*sin(x)*and

*tan(y)*, these are the computational equivalent of driving with one foot on the gas pedal and the other on the brake. That doesn't pose a problem for your average desktop computer that's always plugged in or a laptop computer that you can recharge every couple of hours, but it sure does for an implantable device that needs to run for years on a single battery.

**Drawing Right-Brained**

Over the past few weeks, Iris has been working through the book Drawing on the Right Side of Your Brain. As we drive back and forth from New Jersey to the Berkshires, Iris and I talk about what she's been reading and learning and she sometime reads passages to me. In presenting the basis for her work, the author, Betty Edwards, discusses research conducted on people whose left and right brain hemispheres had been disconnected. In quick summary, there is a strong basis for the belief that the right and left sides of the brain function quite differently from one another, the left (controlling your right side) is the seat of language and structure, the right (controlling your left side) is the seat of visualization, abstraction and creativity. That's the rough of it.

Most of us favor our left brain. Allowing it to dominate how we think. Most formal education processes reinforce this. Indeed, there are many tasks (e.g., speaking) that require left-brained thought. There are other tasks (e.g., visualization and intuition) that favor right-brained thought. Apparently, when most of us try to draw, we tap into our favored left side which immediately translates whatever it is we're looking at into words or objects. We no longer see the pure image of a face, but instead, we see eyes, ears, mouth, nose, chin, etc. We objectify and organize a face and then draw the objects as we've defined them, not the image that we're viewing. The book provides exercises to help overcome the left-brained dominance and liberate the right-brain in guiding drawing.

OK, so that's the very rough and short of it. I can tell you from watching Iris and her drawing that it really seems to work.

**Back to Class**

Now, back to integer arithmetic versions of trigonometric functions.

As I've been looking at ways to do this, I've been googling a bit. It seems that almost everything written about trigonometry is left-brain oriented. People teach trigonometry by teaching formulas:

At this level, you can pretty easily picture the formulas in your head visualizing a right triangle from any one of the corners. However, the text pretty quickly abandon visualization and dive into learning formulas.

sine = opposite/hypotenuse

cosine = opposite/adjacent

tangent = opposite/adjacent

The problem with learning formulas is that you know how to use various functions to accomplish various tasks, but you don't actually know

*how*they work, you can't see them. This is all fine as long as you have a calculator that can perform the functions for you; you fill in the blanks and the calculator does all the work. It doesn't serve you well if you have to actually build the calculator, specially if you can't rely on the traditional methods of doing so. To do this, you need to actually understand how the basic functions work, not just how to string them together into useful applications.

As I've been looking at this, I've come to realize (with the help of some of what Iris has been reading) that most of us live fairly exclusively in either one side of the brain or the other. We may occasionally venture from side to side, but it's a quantum leap; the two are not connected. Here's the rub, true understanding doesn't occur until you can translate from one side of the brain to the other and back again.

For example, you can talk about abstract concepts like love associatively exclusively using simile: love's like... However, although you'd be engaging your left brain to grab the words, you wouldn't be using it to express love. Simile, analogy, metaphor, i.e. associative thinking are all right-brain activities. To really understand love would require expressing it in prose without simili.

Similarly, if you can talk about algebra or trigonometry or calculus only in terms of the formulas and functions, you don't really understand the math. You can repeat it, you can use it, but you don't get it. To really understand it, you would be able to easily visualize the geometries and spacial orientations of the formulas and then describe the effect of the formula in terms of the effected change in geometry.

The funny thing is that you can make a good living writing about love or performing calculations without every understanding either in the manner that I've described above. It's generally not required, because generally speaking, no one does understand what they do at that level.

The words that come to mind are, "What a shame."

Being able to translate new, abstract concepts into clearly articulated prose is powerful. So is being able to visualize the implications of complex formulas. Or for that matter, the instructions on how to install an attic fan.

The good news is that none of this is permanent or irreversible; there's no such thing as someone who's only left-brained or only right-brained, there are just people who favor one or the other and thus tend not to use the other.

Watching Iris and her experience with Drawing on the Right Side of Your Brain, I would suggest that it's a great place to start. Alternatively, if you're learning math or helping someone to learn math, take time to actually draw the effect of various formulas. Visualize the graphical representation on paper.

Well, back to highly-efficient, low-power trigonometry.

Happy Tuesday!

Teflon

I remember feeling really frustrated with trigonometry because the first explanation of the ratio was the formula. I abandoned advanced trig and advanced calculus for those very reasons. Looking forward to doing them in homeschool, since I'm understanding sooo much more than I did, even when teaching it.

ReplyDeleteI wonder if the 'I don't understand!' phenomena that happens with math (or anything else) can be overcome by comfortably waiting (while thinking..?) for the translation from one side of the brain to the other. Like what happens when looking into a picture for hidden pictures.

Faith, I think that's it exactly. It's the soft eyes that see the hidden pattern. The thing is to start teaching with the pattern sources (for example a bunch of sign waves or right triangles) and then play with them looking for the patterns. How are all these similar? What makes them similar? What can I derive from that?

ReplyDeleteThis is so much better than starting with the answer and telling someone, "Let me show you the patter that proves this."