Saturday, August 07, 2010

Our Plan for the Year: A Mathematical Rabbit Trail Part I

In my last post, I covered our plan for the knowledge of God and ended by saying that studying the universe reveals God's divine nature. Under the knowledge of the universe are science, geography, physical development, handwork, and mathematics.

What, you say? Mathematics! I know you hate that detestable subject and never use what you learned in high school in real life. Math is so boring and impractical. How on earth could it have anything to do with God? Only a handful of you love math and I know you'll bear with this rabbit trail to win a few hearts to the Queen of the Sciences!

I know that many of you rebelled against mathematics because you thought you were never going to use it again. However, Mason saw this subject as very useful, but not in the way you are thinking. She believed that, when properly taught, mathematics helped form helpful mental habits like clear and ordered thinking, reasoning powers, insight, readiness, accuracy, intellectual truthfulness, rapid and careful execution, attention, concentration, abstract thinking, and independent work (page 254). While society focuses on the product of math (knowing how to simplify a very long string of letters, symbols, and numbers), she considered how the process of doing mathematics, when properly taught to the fully attentive mind, built mental processes that we ought to do every day.

Why do so many people hate mathematics? Well, Mason believed that success in mathematics depended upon two things: good teaching and attentive students (page 7). Because most of us were so poorly taught, we became bored out of our minds and grew to despise the subject. What do I mean by poor teaching? How many of you did the following in math class? Copied material from the bored without understanding what it meant. Watched the teacher prompt you through a problem because letting students think took too long. Let the smart kid spoon-feed you through your homework. Crammed the night before a test. Reworked a problem fifty different ways without knowing why until you got the right answer.

So what is good teaching? Well, the first problem is learning to mark a problem WRONG without reworking. For some reason, people find this heretical so hold on! Reworking problems teaches children to do sloppy, careless work because someone is going to help them fix it anyway. It gives them the notion that wrong can be mended, but, in the real world, that is a dangerous habit. People die when a walkway collapses because of sloppy calculations or the pharmacist makes a math error. Instead of reworking, teachers ought to figure out what went wrong and clear up the student's misunderstanding before doing any more problems. Mason concluded, "The child must not be let run away with the notion that wrong can be mended into right. The future is before him: he may get the next sum right, and the wise teacher will make it her business to see that he does" (pages 260-261).

I have found the advice of fellow Mason educators very helpful on this issue. If you are a person who assigns every problem because of silly errors and sloppy work, explain to the child that you will only assign half of the problems. For every wrong problem, you will figure out what happened, find strategies to prevent that error, and assign one new problem for every wrong one. That means doing careful work will lead to what every child and teacher wants: more free time! When I started doing this with my math hater (David), his work improved because he had incentive to do it right the first time.

Other elements of good teaching include inspiring ideas, quickening imagination . . . Math . . . inspiring? Imaginative? Are you kidding?


I mean it!

Think back to your most boring day in math class. You were probably looking out the window, daydreaming, trying to be anywhere but there. A walk in the fresh air away from all that chalk dust was not too much to ask. And, our generation was lucky because most of us did not start school until Kindergarten at the earliest--half a day of fun, games songs, play and naps. In today's "early is better" world, our poor kids are already doing worksheets in preschool! Mason thought like you: she thought children under the age of six ought to be outdoors: no worksheets, no books, and no formal lessons. However, that did not mean a vacation from math.

Mason suggested that educators could guide mathematical thinking and language when preschoolers told what they saw in their explorations outdoors (Part I and Part II). Find a tree outdoors and describe it. I'm thinking about oaks near my house. Three tall trees stand on the left side of the driveway, which is west of our house. They are along a brick wall parallel to the driveway. The green leaves are long, thin, and oval. If you pop the top off an acorn, shaped more like a sphere than most, you find a circle.

The great outdoors provides scope for the mathematical imagination. You can talk about quantity, size, shape, and direction. Other topics include seasons, calendars, time, distance direction, geometry, etc. A nature diary where the child draws what inspires them the most will document what their early understanding of mathematics without a worksheet in sight. We can nurture habits like accuracy, attention, and truthfulness in conversations about nature. Concentrating on concrete things they find in the real world will build a great foundation for learning elementary school math.

If you are still not inspired, stay with me. Suppose you and your child are looking for four-leaf clovers, which are very rare, because three-leaf clovers are so common. You have been counting the petals of flowers for several years and your child comments, "Lilies have one petals, irises have three, and buttercups have five. I can't think of any flowers with two or four petals. Do you think flowers always have an odd number of petals?"

You reply, "What about the delphiniums we found the other day?"

"Oh, yeah, they have eight. Yes, but marigolds have thirteen and black-eyed susans have twenty-one."

As we explore the universe, we constantly uncover these patterns. G. H. Hardy wrote, ""A mathematician, like a painter or a poet, is a maker of patterns." How much more exciting mathematics would be if we treated it as the search for patterns?

A week later, your child is on the search for different species of daisies. "Did you know that some daisies have thiry-four petals and others have fifty-five petals? I even found one patch that had eighty-nine petals and another with twenty-one. Isn't that strange?" You point out some exceptions to his theory that there are some flowers with four petals and six petals. "Yes, but have you noticed how the primrose looks like two sets of two like the wings of a butterfly and that mayapples are two sets of three?"

A few weeks later, you are doing a sunflower study. Your child starts counting the spirals and discovers that the number of spirals in one direction is thirty-four and is fifty-five in another direction. Finally, your child writes down the numbers in the patterns in flowers that have been so inspiring:

1, 2, 3, 5, 8, 13, 21, 34, 55, 89 . . .

Wait a minute! There is a pattern! Do you see it? I'm not going to give it away because I want to give you time to think.

Your child sees the pattern and goes online. There are all kinds of websites explaining the mathematics behind the spirals of a sunflower and giving the name of the series. There are many sites for finding this series in nature and a few scoffers who call it flim-flam. From that day on, some of your child's nature study becomes devoted to settle the question in his mind.

Hardy went on to say, "And just as in poetry and painting, the mathematician's patterns must be beautiful. Beauty is the first test. There is no permanent place in the world for ugly mathematics."

After doing this flower study, would you not agree?


Penny said...

I hate math. I would like to love it like you do. For the moment, I'll live vicariously through your eyes. The info on the sunflower is really neat.

Stephanie said...

What a marvelous post. I first heard about Fibonacci numbers when I started homeschooling and I wondered "Why didn't they teach us about cool stuff like this in school??" I was trained to think of math as a form of drill, not as a living, 3-dimensional thing existing in the universe.

When you look at patterns in nature, it really is like seeing God's fingerprints all over the world. :-)

I'm still not in love with math, but I'm gradually learning. No longer a "math atheist" -- maybe an open-minded math agnostic?

poohder said...

I am envious that you have that sort of "relationship" with math. My teachers RUINED it for me.


The Glasers said...

I know you all may have negative episodic memories of math. Maybe we need RDI for math atheists and agnostics!

Phyllis said...

Did you see the Num3ers episode about this?
And, have you read "A mathmatician's Lament"?
I think you would like them both.

Kathleen said...

Once upon an eon ago I took a class called The History and Philosophy of Mathmatics. I learned lots of fun stuff - including this little tidbit. LOL - I even remember my final paper for the class - The Mathmatics of Poetry ;-)