Abstract nonsense. Is that all mathematics is?
Abstraction, frustrating though it might be, is the very engine that makes mathematics a compelling way of talking about order in the world. To abstract means that we set aside irrelevant distinctions and arrive at the common structure governing a whole.
We must abstract from real things. It is in those things, there, among real stuff in the world, that we begin our journey towards higher and more intelligible order.
The challenge for the teacher, then, is to set the right things before the student. Only then can the student also follow in the path of abstraction.
Here the traditional disciplines of the quadrivium reveal a deep wisdom about human nature and our ability to learn. Arithmetic builds on simple counting. Geometry begins when we start to name shapes and distinguish among them. Music already moves and soothes us in the first moments of our lives. Sunrise, sunset, phases of the moon, seasons of the year: in each we grasp a familiar and lasting order.
The quadrivium builds on these real, familiar, tangible experiences. Because the objects are so familiar, students are able to reason and judge confidently. As a result, they acquire the conviction that their studies are placing them in contact with something true, something lasting, something substantial. Such conviction is a contrast to the confusion and discouragement that can arise from a multitude of algorithms and confusing symbols, things that lead many to find math frustrating.
In the quadrivium, mathematical abstraction is founded in the real. It does not obscure. It illuminates.