by Stratford Caldecott
This book was my meaty read for summer 2018. I didn't finish it before becoming swamped by high school planning, so it carried over into 2019. For many years, we have struggled in our homeschool to avoid viewing math as drudgery. I think we are fairly good at creating an environment in which math is often fun with Life of Fred books as our math texts and plenty of math games from books and our shelves.
Yet this environment is only partially meeting my goal. I have always thought the children should also learn to find the beauty and truth in mathematics, that it should somehow connect them to the natural world, even though this beauty is something I only vaguely understand myself.
I hoped this book would show me how to reveal the beauty and truth of mathematics to my children in our homeschool.
The Forward is by Ken Myers (of Mars Hill Audio Journal):
Since the Logos is love, and since all things are created through him and for him and are held together in him, we should expect the logic, the rationality, the intelligibility of the world to usher in the delight that beauty bestows.A substantial part of the book focused on arguments explaining why the study of science and mathematics is enhanced and fulfilled through explicit relationships to the humanities and liberal arts. Among the many voices Caldecott gathers together in his reasoning are those of James S. Taylor in Poetic Knowledge, Bl. John Henry Newman in The Idea of a University, and Josef Pieper in Leisure the Basis of Culture. That last one is on my wish list.
An integrated curriculum must teach subjects, and it must teach the right subjects, but it should do so by incorporating each subject, even mathematics and the hard sciences, within the history of ideas, which is the history of our culture. Every subject has a history, a drama, and by imaginatively engaging with these stories we become part of the tradition.Most of these ideas are not new to me and frankly, I was convinced of this much before I started the book, but Caldecott drew connections throughout history from ancient Greece to modern times that I found helpful. His prose is as elegant as you might hope based on the gorgeous cover of this book.
The purpose of an education is not merely to communicate information, let alone current scientific opinion, nor to train future workers and managers. It is to teach the ability to think, discriminate, speak, and write, and, along with this, the ability to perceive the inner, connecting principles, the intrinsic relations, the logoi, of creation, which the ancient Christian Pythagorean tradition (right through the medieval period) understood in terms of number and cosmic harmony.Homeschooling with Charlotte Mason's philosophy means this relationship of ideas is already integral to our curriculum. We are reading history and science and geography together, allowing the story of humanity to be woven by the student from these different threads. Or rather, allowing the opportunity for these relationships to be developed; each student does his or her own hard work.
Moreover, though we have every intention of our children going to college or trade school and learning how to earn a salary so they can care for a family, either in a domestic church or in the Church, our educational goals are focused on providing the wonder and wisdom for our children to become the people God wants them to be. A job is only a small part of their lives.
The principle remains the same: knowledge is its own end--"worth possessing for what it is, and not merely for what it does." It is not to be valued for the power it gives us over nature, or even for the moral improvement it may bring about in us (even if these things may flow from it). It is to be valued for its beauty. "There is a physical beauty and a moral: there is a beauty of person, there is a beauty of our moral being, which is natural virtue; and in like manner there is a beauty, there is a perfection, of the intellect."The quotes are Newman's from The Idea of a University.
After these basic arguments, Caldecott begins exploring numbers, shapes, and supernatural relationships. For example, he examines the "irrational beauty" of the golden ratio, phi, and the Fibonacci sequence. Supernatural relationships, like that between Father, Son, and Holy Spirit, can be represented geometrically, revealing a greater depth to the relationship.
Then [pi] could be read as describing the relationship between the Persons, a relationship that is infinitely fruitful and never ending. Thus the endlessly flowing numbers of [pi] suggest the super-abundance of God's mercy, the infinite quality of his love, and the unlimited space opened up within the Trinity for the act of creation.These explorations were exactly the kind of material I sought. Much of it is understandable without knowing too much higher level math, but the combination of mathematics and philosophy and theology made many of the discourses difficult to follow. Thales (before Pythagoras) showed how
the perpendicular line drawn from a right angle touching the circumference back to the hypotenuse will always equal the mean proportional between the segments into which it divides the diameter[.]There's a diagram in the book for this one (and many others) that helps a little, but I still often found myself reading sections a second or third time to try to understand exactly what Caldecott meant. I'm certain I could glean even more from the book if I read it again.
In the end, though, the important idea is that these sorts of explorations reveal an inherent perfection of the universe which point us always to the Creator and his relationship with Creation.
Speculations like those I have mentioned in this chapter will appear forced to many. Yet we must return to the central idea that God's archetypal forms or Ideas are inevitably found within nature at every level, reflected with greater or lesser degrees of accuracy. That is not pantheism but Christian Platonism, perfectly compatible with the insights of theology and revelations of scripture.Discussions of frequency, harmonics, and Chladni patterns allows Caldecott to connect a celestial harmony with liturgy, worship, and prayer. He quotes C. S. Lewis (Planet Narnia: The Seven Harmonies in the Imagination of C. S. Lewis by Micheal Ward).
[Celestial harmony] is the only sound which has never for one split second ceased in any part of the universe; with this positive we have no negative to contrast. Presumably if (per impossibile) it ever did stop, then with terror and dismay, with a dislocation of our whole auditory life, we should feel that the bottom had dropped out of our lives. But it never does. The music which is too familiar to be heard enfolds us day and night and in all ages.All of these subjects must come together in our education. According to Caldecott, integrating science with poetry, art, music, and the humanities allows students, all of us, to understand the universe in a more complete way, one which will at the same time, allow for greater understanding in scientific and mathematical disciplines.
Music, architecture, astronomy, and physics--the physical arts and their applications--demonstrate the fundamental intuition behind the Liberal Arts tradition of education, which is that the world is an ordered whole, a "cosmos," whose beauty becomes more apparent the more carefully and deeply we study it. By preparing ourselves in this way to contemplate the higher mysteries of philosophy and theology, we become more alive, more fully human.After reading this book, I have a greater appreciation myself for the beauty of mathematical thought and how the underlying principles of mathematics can reveal universal truths. It is not, however, a book I can simply read to my children or even realistically assign to a high school student. While it's been many years since I was in a college classroom, I have a far greater knowledge base than most high schoolers, and certainly a greater intrinsic interest, and I often struggled while reading the book.
So what I need know is for someone to take the next step. Use Caldecott's philosophy to write a mathematics curriculum or supplement or something I can share with my children.
I have received nothing in exchange for this post; all opinions are my own. I purchased this book at a local bookstore. Links to Amazon are affiliate links.