Math Lessons for Elementary Grades

Dorothy Harrer

New revised and reorganized by grade - excerpts from Steiner's lectures added

Covers Grades 1-6

Softbound

$14.00

After far too many years out-of-print, Dorothy Harrer's beloved Math Lessons is again available - and this edition is even better than the original. Now all of the lessons have been organized by grade and proofread by expert mathematics teachers. Additionally, there are wonderful excerpts from Steiner that have been added; each of them beautifully illuminates the approach you'll find in this book. Overall, this is a book to love and rely on - it is filled with joyful potency.

Contents:

• Excerpts from Rudolf Steiner
• Some thoughts in relation to arithematic teaching
• Numbers and the four processes in First Grade
  • Numbers
  • The number three
  • Two, four, and eight
  • One through five
  • Six
  • Sums
  • We are all one whole class
  • Gnomes and jewels
    • The four processes
  • The family that became one person
• A selection of lessons for the Second Grade
  • The richest number
  • Big smoke and little flame
  • Two shapes
  • The king's advisors
  • A number story for the sixth table
  • The giant who counted with his feet
• A selection for lessons for the third grade
  • Big brother and little sister
  • Plus-Times and Minus-Divideby
  • Early measures
• A selection of lessons for the fourth grade
  • The numbers workers
  • Riddle
  • Have an apple!
  • The tens as helpers
• A selection of lessons for the fifth grade
  • History of numbers
  • The story of Pythagoras
• A selection of lessons for the sixth grade
  • The decimal system
  • Percent means per hundred
  • What is a mill?
  • Fractional percents
  • Introduction to interest
  • More matters of interest
  • This year? Next year? Some time?
  • Simple interest and compound interest
  • Commission
  • Taxes
  • Ratio equivalents

A Steiner-Waldorf Mathematic Resource
Grades 1-8

A Path of Discovery Book

A compilation of mathematic resources taken from A Path of Discovery - Grades 1-8. Revised and with additional material.

Eric Fairman

Comb Bound

$20.00

Eric Fairman's Mathematic Resource offers a beautiful, wonderfully useful overview of the mathematics currciulum for the entire 8 grades. Now, in one volume, you can see not only what to teach in any given grade, but where that teaching is headed and/or how the foundation for it has been laid. This will be of enormous help to homeschoolers and class teachers alike - assessing a student's level and finding ways to augment their studies will become much, much easier.

A Steiner-Waldorf Mathematic Resource also contains new material as well as revisions of past publications.

Contents:

• Grade 1 - Introduction to Numbers and the Four Processes
• Grade 2 - Shapes, Number Patterns and Skills
• Grade 3 - Measurement
• Grade 4 - Review and Fractions
• Grade 5 - Decimals and Fractions
• Grade 6 - Geometry and Business Math
• Grade 7 - Geometry, Geometric Proofs, Logarithmic patterns
• Grade 8 - Platonic Solids
• Appendix: Fraction Dominoes for Grade Four

Active Arithmetic
Movement and Mathematics Teaching in the Lower Grades of a Waldorf School

Henning Andersen

Softbound

$16.00

Is it possible to teach arithmetic in the first years of school in such a way that the majority of pupils do not experience the lesson as a strain? Are there ways of practicing arithmetic that do not adversely affect a child's zest for life . . . and still teach the subject well?

Active Arithmetic describes the teaching of arithmetic in Rudolf Steiner/Waldorf Schools, primarily from the perspective of the lively movement that conveys so much of the lesson. The many different aspects of movement and drawing that teachers use in the course of the arithmetic lessons satisfy both the children's great need of physical activity and their innate curiousity about the world of numbers.

This books is directed not only to teachers of arithmetic in the younger classes, but also to people who work with the same age group outside school using games and movement. Homeschoolers and parents wishing to help support their children's mathematics lessons at home will find this book to be a wonderful resource.

Teaching Mathematics for First and Second Grades in Waldorf Schools

Ernst Schuberth

Temporarily Out-of-Print

Backorders Accepted

No reprint date announced, but we believe we'll see it again sometime in 2008

$12.00

This book is an excellent support and resource for the classroom or homeschool teacher beginning with first grade. It describes in careful detal the elements necessary to giving children a healthy understanding and enjoyment of mathematics. Dr. Schuberth also gives important insights into what can be done to help the child who already at the beginning has difficulty with math. This book is based on deep and thorough penetration of mathematics and on a a lifetime of experience with children and with an anthroposophical appraoch to child development.

Teaching Mathematics

Roy Wilkinson

Softbound

$13.95

At long last, teachers, parents, and most importantly, the students in their charge can again benefit from Roy Wilkinson's insights into teaching mathematics.

Here, in his typically concise and complete style, Wilkinson has shared the basics of the Waldorf approach to teaching mathematics in Grades 1 through 8. Each page is packed with direction, examples, and a world of understanding. This is an outstanding resource for anyone wanting to know more about the art of teaching as it applies to the world of numbers.

Contents:

• Class 1 & 2: From the whole to the parts - Number symbols - Pictorial introduction to the four processes - Rhythms - Number games - Geometry in drawing and movement
• Class 3 - Measurement - Derivation of terms
• Class 4 - Fractions - Factors - Ratio and proportion
• Class 5 - Families of numbers - Decimals
• Class 6 - Squaring, cubing, square and cube roots - Geometrical figures - Pi - Percentage and interest
• Class 7 - Algebra - Graphs - Golden ratio - Progressions - Pythagorean theorem
• Class 8 - Platonic solids - Algebraic fractions

Teaching Mathematics in Rudolf Steiner Schools for Classes I-VIII

Ron Jarmon

Softbound

$65.00

Ron Jarmon has created a beautiful overview (complete with a lot of how-to's and specific examples!) of the Waldorf mathematics curriculum for grades 1-8. Whether you are teaching at home to one or a few children, or in a large classroom, you'll find Jarmon's presentation will not only set you clearly on the path of what your students need to learn, but will also enliven your own interest in the world of numbers. This is an exciting presentation that:

• covers Grades 1-8 thoroughly, so you can plan easily
• shows how to introduce arithmetic in the early years
• integrates the teaching of mathematics with everyday life
• gives examples so that you can create your own exercises
• shows how to enable your students to easily grasp the basic processes without endless practice and worksheets
• clarifies the importance of testing to help pupils assess themselves
• relates mathematics to other subjects
• introduces principles and skills in accord with students' development

Geometry Lessons in the Waldorf School
Freehand Form Drawing and Basic Geometric Construction in Grades 4 and 5

Ernst Schuberth

Softbound

Includes CD-ROM with additional exercises and color plates

$18.00

This book is the second volume* of mathematician and Waldorf teacher Ernst Schuberth’s Geometry Lessons in the Waldorf School. With an abundance of black and white drawings and a clear, descriptive text, Schubreth covers the free form drawing and basic constructions taught in the Waldorf school fourth and fifth grades.

A real bonus is the accompanying CD Rom that has offers additional exercises and color plates for the teacher to study.

*The first volume is Mathematics Lessons for the Sixth Grade.

Geometric Drawing and the Waldorf School Plan

Hermann von Baravalle, Ph.D.

Softbound

$7.95

Hermann von Baravalle was one of the original teachers at the Waldorf School in Stuttgart. Steiner speaks very highly of him as a scholar and as a person able to pioneer the work of reuniting science and mathematics with the joy of art and a depth of spirt.

Geometric Drawing delves into all manner of ways teachers and students can explore the world of form -- it is actually exciting to page through the book and rediscover various geometric progressions and their transformations into related forms.

This is ideal for the Waldorf 5th grader -- or anyone else wanting to extend or refresh their understanding of geometric construction.

String, Straightedge and Shadow
The Story of Geometry

Julia E. Diggins

$16.95

2003 has become one of the best years in my memory for outstanding new releases and especially outstanding republication of long out-of-print treasures. String, Straightedge and Shadow is one such treasure - and it is so very good to see it returning to print.

Geometry was the one high school mathematics class in which I did not have to struggle - in large part because I happened to be in a school district that chose to teach it through theorems and proofs (this was like sitting in Pythagoras's classroom - truly wonderful!) with practical explanations similar to (but not as beautifully extensive as) those in String, Straightedge and Shadow. As a result, I can personally attest to the wonders this book holds for any student fortunate enough to encounter it.

Since the 1960s, when it went out of print, String, Straightedge and Shadow has made its way into classrooms and homeschools via multi-generation photocopies - it is so highly valued that even without the beautiful illustrations, its contents are treasured by anyone who has used them.

The fabric of the story is woven out of archeological and historical records and of legends about the major men of mathematics. By reconstructing the events as they might have happened, Julia Diggins allows us to easily follow the pattern of reasoning that leads to an ingenious proof of the Pythagorean theorem, an appreciation of the significance of the golden mean in art and architecture, and the construction of the five regular solids.

Using only three simple tools - the string, the straightedge, and the shadow - men discovered the basic principles and constructions of elementary geometry more than two thousand years ago. This engrossing book reveals how these discoveries were made and shows how they were related to the early civilizations of Mesopotamia, Egypt and Greece.

Whether you are learning geometry for the first time, teaching it to students at home or in the classroom, or are parents helping your children through their first geometry course, this is a must-read for everyone! You will be fascinated with the graphic illustrations and written depiction of how the knowledge and wisdom
of so many cultures helped shape our civilization today.

Drawing Geometry
A Primer of Basic Forms for Artists, Designers and Architects

Jon Allen

Softbound

Regular Price: $20.00

Introductory Price: $18.00

Teachers, parents, artists and design professionals can all find something to love in Jon Allen's Drawing Geometry. Here, in clean, clear drawings with step-by-step instructions, are the secrets of constructing a myriad of plane geometric forms. If you've never had the joy of discovering how a compass can transform paper and pencil into blossoming flowers of form, this book will introduce you to the magic and wonder of it all. If you've already mastered geometric construction, you'll love Allen's book even more - here in one volume lie the keys to the kingdom.

Truly a delight - do enjoy, whether with your students or for yourself!

Mathematics Around Us

A teacher's resource book for Waldorf Class 7 - covering mathematics in nature, Pythagoras and numbers

Lavishly illustrated in full color

John Blackwood

Softbound

$25.00

John Blackwood's new book makes mathematics so exciting I wish I could go back and study it again, this time with the life and energy Blackwood pours into it.

Here you will find lesson after lesson on the subjects covered in Grade 7 of the Waldorf curriculum: a tour de force review of geometric contsruction, from perpendicular angles to generalized spiroids; the harmonies and proportions of nature (spirals, Fibonnaci sequences, Golden angles, more); the nature of numbers themselves (numeric representation, binary numbers, units of measure, set theory, the magic of Pythagoras' Theorem and Bhaskara's demonstration).

What makes Blackwood's guide so outstanding is that in addition to really clear and easy-to-follow explanations of every single thing to be covered in these lesson blocks, practically every page has one or more main lesson page, colored drawing, or photograph that will inspire the master teacher and leave a clear, richly laiden path for the new teacher.

Here is a curriculum book to love - may it see an abundance of use!

Mathematics in Space and Time
Grade 8 Curriculum Resource

John Blackwood

Softbound

$25.00

From the author who brought us Mathematics Around Us comes another outstanding, even beautiful, mathematics teacher's resource, this time for Grade 8 in the Waldorf curriculum.

John Blackwood guides us through a thorough exploration of the Platonic Solids with mathematical precision and an artist's eye for color and form. It is heartstoppingly wonderful to see and read his exposition of these forms -- I'm quite sure Plato must be smiling.

He then completes the block with lessons on Rhytms and Cycles that brings alive the rhythmic relationships of both space and time. You'll love it. More importantly, so will your student(s). Blackwood's work is one of the most valuable contributions to the teaching of the Waldorf curriculum to have ever been published.

Perspective Drawing

For the 7th Grade as taught in Waldorf Education

Hermann von Baravalle, Ph.D.

Softbound

$7.95

Perspective drawing stands between geometric law and artistic skill and also between plane geometric construction and solid geometry. Accordingly, it is taught in Waldorf Education during the 7th grade year, and offers a middle ground between the study of two-dimensional forms in the 6th grade and the three-dimensional forms of 8th grade.

Contents:

• Preface
• Perspective Divisions and Perspective Series
  • The Perspective Middle
  • Perspective Divisions in 4 and 8 Parts
  • Perspective Trisection
  • Perspective Extrapolation
  • Perspective Series of Horizontal Lines
• Perspective Views with Two Vanishing Points
  • Inside Perspectives
  • Construction of Staircase
• Roof Construction in Perspective
  • Pointed Roofs
  • The Gable Roofs
  • The Mansard Roofs
  • The Combined Gables
• The Construction of Shadows
• Perspective of Curves
• Perspective Views from Nature

NEW! - 3rd Edition
Making Math Meaningful
A Middle School Math Curriculum for Teachers and Parents

Jamie York

Softbound - 189 pages

$22.95

New in Third Edition: Reworking of 7th grade percents unit, major reworking of 7th grade ratios unit. Two new sections for 8th grade, calculating the Area of Four Types of Triangle, calculating the area of a Cone and Pyramid and much more!

In addition to the original text (see original review below), Jamie York has made the following additions to this 2nd edition of Making Math Meaningful:

• A section on Simple Interest (6th grade Business Math).
• A wonderful drawing exercise titled The Metamorphosis of a Limaçon (6th grade Geometry).
• A section on The Ratio in a Square, which discusses the "great Pythagorean mystery" (7th grade Ratios).
• A section on The Four Ways of Expressing pi as a Ratio (7th grade Irrational Numbers).
• A reworking of the section on The Square Root Algorithm (7th grade Irrational Numbers).
• Formulas showing how Pythagoras and the Arabs calculated Pythagorean Triples (7th grade Geometry).
• Heron's Formula for the Area of a Circle (8th grade Arithmetic).
• A new table in the appendix listing the Prime Numbers up to 2000.
• An entirely new appendix, titled Ideas for Fifth Grade, which includes: Square and Triangular Numbers; Perfect, Abundant and Deficient Numbers;
  Making a Right Angle with a Rope; The Unitary Method; The Sum of Two Prime Numbers; Sums and Differences of Square Numbers, and more.
  

Original review: York's original goal in writing this was primarily to create a math curriculum guide for his fellow Waldorf teachers. However, his results go far beyond Waldorf schools - public school teachers, homeschoolers and parents which to help their children will also find that the thorough clarity of this book can set them and their children free in the world of numbers.

This is the math book we've been waiting for - a curriculum guide that is thorough, expert, accessible even to those of us who are "math challenged", interesting and inspiring even to those who are mathmatically expert - and best of all, covers the grade levels making the transition from the four processes to higher mathematics - grades 6, 7 and 8.

Here's what you'll find:

• Jamie York's thoughts on teaching math - based on his experience as a Waldorf math teacher and his training in mathematics, this is must-read material for anyone interested in or responsible for teaching mathematics to pre-teens. He discusses all aspects of mathematics that apply to grades 6, 7 and 8 as well as giving tips on using the book, discerning what must be taught from among all the things that could be taught, and (I love this!) planning the math block!
• A summary of all math topics covered, including the mathematical skills that must have been reached in grade 5 in order to proceed with the middle school curriculum.
• Sixth grade arithematic and geometry; 7th grade arithmetic, geometry and algebra; 8th grade arithmetic, geometry, algebra and computers
• All subject matter in this curriculum is presented in an accessible, step-by-step way that teaches as it explains, allowing the weak math teacher to struggle (successfully!) with the concepts and really learn them before stepping into the classroom.
• Appendices with classic problems, resources and in-depth explanations.

This book is an outstanding addition to the world of education! Recommended without reservation!!!!

Fun with Puzzles, Games, and More!
Making Math Meaningful

Randy Evans, Mick Follari and Jamie York

Grades 5-12

Spiral bound

$12.95

From my perspective as a bookseller, awaiting this book has been a lot like going on a long car trip with the children in the back seat: "Are we there yet?" became "Do you know when the math puzzle book will be ready?" I didn't, but I can tell you that we are as delighted as can be that it is available.

And, what a happy treat it is! Each game, puzzle or problem is loaded with humor, interest and, most important, fun. If your math lessons are getting a bit morose, reach into this little treasure chest of good things and pull out some relief. Your students will never know what hit them: they'll be so busy laughing and cheering that they won't have any idea about how much they learned.

But learn they will. And smile. And enjoy.

Thanks to all the authors for making this possible.

Making Math Meaningful
A 6th Grade Student Workbook
Teacher's Edition with answer key
(student edition below)

Jamie York

$13.95

Topics covered in the 6th Grade workbook.

• The review topics from previous years are multiplication, short and long division, fractions, decimals, measurement, unit cost, factors, and
  estimating.
• The new topics for sixth grade are casting out nines, exponents (powers), square roots, divisibility, formulas, the metric system, converting decimals to fractions and fractions to decimals, angle measure, percents, mean/median/mode, pie charts, line graphs, area and perimeter, business math (discount, profit, etc.), ratios, rates (speed, wages), exchange rate, and complex fractions.
• Mental math. York strongly believe that it is as important as ever for children to develop the ability to do calculations in their head. Each worksheet therefore begins with a section on mental math, focusing on four areas (each one of which is listed and described in Making Math Meaningful: A Middle School Math Curriculum):
  1. Multiplication facts, with selected facts from the 13 through 18 tables, and the 25's table (i.e. 15·5=75);
  2. Powers with base 2 through 5 (i.e. 34 = 81);
  3. Fraction to decimal conversions (3/8 = 0.375);
  4. Math tricks, as described in A Middle School Math Curriculum, Appendix B.

6th Grade - The Year for Strengthening Skills. While there are new topics to be introduced in sixth grade math, much of the year is an essential review, or a furthering of material introduced in earlier years. The challenge is to weave in the review in such a way that
there is always something new. Therefore each worksheet includes several review problems. The goal is for the students to enter seventh grade feeling that division, fractions, and decimals are all easy, and for them to be excited about learning math.

What is the intention of this workbook? The sheets found in this workbook are designed for use as homework
assignments during the sixth grade year. The whole thing emerges from a carefully thought out, developmentally-based math curriculum, which is fully described in my book, Making Math Meaningful - A Middle School Math Curriculum for Teachers and Parents, a book that makes available the material I have discovered and developed while teaching middle school math for several years in a Waldorf school. This material is both exciting and age-appropriate for developing the minds of this age group. It is an alternative to the normal "they-need-all-this-stuff-in-order-to-stay-ahead" approach.

Who can use this workbook? While most of the author's teaching experience is within the Waldorf school system, the curriculum that York present in A Middle School Math Curriculum, which is what this workbook is based upon, can be effectively used by any teacher or homeschooling parent wishing to bring meaningful, age-appropriate material to their students.

Required reading. It is necessary to read Making Math Meaningful - A Middle School Math Curriculum for Teachers and Parents in order to fully understand the
material in this workbook.

Making Math Meaningful
A 6th Grade Student's Workbook

Jamie York

$8.95

Hands-on practice sheets that engage both student and teacher as they help guide the student through the essentials of 6th grade math. A long-awaited boon to classroom and homeschool teachers!

Topics covered in this 6th grade student's workbook:

• The review topics from previous years are multiplication, short and long division, fractions, decimals, measurement, unit cost, factors, and
  estimating.
• The new topics for sixth grade are casting out nines, exponents (powers), square roots, divisibility, formulas, the metric system, converting decimals to fractions and fractions to decimals, angle measure,
  percents, mean/median/mode, pie charts, line graphs, area and perimeter, business math (discount, profit, etc.), ratios, rates (speed, wages), exchange rate, and complex fractions.
• Mental math. York strongly believe that it is as important as ever for children to develop the ability to do calculations in their head. Each worksheet therefore begins with a section on mental math, focusing on four areas (each one of which is listed and described in A Middle School Math Curriculum) :
  1. Multiplication facts, with selected facts from the 13 through 18 tables, and the 25's table (i.e. 15·5=75);
  2. Powers with base 2 through 5 (i.e. 34 = 81);
  3. Fraction to decimal conversions (3/8 = 0.375);
  4. Math tricks, as described in A Middle School Math Curriculum, Appendix B.

Making Math Meaningful
7th Grade Workbook
Teacher's Guide

Jamie York

Spiral Bound

$19.95

The beginning of abstract thinking. Seventh grade is an important year academically. This is the year when the students start to develop abstract thinking (through algebra, physics, essay writing, etc.), and when study habits are developed. Getting homework done regularly (even in small amounts), and keeping a well-organized notebook, are both very important. It is relatively common for a student to enter seventh grade weak in math, but then to "wake-up" during seventh grade, and, in the end, to enter high school quite strong in math.
This Teacher's Edition contains answer keys!

What is the intention of this workbook? The sheets found in this workbook are designed for use as homework assignments during the 7th grade year. The whole thing emerges from a carefully thought out, developmentally-based math curriculum, which is fully described in my book, Making Math Meaningful - A Middle School Math Curriculum for Teachers and Parents, a book that makes available the material I have discovered and developed while teaching middle school math for several years in a Waldorf school. This material is both exciting and age-appropriate for developing the minds of this age group. It is an alternative to the normal "they-need-all-this-stuff-in-order-to-stay-ahead" approach.

Required reading. It is necessary to read Making Math Meaningful - A Middle School Math Curriculum for Teachers and Parents in order to fully understand the material in this workbook. Teacher's Guides and Student Workbooks are also available for 6th, 7th and 8th grade.

Who can use this workbook? While most of my teaching experience is within the Waldorf school system, this workbook, as well as my whole math curriculum, can be effectively used by any teacher wishing to bring meaningful, age-appropriate material to their students.

Making Math Meaningful
7th Grade Student Workbook

Jamie York

Softbound

$15.95

The 7th Grade Student's Workbook. Please see information for the 7th Grade Teacher's Guide, above. The Student's Workbook is saddle-stitched, lay-flat and includes everything as covered above with the exception of the teacher's answer keys. There is also a thoughtful short note to the student on the back of the workbook which serves to set the tone for the upcoming year.

Making Math Meaningful
8th Grade Workbook
Teacher's Guide

Jamie York

Spiral Bound

$19.95

The seven units in this eighth grade workbook are:

Number Bases
Pythagorean Theorem
Mensuration
Percents & Growth
Proportions and Dimensional Analysis
Algebra
End-of-Year Review
This Teacher's Edition contains answer keys!

What is the intention of this workbook? The sheets found in this workbook are designed for use as homework assignments during the 8th grade year. The whole thing emerges from a carefully thought out, developmentally-based math curriculum, which is fully described in my book, Making Math Meaningful - A Middle School Math Curriculum for Teachers and Parents, a book that makes available the material I have discovered and developed while teaching middle school math for several years in a Waldorf school. This material is both exciting and age-appropriate for developing the minds of this age group. It is an alternative to the normal "they-need-all-this-stuff-in-order-to-stay-ahead" approach.

Required reading. It is necessary to read Making Math Meaningful - A Middle School Math Curriculum for Teachers and Parents to fully understand the material in this workbook. Teacher's guides and Student Workbooks are also available for 6th, 7th, and 8th grade.

Who can use this workbook? While most of my teaching experience is within the Waldorf school system, any teacher, who wishes to bring meaningful, age-appropriate material to their students, can use this workbook.

Making Math Meaningful
8th Grade Student Workbook

Jamie York

Softbound

$15.95

The 8th Grade Student Workbook contains everything that is in the 8th Grade Teacher's Guide, above, without the answer keys. There is also a thoughtful short note to the student on the back of the workbook which serves to set the tone for the upcoming year.

Mathematics Lessons for the Sixth Grade

Ernst Schuberth

Softbound

$17.00

To accompany the outstanding mathematics curricula of Jamie York, we are pleased to add Dr. Ernst Schuberth's approach, which we think you will find quite complimentary to the Making Math Meaningful series. Dr. Schuberth is a German Waldorf teacher of many years experience, and is one of the leaders in the field of education. His approach to teaching mathematics is European, not American, which means you will find less hands-on and more theory. I have always believed that if there were some way to synthesize the European educational perspective with that of the American, we'd have something approaching the best possible education for our children. With both Making Math Meaningful and Mathematics Lessons for Sixth Grade, you can create your own synthesis. Our advice to Americans would be to rely on Making Math Meaningful for the base of the education, and add perspectives from Dr. Schuberth when the student shows and interest and you sense an opening.

Dr. Schuberth provides an overview of the Waldorf elementary mathematics curriculum and provides a thorough presentation on the sixth grade where banking formulae, interest calculation, and elementary economics are considered. The book has weekly lesson plans, many problems, and solutions.

Introduction of Advanced Arithmetical Operations for Waldorf School 7th Grades
The Calculation of a Square Root & Higher Roots

Ernst Schuberth

Spiral-bound

$12.00

An experienced math teacher guides the reader through the calculation of square root and higher roots. Help for all teachers of mathematics!

Finding the Path
Themes and Methods for the Teaching of Mathematics in a Waldorf School

Bengt Ulin

Softbound

$15.00

The presentations in this book are built on experiences from mathematics teaching in grades 7-12. The book does not offer a pedagogical collection of recipes but rather how one might engage the pupils. The presentation is a pedagogical handbook. It addresses such topics as the history of mathematics, Fibonacci numbers, nature’s geometry, the step from arithmetic to algebra, and gaining confidence in thinking. Mr. Ulin takes the reader into the wonders of mathematics. This is a resource for those seeking to understand the magic of the world of numbers and how to present them to students.

Mathematics in the Classroom: Mine Shaft and Skylight
Memories and Reflections of a Teacher

With an Introduction to Projective Geometry and Inversion

Amos Franceschelli

Spiral Bound

$18.50

Amos Franseschelli was a beloved teacher of junior high and high school mathematics at the Rudolf Steiner School in Manhattan. After his retirement, he put together this volume, as sort of a composite of things he most wanted to pass along. It is a deep and deeply useful text, one that elucidates and inspires. Anyone teaching mathematics to high school students can take much from this book back into the classroom. The first section of this book is addressed to teachers at all levels and lovers of mathematics. The second part describes an eleventh grade projective and inverse geometry class, and the third part is skits and examples of how humor can be brought to mathematics. What could be better?

Contents:

BACKGROUND

• Facets of the Mathematical Diamond - Musings of a Teacher
• What Is Mathematics?
• Appreciative Thinking
• Mathematics in the Classroom: Mine Shaft and Skylight

PROJECTIVE GEOMETRY AND THE GEOMETRY OF INVERSION - An Eleventh Grade Geometry Block

• Act I: The Three Aspects of the Three Fundamental Elements
• Act II: The Elements of Infinity
• Act III: The Principle of Duality
• Act IV: Perspectivities and Projectivities
• Act V: Projective Generation of Conics - Prologue

INVERSIVE GEOMETRY

• Introduction
• Inversion in a Plane
• Time-Saving Approximations in Locating Inverses
• Inversion of Lines and Curves: Basic Techniques
• Classroom and Homework Exercises
• The Simple Law of Inversion
• Some Properties of Inversion
• Further Exercises
• An Alphabetical Interlude
• Inversion: Brief Summary and Aims
• Last Exercises

THREE SKITS AND THE END OF A FOURTH ONE

• "Keep Your Balance" - An Algebra Skit for Grade 7
• "Change Ringin - A Mathematical Skit for Grade 9
• "Light Geometrizes - A 10th Grade Geometry Skit
• "The Two Parallels" - Concluding Poem

Geometry and the Imagination
The Imaginative Treatment of Geometry in Waldorf Education

A. Renwick Sheen, Ms.C. (Tech.)

edited by David Mitchell
Newly revised edition

Softbound

$24.00

Geometry was my favorite of all math subjects - in large part because I was taught through drawing and proofs, not through the formulaic method that seems to be so popular right now. Mr. Sheen bases his exploration of geometry on the tenet that experience through drawing is of great value to a student, and I couldn't agree more. His method of presentation is concerned with experience through drawing, which leads to a stimulation of the child’s imagination and understanding of logical relationships. This book is particularly valuable for teachers of grades 6-8 and is richly illustrated. New revised edition.

Solid Geometry
Geometry of the Platonic Solids and Geometry of the Cylinder, Sphere, and Cone

Harry Kretz

Spiral-bound

$15.00

This book addresses the geometry taught in eighth grade in Waldorf schools, with suggestions as to imaginative ways to introduce these concepts to the students.

Space and Counterspace
An Introduction to Modern Geometry

Louis Locher-Ernst

Softbound

$25.00

This classic examination of Projective Geometry offers a detailed exposition of the topic from a variety of perspectives and will be a great aid to Waldorf teachers and others seeking to penetrate the thinking behind this important topic.

Geometry for the Waldorf High School
Rough Copy from a Teacher's Notebook

Herb Swanson

Spiral-bound

$22.00

Notes and lesson ideas for the teaching of geometry on the high school level, including studies on the conic sections, descriptive geometry, perspective geometry, the pentagon, harmonic sets, quadrilaterals, the golden section, gnomons, platonic solids and more. An exhilarating resource from someone who's "been there, done that"!

Trigonometry
A Practical Introduction

Teacher's Manual

Harry Kretz

Spiral Bound

$12.95

Harry Kretz is also the author of Triangle, Circle and Soul

In the Waldorf curriculum, Trigonometry is taught in 12th grade. Harry Kretz has created a beautiful course that is a nearly perfect conclusion to a student's Waldorf education.

In his lessons, he uses all sorts of commonly available materials - making the point that expensive laboratories are not necessary for teaching basic scientific principles. The simplicity of the lessons is of the sort that leads to deep thinking and sound understanding - as well as genuine student interest in the subject.

In his own words:

By practical, I mean an approach that involves the student's whole self so as to develop clarity of thought and an enthusiastic involvement in challenging, meaningful activities. If students work outdoors, as well as at their desks, measuring distances and angles, the topics under discussion are better understood. But if time is spent outdoors, I suggest that the course be scheduled in the fall or spring.

The manual covers:

• Right Triangle Calculations
• Oblique Triangle Calculations
• Problems in Surveying
• and has many helpful notes and suggestions

A Beginner's Guide to Constructing the Universe
The Mathematical Archetypes of Nature, Art & Science

Michael S. Schneider

Softbound

$18.95

Looking for ways to share the glorious beauty of mathematics with your students? Or to inspire yourself with the wonder of creation? This is your book.

Michael Schneider leads us on a spectacular, lavishly illustrated journey along the numbers one through ten to explore the mathematical principles made visible in flowers, shells, crystals, plants, and the human body, expressed in the symbolic language of folk sayings and fairy tales, myth and religion, art and architecture. Here is a comprehensive guide to the patterns that recur throughout the universe and underlie human affairs.

Among the many things you will see and learn are:

• Why cans, pizza, and manhole covers are round.
• Why one and two weren't considered numbers by the ancient Greeks.
• Why squares show up so often in goddess art and board games.
• What property makes the spiral the most widespread shape in nature, from embryos and hair curls to hurricanes and galaxies.
• How the human body shares the design of a bean plant and the solar system.
• How a snowflake is like Stonehenge, and a beehive like a calendar.
• How our ten fingers hold the secrets of both a lobster and a cathedral.

And there's much, much more. This is a resource you'll turn to again and again - a dazzling revelation of the beauty of creation.

Flatland
A Romance of Many Dimensions

Edwin A Abbott

Softbound

$2.00

Flatland is not only one of the very first science fiction books ever written (1884), but it is such an enjoyable and excellent introduction to the concept of dimensions and their effect on our perceptions, that it remains high on the list of books recommended for curious students of mathematics. I would personally recommend it for precocious 7th or 8th graders as well as high school and college students. Every mathematician I've ever met has read it, remembered if fondly and recommended passing it along to the younger generation.

Here's the story line:

A. Square is a mathematician and resident of two-dimensional Flatland, where women -- thin, straight lines -- are the lowliest of shapes, and where men may have any number of sides, depending on their social status. [n.b.: the sexism alone can be guaranteed to spark hours of lively discussion!]

Through strange occurrences that bring him into contact with a host of geometric forms, Square has adventures in Spaceland (three dimensions), Lineland (one dimension) and Pointland (no dimensions) and ultimately entertains thoughts of visiting a land of four dimensions -- a revolutionary idea for which he is returned to his two-dimensional world.

There are charming illustrations by the author throughout. Who says mathematics can't be fun?

The Power of Limits
Proportional Harmonies in Nature, Art, and Architecture

György Doczi

$24.95

image size deceptive - actually an oversize trade paperback

The Power of Limits is an exposition of the rhythms and harmonies of both natural and artistic forms. I have used its contents for meditative contemplation for many years now, and have not come even close to exhausting the insights and revelations Doczi's remarkable book contains. Whenever I get frustrated at how slowly postive change comes about, I reach for The Power of Limits so that I can be reminded of the grace and beauty that limitation brings the world. Steiner defined art as "the impress of the spirit upon matter." Doczi shows us what that means.

Mathematics Colloquium
Working Papers in Progress

AWSNA High School Research Project
October 2000

Spiral-bound

$26.00

This book represents the efforts of the AWSNA High School Research Project to review and reveal the way in which the Waldorf mathematics curriculum exercises and develops various thinking capacities in the adolescent. The participants offer reflections and recommendations on each of the high school grades 9-12 and present the capacities that the teacher of mathematics can develop in each of the grades. One section of the book deals with remedial work in mathematics while another offers recommendations to lower-school teachers, grades 6-8. Several relevant papers are in the Addendum.

Projective Geometry

Lawrence Edwards

Softbound