New England Mathematics Journal

The New England Mathematics Journal is published twice a year in November and May. Regular membership in a section of ATMNE includes a subscription to the New England Mathematics Journal and to the ATMNE Newsletter. Individual copies of the New England Mathematics Journal may be purchased as available for $5.00, payable with request by check made out to ATMNE. Inquiries should be sent to the Journal Editor.

Materials in the New England Mathematics Journal may not be reproduced without permission. To receive permission to reprint articles, please contact the Editor. When articles are reprinted, credit should be given to the New England Mathematics Journal and the author(s). Two copies of the publication must be sent to the Editor. Opinions expressed are those of the authors and are not necessarily those of the ATMNE.

Manuscripts submitted for publication should be addressed to the Editor and should not exceed ten pages. They must be typed, on one side only, and double-spaced with wide margins. Illustrations must be computer generated and appear within the article. The style guide is the Publication Manual of the American Psychological Association (APA), 4th edition, 1994. Manuscripts are accepted for review with the understanding that they are original, have not been published before, and are not under consideration for publication elsewhere. Since all manuscripts are subject to a review process, author identification should appear only on a separate cover sheet. A total of eight copies, including the original and/or photocopies, should be submitted. For the time being, articles are being accepted by invitation only for the focus issues.

The New England Mathematics Journal is edited and published at 229 Main Street, Department of Mathematics, Keene State College, Keene, NH 03435-2010

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New England Mathematics Journal

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May 2005 Volume XXXVII, Number 2

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New England Mathematics Journal

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November 2004 Volume XXXVII, Number 1

Editorial
Beverly J. Ferrucci

Motivating students with creative problems form the basis of all the articles in this issue. The problems are geared toward three different levels of students: pre-service and inservice teachers, precalculus students, and elementary school teachers.

Mary Sullivan’s article illustrates how a pattern problem enhanced the algebraic understanding of pre-service and inservice teachers. Her fundamental belief is that teachers’ learning of algebra focuses on symbolic processing and solving problems that use symbolic processing. A rich, dynamic problem by Bezuszka and Kenney is thoroughly discussed throughout the article.

In his article Robert Galloway presents a challenging problem that provides an application of graphing technology to the simulation of motion. This classroom activity is designed for precalculus students and utilizes the law of cosines to derive an equation relating a harmonic motion. The article concludes with a suggested activity for further exploration.

Marta Samson describes a mathematics lesson on tangrams that she finds very rewarding for teaching her students in Poland. She places the responsibility of designing interesting and exciting mathematics lesson on classroom teachers. Detailed steps for constructing a tangram and an additional tangram worksheet are also included in the article.

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New England Mathematics Journal

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May 2004 Volume XXXVI, Number 2

Guest Editorial
David M. Burton
University of New Hampshire
Durham, NH

The current year marks the hundredth anniversary of the Association of Teachers of Mathematics in New England. In honor of its centennial, this volume of the New England Journal of Mathematics is devoted to aspects of mathematics history over the last hundred years, or roughly, over the twentieth century.

Future historians may well look on the last century as another “golden age” in mathematics. There was unprecedented growth and diversity in the subject as a multitude of new fields and subspecialities were born and developed; the new notions covered the spectrum from Alexander’s horned sphere to Zermelo-Frankel set theory. At the same time, our mathematical vocabulary was enriched by a bewildering array of technical terms such as fractal, Turing machine, foliation, wavelet, Markov chain, Penrose tile, and gauge field.

Among the crowning achievements of the previous century was the resolution of some outstanding problems in mathematics: the Beiberbach Conjecture, the Burnside Problem, the Four Color Problem, the Mertens Conjecture, Fermat’s Last Theorem, and the Kepler Sphere-Packing Problem. The monumental task (12,000 journal pages over a 30-year period) of classifying all finite simple groups was laid to rest; with it came the discovery of the largest such group, the so-called Monster Group. A recent triumph is the apparent confirmation of Catalan’s assertion that among all powers of the positive integers, the only pair of consecutive values is 8 and 9. And now, the century-old Poincare Conjecture, a result joining topology and geometry, also appears to have been confirmed.

The continued refinement of fast computers helped to change the mathematical landscape. Not only did they become valuable research tools in suggesting possible results, but they also led to the creation of new fields such as the theory of algorithms, public key cryptography, coding and complexity theory. The solutions of both the Four Color Problem and Kepler’s Sphere-Packing Problem relied heavily on the raw power of the computer. This evoked considerable controversy at the time as to which physical tools are legitimate in creating mathematically rigorous proof. The notion of “almost certain proofs” – as witnessed in the idea of “probable primes” – was a further departure from the traditional standards of formal demonstration.

Another prominent feature of the twentieth century was the increased participation of women in the mathematical enterprise. Where in the century’s first two decades women were awarded about 13% of American doctorates, their proportion had reached 29% in the last decade, with an incidence of 34% in the academic year 1998-1999. Doctoral recipients early in the century generally found employment in women’s colleges or undergraduate institutions which emphasized teaching to the exclusion of research; today’s Ph.D.s have gained access to the highest ranks at the nation’s leading research-oriented institutions, although as yet in disproportionately low numbers. By the last quarter-century, women had also risen to the presidencies of some of our leading professional organizations, notably the Mathematical Association of America (1979) and the American Mathematical Society (1982).

The following articles center around mathematics history and its proponents over the last hundred years. Several of them have been written by or about eminent New Englanders. The issue begins with “A Conversation with Father Stanley Bezuszka, S.J.” by Beverly J. Ferrucci, a report on a recent interview with Boston College’s respected mathematics teacher and historian Father Stanley Bezuszka, S.J.

In “David Eugene Smith: Mathematics Educator Extraordinaire”, Frank Swetz describes the career of the most influential figure in the fields of the history and teaching of mathematics during the early decades of the twentieth century. Smith is remembered by today’s readers mainly through the donation to Columbia University of his collection of three thousand portraits of a thousand individuals associated with mathematics.

Ed Sandifer provides us with the piece “Gone With the Slide Rule: The Story of the Polar Planimeter.” In it, he describes a simple device – now technologically obsolete – for measuring a plane area while tracing its boundary.

“Raymond Clare Archibald: A Euterpean Historian of Mathematics” by Jim Tattersall and Shawnee McMurran focuses on the career and accomplishments of Brown University’s Raymond Clare Archibald. This renowned scholar was one of the mid-century’s most devoted historians of the mathematical sciences.

The concluding article, “Twentieth Century Histories of Mathematics” by David Burton considers some of the texts frequently used in the college setting, and their evolution from early-century works of modest size without exercises or photographs to the modern large-scale surveys.

As guest editor of this Focus Issue, I would like to thank the authors of its articles for sharing their efforts with us. It is hoped that they will encourage further reading in the history of mathematics and inspire the introduction of historical material into the classroom experience. I have enjoyed working not only with these writers, but also with the permanent editor of the Journal, Beverly Ferrucci.

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New England Mathematics Journal

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November 2003 Volume XXXVI, Number 1

Editorial
Beverly J. Ferrucci

More than 40 countries participated in 1995 in an assessment of mathematics and science achievement at the fourth, eighth and twelfth grades. The international study, named the Third International Mathematics and Science Study (TIMSS), was the largest, more comprehensive assessment of student achievement ever conducted. Four years later in 1999, the population of students originally assessed as fourth-graders had advanced to the eighth grade and were retested for the Third International Mathematics and Science Study – Repeat (TIMSS-R). In both the TIMSS and TIMSS-R assessments, Singapore was ranked at the top for the performance of eighth graders in mathematics. The article by Berinderjeet Kaur, Beverly Ferrucci, and Jack Carter reviews the performance of eighth graders from Singapore and the United States according to International Benchmarks. The authors also examine the performance of the eighth graders in the content area: Fractions and Number Sense. Some factors that may have led to the high achievement of eighth graders in Singapore are also explored.

The article by Craig Sheil discusses a manipulative that future educators may utilize to make learning more meaningful and interactive for students. The manipulative, Pentablocks™, consists of a five-pointed star, a regular pentagon, two rhombuses, and two golden isosceles triangles. Together, these shapes give students a unique way to discover and explore mathematics, such as tessellations, similarity, angle measurement, mirror and rotational symmetry, and congruence. This article describes the Pentablocks™ family and its interesting relationships as well as provides a creative activity that can engage students’ thinking and learning of mathematics.

Exploring the use of games in mathematics teacher education classes is the main topic in the article by Gertrude Toher. The author describes an undergraduate teacher education project in which her students play original games. In an effort to make the projects have strong, meaningful connections to Principles and Standards, she made three substantial changes to her students’ original experience. These included overtly tying the games to student recollection of good mathematical learning, replacing mini-lectures with focused discussions, and relating the experience to the teaching-learning model in Principles and Standards. The article concludes with teacher observations and student feedback for the continued use of the revised instructional sequence.

Jennifer Edwards highlights spreadsheets and their use in an elementary classroom in her article. She discusses possible benefits of using spreadsheets in a classroom and includes specific topics for their use at the elementary level. A sample lesson is also shared. In addition, the article provides an overview of other lessons and concludes with a list of resources where teachers can find more information and lessons on spreadsheets.

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New England Mathematics Journal

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May 2003 Volume XXXV, Number 2

Guest Editorial
James Morrow
Mount Holyoke College
South Hadley, MA 01075

Geometry has awakened! From elementary education to scholarly research to commercial application, the field of geometry has exploded. I’ll very briefly describe some of these exciting developments.

Paper folding has become a way of learning geometry using not just the visual, but the neglected and powerful kinesthetic sense. Folding, as a construction tool, has fascinating comparisons to ruler and compass. Methods in computer graphics and aerial photography, both of which rely on projective geometry, improve our ability to depict the seeming three-dimensional world we live in onto the flat surfaces we view. In order to design drugs, new and faster ways to characterize and recognize the geometric shape of molecules are currently being developed. More surprisingly, perhaps, is the current active mathematical research being done on folding and the use of folding techniques to compactly transport optical mirrors into space, where they can be unfolded for use in collecting data.

Spurred by advances in technology and research, geometry’s interactions with both visualization and computation have created opportunities and responsibilities for educators. The five articles in this issue explore ways of engaging students in geometry and guide us towards goals we should have, what we should teach, and how we may teach more effectively.

What will citizens of the world need to know in 2030? Well, I certainly don’t know, but I would bet that being able to think clearly, flexibly, and from different perspectives will be important skills. For this, we will need to be able to imagine possibilities, make conjectures, and test ideas. To ‘envision’ involves more than just the visual, but simple diagrams and sketches are amazingly helpful and the construction of visual images, as well as just looking at images, is a marvelous stimulation to the imagination. The geometry we bring to students to investigate through making conjectures, posing problems, representing ideas, varying images with computer software can and should make the point that what is presented to us needn’t be the way that things must be – from mathematics to science to politics.

This focus issue has contributions by a diverse and wonderful group of people, each passionately committed to broadly educating people. Each author presents a unique perspective, yet each article complements the others, drawing from experience in teaching, thinking, and searching for wisdom and knowledge with a particular interest in ideas geometric. And, each works to develop her/his students’ visual imagination.

Gorini’s article starts us off by situating us in college classrooms. She shows the myriad ways in which geometry appears in mathematics, in sciences, and elsewhere in the curriculum; she guides us through the use of geometry and visual imagination in calculus, abstract and linear algebra, statistics, the arts and science. Her article details the many geometric challenges that a student may face in undergraduate education and thus what precollege education must prepare students for.

Malloy focuses on elementary and middle school education and on the preparation that teachers should get in order to teach well. With rich illustrations from her course for prospective elementary and middle school teachers, she stresses that students must own the ideas they explore, for only through truly owning ideas can students’ understanding develop and their confidence to boldly imagine grow.

Walter shows the development of visual imagination through problem posing. Her article portrays the power of the visual imagination to pose variations on a static sketch, and to pose alternate ways of viewing and thinking about such a sketch. She illustrates how posing many problems generated from a single problem can create a rich learning environment from a mundane situation.

Scher’s article shows how students’ imaginations, when stimulated by using a computer mouse to vary a sketch while preserving relationships, may produce visualizations and descriptions that a teacher might not expect. He illustrates ways to use dynamic geometry software and cautions us to expect the unexpected. Scher proposes that students spend considerable time playing with a sketch made dynamic by computer software. Implicit is the idea that a solid understanding, confidence in their ability to learn, and a desire to learn will enable people to, as they need them, learn ideas that they weren’t formally exposed to in their education.

Gaddis starts her piece by dealing with the broad question of what geometry is, what it can be, and what it should be. She then vibrantly describes how she has her students imagine such concepts as “lines” on cones and spheres and projections from spheres to planes. She also suggests how to choose from among the many possibilities of what to teach, and describes how students interact with concepts as they construct fundamental ideas.

The word proof, per se, is, perhaps strangely, little mentioned in this focus issue on geometry. Rather than formal proof, the articles in this issue refer to visual reasoning, plausibility arguments, experimental evidence, and inductive reasoning from examples, thus showing proof as part of a continuum that contains geometric reasoning, convincing argument, and explanation – and as a jumping off point for the discovery of new knowledge.

I hope that this focus issue is also a springboard for further thinking about teaching and learning geometry; I would love to hear your comments, questions, and the thoughts you have about extending the discussion! I offer many thanks to the authors for their insights and care and to Beverly Ferrucci for the wonderful opportunity to be a part of this endeavor.

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New England Mathematics Journal

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November 2002 Volume XXXV, Number 1

Editorial
Beverly J. Ferrucci

The articles in this issue involve many different areas of mathematics. They include topics from geometry, calculus, statistics, modeling, and mathematical proofs.

Robert Galloway presents an interesting class activity that shows the integration of mathematics with technology and further demonstrates the links among geometry, calculus, and mathematical modeling. His activity was the result of an inquiry by one of his precalculus students concerning the approximation of the circumference of a circle using inscribed regular polygons. The author also poses interesting extensions at the conclusion of the article for interested teachers and students.

A geometric perspective on the sample standard deviation is illustrated in the article by Bradford D. Allen. He presents the sample standard deviation as an averaged Euclidean distance between two vectors. In doing so, he believes that such a concrete and intuitive interpretation of the sample standard deviation will enhance students’ understanding of this statistical concept.

The article by Thomas Baird examines reasoning and proof at the secondary level. It includes classroom examples and further recommendations for a transitional approach to proof within the existing mathematics curriculum.

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New England Mathematics Journal

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May 2002 Volume XXXIV, Number 2

Editorial
Michael Shaughnessy

In the opening article of this volume of the New England Mathematics Journal, Richard Scheaffer makes the pronouncement that Data are Hot! I would go one step further and suggest that the Teaching and Learning of Data and Chance are Hot! For over thirty years there has been a gradual but consistent push from a growing number of mathematics educators for the inclusion of data and chance as part of the mainstream school mathematics curriculum in the United States. The developmental seeds for the recent recommendations in Principles and Standards for School Mathematics (NCTM, 2000) on the teaching and learning of data and chance can be traced through earlier documents such as the NACOME (1975) report, the Agenda for Action (NCTM, 1980), and the Quantitative Literacy Materials, (Landwehr & Watkins 1985, 1995), just to mention a very few of the many past influential policy documents and curriculum materials that have promoted the inclusion of data and chance in school mathematics. Data and chance now have equal billing in the PSSM curriculum standards, along with number, algebra, geometry, and measurement! Of course, this in no way implies that data and chance will automatically experience equal billing in K-12 classrooms. Data and chance are relative newcomers to the school curriculum in the United States, and teachers and curriculum supervisors will continue to wrestle with the place of data and chance in classrooms for many years to come. The articles in this issue of the New England Mathematics Journal are intended to provide information, provoke discussion, and suggest areas for research, as the teaching and learning of data and chance continue to grow and develop in our classrooms. There is something in this issue for everyone, teachers, supervisors, and researchers alike.

Dick Scheaffer provides us with an overview of the importance of including data and chance in the mathematical education of our current and future mathematics teachers. He makes general curriculum recommendations for teachers at each of the elementary, middle, and secondary grade bands, and then proceeds to provide even finer grain recommendations for specific topics and activities for our teachers at each grade band. Dick adds his own slant on the difference between data analysis and mathematics. “Data analysis is about context; mathematics is about pattern and logic free of context.” Whether it is always this cut and dried a difference between the two is somewhat debatable, but Dick’s point about context as the driving force for data analysis is clearly well taken.

The paper by Ed Mooney, Graham Jones, and Cynthia Langrall provides some insight into how students reason about graphs and make predictions based on data. The authors present us with some cases where students’ reason about graphs and make predictions based on data. The authors present us with some cases where students’ reason about data in several ways, from what they call ‘idiosyncratic reasoning’ about graphs to ‘analytic’ reasoning about graphs. Their paper suggests that it is important for us to provide students with data analysis experiences throughout their entire school experience, so that steady growth in statistical reasoning can occur over time. Their paper also implies that we need to build data analysis experiences for students upon their current thinking, even though that thinking might be quite idiosyncratic at the start.

Susan Friel reports on her observations of students’ strategies while they were partitioning a data set, and comparing distributions that they themselves had formed while using a new software tool for middle school students. Tinkerplots (Konold and Miller, 2001). While using Tinkerplot students not only were reading information from and across graphs, but also were creating their own graphs in a dynamic environment. Friel, curious to learn what students do in this Tinkerplots environment, shares some observations of grade 8 students’ interactive uses of the tool while they worked with a data set.

In the article by Jane Watson we are presented with several classroom vignettes that occurred during her work with students and teachers. These vignettes disclose some students’ beliefs about data and chance, beliefs that were somewhat surprising even to her. The lesson: as teachers of data and chance we need to be ready to put students into situations in which their beliefs are challenged and they are forced to think through some difficult issues for themselves.

Cliff Konold puts forth a case for allowing students to construct their own representations of data using moves like stacking, ordering, and separating within the Tinkerplots environment. Konold argues for the importance of allowing students to construct their own data representations prior to being taught traditional graphical representations such as scatterplots or bar plots or boxplots. His thesis is that concepts like covariation can become more transparent to students when they build their own representations of data than when a graph created by an outside source is just presented to them fait accompli. Read Konold’s chapter, and you be the judge.

In the final article Maxine Pfannkuch, Amanda Rubick, and Caroline Yoon share several stories of the thinking of middle school age students about variation as they worked on some statistical investigations. This article identifies a number of components of the process of thinking about variation, such as noticing variation, dealing with representations of variation, explaining variation, or attempting to measure variation. Their chapter helps us to learn more about how students do or do not deal with certain components of variation.

As guest editor of this issue of the New England Mathematics Journal, I want to thank the authors for sharing their work, and their reflections, on the teaching and learning of data and chance. I hope you, the reader, will find these articles as thought provoking and useful as I have!

Michael Shaughnessy
Department of Mathematical Sciences
Portland State University
April, 2002

REFERENCES
Konold, C., & Miller, C. (2001). Tinkerplots, version 0.42. Data analysis software for the middle school. Amherst, MA: University of Massachusetts.
Landwehr, J.M. & Watkins, A.E. (1985, 1995). Exploring Data. Palo Alton: Dale Seymour Publications.
National Advisory Committee on Mathematical Education (1975). Overview and Analysis Of School Mathematics Grades K – 12. The NACOME report. Washington D.C.: Conference Board of the Mathematical Sciences.
National Council of Teachers of Mathematics (1980). An agenda for action. Reston, VA: The Council.
National Council of Teachers of Mathematics (2000). Principles and Standards for School Mathematics. Reston, VA: The Council.

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New England Mathematics Journal

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November 2001 Volume XXXIV, Number 1

Editorial
Beverly J. Ferrucci

Thomas DeFranco, Jean McGivney-Burelle, and Mary Truxaw describe a preliminary investigation of how fraction necklaces and related activities can help students develop a deeper understanding of fractions. The article features fraction necklaces that were created and worn by fifth grade students as they participated in classroom activities involving fractions. The article goes on to present a theoretical background of imagery and examines the impact the fraction necklaces had on the students' procedural and conceptual understanding of fractions.

The study by Indira Chacko and John Crow utilizes new instruments to measure academic self-concept in English and mathematics of high school students in Zimbabwe.The study attempted to validate both instruments and to ascertain whether there is a relationship between academic self-concept and achievement in the two subject areas. Students from an all girls' school, an all boys' school, and a co-ed school were participants in the study.

Lynn A. Fisher's article discusses the confusion some students experience because of the closely-related names of four concepts in calculus: The average rate of change, the Intermediate Value Theorem, the Mean Value Theorem, and the average value of a function. The article includes an exercise with student-generated data and models that helps students to analyze and synthesize an understanding of the four principles. In addition, the exercise provides students an opportunity to invent their own uses of mathematics, examine functions with multiple representations, and to verbally communicate their conclusions.

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New England Mathematics Journal

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May 2001 Volume XXXIII, Number 2

Editorial
Beverly J. Ferrucci

Ethnomathematics, multivariate calculus, and cryptography are the main themes for the articles in this issue. One discusses the current trend of culture within mathematics education and the other two deal with topics from calculus and discrete mathematics.

An analysis with respect to symmetry patterns of carriage friezes on Brazilian trucks is the focus for Rogeria Gaudencio, Romulo Marinho, and John A. Fossa's article. They present a stimulating article on these friezes and which are decorative pattern strips often with a geometric motif. They discuss the mathematics necessary for analyzing the friezes and conclude with thoughts on the pedagogical value of such an activity from an ethnomathematics perspective.

Jack Carter presents a dynamic study on the investigation of students' ability to identify a function of two variables and its first order partial derivatives from three-dimensional plots. His theoretical framework draws upon previous studies that documented students' difficulties in understanding graphical representations. He concludes the article with a discussion of technology-intensive calculus courses and presents rich possibilities both for classroom use and for future investigations.

The challenges of major cryptographic systems and an analysis of how mathematics contributes to their development are presented in the engaging article by Queena Lee-Chua. She demonstrates ways in which the study of cryptography draws not only from various branches of mathematics (number theory, matrices, statistics) but extends to interdisciplinary areas such as history and finance. She provides several examples and maintains that cracking a code can be as much of an adventure for an elementary school student as for a college mathematics major.

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New England Mathematics Journal

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November 2000 Volume XXXIII, Number 1

Editorial
Beverly J. Ferrucci

A wide range of interesting mathematical topics and applications are discussed in this issue. They include a software application for creating interactive mathematical notebooks and a German research study on a child's ability to perform multiplicative reasoning. It also contains a teaching method that helps to foster the belief that mathematical ideas are interrelated by promoting the use of different thinking styles with a mathematics class.

In Miroslaw Majewski's article, he explores MathView, a powerful computer program used to perform mathematical calculations. In addition, MathView can be used as an authoring tool for creating interactive mathematical notebooks that can be used on personal computers as well as on the Internet. He includes MathView examples in his article and completes it by showing how experimenting with mathematical objects in MathView can improve the teaching and learning of mathematics.

Marianne Franke and Silke Ruwisch investigate a child's ability to perform multiplicative reasoning in a real world setting with the use of a dollhouse. They utilize projects, real world situational contexts, poster tasks, and word problems as a means of presenting applied mathematics problems at the elementary school level. Their analyses involve a comparison of action patterns, arithmetical strategies, and working with remainders.

The article by Eric Pandiscio discusses a method of engaging students in successful problem-solving experiences by providing them with carefully-designed and selected tasks or problems that have the potential for having multiple solutions. He demonstrates a way in which one dynamic problem can be used to achieve a deeper appreciation for the way in which several mathematical concepts can be embedded within a single mathematical challenge. Three mathematically and cognitively distinct solutions to a given problem are also included.

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May 2000 Volume XXXII, Number 2

Guest Editorial
M. Kathleen Heid
The Pennsylvania State University

This special issue of the New England mathematics Journal is devoted to matters of concern arising from the NCTM's Year 2000 revision of its Standards, Principles and Standard for School Mathematics. These new Standards paint an updated vision of the mathematical content and processes that ought to guide the teaching and learning of mathematics at the start of the 21^st Century. In the course of constructing Principles and Standard, the Writing Groups confronted a series of defining issues. The articles in this volume represent a subset of those issues-ones chosen by members of the writing team of Principles and Standards to be of particular importance to school mathematics of the early 21^st Century.

The volume begins with an article by Joan Ferrini-Mundy (the chair of Writing Group for Principles and Standards for School Mathematics), and Gary Martin (Project Director for the Standards 2000 Project). Throughout the Project's denouement, Joan and Gary were constantly assessing the extent to which the document said what it needed to say and was appropriately responsive to the field. In their article, Developing Principles and Standards for School Mathematics: The Role of Feedback and Advice, they describe the careful process of gathering reactions and advice from the field in the midst of public controversy about the directions of school mathematics.

The organizing feature of Principles and Standards for School Mathematics is the identification of ten Standards cutting across four Grade Bands: PreK-2, Grades 3-5, Grades 6-8, and Grades 9-12. For each grade band, this special issue of the NEMJ contains two articles, each written by a member of the Writing Group for that grade band. The topics of the articles were ones chosen by the authors as an issue of particular importance to those concerned with the teaching and learning of mathematics at the targeted grade levels.

Principles and Standard took an important step in giving early childhood mathematics separate and special attention. The first PreK-2 article, Mathematics of Children, Mathematics for Children by Alfinio Flores, describes the mathematics and mathematical thinking that ought to characterize the mathematical experiences of our youngest school children. One of the important stands that Principles and Standards took was to include a technology principle. The principle states: Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students' learning. In Technology, Mathematics, and the Young Child, Douglas Clements makes the case that technology is necessary and appropriate for young children, and he describes two types of computer environments that can contribute importantly to the mathematics of young children.

The articles written about mathematics in grades 3 through 5 develop general issues that influence the teaching and learning of mathematics in the intermediate grades. The first article centers on arguably the strongest stand taken in the Grades 3-5 Standards. Susan Jo Russell's article, Developing Computational Fluency with Whole Numbers in the Elementary Grades, reflects the hard thinking that underpinned the treatment of computation in the Grade 3-5 Grade Band chapter. Susan Jo casts the issue in light of balance, noting that the problem was "how to balance the need for both skills and understanding, how to make sure students develop both procedural competence and mathematical reasoning." In the second article, Who Should Teach Mathematics to Elementary Students: A Case for Mathematics Specialists, Barbara Reys raises a concern about the mathematical preparation needed by teachers in the upper elementary grades. Drawing on models from Sweden and South Korea, her article makes a call for mathematics specialists to teach mathematics to children in grades 4-6.

Algebra and geometry at the middle grades level are not the high school courses moved to earlier grades. Themes taken up in the two middle grades articles are ones of the nature of the algebra and geometry experiences students should have in grades 6 through8. Diana Lambdin, in Algebraic Thinking in the Middle Grades, paints a picture of algebraic thinking in the middle grades and differentiates it from the current focus on symbolic manipulation technologies, and that it is shifting towards functions, modeling, and multiple representations.

In a New Look at Geometry in the Middle Grades, Carol Malloy offers her analysis of geometry experiences for middle grades students that would capture for them the excitement and importance of school geometry. Her article addresses two important issues that are often in conflict:"What geometry should all students experience?" and "What preparation is needed for high school geometry?"

The last pair of articles address the teaching of mathematics in grades 9-12. M. Kathleen Heid's article, Function and Transformation as Central Related Concepts in High School Mathematics, describes some over-arching concepts that provide and interconnectedness to high school mathematics. She describes how the related concepts of function and transformation can play featured and recurring roles as high school students learn algebraic, geometric, and statistical concepts. Perhaps one of the currently most underplayed topics of importance in high school mathematics is that of statistical thinking. In her article, A Case and Pace for Statistics in High School, Sue Eddins takes on the challenge of characterizing why and how statistics will take its rightful place as an area of study for all high school students.

The design of this special issue was to bring before classroom teachers some of the themes and issues that permeate nationwide discussion of the future of school mathematics. The issues are representative not of the recommendations make in the Principles and Standards but of the thinking of some of its writers. It is my hope that it will generate some food for thought for the readers of the New England Mathematics Journal.

Finally, I want to thank the writers of these articles for their generosity and insight and Beverly Ferrucci for her infinite patience.

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New England Mathematics Journal

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November 1999 Volume XXXII, Number 1

Editorial
Beverly J. Ferrucci

The articles in this issue present a wide range of interesting topics and applications of mathematics. They include an Australian research study on the role of manipulatives in the elementary school mathematics curriculum, a trigonometric approach to football, and a description of an interactive mathematics class. Other articles illustrate problems that require the use of recursion and deductive reasoning.

Peter Howard and Bob Perry present a study that raises issues for mathematics educators about the appropriate role of manipulatives within the school mathematics curriculum. They consider the place of manipulatives within current approaches to the teaching and learning of mathematics and the complex role that manipulatives play in students' learning in the elementary mathematics classroom. Their study includes baseline data about Australian elementary teachers' use of manipulatives and the relationship of this use to the teachers' beliefs about mathematics, mathematics learning and mathematics teaching.

Challenging problems that involve the use of recursion are presented in the article by Thomas Koshy. He demonstrates ways in which recursion can be employed to solve problems from number theory, algebra, geometry, and combinatorics. He concludes by demonstrating that such problems provide an excellent opportunity for pattern recognition, conjecturing, data collection and analysis, and mathematical induction.

The article by Betsy McShea and Maureen Yarnevich looks at the use of realistic solutions that arise in the game of football as a means of using mathematics to gain a different perspective into the inner workings of athletics. It focuses on how to integrate the law of cosines, the Pythagorean Theorem, trigonometric functions and the quadratic formula into sports scenarios.

Arthur Johnson presents a series of problems called conclusion quickies and mystery problems that require deductive reasoning. Conclusion quickies are brief problems that may have many different conclusions, while mystery problems require more time and allow students to use pertinent facts to form valid conclusions. In addition, the article contains a list of references for both types of problems that can be used effectively by middle and high school students.

Interactive mathematics is addressed in the article by Kevin Vetre. He writes that students should be shown the relevancy of mathematics courses and uses student-requested projects to demonstrate this relevancy. He describes and discusses specific projects from his secondary mathematics classes that involve interesting materials such as architect's blueprints and multilevel spiral ramps.

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Comments, questions, and suggestions regarding the web page of the Association of Teachers of Mathematics in New England are most welcome. Please direct them to William Bowdish.

New England Mathematics Journal

Table of Contents

May 1999 Volume XXXI, Number 2

Guest Editorial
Karen J. Graham
University of New Hampshire
Durham, NH 03824

It has been over a decade since NCTM's release of the Curriculum and Evaluation Standards for School Mathematics in 1988. Much discussion, debate, curriculum and research initiatives followed its release. Attention is currently focused on the discussion draft of NCTM's updated Standard 2000. Research supports the notion that teaching is a very complex process, at any single moment the teacher is playing a variety of roles and major decisions need to be made at various junctions about the mathematical content (tasks) that are to be use, how to promote discourse, and how to structure the environment to facilitate maximum learning of mathematics by all students. Time for teacher reflection and analysis are critical to this process. This issue contains examples of the teacher as researcher and writer as a way to think about this process of analysis and reflection.

Teacher Research and the Professional Disposition of Teaching by Geoff Mills gives a definition and model for the process of teacher research. It provides an example of how the model was applied in a middle school in Oregon that was reflecting on state assessment results. In addition, the article contains a valuable list of Internet resources which might provide support and guidance for the teacher researcher.

Envisioning New Practices through Teacher Narratives by Deborah Schifter reports on the results of an NSF-funded project, the Mathematics Process Writing Project, designed to support teachers writing and reflection on their own mathematics teaching and that of others. The project serves as a valuable model on which to build activities in preservice and inservice programs for teachers.

Pat Tinto and her colleagues report on another NSF-supported project in their article, Is it worth making changes? Improving Instruction through Teacher Research. The examples describe how teachers in the project used classroom research to rethink their practice. The research projects conducted by the teachers are examples of Mills' teacher research model in action.

Crossing Boundaries: Merging the Roles of Teacher and Researcher by Ann Enyart and Laura Van Zoest provides an example of a collaborative effort between a university researcher in mathematics education and a classroom teacher. It includes aspects of the projects reported on in the three previous articles and ideas on how to get started.

Finally, Richard Zang in his article, Teachers as Classroom Researchers in Mathematics Education, provides a story of his own personal involvement with classroom research at the collegiate level. He states that "good teaching requires all faculty to be scholars" and his work extends the teacher as researcher model beyond the precollege classroom.

I would like to thank ATMNE for the opportunity to serve as guest editor of this issue. In addition, I would like to thank the authors for their thoughtful contributions to this edition. I hope that you will find the articles stimulating and helpful.

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New England Mathematics Journal

Table of Contents

November 1998 Volume XXXI, Number 1

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Comments, questions, and suggestions regarding the web page of the Association of Teachers of Mathematics in New England are most welcome. Please direct them to William Bowdish.

New England Mathematics Journal

Table of Contents

May 1998 Volume XXX, Number 2

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Comments, questions, and suggestions regarding the web page of the Association of Teachers of Mathematics in New England are most welcome. Please direct them to William Bowdish.

New England Mathematics Journal

Table of Contents

November 1997 Volume XXX, Number 1

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Comments, questions, and suggestions regarding the web page of the Association of Teachers of Mathematics in New England are most welcome. Please direct them to William Bowdish.

New England Mathematics Journal

Table of Contents

May 1997 Volume XXIX, Number 2

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Comments, questions, and suggestions regarding the web page of the Association of Teachers of Mathematics in New England are most welcome. Please direct them to William Bowdish.

New England Mathematics Journal

Table of Contents

November 1996 Volume XXIX, Number 1

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Comments, questions, and suggestions regarding the web page of the Association of Teachers of Mathematics in New England are most welcome. Please direct them to William Bowdish.