Part 1 (Basic Overview: Classical Education and Mathematics)
Part 2 (Day 1 Notes: Cosmos!)
Part 3 (Fibonacci)
Part 4 (Day 2 Notes: Playing With Cosmos (Poetry))
Day 3: Worship. Attention. Prayer.
“Pay attention. Be astonished. Tell about it.” ~Mary Oliver
“The reason you study math, science and art is so that your imagination will be filled with wonder and awe at the Creator of the most mind blowing project ever: the world. And whether you are learning to read music or playing an instrument, whether your hand is holding a pencil or gesturing in the theater, you are training yourself for the warfare of worship. You are teaching your body gratitude; you are teaching your soul thanksgiving. There is hardly an adequate evaluation of your progress, but the best grade you can receive is the outworking of a thankful heart. If you have truly learned Algebra, if you have mastered the story of Western Civilization, if you can tell me the names of the constellations that whirl about our heads, then you will do it with laughter in your voice, you will do it with joy in your heart and gratitude in your bones. Worship is the point of learning because worship is the point of life.” Toby Sumpter, in response to the questions ‘Why are you in school? Why are you reading this page? Why are you reading Mein Kampf?’ This is an excerpt from Veritas Press’ Omnibus III Textbook. Read the whole link; it’s excellent.
Attend (from Latin Attendere: to bend toward)
:: Lectio divina: paying attention by Katherine Pershey @ Simple Mom
“One of my favorite definitions of prayer is that it is the practice of paying attention. Not merely that you must pay attention while you’re praying, but that prayer itself is the act of attending: to God, but also to the beauty – and ugliness – before us.”
:: Stratford Caldecott, Beauty in the Word: Rethinking the Foundations of Education (from the passage titled ‘Attention’):
"[T]he important thing, the real goal of study, is the 'development of attention.' Why? Because prayer consists of attention, and all worldly study is really a stretching of the soul towards prayer. 'Never in any case whatever is a genuine effort of the attention wasted. It always has its effect on the spiritual plane and in consequence on the lower one of the intelligence...'"
"Attention is desire; it is the desire for light, for truth, for understanding, for possession. It follows, according to Weil, that the intelligence 'grows and bears fruit in joy,' and that the promise or anticipation of joy is what arouses the effort of attention: it is what makes students of us."
:: Anthony Esolen, Ten Ways to Destroy the Imagination of Your Child:
“The sky suggests the vastness of creation and the smallness of man’s ambition. It startles us out of our dreams of vanity, it silences our pride, it stills the lust to get and spend. It is more dangerous for a human soul to fall into than for a human body to fall out of…
A child that has been blared at and distracted all his life will never be able to do the brave nothing of beholding the sky. He will not be able to ask, with the Psalmist,
‘When I consider the heavens, the work of thy fingers, the moon and the stars, which thou hast ordained; What is man, that thou art mindful of him? And the son of man, that thou visitest him?’”
:: From the Educational Plan of St. Jerome Classical School in Washington DC, as quoted in Beauty in the Word, page 98:
“Religion is not just one subject within the curriculum, but the key to its unity and integration. The cosmos is an ordered, unified whole because it is created in Christ—‘in whom all things hold together’ (Col. 1:17). Belief in God as our Father and the world as His beautiful and rational creation binds faith and reason, nature and culture, art and science, morality and reality in to a coherent and integrated unity. This unified view reaches its summit in worship, which is the highest form of knowledge and thus the end and goal of true education.”
:: Stratford Caldecott, Beauty for Truth's Sake: On the Re-enchantment of Education (pages 129-130):
“Liturgy therefore starts with remembrance. We do not make ourselves from nothing. To be here at all is a gift, and a gift (even if we are at times only obscurely aware of the Giver) evokes a natural desire to give something back to someone. We have only what we have received, but included in that gift is the capacity to transform what we now possess into something that is truly our own. Furthermore, the more grateful we are, and the more conscious of the greatness of the One, the source who gave us existence, the more beautiful we will try to make the gift. That is partly why liturgy has always inspired art. As I once heard an art historian say, “The fine arts were born on the altar.”
:: Nine Throw-Away Ideas With Which to Think by Andrew Kern @ CiRCE Institute (Go read the whole post! I love the idea that questions are really gaps in form that students strive to fill.):
“Because truth is musical, we encounter a sixth wonder: form enables us to discover truth better than analysis or induction. In no way is this meant to dismiss analysis or induction. Rather, it is to restore them to their exalted place: to test our hypotheses, which are always deduced from formal leaps.
But truth is formal. And when we learn to think musically, we learn to anticipate gaps in the form and what might fill them. Some examples:
The asteroid belt was believed to be where it was long before it was discovered because a mathematical formula had predicted a planet at that distance from the sun. There was a dissonance in the music, a gap in the calculations, and the asteroid belt filled it.
When we listen to a song or composition, the composer creates a tension by creating a gap in the form that our very soul strives to fill. When he brings about the resolution, we feel joy. The same thing happens on a math equation.
A poet will adopt a form and find that he needs more content to fill in a verse. This will generate ideas that would not otherwise have been discovered…
…But formality, (that is) love of harmony, enables anticipations that analysis misses.”
Rhetoric. “Bear fruit in wisdom.”
After the input of information and experience (grammar) and the processing (dialectic—asking how and why questions, finding relationships, comparing and contrasting, and using analytical subjects such as algebra and formal logic), we arrive at the stage of original output (rhetoric—speaking, writing, creating, integrating, performing, teaching).
“Rhetoric is the art of expression. During the rhetoric stage the student learns to express himself or herself with fluency, grace, elegance, and persuasiveness.” ~Susan Wise Bauer
“Wisdom is the ability to make judgments.” ~Andrew Kern
We talked about poetry as cosmos on day 2, and participants were encouraged on day 3 to share the Fibonacci poems they had written as an expression of rhetoric. As the poems were being shared, truth became manifest—that gaps in form move a person to fill the space with beauty or creativity that had not previously existed.
I was given permission to share a few from practicum.
From Mindy Pickens:
God
Me
Journey
Heart in hand
The time that is trod
Brings my soul to humbly applaud
The Creator, King, Artist, Source who had it all planned.
Sperm
Egg
Baby
A person
Uniquely ablaze
Under the constant gaze of God
Whose Love chose to die, to save each of them, you and I.
Pop
Star
Bieber
Annoying
Your pants are too low
Baby, baby, baby oooooooh, like baby, baby
You thought she'd always be around, but you are a girl.
And from Sarah Owens:
Clothes
There
Always
Piled high
Will it ever end
Evidence of little blessings
On those days we all have had let us remember this.
If you would like to share a Fibonacci poem, please feel free to add it in the comments!
[Another example of using form to create beauty that didn’t previously exist: The Simplest Periodic Table We’ve Ever Seen @ Popsci (lovely!!)]
Quadrivium. Laws. Music.
“Math teaches you to see what other people see. It teaches you to see what another author has written down. When we read, we don’t see the words ‘a’ or ‘the.’ Math makes you stop and say, I have to see the decimal, I have to see the exponent. Math is just good practice for being a human being who sees the world. Just think how an artist can see color difference, shapes, colors. Our kids should see a math formula better. If someone would just show them. It is the same as artistic endeavors. If you can see the numbers, if you can see the operations, if you can see the laws, it will all change your ability to see complex ideas.” ~Leigh Bortins
“But mathematics is the sister, as well as the servant, of the arts and is touched with the same madness and genius.” ~Harold Marston Morse
“The principles of number and space are imbedded in created reality, the way the universe works and the way we think. It is the beauty and power of this reality that should be the primary motivation for studying and understanding mathematics, but in most cases it is not. Since utilitarianism governs most of math instruction (K-12), there is a tendency to focus on dictating rules without the requisite understanding, but it is in understanding why a principle works that a student is (1) introduced to the beauty of mathematics and (2) learns to master its unique symbolic language. And, in understanding the laws of mathematics, one becomes comfortable in the world of God’s making and how man has developed it. We don’t trump utility with beauty because both go together. They are two sides of the same coin. Mathematics is a unique tool of wonder.” James D. Nickel, author of Mathematics: Is God Silent?
“By concentration on what, and leaving out why, mathematics is reduced to an empty shell. The art is not in the “truth” but in the explanation, the argument. It is the argument itself which gives the truth its context, and determines what is really being said and meant. Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity—to pose their own problems, make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs—you deny them mathematics itself. So no, I’m not complaining about the presence of facts and formulas in our mathematics classes, I’m complaining about the lack of mathematics in our mathematics classes.” (From A Mathematician’s Lament by Paul Lockhart, page 5)
We talked about the four laws that Foundations students memorize: Commutative, Associative, Identity, and Distributive.
(I’ve linked the Khan Academy videos for each law. If you are not familiar with the free online resource of Khan Academy, you need to be. I cannot recommend it highly enough!)
I had an epiphany when attending the Salem practicum where two friends of mine spoke. The afternoon math lesson on day 2 (complex fractions from Saxon 8/7, in the practicum handout) was a complete revelation. I remember being taught that in order to divide fractions, we multiply the dividend by the reciprocal of the divisor. But I had no idea why. Turns out, it’s because of the identity law. (Which corresponded perfectly with the video of Leigh and Lisa for day 2, below.)
Two things:
1. Short cuts, the faster ways to solve problems, are important—but only after students understand why they work. We do students no favors by focusing on speed and ease at the expense of true understanding. Mathematics is not the art of “git ‘er done.”
2. I remember thinking (I’m ashamed to admit), “Why on earth do these students memorize the identity law? It’s so obvious and ridiculously simple.” It turns out that the principle is easy to state, but becomes much more complex in practice. Students need to have the basic idea so deeply internalized that they are able to see it and use it as they progress through to much more advanced mathematics.
Another person asked (on day 2) about why a negative times a negative equals a positive. Turns out, it’s because of the distributive law!
For a visual/kinesthetic explanation for younger kids, I found (at MathForum.org) a great teaching tool. Imagine yourself standing on a number line. If your first factor is negative, face toward the negative numbers on the number line. If your second factor is negative, walk backwards (towards the positive numbers).
And a joke for you:
What did the Zero say to the Eight?
Nice belt!
“In music we glimpse the grammar of creation itself, from the harmony of the planetary and subatomic spheres to the octaves of human experience and the cycles of growth in plants and animals. Modern writers as varied as Schopenhauer and Tolkien have seen the world as a kind of ‘embodied music,’ and of course the notion is ubiquitous among the ancients. Music in turn is a play of mathematics, coherent patterns of number and shape in time and space, expressed in rhythm and timbre, tone and pitch. It is the closes most of us get to seeing and feeling the beauty of mathematics.” (Stratford Caldecott, Beauty in the Word, pages 57-58)
“Music is the pleasure the human mind experiences from counting without being aware that it is counting.” ~Gottfried Leibniz
“Music is a secret arithmetical exercise, and the person who indulges in it does not realize that he is manipulating numbers.” ~Gottfried Wilhelm Leibniz
“Notes that are in whole-number ratios to each other sound good together. These rations can be displayed visually by an instrument called a harmonograph, in which each vibration is conveyed by pendulum to a pen and paper. Harmonic or resonant patterns can also be displayed on a plate covered in sand that is made to vibrate at certain frequencies by being connected to a sound system. Either way, sounds made by notes that harmonize together turn out to be visually, as well as audibly, beautiful: (followed by image).” (Stratford Caldecott, Beauty for Truth’s Sake, page 92)
(Very soon after I read that passage, a friend shared the following video. I love synchronicity!)
(I have one more post coming up with general and various quotes, verses, links, and resources…)
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