High School Mathematics: Projective Geometry
“Projective geometry has the capacity to open minds and broaden thinking. I learned about things like perspective and duality, and all of this came together at infinity to create an understanding that I knew I didn’t have at the start of this block.”
Problem Solving and Perspective
The central point of mathematical activity in the Waldorf high school is problem solving. The important thing is learning how to solve problems, not what the answer is. With this as the focus, high school mathematics builds on both bases of mathematics: inspiration (induction) as a beginning and logical conclusion (deduction) at a later stage in the mathematical activity.
The most important aim is to develop the students ability to think with a wide range of approaches until they get to the logical conclusion, and to give them confidence in themselves and in their thinking. Another important goal is to prepare the students to apply calculations methods to everyday life and also to give them the foundation for further education.
Geometry is the mathematical discipline that deals with the interrelations of objects in the plane, in space, or even in higher dimensions. More than any other mathematical discipline, the field of geometry ranges from the very concrete and visual to the very abstract and fundamental. In one extreme, geometry deals with very concrete objects such as points, lines, circles, and planes and studies the interrelations between them. On the other side, geometry is a benchmark for logical rigor, the elegance of axiom systems, logical chains of proof, and the parallel world of algebraic structures.
In tenth grade, students study the projective properties of geometric figures
In high school, children reach a new stage of development where an individual’s inner life confronts the outer world in a relationship that still has to find a form. In an integrative education, even geometry has its place in the deep work of young adults. The deeper concepts of mathematics around perspective, infinity, transformations, angles, boundaries, and duality lead to new insights and broader understanding of not just geometry, but of life.
Introduction to Projective Geometry from a student’s main lesson book:
The Euclidian geometry we have worked with up until this point has dealt with the finite, the measureable. In the consciousness of the ancient Greeks, even the realm of the gods was considered in finite terms. Of course this finite or measureable nature implies ideals; for in actuality we can never be exact. As soon as we try to represent a point or line on paper, it is only an approximation, or rather a two-dimensional representation of the ideal. A point, as defined by Euclid is that which has no part, and a line is breathless and thus can never actually exit in the physical.
Projective geometry takes the elements of Euclid but stretches them in space toying with the idea of infinity. This geometry has seen application in the perspective drawings done already during the Renaissance by such artists as DaVinci and Durer. Projective geometry challenges Euclid’s elements asking us to see points as lines of infinity and whole planes becoming points. The mysteries of infinity order the random and obscure the ordered.
This block is an exploration of space, projecting lines and points to infinity with geometric nets and conic sections, observing the phenomena as they occur. We can wrestle with the ideas, but this course also gives us the opportunity to step back and relish the beauty and magic of these lines and points as we strive for exactness and perfection.
Why Do Geometry By Hand?
Students need to develop an intuitive understanding of geometric relationships and how to manipulate them. Learning how to do geometric proofs with compass and straightedge is an essential part of developing that knowledge. That knowledge will be used by an architect in many ways, from the creation of complex computer models to hand-sketching. In fact, one of the first things they teach in architectural perspective drawing class is how to use basic geometric principles we all learned in 10th grade geometry to quickly draw realistic and correctly-proportioned perspective images.
The relationship between mind and hand through pencil and paper is very direct (same with sculpting clay, for that matter). You lose that direct connection when a computer interface is involved. Once you know and have intuitively internalized the principles, the computer allows you to magnify that knowledge in practical applications.
I insist on seeing a demonstration of hand-drawing skills even for prospective employees who will only be doing computer drafting or modeling. What they can do with a pencil shows me in a very direct way how their brains work and whether or not they really understand what they’re doing when they try to graphically represent spatial concepts and systems,
So, yes, I think it’s important that students still learn how to do geometry the old fashioned way. Even though a computer will automate a lot of the calculation and construction for you, you still need to understand the geometric principles at work in order to use them. – Archinect