Mathematical skills are just as important in modern society as being able to read. We need numerical tools to navigate through our daily lives in many ways that we take for granted, such as comparing our speed to speed limit signs, making change and making sure that the checker at the store got the amount right. Innumeracy imposes as many personal challenges as illiteracy.
Geometry is as important as any other mathematical skill. Everyone over the age of two understands that a mountain 50 miles away is actually much taller than a telephone pole nearby: that’s geometry. Nature even agrees – recent research shows that some geometric sense appears to be hardwired in the brain.* Many people see the world as shapes, and entire art movements are based on geometric shapes.
If geometry is so useful and important, why is it taught halfway through high school, and shrouded in esoterica like constructions and proofs? The only answer seems to be that it has always been taught this way. As students have been asking for centuries, “Is this necessary? If so, then why?”
The answer is that Math in school is not really about math: it’s about empowerment. Empowerment in this case includes not only learning how to understand shapes and lines, but also how to think and reason. The innate geometric sense shared by everyone provides a good intuitive basis for more abstract reasoning. This is one of the first times that students are exposed to formal reasoning, learning how to make assertions, and how to demonstrate truth or falsity. It is easier to use geometry to teach reasoning of this type than doing the same thing with algebra or formal logic – geometry is more concrete and intuitive.
Geometry also provides an opportunity for students to excel at math after having trouble with algebra. Everyone has a unique learning style, with some preferring to learn via pictures, others via written text, and still others via lecture or a combination of all three.
Having established that teaching geometry is important, we must still answer the question as to what type of geometry to teach. The immediate and easy answer is Euclidian geometry, which is the geometry of the ancient Greeks. This does not mean that it is obsolete; quite the contrary. It is used today at least as much as it was in ancient times. This is the geometry of our intuition, with applications in both the real world and in science classes.
Geometry is a challenging subject, and some find it discouraging, especially at first. But the feeling of accomplishment after completing the course can provide the confidence to take on other academic challenges.
After developing our geometric intuitions, we can explore other worlds, something not done often enough in traditional geometry classes. Other geometries require a suspension of normal intuitions, and a willingness to leave our intuitions behind. That is best done after developing a comfort level with traditional geometry. After that there are wonderful worlds to explore – we can deform teacups with topology, we can build entire worlds using fractals, we can explore mathematical monsters with chaos theory, and we can make pictures on walls using projective geometry. All in its time. As the great geometer Archimedes said, “Give me a place to stand, and I can move the world.” That’s geometry too.