
New England Mathematics Journal
An official publication of the Association of Teachers of Mathematics in New England (An Affiliate of the National Council of Teachers of Mathematics)
Editor
Beverly J. Ferrucci, EdD., Ph.D.
Department of Mathematics
Keene State College
229 Main Street
Keene, NH 034352010
The New England Mathematics Journal is published twice yearly (May and November). Regular membership in a New England section of the Association includes a subscription to the New England Mathematics Journal.
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).
Manuscripts submitted for publication should not exceed ten pages. The style guide is the current edition of the Publication Manual of the American Psychological Association. 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.
The New England Mathematics Journal is edited and published at 229 Main Street, Department of Mathematics, Keene State College, Keene, NH 034352010.
Preview Issues of the New England Mathematics Journal
Table of Contents May 2015 Moving Principles into Actions: Understanding the Challenges and Promise of Principles to Action  Part I
Table of Contents May 2014 Classroom Assessment to Achieve the Common Core Standards for Mathematical Practice
Table of Contents May 2013 Mathematics Coaching: Implications for Change
Table of Contents May 2012 Envisioning Effective Implementation of the Common Core State Standards
Table of Contents May 2011 Exploring the Richness of Geometry via Technology
Table of Contents May 2010 Empowering and Supporting New Teachers of Mathematics
Table of Contents May 2009 Approaches to Equity in Mathematics Education
Table of Contents May 2008 Mathematics and Language
Table of Contents May 2007 Differentiating Instruction in Mathematics Education
Table of Contents May 2006 Different Perspectives on Mathematics Assessment
Table of Contents May 2005 Teacher Mentoring and Learning to Teach Mathematics
Table of Contents November 2004
Table of Contents May 2004 Perspectives and Reflections on the History of Mathematics
Table of Contents November 2003
Table of Contents May 2003 Learning and Teaching Geometry
Table of Contents November 2002
Table of Contents May 2002 Focus Issue on the Teaching and Learning of Data and Chance: Cases, Reflections, Suggestions
Table of Contents November 2001
Table of Contents May 2001
Table of Contents November 2000
Table of Contents May 2000 Millennium Focus Issue  Perspectives on Principles and Standards
Table of Contents November 1999
Table of Contents May 1999 Focus Issue on Teachers as Researchers
Table of Contents November 1998
Table of Contents May 1998 Focus Issue on Algebra
Table of Contents November 1997
Table of Contents May 1997 Focus Issue on Assessment
Table of Contents November 1996
Table of Contents
May 2015 Volume XLVII Part I
Guest Editorial
Moving Principles into Actions: Understanding the Challenges and Promise of Principles to Actions
Robert Q. Berry III
Page 3
Supporting Productive Struggle in Mathematics Classrooms
Margaret Smith and Jeffery Ziegler
Page 7
Teaching for Representational Competence in Mathematics
DeAnn Huinker
Page 18
One District’s Journey to Promote Access and Equity
Matthew R. Larson and Delise Andrews
Page 31
Assessment as the Cornerstone of the Mathematics Classroom
Daniel J. Brahier
Page 41
Professionalism: The Newest Principle
Steven Leinwand
Page 53
Membership
Page 64
Table of Contents
May 2014 Volume XLVI
Guest Editorial
David C. Webb
Page 3
Reasoning and Representation in the Elementary Grades: Two Perspectives on Assessing the Standards for Mathematical Practice
David C. Webb
Page 6
The Role of Curriculum Design and Language in Informal Assessment
Jeffrey Choppin
Page 25
Assessing and Using Students’ Prior Knowledge in ProblemBased Instruction
Gloriana González, Anna DeJarnette, Jason Deal
Page 38
Using Tasks to Access and Support Student Modeling, Use of Mathematical Structure, and Argumentation
Kevin J. Reins
Page 50
Table of Contents
May 2013 Volume XLV
Guest Editorial
Neelia Jackson
Page 3
Math Coaching: Reflecting on Lessons Learned
Suzanne Landers Zavatsky
Page 10
The Top 5 Rules for All Coaches
Lorrie Quirk
Page 22
District Level Coaches Provide Crucial Support for a Successful Coaching Model
Monica L. Leon
Page 28
Coaching in a Time of Change
Connie Henry
Page 34
Challenging Assumptions About Coaching
Lucy West and Antonia Cameron
Page 41
Instructional Coaching: An Introduction and Theoretical Framework
Jodie Flint and Duana Swenson Howerton
Page 62
Table of Contents
May 2012 Volume XLIV
Guest Editorial
Bradford R. Findell
Page 3
Get to Know Your States’ Website on the Common Core State Standards for Mathematics
Beverly J. Ferrucci
Page 13
Ten Essential Mindsets for Tilling the Soil For the Common Core State Standards for Mathematics
Steven Leinwand
Page 21
CCSSM Adoption is an Adaptive Challenge: The Example of High School Algebra I
Life LeGeros
Page 34
Up and Running With the Common Core State Standards
Anne M. Collins
Page 47
Our Journey to “The Core”
Christine K. Newman
Page 56
Membership
Page 64
Table of Contents
May 2011 Volume XLIII
Editorial
Sharon McCrone
Page 3
Can a Kite be a Triangle? Bidirectional Discourse and Student Inquiry in a Middle School Interactive Geometric Lesson
Paul Woo Dong Yu and Dante A. Tawfeeq
Page 7
Making Conjectures in Dynamic Geometry: The Potential of a Particular Way of Dragging
Maria Alessandra Mariotti and Anna BaccagliniFrank
Page 22
Mirrors and Dragonflies – Dynamic Investigations with GeoGebra
Markus Hohenwarter
Page 36
What’s 2 Got to Do with It? Using Dynamic Geometry Environments to Find Surprising Results and Motivate Proof
Tami Martin, Craig J. Cullen, and Roger Day
Page 49
Serious Play with Dynamic Plane Transformations
James King
Page 65
Membership
Page 80
Table of Contents
May 2010 Volume XLII
Guest Editorial
Michaele F. Chappell, Middle Tennessee State University
Page 4
Showing Your Students You Care: Seeing the Individual Trees in the Classroom Forest
Randolph A. Philipp and Eva Thanheiser
Page 8
Plan B: If I Knew Then What I Know Now
Danny Bernard Martin and Aleksandra Mironchuk
Page 20
Developing Mathematics Content Knowledge Along New Paths: Shifting from Student to Teacher
Mary C. Enderson, Jacob T. Klerlein, and Jason D. Johnson
Page 31
How Elementary School Principals with Different Leadership Content Knowledge Profiles Support Teachers’ Mathematics Instruction
Barbara Scott Nelson
Page 43
Navigating the Peaks and Valleys of Teaching
Cathy Seeley
Page 64
Table of Contents
May 2009 Volume XLI
Editorial
Linda Dacey
Page 3
Building a Nurturing Visual MathTalk TeachingLearning Community to Support Learning by English Language Learners and Students from Backgrounds of Poverty
Karen Fuson, Todd Atler, Sheri Roedel, Janet Zaccariello
Page 6
Voices, Power, and Multiple Identities: African American Boys and Mathematics Success
Robert Q. Berry and Oren L. McClain
Page 17
Meeting the Needs of Special Education Students in Mathematics Classrooms using SchemaBased Instruction
Asha K. Jitendra
Page 27
SelfRegulation and Mathematical Problem Solving
Marjorie Montague
Page 34
Mathematically Promising Students Deserve Equity, Too
Linda Sheffield
Page 43
Achieving Equity for All Students
Anne Collins
Page 52
Membership
Page 64
Table of Contents
May 2008 Volume XL, Number 2
Editorial
Tim Whiteford
Page 3
Mathematics as a Second Language: Reading, Writing, and Doing Mathematics
Mahesh Sharma
Page 6
Writing, Mathematics, and Meaning
Joan Countryman
Page 20
Developing Mathematics Vocabulary in Context
Miki Murray
Page 24
The Role of Language in Mathematics Achievement in Deaf Students
Ellen B. Metzger
Page 42
Linguistic Registers in the Mathematics Classroom
Tim Whiteford
Page 54
Membership
Page 64
Table of Contents
May 2007 Volume XXXIX
Editorial
Carol Ann Tomlinson and Catherine Finlayson Reed
Page 4
Using Rich Problems for Differentiated Instruction
Eric Hsu, Judy Kysh, and Diane Resek
Page 6
Differentiating Mathematics For Young Students
Megan Kelly Murray
Page 16
Differentiation In Algebra I: A Tiered Graphing Lesson
Catherine Finlayson Reed
Page 26
Using Tiers to Differentiate Instruction On Properties of Two  Dimensional Shapes
Shannon O. S. Driskell and Janet M. Herrelko
Page 36
Differentiating Instruction Using Technology: Some Positive Results In Student Achievement
L. Riggs, S. Thomas, and M. McHenery
Page 46
Differentiation Through Instructional Design For Struggling Students
Bernadette Kelly and Douglas Carnine
Page 54
Membership
Page 64
Table of Contents
May 2006 Volume XXXVIII, Number 2
Editorial
William S. Bush and Maggie McGatha
Page 4
A Cognitive Perspective On Formative Assessment
Michael T. Battista
Page 6
Improving Mathematics Instruction Through Formative Classroom Assessment
George W. Bright and Jeane M. Joyner
Page 23
Assessing Teacher’s Knowledge: Diagnostic Mathematics Assessments For Elementary and Middle School Teachers
E. Todd Brown, Maggie McGatha, Karen Karp
Page 37
Mathematics Educators and LargeScale Assessment: Eight Lessons We Can Teach
Lew Romagnano
Page 51
Table of Contents
May 2005 Volume XXXVII, Number 2
Editorial
Jian Wang
Page 4
A Distributed Model for Teacher Mentoring: Broadening the Learning Community
Jeffrey Frykholm
Page 10
Using Mentoring and Professional Development Approaches to Educate Urban Mathematics Teachers
Pamela FraserAbder
Page 20
Mentoring in Preservice Mathematics Teacher Education
Denise S. Mewborn
Page 30
Factors That Affect Mathematical Discussion Among Secondary Student Teachers and Their Cooperating Teachers
Blake Peterson, Steven R. Williams, Vari Durrant
Page 41
Challenges and Opportunities of Peer Mentoring: A Case Study of MentorMentee’s Conversations on Mentee’s Teaching
Mary Sowder
Page 50
Thoughts on Teacher Mentoring
Louise M. Lataille
Page 60
Membership
Page 64
Table of Contents
November 2004 Volume XXXVII, Number 1
Editorial
Beverly J. Ferrucci
Page 3
Deepening Teacher Understanding: The Case of Algebra
Mary M. Sullivan
Page 4
The Harmonics of a Piston
Robert Galloway
Page 26
Interesting Things to do in a Classroom with only the Students, One Teacher, and a Chalkboard
Marta Samson
Page 32
Membership
Page 48
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: preservice and inservice teachers, precalculus students, and elementary school teachers.
Mary Sullivan’s article illustrates how a pattern problem enhanced the algebraic understanding of preservice 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.
Table of Contents
May 2004 Volume XXXVI, Number 2
Editorial
David Burton
Page 3
A Conversation with Father Stanley Bezuszka, S.J.
Beverly J. Ferrucci
Page 7
David Eugene Smith: Mathematics Educator Extraordinaire
Frank Swetz
Page 14
Gone with the Slide Rule: The Story of the Polar Planimeter
Ed Sandifer
Page 22
Raymond Clare Archibald: A Euterpean Historian of Mathematics
Jim Tattersall and Shawnee McMurran
Page 31
Twentieth Century Histories of Mathematics
David Burton
Page 42
Membership
Page 48
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 ZermeloFrankel 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 SpherePacking Problem. The monumental task (12,000 journal pages over a 30year period) of classifying all finite simple groups was laid to rest; with it came the discovery of the largest such group, the socalled 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 centuryold 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 SpherePacking 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 19981999. 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 researchoriented institutions, although as yet in disproportionately low numbers. By the last quartercentury, 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 midcentury’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 earlycentury works of modest size without exercises or photographs to the modern largescale 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.
Table of Contents
November 2003 Volume XXXVI, Number 1
Editorial
Beverly J. Ferrucci
Page 3
TIMSS & TIMSSR: Performance of Eighth Graders from Singapore and the United States in Mathematics
Berinderjeet Kaur, Beverly J. Ferrucci, and Jack Carter
Page 5
PentaBlocks™: Using this Manipulative in the Classroom
Craig Sheil
Page 26
Using Mathematical Games in Teacher Education
Gertrude R. Toher
Page 35
Spreadsheets in the Elementary Mathematics Classroom
Jennifer Edwards
Page 45
Membership
Page 56
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 fourthgraders had advanced to the eighth grade and were retested for the Third International Mathematics and Science Study – Repeat (TIMSSR). In both the TIMSS and TIMSSR 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 fivepointed 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 minilectures with focused discussions, and relating the experience to the teachinglearning 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.
Table of Contents
May 2003 Volume XXXV, Number 2
Editorial
James Morrow
Page 3
Further Steps: Geometry Beyond High School
Catherine A. Gorini
Page 6
Teaching and Learning Geometry Through Student Ownership
Carol E. Malloy
Page 16
Problem Posing in Geometry: Questions I Wish Someone Had Asked!
Marian Walter
Page 28
What to Expect When Geometry Becomes Interactive
Daniel Scher
Page 36
Thinking Through Geometry in the Undergraduate and Teacher Preparation Curriculum
Kelly Gaddis
Page 44
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 threedimensional 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.
Table of Contents
November 2002 Volume XXXV, Number 1
Editorial
Beverly J. Ferrucci
Page 3
Converging on the Circumference of a Unit Circle
Robert Galloway
Page 5
A Geometric Perspective on the Sample Standard Deviation
Bradford D. Allen
Page 19
Reasoning and Proof in Your Middle School and High School Classroom
Thamas Baird
Page 28
Membership
Page 40
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.
Table of Contents
May 2002 Volume XXXIV, Number 2
Editorial
Michael Shaughnessy
Page 3
Data Analysis in the K12 Mathematics Curriculum: Teaching the Teachers
Richard L. Scheaffer
Page 6
Making Inferences and Predictions from Data in the Elementary and Middle Grades
Edward S. Mooney, Graham A. Jones, and Cynthia W. Langrall
Page 26
Wooden or Steel Roller Coasters: What's the Choice?
Susan N.Friel
Page 40
When 2+2≠4 and 6+6≠12 in Data and Chance
Jane Watson
Page 56
Teaching Concepts Rather Than Conventions
Clifford Konold
Page 69
Statistical Thinking: An Exploration into Students' VariationType Thinking
Maxine Pfannkuch, Amanda Rubick, and Caroline Yoon
Page 82
Membership
Page 100
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 K12 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.
Table of Contents
November 2001 Volume XXXIV, Number 1
Editorial
Beverly J. Ferrucci
Page 3
Imagery and Motion: Promoting Students' Understanding of Fractions
Thomas De Franco, Jean McGivneyBurelle, and Mary Truxaw
Page 4
A Study of the Relationship Between Academic SelfConcept Achievements of OLevel Students in English and Mathematics
Indira Chacko and John Crowe
Page 20
The Calculus of the Middle
Lynn A. Fisher
Page 35
Editorial
Beverly J. Ferrucci
Thomas DeFranco, Jean McGivneyBurelle, 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 selfconcept 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 selfconcept and achievement in the two subject areas. Students from an all girls' school, an all boys' school, and a coed school were participants in the study.
Lynn A. Fisher's article discusses the confusion some students experience because of the closelyrelated 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 studentgenerated 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.
Table of Contents
May 2001 Volume XXXIII, Number 2
Editorial
Beverly J. Ferrucci
Page 3
Symmetry Patterns in Carriage Friezes on Brazilian Trucks
Rogeria Gaudencio, Romulo Marinho, and John Fossa
Page 4
Improving Geometric Interpretations in Multivariate Calculus
Jack Carter
Page 27
Matrices, Frequencies, Congruences, and Codes
Queen LeeChua
Page 38
Membership
Page 48
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 threedimensional plots. His theoretical framework draws upon previous studies that documented students' difficulties in understanding graphical representations. He concludes the article with a
discussion of technologyintensive 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 LeeChua. 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.
Table of Contents
November 2000 Volume XXXIII, Number 1
Editorial
Beverly J. Ferrucci
Page 3
Learning and Teaching Mathematics by Experimenting in MathView
Miroslaw Majewski
Page 6
A Dollhouse as a Situational Context for Multiplicative Reasoning
Marianne Franke and Silke Ruwisch
Page 18
Multiple Solutions to a Single Problem
Eric Pandiscio
Page 30
Membership
Page 42
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 problemsolving experiences by providing them with carefullydesigned 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.
Table of Contents
May 2000 Volume XXXII, Number 2
Guest Editorial
M. Kathleen Heid
Page 3
Developing Principles and Standards For School Mathematics: The Role of Feedback and Advice
Joan FerriniMundy and W. Gary Martin
Page 6
Mathematics of Children, Mathematics for Children
Alfinio Flores
Page 18
Technology, Mathematics, and the Young Child
Douglas H. Clements
Page 28
Developing Computational Fluency with Whole Numbers in the Elementary Grades
Susan Jo Russell
Page 40
Who Should Teach Mathematics to Elementary Students? A Case for Mathematics Specialists
Barbara J. Reys
Page 55
Algebraic Thinking in the Middle Grades
Diana Lambdin
Page 65
A New look at Geometry Taught in the Middle Grades
Carol E. Malloy
Page 78
Function and Transformation as Central Related Concepts in High School Mathematics
M. Kathleen Heid
Page 88
A Case and a Place for Statistics in the High School
Susan K. Eddins
Page 99
Membership Information
Page 108
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 issuesones 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 FerriniMundy (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: PreK2, Grades
35, Grades 68, and Grades 912. 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 PreK2 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 35 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
35 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 46.
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 912. M. Kathleen Heid's article, Function and Transformation as Central Related Concepts in High School Mathematics,
describes some overarching 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.
Table of Contents
November 1999 Volume XXXII, Number 1
Editorial
Beverly J. Ferrucci
Page 3
Learning Mathematics and Manipulatives: Not Just Child's Play
Peter Howard and Bob Perry
Page 5
Can Recursion Be Taught in Schools?
Thomas Koshy
Page 16
A Trigonometric Approach to a Quarterback's Passing Game
Betsy McShean and Maureen Yarevich
Page 26
Spadework Prior to Formal Deduction
Arthur Johnson
Page 37
Interactive Mathematics
Kevin Vetre
Page 48
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 studentrequested 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.
Table of Contents
May 1999 Volume XXXI, Number 2
Guest Editorial
Karen Graham
Page 3
Teacher Research and the Professional Disposition of Teaching
Geoff Mills
Page 5
Envisioning New Practices Through Teacher Narratives
Deborah Schifter
Page 18
Is it Worth Making Changes? Improving Instruction Through Teacher Research
Patricia P. Tinto
Page 30
Crossing Boundaries: Merging the Roles of Teacher and Researcher
Ann Enyart and Laura Van Zoest
Page 40
Teachers as Classroom Researchers in Mathematics Education
Richard A. Zang
Page 50
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 NSFfunded 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 NSFsupported 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.
Table of Contents
November 1998 Volume XXXI, Number 1
Editorial
Beverly J. Ferrucci
Page 3
A Comparison of Problem Sets in American and Chinese\Mathematics Textbooks
Yeping Li and Jack Carter
Page 5
Another Development of Basic Trigonometric Relations
Paul Maiorano
Page 21
Approximating the Standard Normal Distribution Table
Irwin Kabus
Page 25
Punnett Squares for Multiplying and Factoring Polynomials
Carol M. Lerch
Page 34
Homework Review Model: A Means of Advancing Students' Understanding
Regina M. Panasuk and Joseph W. Spadano
Page 45
Strategies for Successful Math Teaching on a Block Schedule
Tracy J. Shannon
Page 54
Table of Contents
May 1998 Volume XXX, Number 2
Guest Editorial
Thomas C. Defranco
Page 3
Part I: What is Algebra?
Teaching of Algebra, K12
Stephen Willoughby
Page 5
The Big Ideas of Algebra
John A. Dossey
Page 18
Algebra: From the Past to the Future
Gail Burrill
Page 24
Algebra in a Technological World
Dwayne Cameron
Page 35
Part II: The Teaching of Algebra
Visual Patterns: A Powerful Connection Tool in Algebra
Mike Shaugnessy
Page 39
Algebra in the Millde Grades
Elizabeth A Phillips, John P. Smith III, Jon Star, Beth HerbelEisenmann
Page 48
Algebra: A Gatekeeping Course That Must Go!
Johnny W. Lott
Page 61
Algebraic Reasoning with Technology in the College Curriculum
Rose Mary Zbiek
Page 74
Table of Contents
November 1997 Volume XXX, Number 1
Editorial
Beverly J. Ferrucci
Page 3
Meaningful Population Parameters
John Judge
Page 4
AllFemale Math Classes: An Opportunity for Classroom Teachers and Researchers to Collaborate
Bonnie Wood
Page 10
Exploring a Fibonacci Puzzle
Regina M. Panasuk and Mary M. Sullivan
Page 18
Digital Roots of Figurate Numbers
Thomas Koshy
Page 28
Geometric Scaling and the Volume of a Pyramid
William J. Leonard
Page 36
The HardyWeinberg Law in population Genetics
Mitchell Preiss
Page 51
Table of Contents
May 1997 Volume XXIX, Number 2
Guest Editorial
Robert Kenney
Page 3
What Does It Mean To Be Fair?
Jean Kerr Stenmark
Page 4
Vermont's Mathematics Assessment Program
Page 10
Clear Expectations=Quality Work: Using Projects and Rubrics in Geometry Class
Joanne Finnegan and Lynn Hier
Page 11
Rhode Island's Mathematics Assessment Program
Page 19
Professional Development in Performance Assessment
Judy WilsonDroitcour and Anne Seitsinger
Page 20
New Hampshire's Mathematics Assessment Program
Page 25
Teaching to the Test: The Positive Side
Judith Blood
Page 26
The Connecticut Mastery Tests and Academic Performance Test
Page 29
One High School's Response to Connecticut's State Program
George E. Parker
Page 30
Maine's Mathematics Assessment Program
Page 33
Flock Talk
Dan Hupp
Page 34
Massachusetts' Mathematics Assessment Program
Page 37
A Classroom View of the Implementation of the Massachusetts Curriculum Frameworks and the State Assessment Program in Middle School Mathematics Anne M. Collins
Page 38
Table of Contents
November 1996 Volume XXIX, Number 1
Editorial
Beverly J. Ferrucci
Page 3
Some Interesting Aspects of Euler's Phi Function
Sandra M. Pulver and Mitchell P. Reiss
Page 4
Exploratory Data Analysis for the Classroom: Total Hit Statistics in Baseball
Bonnie H. Litweiller and David R. Duncan
Page 12
Assessment to Enhance Teaching and Learning
Regina Panasuk and Shelley Rasmussen
Page 20
Draw Your Way Through Algebra Andrew Freda
Page 29
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