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From Donald Duck to Flatland

By Bob Hazen
Printed in Practical Homeschooling #28, 1999.

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Bob Hazen


A comic book helped make me a math teacher. When I was in 4th grade or so in the early 1960's, Walt Disney published a comic book entitled Donald Duck in Mathemagic Land. Disney also later made a movie (now a video) of the same name, but as usual, the book was better than the movie. In a dream, Donald goes to Mathemagic Land, where tree trunks grow out of the ground in a square frame shape (get it? - square roots!). Donald meets a wise guide who shows him the mathematical aspect of almost every conceivable area of life, from compound interest to arithmetic shortcuts to engineering, accounting, architecture, and music. In the end, Donald appreciates math, learns the magic of exponents, and finds how to use a chessboard to pay off a debt he owed Uncle Scrooge.

I remember re-reading this comic book countless times, fascinated with the math tricks, the cool shapes, and the clever ways in which math kept popping up everywhere in life. In that era, long before the availability of handheld calculators, my brother and I were inspired by Donald's chessboard problem to determine the value of 264 - by hand. I remember filling up sheet after sheet of notebook paper determining that since 23 was 2 x 2 x 2 = 8, then 24 was 16, 25 was 32, and so forth. We eventually got out to the millions somewhere, where we got bogged down with cumbersome and lengthy results (and different answers as well). But I learned first hand - from a comic book - the power of exponential growth.

Such is the power of literature, even for the field of mathematics; the power to capture imaginations and stimulate interest.

Since those comic book days, other math-related books have come along every so often that still capture my imagination and appreciation of things mathematical. While there are a good handful or two of books on my favorites list, there are surprisingly very, very few on a "must-read" list for homeschooling parents.

One of them is Samuel Blumenfeld's How to Tutor, with its chapter on arithmetic. Blumenfeld's discussion of the development and genius of the base ten place-value system provides invaluable insight for any homeschooling parent. His further discussion of the power of memorization and the difference between "working knowledge" versus "intellectual understanding" of arithmetic are important points that are too often lost (although more so outside the boundaries of homeschooling). The rest of this particular chapter gives valuable, practical tips on how to effectively teach elementary arithmetic, from basic facts to fractions and measurement units. While I would supplement his suggested steps with other "stuff" I write about in this column (such as skip-counting musical tapes, math games, and the use of higher math to teach and reinforce basic arithmetic), his book is sound advice for teaching the necessary, essential mental-math and pencil-paper skills of arithmetic.

My Favorite Math Books

Occasionally, readers ask me if there are books that incorporate and extend the kinds of practical principles I have been writing about in my columns. Unfortunately, I am not aware of such books. That is precisely why I started writing the columns in this magazine - because I did not know of books that identified and explained practical, flexible, and powerful principles that unified mathematics - especially elementary mathematics - across grades and across topics.

Nevertheless, at this point, I would like to recommend some of my favorite math books that you might want to add to your home library.

At the early childhood end, The M&M's Chocolate Candies Counting Book is a wonderful book that covers the traditional counting sequence of 1 through 12, with the added motivation of working with M&M candies that readers (presumably!) get to eat at the end. As with the standard battery disclaimer, M&M's are not included with the book.

As children progress into the upper elementary years, puzzle books and "arithmetrick" books can provide important additions to a child's mathematical stimulation and understanding. Most of the books I have are old and tattered and out-of-print, but new books like this are frequently published by various sources too numerous to mention, usually with titles along the lines of Arithmetic Oddities or Mathematical Shortcuts and Tricks - or something with the word "brainteasers" in it. Don't forget maze books either; they are simply geometric-shape problem-solving books!

For students in grades 5 and up (even to high school!), Rapid Math Tricks and Tips provides a wealth of mental math shortcuts that help students gain better number sense. Author Edward Julius encourages readers to think of numbers in more than one way - for example, 58 can be thought of as plain 58, as (60-2), as (50+8), as (29x2), or as 58.0, or 58.00, or 58.000. You can learn to multiply a 2-digit number by another 2-digit number in your head, or how to square any number in the 50's and 90's almost instantly. While these are "mere" tricks and shortcuts, they work because they draw upon legitimate and important mathematical principles that foster better and faster number sense for children who practice them. In fact, what children are practicing in many of these "tricks" is the ability to decompose (take apart) and recompose (put back together) numbers. This mental flexibility increases a child's skills and confidence.

In the past few years, I have worked with two mathematically advanced sixth grade homeschooled students for whom I decided to make the traditional Algebra I course a two-year experience. They were ready for the substance of Algebra I, but I wanted to deepen their understanding of algebra by using a two-year schedule that left lots of room for enrichment, including regular work with Rapid Math Tricks, which sharpened their number sense and mental math skills while they studied the concepts and procedures of algebra.

Another valuable math enrichment tool is Geometric Constructions from Dale Seymour Publications, a workbook in which students are shown how to use the standard compass and straightedge only (no rulers!) to create simple (at first) and elaborate (later) patterns with circles, triangles, squares, diamonds, pentagons, and hexagons.

Moving into a broader perspective of mathematics, three other books come to mind. One has the simple title of Mathematics, from the Time/Life Science Library. This book covers the history of mathematics from ancient Greece, Egypt, and India to modern times, along with some of its practical and surprising elements - and many of mathematics' colorful historical characters. Each of the eight chapters has two sections - one a substantial but readable chunk of text on that chapter's topic, and the other a colorful picture essay with notes and comments. Both sections stimulate the imagination with pictures or stories: Why were square roots of certain numbers initially disliked? Why is a family with four children less likely to have two boys and two girls and more likely to have three of the same sex? While this book may be for the more math-serious student, it is also the kind of book that can be left available for perusal by adults, children, and other passersby. I have seen this particular book in used book stores and at libraries, so its ready accessibility may continue to stimulate children with the wonder of mathematics.

The second book with this broader perspective is the irreverent and enjoyable Math Curse. This colorful picture book is a hilarious tour through a normal day of a girl who realizes, as her math teacher said, "If you think about it right, everything is a math problem." If you think about it, of course, she's right.

The third broader-perspective book is considered a classic in mathematics education that could be enjoyed by children who like math as well as by those who need enrichment or novelty. Edwin Abbot's Flatland is the story of flat, two-dimensional creatures who live in a flat, two-dimensional world. While Part I of the book is a somewhat tedious description of how Flatland works (from the Flatlanders' perspective), Part 2 presents a curious and absorbing phenomenon. The narrator of the story is a circle who is one day visited by a three-dimensional sphere, who "lifts" the circle out of and above Flatland to took down upon his flat world from our own normal 3-D perspective. The circle is amazed to find he can see inside his flat, 2-D friends below, and he also has a difficult time trying to explain his 3-D experience after his sphere friend drops him back into Flatland.

Author Abbott was a scholar and theologian of the early 1900's, and his allegory of three-dimensional beings who visit two-dimensional creatures is analogous to divine and angelic appearances of the Bible. It is entirely possible that our Creator God may inhabit a higher-dimensioned reality than his created 3-D world in which we live and move and have our being. While caution is always necessary on such speculative topics as this, Flatland provocatively connects geometry with the biblical worldview that we live in a marvelously consistent universe that is still subject, in some mysterious way, to penetrations, revelations, and visitations from our Creator and His messengers - and occasionally His enemies. You may be able to find Flatland at libraries, or at used book stores for a nominal price.

Finally, you can also find a number of other excellent "math-type" books available at a good bookstore. On a recent visit, I was pleasantly surprised at how many quality math books I found - stories, puzzles, mazes, arithmetricks, and engaging activities. There were far too many books to even mention here, so check this out yourself. If budget is an issue, check with your local library to see if they carry any books you liked at the store - and if they don't, request that the library acquire them. Along with other resources I have mentioned in previous columns - such as math games and musical skip counting tapes - these types of quality literary mathematical books can play a significant role in providing a mathematically rich environment for our children. Perhaps some of these books will become your children's favorites as well.


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