Thomas Treloar of Hillsdale College gives a very helpful overview of the role of mathematics historically and in classical education. I think more should be said about the difference in character between algebraically-based modern mathematics, which trains us to reason without being distracted by the real things that gave rise to the problem, and ancient mathematics, which relied so heavily upon the imaginative presentation of figures and even numbers in its reasoning. The Euclid session of our Academic Retreat for Teachers is almost always the most popular in part because teachers find math so much more understandable when clear reasoning is applied to a diagram that gives them an intuitive connection.
In a related context, I can’t recommend highly enough Paul Lockhart’s soon-to-be-released book, Measurement. He does not take a Euclidean approach, and his philosophy borders on Kant, but the way he opens the delights of mathematics in a way suitable for high school freshmen is tremendous. Through his approach, he is able to introduce calculus early. The techniques of calculus are not conceptually difficult; in fact, they are strongly connected to the imagination. Lockhart introduces these early and naturally. The only reason that calculus has been relegated to upper level high school is because of its preferred use for transcendental functions, which are conceptually very difficult.