Making Math Meaningful

A Scource Book for Teaching Math in Grades One through Five

Nettie Fabrie, Wim Gottenbos, and Jamie York

Softbound, large format

$24.95

Add a reviewThis newest contribution to the math teacher's paintbox has been in the making for at least three years, and longed for by elementary school teachers for decades before that. At last, it is here and it is just what the math teacher, whether in the class or at home, ordered.

This newly released Making Math Meaningful® A Source Book for Teaching Math in Grades One Through Five continues with the same meticulous detail as all the middle school math series. The major concepts contained in the lower grades book are:

- Principles of Waldorf Education
- Working with "struggling" students
- Will our children be prepared?
- Golden rules for teaching math
- Topics in the math classroom
- Ideas for manipulatives, movement, main lesson books, etc.
- The art of teaching math

What makes this book different from other math books for Waldorf teachers?

- Detailed curriculum guidelines for first through fifth grade.
- Overview of child development in each grade.
- Specific ideas for every math main lesson.
- Step-by-step progressions for learning the arithmetic facts, times table, basic addition, and subtraction, long multiplication and long division, etc.
- General ideas and suggestions for teaching math successfully in the lower grades.

This is a book that everyone, experts and novice teachers alike, will love - and the best part is that your students will benefit from your teaching in powerful and joyous ways.

Math Lessons for the Elementary Grades

Dorothy Harrer

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

Covers Grades 1-6

Softbound

$14.00

Add a reviewAfter 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

King Maximo and the Number Knights

Whether you are homeschooling or teaching a class, *King Maximo and the Number Knights *is a delightful, engaging way to teach the basics of numbers to young children. The story, written as a series of 12 lesson-chapters, retells how King Maximo sent his 12 number knights out into the world charged to discover the greatest of the numbers. The knights go out and then return and meet in King Maximo's great hall. As each tells the tale of his adventures, we find that each knight has a different opinion of what is, truly, the greatest number.

Included with each chapter is a shield that evokes the wonders of the number that chapter's Number Knight found to be the greatest. These shields are just right for the artistic element of each lesson, and are sure to engage the students as they also convey much of the essence of the number they represent.

*King Maximo and the Number Knights* is another gift by master teacher Howard Schrager, one that every child lucky enough to hear these tales and draw the number shields will carry within through a lifetime.

Outstanding! For Grade 1 in the Waldorf curriculum.

Multiplicando

From the mind and heart of a truly great Waldorf teacher comes a teaching story that I believe the world has been waiting for for a very long time. Even if *Multiplicando* weren't teaching, it would still be a great children's story, with warm and funny characters, interesting adventures and one surprising discovery after another on each page. The fact that this riveting story *also* teaches multiplication and its relationship to the other 3 mathematical processes, and teaches all this so clearly and thoroughly, makes *Multiplicando* a gift for the ages.

Divided into twenty-two chapters, *Multiplicando*, follows the humorous adventures of (who else?) Multiplicando, Jester to the Court of King Divide and His Queen, the Highness Minus. Using these stories over several weeks, teachers and parents can lead their students into the mysteries of multiplication, and come to see how that relates to division, subtraction and addition. Students will come away with a story they will remember well enough to look back upon as adults, and with a clear understanding of multiplication and the nature of all 4 processes.

Additionally, at the back of the book the author has shared all sorts of teaching tips and tricks, so many that its a bit like finding the King's treasure house.

*Multiplicando *is a treasure and a joy. I would wish every student could have such an introduction to mathematics.

A Steiner-Waldorf Mathematic Resource

Grades 1-8

Eric Fairman

A Path of Discovery Book

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

Comb Bound

$23.50

Add a reviewEric Fairman's Mathematic Resource offers a beautiful, wonderfully useful overview of the mathematics curriculum 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

A Children's Anthologia - A Gathering of Flowers

Follow the winding curriculum road with a verse at hand for each new turn! Alan's poems for classes 1-8 go to the heart of the subject with verses humorous, serious, charming, and poignent. Each one of them is "just right" for the age group and subject addressed; each one is sure to add a dash more beauty and depth to the lesson.

Verses cover language arts, mathematics, social studies, science and the festivals. Includes the 12 Greek Zodiacal Rhythms, the 7 Meters and 7 Speech Styles.

Active Arithmetic

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

Henning Andersen

Softbound

$16.00

Add a reviewIs 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?

In many ways it was the answers to these questions that first engaged what became my passionate commitment to Waldorf education. My own grade school struggles with arithmetic laid the foundation for a lifelong quest to find "a better way." When I discovered Rudolf Steiner's lively approach to teaching mathematics with rhythm, art, even stories and song, it resonated so deeply in my heart that helping this approach to education move forward became my life's work.

In my time, arithmetic was taught pretty much solely with pencil and paper along with some verbal and written examples from the teacher and textbooks. I was not a child who found much joy in that, nor was I terribly interested in remembering abstract details for which I could find no connection in daily life. I know that had I been taught using Waldorf methods, I would have come to love mathematics much earlier in life, and would have dreaded math class much less.

*Active Arithmetic *describes the use of lively, rhythmic movement as part of the mathematics education in the early grades (grades 1-3, approximately). Anyone who has seen these or similar exercises carried out in a classroom or homeschool knows the happiness they bring and can testify to the lifelong learning they foster. My own children, now adults, all still remember what I call the "body mnemonics" of the clapping, singing, dancing, and even game playing through which they were taught the basics of arithmetic. Comparing their experiences to my own is like comparing a gray, chilly day to a garden bathed in summer sunlight. It's small wonder that they understand arithmetic so very well, and remember it so accurately.

*Active Arithmetic* is an exciting resource that clearly describes much of what Waldorf education offers children in Grades 1-3. I can't recommend it highly enough for teachers and homeschoolers; I also recommend it for parents wishing to supplement their children's more traditional math education. Making learning fun is always a gain for all.

Teaching Mathematics for First & Second Grade in Waldorf Schools

This book is an excellent support and resource for the classroom or homeschool teacher beginning with first grade. It describes in careful detail 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 approach to child development.

Teaching Mathematics

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

Geometry Lessons in the Waldorf School - Volume 2

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

Add a reviewThis 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.

First Steps in Proven Geometry

This is a beautiful introduction to the harmony and wonder of Pythagorean geometry which, to my my way of thinking, is a way of understanding the paths of accurately knowing truths that are present just below the surface of immediate perception. I just love it (and do wish I felt the same way about algebra, honest!).

Schuberth's presentation is fresh, lively and artistically wonderful. I can't imagine a student not discovering one thing after another to be interested in and pursue, this is just so packed with the spirit of joyful discovery. Teachers and parents who use it to guide their middle school students along the pathways of geometric proof will be offering them a lifetime of of interest in logic and spatial relationships. This book is a gift.

Contents:

- Foreword
- Introduction
- Angles in Regular Polygons
- Triangles
- The Right Triangle and the Circle: The Theorem of Thales
- The Rules of the Right Triangle
- The Principles of Congruence and Basic Assignments Using the Triangle
- The Important Lines of a Triangle
- The Platonic Solids
- Teaching Projection and Shadow Drawing in the Upper Elementary Grades
- Geometry and Mineralogy
- Outlook
- Endnotes
- Bibliography

Geometric Drawing and the Waldorf School Plan

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 spirit.

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, Straight-Edge and Shadow

The Story of Geometry

Julia E. Diggins

Softbound

Wonderful duotone illustrations by Corydon Bell

$16.95

Add a review2003 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

$25.00

Add a reviewTeachers, 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!

Triangle, Circle and Soul

Harry Kretz has taught geometry in Waldorf schools for many years, pondering why the study of points, lines, planes, and three dimensional forms so powerfully holds the students' interest and desire for further exploration.

Moses wrote that God created man, a living soul. Plato wrote that God geometrizes. If the sould is thought of as the threefold principle of life - thinking, feeling, and action - and the triangle is a form with three sides, is there a connection? The author's dream experience seems to say 'yes.'

A fascinating exploration - full of things to ponder. I've had a few "geometric" dreams in my life (many years ago now) - I still remember them vividly. They were among the handful of dreams I've had where I felt myself immersed in understanding, where things were explained that I couldn't quite "get" in broad daylight. Almost uniquely, this understanding stayed with me after waking, and is with me now. Harry Kretz had many more such experiences -- I think his sharing of them can offer much which enriches.

Woolly Thoughts

Unlock Your Creative Genius with Modular Knitting

Pat Ashforth and Steve Plummer

Softbound

$8.95

Add a reviewHere is a book to delight knitters and teachers of mathematics - all at the same time!

How do they do this? Well, they are both mathematics teachers, amazing in their ability to make everything clear and fun. Right from the start, they forge a pathway to knitting heaven by having you take out whatever yarn you like, your choice of needles, and a small calculator. What happens next is a guided tour through a mosaic of shapes and colors -- all of them easy, all of them fascinating and beautiful. With Pat and Steve at your side, you can truly forget about conventional knitting patterns and follow your imagination where ever it leads you. Your students will begin absorbing and delighting in geometric and mathematical relationships without even noticing that's what they are doing. They will also being to love the time they spend knitting simply because they can follow their own inspiration so easily.

Happily and heavily illustrated throughout, with a section of color plates showcasing some of Pat's creations, *Woolly Thoughts* brings you everything you need for a lifetime of discovering the joys knitting and mathematics. This is a book that students and teachers will love equally - each will discover their knitting and their mathematical skills just get better and better, one stitch at a time!

A beautiful way to unite handwork with mathematics!

Perspective Drawing

For the 7th Grade as taught in Waldorf Education

Hermann von Baravalle, Ph.D.

Softbound

$7.95

Add a reviewPerspective 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

Mathematics in Nature, Space and Time

John Blackwood

Softbound

Formerly published as *Mathematics Around Us *and *Mathematics in Space and Time*

$29.95

Add a review*Mathematics in Nature, Space and Time *brings teachers (and thereby students) lesson after lesson from the Grade 7 Waldorf mathematics curriculum: a *tour de force *review of geometric construction, 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 laden path for the new teacher.

John Blackwood then takes on the 8th grade Waldorf mathematics curriculum and guides us through an 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 Rhythms and Cycles that brings alive the 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.

Making Math Meaningful

A Middle School Math Curriculum for Waldorf Teachers - 3rd Edition

Jamie York

Softbound - 189 pages

$26.95

Add a review**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 mathematically 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!!!!

Making Math Meaningful - Fun with Puzzles, Games, and More! - 3rd Edition

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 now 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

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): Multiplication facts, with selected facts from the 13 through 18 tables, and the 25's table (i.e. 15·5=75);
- Powers with base 2 through 5 (i.e. 34 = 81);
- Fraction to decimal conversions (3/8 = 0.375);
- 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 Workbook

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) : Multiplication facts, with selected facts from the 13 through 18 tables, and the 25's table (i.e. 15·5=75);
- Powers with base 2 through 5 (i.e. 34 = 81);
- Fraction to decimal conversions (3/8 = 0.375);
- Math tricks, as described in A Middle School Math Curriculum, Appendix B.

Making Math Meaningful - 7th Grade Workbook

**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

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

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

Mathematics Lessons for the Sixth Grade

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 the Waldorf School 7th Grade

Finding the Path

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

Bengt Ulin

Softbound

$15.00

Add a reviewThe 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.

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

Add a reviewGeometry 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.

Making Math Meaningful - A 9th Grade Workbook

In middle school, the students are introduced to some basic algebra concepts. Now, in ninth grade, they are ready for a full year of algebra studies. This important year of math forms the foundation for all future mathematics study -- after all, algebra is the "language of mathematics." There is a good deal of challenging work and learning to be done!

The units covered in this workbook are:

- Pre-Algebra
- Algebra Basics
- Exponents and Polynomials
- Factoring
- Word Problems
- Midyear Review
- Fractions and Square Roots
- The Quadratic Formula
- Logarithms
- Possibility and Probability

Making Math Meaningful - Answer Key to Our 9th Grade Workbook

For the teacher: Answers to all problems in the Making Math Meaningful 9th Grade Workbook.

Making Math Meaningful - A 10th Grade Workbook

After a full year of algebra studies, the student enters tenth grade ready to take a big step forward: learning how to write deductive proofs. There is a great variety of topics to be explored in this workbook. While it is possible to cover most of these topics in the tenth grade year, a class may instead spend most of the year on the geometry topics, and then perhaps return in eleventh grade to a couple of the topics found in the second half of this workbook. Either way, this workbook promises to increase the students' sense of wonder at the power and beauty of mathematics.

The units covered in this workbook are:

- Geometry Basics
- Circle Geometry
- Triangle Geometry
- Proofs!!
- Mensuration
- Algebra Review
- Logarithms, Part II
- Exponential Growth
- Sequences & Series
- Math & Music

Making Math Meaningful - Answer Key to our 10th Grade Workbook

For the teacher: Answers to all problems in the Making Math Meaningful 10th Grade Workbook.

Making Math Meaningful - An 11th and 12th Grade Workbook

Algebra II, Precalculus, and some Calculus

Andrew Starzynski and Jamie York

Softbound

$19.00

Add a reviewBy waiting until this point to begin Cartesian geometry, the eleventh grader is ready to fully absorb this important topic, and this workbook does it in depth with four separate units. Through a year and a half of study, this workbook weaves together Cartesian geometry, trigonometry, complex numbers, and other key topics found in traditional algebra II and precalculus courses, and ends with two great units of calculus. Any student completing this book is poised to study anything in college!

The units covered in this workbook are:

- Cartesian Geometry, Part I
- Trigonometry, Part II
- Complex Numbers, Part I
- Cartesian Geometry, Part II
- Possibility & Probability, Part II
- Trigonometry, Part III
- Cartesian Geometry, Part III
- Logarithms, Part III
- Complex Numbers, Part II
- Trigonometry, Part IV
- Cartesian Geometry, Part IV
- Calculus, Part I
- Calculus, Part II
- Problem Solving Exercises

Making Math Meaningful - Answer Key to Our 11th and 12th Grade Workbook

For the teacher: Answers to all problems in the Making Math Meaningful 11th & 12th Grade Workbook.

Solid Geometry

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

Harry Kretz

Spiral-bound

$15.00

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

Projective Geometry

The exercises in this book reveal the secrets of space. With over 200 instructive diagrams, Lawrence Edwards presents a clear and artistic understanding of the intriguing qualities of non-Euclidean geometry.

The elements of geometry are points, lines and planes, together with Space and Nothing. In Euclidean geometry, we observe these as if we are point-centered beings concerned with extensive measurement. In non-Euclidean, or projective geometry, we are 'planar' beings observing spatial relations, free of measurement, as dynamic and transformational. According to the author, this is a polar opposite vision of intensive space, and it reveals remarkable secrets. Edwards invites the reader to discover these secrets through practical exercises in creative geometry.

Space and Counterspace

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

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

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, and Science

Michael S. Schneider

Softbound

$18.95

Add a reviewLooking 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

*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 it 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

Softbound, Large format, profusely illustrated

$29.95

Add a reviewOne of the delights of life is the discovery and rediscovery of patterns of order and beauty in nature - the designs revealed by slicing through a head of cabbage or an orange, the forms of shells and butterfly wings. These images are awesome not just for their beauty, but because they suggest an order underlying their growth, a harmony existing in nature. What does it mean that such and order exists; how far does it extend?

*The Power of Limits* begins with such simple discoveries of harmony and goes on to investigate and measure hundreds of patterns -- ancient and modern, minute and vast. Doczi's discovery, vividly illustrated here, is that certain proportions occur over and over again in all these forms. Patterns are also repeated in how things grow and are made -- by the dynamic union of opposites -- as demonstrated by the spirals that move in opposite directions in the growth of a plant.

The joining of unity and diversity in the discipline of proportional limitations creates forms that are beautiful to us because they embody the principles of the cosmic order of which we are a part; conversely, the limitlessness of that order is revealed by the strictness of its forms. The author shows how we, as humans, are included in the universal harmony of form, and suggests that the union of complementary opposites may be a way to extend that harmony to the psychological and social realms as well.

I personally see *The Power of Limits *as a revelation of the beauty and wisdom of limitation itself. Each form is limited by its inner orderly nature as well as outer, imposed limitations. Were such limitations not to exist, undifferentiated growth would manifest in everything and, frankly, beauty would never result; nor would function. And there would be no possibility of and form of self-awareness arising from such a cacophony as would result without limits. In other words, the harmony, order, and beauty that abounds everywhere would be impossible, as would human beings and nature itself as we know them.

I have lived with this book for almost 40 years and still find it one of the most profound studies ever published. It is as revealing and wise as the forms it studies - I hope many, many people are able to experience it.

Mathematics Colloquium

Working Papers in Progress

AWSNA High School Research Project #3, October 2000

Spiral-bound

$26.00

Add a review

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.