God the Geometrician

SnowCrystals.com

"I had been reading one of the driest passages imaginable from the Scriptures where Israel came out of Egypt and God arranged them into a diamond-shaped camp. He put Levi in the middle and Reuben out in front and Benjamin behind. It was a diamond-shaped moving city with a flame of fire in the middle giving light. Suddenly it broke over me:

God is a geometrician, He’s an artist! When He laid out that city He laid it out skillfully, diamond-shaped with a plume in the middle, and it suddenly swept over me like a wave of the sea: how beautiful God is and how artistic and how poetic and how musical, and I worshiped God there under that tree all by myself. " A.W. Tozer

I was never very good at geometry myself, yet the past several years I've been intrigued with the grand thought that God is certainly a geometrician. Tozer sites one evidence in his quote above, and it is a delight to keep a running list of other places His geometrical masterpieces are found.

Put snowflakes on the list. After our recent blizzard, it is only natural that we should be thinking on these little prismatic wonders! Each flake is a perfect hexagon, decorated with ridges, dendrites, and endless combinations of symmetrical artistry. Enjoy reading about them in the Guide to Snowflakes and take time to ooh and aah over the gorgeous photo gallery of snow crystals here.

Beauty and Mathematics

"One thing have I desired of the Lord, that will I seek after; that I may
dwell in the house of the Lord all the days of my life, to behold the beauty of
the Lord and to inquire in His temple." Psalm 27:4

One way to behold the beauty of the Lord is to examine His creation mathematically, for the natural world speaks loudly of the Creator's remarkable orderliness. Take for example, the sequence of numbers known as the "Fibonacci Numbers":

1, 1, 2, 3, 5, 8, 13, 21, 34, 55..........

In this sequence, each number in line is the sum of the preceding two numbers. We can trace these numbers in nearly everything that we would call "beautiful" in the world of nature: the spirals on a pineapple, the whorls at the center of a sunflower, the pattern of a pine cone. As you see the predictable pattern in growing things, you begin to surmise that nothing is truly random. That is a comforting thought to me! Foundational structure provides much safety.

Comprehending patterns visually can lead to the next step, which is to apply knowledge of the seen to the unseen. The patterns we can see become a hook onto which we can hang our understanding of things invisible. The workings of the cell, DNA strands, atoms, the concept of aerodynamics-- all of these things we can understand to some extent because we are acquainted with similar, concrete models of the same sort.

"For by Him all things were created....visible and invisible...." Colossians 1:16

The ability to think abstractly is a strictly human attribute and mathematical patterns such as the Fibonacci Numbers give us a starting point onto which we can connect deeper understanding. And it is here that I have only recently made the startling discovery that mathematics is really NOT just about numbers. It is more accurately the science of patterns. How I wish I would have known this years ago!! To have been able to trace the foundational patterns of the universe to the God who made it would have perhaps given me a fresh outlook on all of the math assignments I so dutifully endured but failed to enjoy.

*Music has a definite, mathematical pattern.

*Linguistics involves the study of grammatical pattern.

*Genetics link DNA patterns with physical attributes.

*Chemists understand the orderly pattern of the periodic table of elements.

*Astronomy incorporates the study of orbital patterns of moving objects in space.

*This list could go on endlessly and become a mathematical object lesson for the word "infinity"!

I think that perhaps the most fascinating thing about all of this is that humans recognize beauty within the boundaries of orderly pattern. We perceive only some patterns as beautiful, and those are usually linked to the Fibonacci numbers. Look again at that sequence: take two adjacent numbers and make a ratio; for example 3:5. Now, do the math:

5 divided by 3 = 1.6

The 1.6 quotient has become the famous "golden ratio." Our bodies were designed according to this ratio. The Greeks built the Parthenon in keeping with this pattern. Even the 3x5 inch index card you use for a bookmark is cut to pleasing proportion according to this principle. Whether it is a building a painting, a face, or a song--humans seem innately attracted to anything designed in the 3:5 ratio. Conversely, the things we would dub grotesque or deformed fall outside of the Golden ratio. Beauty has a design, a predetermined orderliness and life becomes endlessly fascinating as we uncover layer after layer of these divinely-conceived patterns.

Here are some beautiful words by a mathematician:

"The mathematician's patterns, like the painter's or the poet's, must be
beautiful, the ideas, like the colors or the words, must fit together in a
harmonious way. Beauty is the first test; there is no permanent place in
the world for ugly mathematics....It may be very hard to define mathematical
beauty, but that is just as true of beauty of any kind--we may not know quite
what we mean by a beautiful poem, but that does not prevent us from recognizing
one when we read it. " --quote by G.H. Hardy from his book, A Mathematician's Apology.

On the Teaching of Math

The world appears to be split into two types of people: symbolic thinkers who seem to be born with the "math gene" and concrete thinkers who more appreciate words. The language and word lovers are often left scratching their heads when a symbolic thinker makes a mathematical explanation, whether orally or in writing. Listen to what Jacques Barzun says about this:

"....from the very first grade one child "loves" math and another hates it. This is the beginning of the school's difficulty. (SNIP) What is seldom noticed is that the trait in students who favor math is also the cause of much poor teaching. For the mind that feels at home with math is likely to lack the qualities that make a good teacher. (gasp!!) SNIP

....the math or science teacher may fail, although well prepared and of course well intentioned. He or she, moving easily among abstractions and their symbols, has no idea how the "other mind" works---does not perceive how strange the square root of minus one seems to the concrete-minded (SNIP) It is a case of two minds out of tune with each other. And a proof that the disparity is not accidental or occasional is that the same fault is found again and again in math and science textbooks. The author assumes a kind of scientific common sense which will make things clear at once. Accurate with symbols, as writer or teacher he often uses words carelessly, failing to see that to the concrete-minded words have distinct connotations."

Barzun follows this with an example from a textbook, and shows methodically how the poor wording would befuddle any word lover.

Being myself a word lover and a recovering math-o-phobic, I've given serious consideration to this dilemma. As a student who was befuddled by murky explanations, I now sit in the seat of a teacher who wants to make math come alive for my students. My solution: DO NOT RELY SOLELY ON A TEXTBOOK.

Most people who know me know that I am allergic to textbooks :) I use them more in math than in any other subject, but try not to over- rely on them. Steve Demme has taught me that "teachers teach~textbooks drill." I think there is a lot of wisdom in those words.

So how do you wean yourself from dependence on a textbook? The first thing is to set apart one day a week to do math in a different way; no textbook allowed. Friday is that day at our house.

Here are some ways to use that time:

*Play games. Lots of games: dice games, strategy games, board games, card games. Remember that math is not just about numbers; it is the study of patterns. Games are invaluable in imparting the art of logical thinking. It's fun, too!

*Choose math stories to read together. Look at Julie's list of literature books that qualify as "math" here. Some of our favorites have included The Number Devil, Math Smart Junior, books by Greg Tang , and Sir Cumference series books.

*Keep a math journal. Write where and how you saw math being used in your everyday life.

*Narrate. Some people are surprised that you can use narration in a math lesson, but it is very effective. To have a child explain HOW he works a math problem brings it out of the realm of symbols and into the world of words. This can be done as either an oral or written narration. For very small children, you can switch roles. Pretend you are the student and ask them to be the teacher and explain their current math concept to you. They won't even know they are doing school! Today, one of my highschoolers explained very lucidly how to approach an algebra problem using the correct order of operations. Narration works for young or old.

*Play with math. My husband is a true symbolic thinker and it is second nature for him to pose math problems at the table as though he were thinking out loud. This week, he mentioned to us that he had determined that his car gets 2 more miles per gallon if he uses regular unleaded gasoline rather than ethanol. "Yes, Daddy, but doesn't it cost 10 cents per gallon more?" And then ensued a real brain teaser. What is the difference in price between a tank of ethanol vs. a tank of unleaded? How many miles can you drive on each tank? Which is the better value? Keeping alert to these kind of applications brings math down to earth.

*Build things. Buy bridge building kits. Make geodesic domes out of rolled newspapers (that is what you see in today's picture). Construct architectural models using marshmallows and toothpicks.

*Buy a few age appropriate math computer games. My kids enjoyed Cluefinders when they were younger. There are probably better ones now.

Next time....stimulating your own passion for the subject of math.

My Math Makeover

If you would have told me 10 years ago that I would be teaching a math class at a cooperative school, I would have laughed at you. I have been, for most of my life, a math-a-phobic. It's true I managed to get decent grades in my math classes at school, but that was the result of hard work, drudgery, a lot of prayer, and a little luck.

When I began to homeschool my children, I started to think about my attitude toward math. I did not want to pass along a math aversion to the next generation. My hope was that my own kids would not dread or fear this subject and that they would connect its orderliness to the God who designed an orderly universe. I was not so ambitious, however, to set myself a goal of creating math LOVERS.

I began my "math makeover" with a book by Marilyn Burns, entitled Math: Facing an American Phobia. It was a fantastic start, written by a math educator who finds delight in real-world applications rather than in rows of "right" answers on a worksheet. For me, she put to death the myth that there is only ONE way to come up with the answer to a math problem. If you had to quickly tabulate the sum of 37 plus 50 would you first add 40 to 50 and then subtract 3? Or would you make an addition problem in your mind, lining up the 37 above the 50 as you would have written it on the blackboard at school? Different strokes for different folks. There is nothing wrong with either way of doing it. I think Marilyn gave me a warm little boost as if to say, "You can do this, and although it may never become your passion, you can learn to enjoy math."

At about this time I had to select math curriculum for my oldest child. I remember being so overwhelmed with all the choices-- interviewing veteran teachers, poring over reviews, looking at a myriad of samples. I ended up making what would be a good choice for our family of 4 children: Math-U-See. There are many things to like about Math-U-See, but what contributed to my "math makeover" were the video segments in which Steve Demme teaches the teacher HOW to present the concepts. This man has a bag of slick math tricks, jokes, and quirky explanations that hold your interest and make learning painless, if not fun. This curriculum was a perfect fit for all but one of my children (and that is another story), but what happened as I used it was that I gained confidence in my ability to explain math concepts.

People who write math books have what is playfully called the "math gene", and because they do, their explanations appeal mostly to other kindred spirits who share the "math gene", while language and word-lovers are left scratching their heads. Steve Demme's jokes and stories and games added the language dimension that helped me immensely.

What I didn't know at the time is that the curriculum really did not matter very much. It was my attitude that mattered. A good teacher uses curriculum only as a tool; her enthusiasm and confidence can make cake out of dry bread crumbs.

I have more to say about my previous mention of symbolic thinkers (math geeks) versus language lovers (word geeks), which I will address separately in another post. It took me a few years to realize that there is a complementary, even synergistic relationship between the two. I'll save that until next time.