Sunday, 30 August 2009

Education and Architecture

It seems to me that one of the things a teacher can do to "re-enchant" education is draw his students' attention to the world we live in, and to the world we have built around us - the language of form in architecture. We can connect this easily with mathematics, geometry, astronomy, music, and history. An article by philosopher John Haldane in The Catholic Herald for 30 August 2009 even gives us a way to connect it with biology.

The architects of classical antiquity and of renaissance and later neo-classicism were resolute humanists. For them, man and measure go together, either with man being the measure of all things, or else in his measuring cosmic order through his ability to discern mathematical sequences, ratios and parallels.

The principle architectural expression of this abstract ordering is the fa├žade or wall divided into parts: surface and spaces defined by classical geometry. When competently conceived and executed such schemes are undeniably pleasing, like the rhythm of a well-ordered rhyme. But the inspiration is less to do with what is observed in nature than with what is reasoned to through mathematics and philosophy. The Gothic, by contrast, takes its key from the living world of ordered growth. From beneath the earth the germinated seed breaks through, first establishing a stem, then branching, next putting out leaves and buds, then in turn producing flowers and fruits. The order is not one of mathematical design but of organic progression; and it pre-exists invisible but immanent within the seed.

Like plants, Gothic buildings grow out of the earth and are developed upwards, drawing material from below but reaching for the light. They represent a recognition of the order of nature and an identification with it; acknowledging and seeking to imitate divine design.
Of course, the "living world of ordered growth" is itself a mathematical order, and the Gothic masons probably understood it as such. Nevertheless, Haldane is absolutely right that in a Gothic cathedral we easily imagine ourselves "standing beneath the canopy formed in an avenue of over-lapping trees, or imagine the undersides of the leaves of those trees." And it is surely true that "the power of England's Gothic cathedrals to prompt wonder at the order of nature and to encourage speculation about the source of that order is not altogether diminished. Hundreds of years on from their first appearance they remain sources of theological inspiration and reminders of the possibility of integrating intellect, imagination and sense in the embodiment and recognition of religious meanings."

Born in Britain but living in California, the architect Christopher Alexander believes that architecture should be rooted in a profound understanding of the human person as spiritually transcendent, yet intimately related to the cosmos. His book series The Nature of Order (2004) opens with an assault on the mechanistic idea of order, which he traces back to Descartes in the seventeenth century. He argues against this idea that matter and space possess degrees of life, because the elements of which they are made relate to each other as mutually supporting “centers”, making the whole more than the sum of its parts. He illustrates this by means of the patterns in a Turkish carpet and the architectural and decorative features of buildings like the Alhambra and Chartres Cathedral.

Alexander defines fifteen structural features that correlate with degrees of life. This enables him to become quite practical in his recommendations. The examples he chooses are eclectic, ranging from mud huts to palaces, from Shaker furniture to Persian glassware, from electrical discharges to cell walls, from the branches of plants to the cracking of mud and the formation of crystals and feathers. He suggests that his approach offers a way beyond Hume’s fact-value distinction. Our feeling-response to things, properly discerned, is an objective measure of their structural wholeness. The implications of all this for education remain to be explored.

Take a look at the "50 most extraordinary churches in the world" and see if you agree with your friends on which are the most beautiful!

Friday, 28 August 2009

The beauty of mathematics

For many of us, “mathematics” and “beauty” sit oddly together. We may remember math as boring or even frightening, but hardly beautiful. Yet math is the key to science, and science dominates our age. And there is another way to look at it.
The single most compelling reason to explore the world of mathematics is that it is beautiful, and pondering its intriguing ideas is great fun…. To study the deep truth of number relationships feeds the spirit as surely as any of the other great human activities of art, music, or literature. -- Calvin Clawson
This quotation is taken from p. 239 of a wonderful book called A Passion for Mathematics: Numbers, Puzzles, Madness, Religion, and the Quest for Reality, by Clifford A. Pickover. It is full of games and quotes and ideas that parents and teachers will find useful to get kids of all ages and all backgrounds interested and involved with maths and geometry. Pickover himself believes that “mathematics is the loom upon which God weaves the fabric of the universe” (p. 53).

Looking back on the maths classes I sat through as a kid, I can't help wishing I had been taught the subject not as a collection of seemingly arbitrary rules and procedures but (1) historically (starting with Pythagoras), (2) aesthetically (in relation to music, painting, architecture), and (3) symbolically (with a view to qualities, meanings, analogies inherent in numbers and shapes) - not to mention (4) playfully.

The religious and secular use of numbers are related, as I tried to show in The Seven Sacraments (Crossroad) as well as in Beauty for Truth’s Sake. The numbers particularly prominent in Christian tradition are 1, 2, 3, 4, 7 10 and 12. One is the source of all other numbers, 2 is the beginning of multiplicity, there are three divine Persons and three theological virtues, four cardinal virtues and four Gospels, seven sacraments as well as seven days of creation, ten commandments, twelve apostles and twelve tribes. And all these numbers are interrelated: 7 is 3 plus 4, 10 is 3 plus 7, 12 is 3 times 4. So in a way the fundamental structural numbers only go as high as 4, and all the others are made up from these. That takes us back to the sacred Tetraktys of the Pythagoreans, whose influence on the Christian tradition has been underestimated.

There is a fascinating article by Karen Kilby called MATHEMATICS, BEAUTY AND THEOLOGY that I recommend to your attention if the subject interests you. (Just follow the link.) And I also want to mention "The Curriculum of Beauty" by David H. Albert and Joyce Reed in Life Learning magazine. Albert writes:

Our children have within themselves, or so I am led to believe by my experience of them, an inner yearning for the beautiful, a potential wonderment and a delicious longing and love and trembling waiting to be empowered on its quest. This yearning is not likely to be fulfilled in a high school hallway or on the shopping mall checkout line. So what if we were to set as our task – as parents and as educators – acquainting our children with the beautiful without and the cultivation of the beautiful – the yearning – within? How might we go about our homeschooling lives differently if we were to conceive of what we are doing as primarily an aesthetic task?

Thursday, 27 August 2009


Education is a never-ending process – or should be. And what we learn depends largely on what catches our imagination. Not long agoI learned something by chance that changed the way I look at the world. Like many people, I grew up with the idea that the light by which we see things emanates from a source, bounces off the thing I am looking at, and collides with a cell in my retina. Not true, apparently. When a photon hits something, it is absorbed. This energy is then radiated back again as light: not the same photons, but new ones. In a way, then, the whole world is glowing. The leaves on the trees are alight. Your eyes are (almost literally) shining like stars.

Such a world is much more alive, more beautiful, it seems to me, than the passive, lumpen world I previously inhabited. Matter is active in revealing its colours and shapes to our eye. I think St Denys would have appreciated this. The way light is passed on by matter is an analogy for the way spiritual illumination is passed down his hierarchy of angels: each angel makes the light his own, and illuminates the others by his own gift. It also echoes the pattern of the Trinity, the supreme three-dimensional act of love. In love each gives to the other, each receives from the other. When we see anything, it is because a photon has been received, and a photon given. Light is gift.

Tuesday, 4 August 2009

Reconnect with nature

The word “ecology” was coined only as recently as 1873, by the German zoologist Ernst Haeckel. He based it on the Greek word oikos meaning “house, dwelling place, habitation” (plus, of course, logos). But though the scientific study of ecology, referring to the complex inter-relationships of biological entities with each other and with their environment, is a modern development, the traditional worldview has a great deal to say on the matter. The medievals did not possess posters showing the fragile earth floating in a dark sea of space, but they were deeply aware of the inter-relatedness of the natural world, and of man as the focus or nexus of that world, which they expressed in the doctrine of correspondences. It was, of course, more poetic than scientific in its formulation, but it expressed a profound insight that remains valid, and the present ecological crisis could only have developed in a world that has forgotten it, or forgotten to live by it. The cosmological principles underlying the quadrivium, even today, can help us learn to dwell more wisely in our common home.

James S. Taylor’s Poetic Knowledge: The Recovery of Education encapsulates the wisdom of the Integrated Humanities Program taught at the University of Kansas in the 1970s and 80s. Taylor writes, for example:

“The implications for education, especially for learning in the poetic mode, as a result of the Cartesian revolution should not be difficult to trace since modern education is dominated, in one way or another, by its influence. Any school of philosophy that claims that one particular science (mathematics and scientific method in the case of Descartes and his followers) must be applied to all other subjects of knowledge, will impose a formal rigor upon the entire curriculum, eliminating even the contemplative nature of mathematics and science. Sooner or later, it is all reduced to ‘facts.’ This approach bypasses the contemplative nature of knowledge, leaving the student disconnected from his nature and the nature out there.”
It is not Taylor’s conviction – and it is certainly not that of Beauty for Truth's Sake – that we can or should do without “rigor” in science or in education; just that we must keep things in right proportion. Contemplative, or poetic, or intuitive ways of knowing can give profound insights into reality. Any education that fails to integrate such knowledge is liable (as most of our schools demonstrate) to produce men and women alienated from their own full humanity and from nature itself.

Monday, 3 August 2009

Sacred geometry

The vesica piscis is one of the most important figures in sacred geometry, and symbolically speaking an image of the source of life. It may best be described as a lens shape formed by intersection of two circles with the same radius, intersecting in such a way that the center of each circle lies on the circumference of the other. The name in Latin means the bladder of the fish. The shape is also called, somewhat more poetically, mandorla ("almond" in Italian).

The Mandorla halo, often seen in religious art enclosing Christ or his Mother, is also often taken to represent the womb, and to have been associated with the power of giving birth, since it was used to generate many other numbers and forms, beginning with three and four, the triangle and the square. In Christian art, the area where the two circles overlap may be taken to represent the Incarnation, the beginning of the union of divine and human natures.

The figure was the subject of intense mystical speculation among the Pythagoreans and their successors. The mathematical ratio of its width to its height is the square root of 3, the ratio 265:153 being the best possible approximation to this square root using small whole numbers. As a result the number 153 was sometimes called “the number of the fish”. It is probably no coincidence that the number 153 also appears in the Gospel of John (21:11) as the number of fish Jesus caused to be caught miraculously after his resurrection. The writer of the Gospel would hardly have recorded the number if it had not been believed to be highly significant, and its occurrence is suggestive of a Pythagorean or Platonist influence.

But why as a Catholic do I speak of “sacred geometry”, a term so associated with the New Age movement? I believe that, as the Pope says in The Spirit of the Liturgy, the “mathematics of the universe… comes from the Logos, in whom, so to speak, the archetypes of the world’s order are contained.” And I agree with Michael S. Schneider that children miss out if they are exposed to numbers merely as quantities instead of qualities, each with a distinct character, connected with all others in endlessly fascinating patterns. Such patterns can awaken us to a world of wonder and beauty, and point to the underlying harmony between science, art, and religion.

As you might see from the Comments, Charlotte Ostermann and others are adapting some of these ideas for Catholic homseschoolers. If you are interested to find out more, add a comment of your own, or get in touch directly. In the sidebar you'll find a section where I plan to collect useful ideas for what to do with younger kids to get them interested in all this - not just in geometry, but in the whole process of learning and the development of a vision of how everything connects together.