Geometry Problem and Awareness of Problem-Solving

I was making something today that another blogger had written about in a private document, and three geometry problems came up in the course of making them.  The first problem was to draw a triangle inside a circle, thus:

A triangle in a circle

As maybe you my readers know, this problem generates the mandorla, which is the sideways almond shape in the middle of the geometry problem, across the triangle’s base.  The Mandorla, or double-vescia, is a shape associated with the feminine energies of the universe, and I’m told it generates a length that represents the square root of 3.

The second problem was to create a six-pointed star.  Like the previous math problem, it’s based on the ad triangulum form of geometry according to medieval theories of mathematics, and it’s again based on a square root of 3. I did it this way:

Drawing a six-pointed star

The third problem involved something which I do using a method that I learned from a brother of the lodge, which I can’t show here, because I feel it touches on my oaths and promises, even if it’s something I didn’t learn in lodge itself.

But as I worked through these geometry problems in service to a larger creative goal that whiled away a little extra time on an off-duty Friday, I couldn’t help but think on the underlying meanings of these simple processes. J.K. Rowling gave the first technical geometry problem a meaning as the Deathly Hallows — the Stone of Immortality, the Elder Wand, and the Cloak of Invisibility (though to be fair, her geometry problem is ACTUALLY the circle circumscribed by a triangle – that is, the triangle is on the outside, rather than the inside).  The second geometry problem effectively reminded those who carried it out of Judaism, of the classical (visible) planets, and the inter-relationship of male and female (the upward and downward pointing triangles of Dan Brown fame).

Even as I made the arcs with my compass and drew the lines suggested, my mind was reaching back to the process of learning these three mathematical constructions, and I realized again…

I never use algebra.

Ok, from time to time I calculate a percentage grade using that simple formula everyone learns in algebra I or Algebra I, which is simply a fraction where the percentage works out to x/100, and there’s an equal sign and another fraction, and you solve for x by cross-multiplication.

Occasionally I solve a problem algebraically, ok. Maybe.  But not often.  Nearly all of the problems I solve, I solve through some form of geometry or visualization: comparison of areas, imagining piles of money (though I’ve really screwed that up often enough), and so on.  More and more, I solve mathematical problems visually.

And I find myself wishing that more mathematics teachers recognized this, or at least that my mathematics teachers recognized it.  Because geometry has helped me to be a better artist and a better designer, and algebra hasn’t.

Is there a way to teach math that helps visual learners understand algebra better? Is there a method to teach linear thinkers how to do geometry better? And why don’t we make kids get out the old-fashioned ruler and compass (and maybe a slide rule?) more frequently?

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