this article, the authors illustrate ways in which data simulation and visualization tools have opened up new pathways for learning as they explore several rich mathematics tasks by using Fathom dynamic data analysis software. As their examples suggest, the availability of intuitive, drag-and-drop data visualization and simulation tools blurs the distinctions between algebra, geometry, and data analysis in the secondary school curriculum. (Contains 16 figures and a bibliography.)

Is algebraic manipulation important for students' mathematical understanding? Will students be crippled in subsequent, more advanced courses in mathematics and science if most, or all, algebraic manipulation is removed from high school algebra courses? Champions of the full and unrestricted use of computers and calculators to do algebraic manipulation (e.g., MuMath, Macsyma, HP 28C) argue for a substantial reduction of the time spent teaching algebra manipulation. They would use this time to teach new, modern numerical topics and techniques that foreshadow the study of statistics and computer science. But is the old mathematics bad mathematics? Many conservative mathematicians and mathematics educators argue for the status quo. But have we no way to improve on the old mathematics with new technology? Is it fair to keep students in the dark about modern developments in the field of mathematics.

Discussed are the philosophy and procedures behind the introduction of algebra to students in grade 7 in Australia. Included are the importance of concrete experiences, language development, and the consequences involved in this procedure. (CW)

Provides a brief synopsis of the development of the National Fire Danger Rating System along with an explanation of how it works and possible ways to use it in a modeling or problem-solving unit in a secondary mathematics classroom. (ASK)

This article describes preparation and delivery of high school mathematics lessons that integrate mathematics and astronomy through The Geometer's Sketchpad models, traditional proof, and inquiry-based activities. The lessons were created by a University of Texas UTeach preservice teacher as part of a project-based field experience in which high school students construct a working Dobsonian telescope. (Contains 9 figures.)

Presents a case study of one high school student's experiences in a first-year algebra course. Emphasizes the use of technology and how it changed the mathematics curriculum. (ASK)

Described is a software program with two components. One is muMATH, a package of functions that can perform a wide variety of symbolic manipulations. It is surrounded by, and written in, the accompanying language, muSIMP. A problem is described with a muMATH solution; another example uses the muSIMP language. (MNS)

The 2008-2009 U.S. economic crisis stimulated this exploration of the algebra of mortgages, the bubble in house prices, and how to reduce foreclosure rates. Additional exercises: http://www.nctm.org/uploadedFiles/Journals_and_Books/MT/Extras/MT2009-12-356c.doc

Algebra on the high school, community college, and college levels is often viewed by students as an abstract game—as skills to be mastered and algorithms to be memorized. To move students from this static view, algebra on all levels should be grounded in applications so that students see it as a useful tool in solving real-world problems. Early in their study of algebra, students should be exposed to mathematical modeling, which can link to their interdisciplinary interests as well as illuminate the need for algebra as a tool for making decisions in a democratic society.

The NCTM's Curriculum and Evaluation Standards for School Mathematics (Stan dards) (1989) designates four standards that apply to all students at all grade levels: mathematics as problem solving, mathematics as communication, mathematics as reasoning, and mathematical connections. These and NCTM's other standards are embedded in a vision of technologically rich school mathematics classrooms in which students and teachers have constant access to appropriate computing devices and in which students use computers and calculators as tools for the investigation and exploration of problems.

This article illustrates the fact that unless tempered by algebraic reasoning, a graphing calculator can lead one to erroneous conclusions. It also demonstrates that some problems can be solved by combining technology with algebra.