I raced through The Liberal Arts Tradition because it was so interesting and answered many questions that I have about how to approach the disciplines more classically. One of the primary lessons for me was how differently they viewed math. The overview of the history of math was quick and inviting.
In Book VII of the Republic, when he introduces the central place of the Quadrivium in his educational program, he [Plato] tells Glaucon that the students must not pursue it with an eye toward useful business as shopkeepers do.
In fact, before the seventeenth century, scholars did not consider mathematics as widely applicable to the real world at all.
Thus the original role of the Quadrivium, to lead the mind to the realm of eternal and unchanging truths, was eventually displaced by the amazing power of mathematics to describe the physical world. A traditional liberal arts education should encompass both of these perspectives: the useful and the formative.Below is a good reminder, for all subjects:
Students who encounter mathematics with wonder are far more likely to commit to the rigors of its work.
Here is a whole new way to think about the implications of geometry:
In fact, the story of modernity has at times been told as the rise and fall of the geometric paradigm.From there he goes into the history of Euclid's Elements and why it is such a crucial text, greatly influencing Descartes and philosophers after him.
So, for those searching for a classical liberal arts paradigm for the study and teaching of geometry, the answer is found in a return to Euclid.
While Euclid's Elements has a total of thirteen books, a normal high school geometry class will usually cover only material from books I-IV and VI, representing under 100 propositions.I was a little shocked to read this:
Because historically geometry provided the foundation for the very concept of proof in mathematics, and because its constructions make it a more concrete subject, it should be placed as the next subject in mathematics after elementary arithmetic. (Yes, before algebra 1).The answer regarding how to cover algebra and calculus is also unexpected:
This is not a call to jettison the integrated corpus of material called algebra or calculus; it is instead a call to resituate them in a more connected context by exploring how they grew out of the study of discrete arithmetic or continuous geometry.
Cleaving to the traditional distinction between arithmetic and geometry, the difference between discrete quantity and continuous magnitude will indeed lead to wonder, wisdom, and worship.With quotes it is difficult to go into the deeper philosophical discussion around the idea of the one and the many that is essential to the development of math and geometry and ties into the contributions of Christian thought to mathematics. So, I encourage you to read the book to get the sense of wonder, rich history and essential questions in math. You can also see the outline of how the author's school introduces math and science here (pp 13 - 15).
I might try to summarize the science but you start getting into really deep waters there!
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