Notes, Quotes, Links, and More ~ Part 5

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!

Why a Negative Times a Negative is a Positive: Why negative number products are defined in the way they are.

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…)

Sneak Peek ~ Ilex’s Senior Photo Session (Take 1)

Let’s just say she makes a photographer’s job very, very easy. (Can you believe she is a senior already?!! Do you remember these photos? I took them six years ago!!)

Notes, Quotes, Links, and More ~ Part 4

Part 1 (Basic Overview: Classical Education and Mathematics)

Part 2 (Day 1 Notes: Cosmos!)

Part 3 (Fibonacci)

 

Day 2: Playing with Cosmos (Poetry)

Octoproblem by Kenn Nesbitt (poem at link)

What is the grammar one must know to get the joke? (Grammar students: math facts “pi,” Latin vocab “Octo,” Latin declensions (plural second declension)).

[Because words matter, I discovered that the plural form of octopus is actually octopuses (or occasionally octopodes). Octopus is not a simple Latin word of the second declension, but a Latinized form of the Greek word oktopous, and its 'correct' plural would logically be octopodes. Interesting, no?]

Reviewing Cosmos:

A cosmos is an orderly or harmonious system. The word derives from the Greek term κόσμος (kosmos), literally meaning "order" or "ornament" and metaphorically "world", and is diametrically opposed to the concept of chaos.

Order and Ornament
Truth and Grace
Mathematics and Language

Know form. Add beauty.

Array: put in order and then deck out!

“Structure—a ‘grammar’ that orders every part in its appropriate place—is important not only for the physical sciences, but for every kind of intellectual endeavor. It allows us to do more than weave a fancy from the bits and pieces of our private experience. We can, by the power of structure, weave a whole artistic universe.” Anthony Esolen, Ten Ways to Destroy the Imagination of Your Child

Let’s play with Cosmos!

We are going to try combining language and math today. (I was trying to figure out a way to tie in sentence diagramming (ha!!), but I couldn’t make it work. [After I mentioned this, a parent at the practicum shared a link to her son’s blog wherein he creates a sentence diagram of mathematical notation. It’s fantastic. I’m inspired to try one of my own—on a much, much lower level…] I really wanted to do personality types, but poetry spoke to me.

“After all, science, like poetry, begins with a search for unifying principles, and the unifying factor in creation is its relation to God.” (Stratford Caldecott, Beauty for Truth's Sake, page 29)

“What I want to suggest is that the opposition between the “cultures” of science and the arts can be overcome by teaching science and mathematics themselves at least partly according to the poetic mode. In other words, the best way to teach them is by first awakening the poetic imagination. We need to reestablish—for the sake of science as much as for the arts—a truly humane education that, in Taylor’s words, “begins with the senses, and the discovery and cultivation of harmony and beauty in the soul by way of the sense’s natural affinity for the harmonious, proportionate, and the beautiful in nature and the arts.” If children were from an early age exposed to a “musical” training in the Greek sense, if their poetic sensibility was kindled by training in the observation of nature and the learning of poetry, and if mathematics and science were taught historically, with due attention to the symbolic and beautiful properties of numbers and shapes, then we might even begin to see the birth of that “regenerate science” that Lewis prophesied.” (Beauty for Truth’s Sake, page 45)

“Additionally, because of the nature of poetry, poets are often compelled to stretch our vocabulary, utilizing words and expressions in uniquely sophisticated—but almost always correct—language patterns.” (Andrew Pudewa, 1 Myth, 2 Truths)

“When a carpenter creates, there is a sense in which he destroys the original in order to create something new. When he makes a table, he has to first destroy the tree. The author, on the other hand, does not destroy Hamlet in order to create Falstaff. This is the closest we experience creation out of nothing. Sayers is echoing the teachings of the church fathers who taught that in creating something orderly and beautiful that did not previously exist, the artist is paralleling what God did in the act of creation.” (Imago Dei and the Redemptive Power of Fantasy—Part 1 by Angelina Stanford @ Circe Institute)

Math communicates a lot of meaning through an economy of symbols—like poetry.

For example:

((12 + 144 + 20 + (3 * 4^(1/2))) /7) + (5 * 11) = 9^2 + 0

Some of you, through natural talent and/or practice over time have developed a set of math “eyes” and can immediately see beautiful harmony in this equation. For some of you, this is a fascinating puzzle you are itching to solve. For some of you, this equation strikes your heart with dread.

What you might not see is poetry, but it’s there. Look closely, and take a moment to let it sink in.

“A Dozen, a Gross and a Score,
plus three times the square root of four,
divided by seven,
plus five times eleven,
equals nine squared and not a bit more.”

(From Discover Magazine, attributed to John Saxon)

What grammar do you have to know to understand this poem (get the joke)? Numeracy, notation (^caret * /), operations & order of operations (which we are covering shortly), “dozen, gross, and score,” that any number to the half power = the square root of the number (I didn’t know that), the poetry FORM. Does anyone know what this specific poetry form is called?

Defining/history: A limerick is a short, humorous, often vulgar or nonsense poem. The form can be found in England as of the early years of the 18th century. It was popularized by Edward Lear in the 19th century, although he did not use the term. Even Shakespeare did in fact write limericks which can be found in two of his greatest plays - Othello and King Lear.

FORM :

A Dozen/, a Gross, and/ a Score,
Plus three times/ the square root/ of four
Divided/ by seven
Plus five times/ eleven
Equals nine squared/ and not a/ bit more.

1 stanza (like a paragraph) of 5 lines (counting).
AABBA rhyme scheme (pattern).
Lines 1, 2, 5 have three feet (like measures in music) with three syllables (or beats) each. Lines 3, 4 have two feet with 3 beats each (multiplication).
Usually anapest (ta-ta-TUM), but sometimes amphibrachic (ta-TUM-ta) (rhythm).
(This poem has a silent beat at the end of lines 1, 2, and 5—like a rest in music. And “equals” is squished into one beat.)

Considered easy to compose, historically limericks have been used by the “working class.” Not necessarily a sophisticated form of beauty, but at least there is room for creativity and enjoyment. A chance to play with form.

ETA: I thought I’d give a stab at diagramming that one. What do you think? Would you diagram it differently?

I’ll give you one more:

‘Tis a favorite project of mine
A new value of pi to assign.
I would fix it at three
For it’s simpler, you see,
Than three point one four one five nine.

Let’s try something a little different. Let’s add in some Fibonacci.

The Fibonacci sequence is named after Leonardo Fibonacci. His 1202 book Liber Abaci introduced the sequence to Western European mathematics. It is a number pattern found in nature—such as in branching in trees, the arrangement of leaves on a stem, the fruit sprouts of a pineapple, the flowering of artichoke, an uncurling fern and the arrangement of a pine cone. It also has many practical applications—the Fibonacci sequence is also the foundation of how apparel is sized (called "grading") and it’s used in knitting. There is so much more to say about it, but for now I’ll just tell you that the sequence starts with the numbers 0 and 1. Then every subsequent number is the sum of the previous two numbers. (White board!) So you have 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, and so on.

Gregory K. Pincus, a screenwriter and aspiring children's book author in Los Angeles, wrote a post on his GottaBook blog inviting readers to write "Fibs," six-line poems that used a mathematical progression known as the Fibonacci sequence to dictate the number of syllables in each line.

Your mission, should you choose to accept it, is to write a Fib and share it on the Day 3 (Rhetoric) post coming up!!

Mathematics

 

(When is 4 half of 9? Draw a horizontal line through the middle of IX.)

Operations

What is the definition of a noun? A noun names a person, place, thing, activity, or idea.

Numbers are the nouns of math. Numerals name the idea of numbers.

Operations are verbs.

We are doing something with numbers. Action.

Equations are like linking verbs. They make an assertion. This thing IS this thing. It is the idea of equality. (Harmony, not discord!)

Let’s talk about the vocabulary of operations. We cannot have a conversation without words that MEAN something. Use the correct vocabulary when you are talking about math with your kids! They will pick it up effortlessly, just as they did “ball” or “apple.” If you need a refresher, Understanding Mathematics: From Counting to Calculus is a great place to start. Often (when there are gaps in our own education) we as adults have to start at the grammar stage. And it will take more than reading the definition for these words to become part of our natural vocabulary. We have to use them in conversation. Over and over and over and over again. Repetition. Duration. (Commit to teaching these concepts to a group of adults and BAM! You’ll have intensity. Ha!)

The PURPOSE of the grammar stage, laying these foundations, is so that students have the tools they need to function in the next stage. This isn’t a parlor trick. Or entertainment when the family gets together at Christmas. We are not trying to torture our children with needless repetition. (Piano students who learn their scales to the point of muscle memory have a huge advantage when learning complex pieces of music. The scales are not the end! Basketball drills are not the end. They are a means to a higher purpose—the dialectic and rhetoric process.) It will be laborious if not impossible for our kids to have conversations about math—more complex math—when they get to Challenge if they have to learn the grammar simultaneously. And parents who become tutors—as we transition to a more rhetorical model of math in the Challenge seminar—you will not be able to facilitate discussion if you do not have the vocabulary!

When kids are having trouble with a problem and ask for help, start by asking them to define their numbers and operations. What is this? What is this asking you to do?

“Every math problem provides a micro-example for practicing the skills of learning. The students demonstrate that they have mastered the math terms used (grammar) and that they understand the rules and strategy of the problem so that they can solve the problem (dialectic). Finally, they explain how they solved the problem rhetorically, demonstrating that they understand the algorithm.” (Leigh Bortins, The Core, page 134)

I’m preaching to myself, because this is NOT something that is natural for me! But if I can learn this grammar to present at the practicum, you can learn it to teach your children. And we get to exercise our brains!! And learn more about the nature of God!

Let’s travel back in time to first grade. + - = x or * (asterisk or dot for multiplication) There are 4! ways to denote division including fractions and ratios such as 6:3. (Symbols are “operators”)

Everyone okay so far?

Addend: a number that is added to another in forming a sum.
Sum: The answer to an addition problem

Minuend: a number from which another is subtracted (the number to be diminished or made smaller; musicians think “diminuendo”)
Subtrahend: a number that is subtracted from another (sub = under like submarine)
Difference: The answer in a subtraction problem

Multiplication: The repeated addition of a certain number.
Factors: Numbers being multiplied
Product: The answer to any multiplication problem.

Dividend: The number that is being divided.
Divisor: The number that is doing the dividing.
Numerator: The number above the division sign (a).
Denominator: The number below the division sign (b).
Quotient: The answer of a division operation.

How are we doing?

How about > and < or >? “Does not equal” symbol.

Subtraction is just like addition, but you move backwards on the timeline.

Division is the opposite of multiplication, right?

Exponents (two ways to write). 4^ (caret) 2 = 4x4
Exponents: Just as multiplication is repeated addition of a number, exponents are a shortcut notation when there is a need to multiply the same number together many times. It is just a specific form of multiplication.
Base: The number in an exponent notation to be multiplied.

Roots (symbol) or radicals. Square root of 16 (What number multiplied by itself equals 16?)
Opposite of exponents. Just a specific form of division.

Order of operations: PE MD AS (Please Excuse My Dear Aunt Sally) Parentheses, Exponents. Multiplication and Division (from left to right). Addition and Subtraction (from left to right).

Just as we were challenged on day 1 to express a numerical value in different ways (fraction, decimal, percent, scientific notation—even Roman numerals, tally marks, dots on dice, pictures, etc.), now we can express a single numerical value using the form of operations. As the complexity increases, so does the creativity. We have more ways to express the same value!

12/2
2+1+3
1 x 6
(3.0 x 10^0) + (3.0 x 10^0)
127 - 121
The square root of 36 or 36^(1/2)
1 6/6 + 2 12/6
800% - 200%
3^2 - 3^1

Playing Board Slam is an entertaining way to become comfortable with manipulating numbers and operations.

Write the numbers 1-36 (in rows of 6) on a white board or a piece of paper. (You can go up to 100 or higher if you have math dice with numbers up to 12.)

Roll three dice and write the numbers on the board. Players are challenged to use all three numbers once each, in any order, with any operations (or order of operations), to make up a numerical value. They state the value (and how they got it), and that number is crossed off the board. Players come up with as many numerical expressions as possible. The goal of the game is to cross off as many numbers as possible with one roll of the dice, or cross off all numbers with as few rolls of the dice as possible.

This game can be used at any level. Start with addition and subtraction and work your way up. Adults and older students find the game challenging with exponents (must use one of the given numbers as an exponent, or agree to use a “free” zero), square roots, factorials (4! = 4 x 3 x 2 x 1). Try it for family game night!

Ready to move beyond base 10?

Although the concepts are more simplistic (only halving and doubling), I find Ethiopian math takes me more time to solve.

Here’s a method for more speed:

Mt. Hope Academy @ The Live & Learn Studio ~ July 2013

Food For Thought

Here is a buffet for you—all sorts of good things to digest. Take your time.

::  I Wonder: Work, Art, and the Deeper Meaning of Life by Somer Salomon @ Transpositions:

“Wonder, however, acknowledges first that we exist in a created reality bigger than ourselves.  In this way, Pieper maintained that the proper beginning for both the arts and philosophy is an openness to this created reality - and the highest form of understanding is then received as a gift from the Creator.[3]  Pieper writes that art and philosophy “cannot be accomplished except with an attitude of receptive openness and attentive silence - which, indeed, is the exact opposite of the worker’s attitude of concentrated exertion.”

::  Artists: Don’t Just Work; Be at Leisure! by Somer Salomon @ Transpositions:

Instead, Pieper maintains that the arts are most definitely rooted in leisure. Pieper insists that we have forgotten the true meaning of leisure, from which springs richness, fullness of life, existential meaning, and happiness.  Leisure is not idleness or even relaxation (both of which Pieper ironically says are other forms of work).  Instead, leisure is the openness to the given world, an attitude of considering the things before us in a celebratory spirit.  Ultimately, Pieper maintains, leisure is rooted in the idea of festival!  Festival is humanity’s chance to rejoice in our being and offer thanks for our lives; it is the joyful homage we bring to the Creator for the harmony of his world and our place in it.

:: “If we knew that all our students wished to be corporate executives, would we train them to be good readers of memos, quarterly reports, and stock quotations, and not bother their heads with poetry, science, and history? I think not. Everyone who thinks, thinks not. Specialized competence can only come through a more generalized competence, which is to say that economic utility is a by-product of a good education. Any education that is mainly about economic utility is far too limited to be useful, and, in any case, so diminishes the world that it mocks one's humanity. At the very least, it diminishes the idea of what a good learner is."
-Neil Postman, The End of Education

::  Beyond Newspaper Chewing: Why it Matters What is Read in High School (Part I of II) by Russell Kirk @ Crisis Magazine (Incredible article. Click on the link to read it in full. Here is a small taste.):

“Genuine relevance in literature, on the contrary, is relatedness to what Eliot described as “the permanent things”: to the splendor and tragedy of the human condition, to constant moral insights, to the spectacle of human history, to love of community and country, to the achievements of right reason. Such a literary relevance confers upon the rising generation a sense of what it is to be fully human, and a knowledge of what great men and women of imagination have imparted to our civilization over the centuries. Let us be relevant in our teaching of literature, by all means—but relevant to the genuine ends of the literary discipline, not relevant merely to what will be thoroughly irrelevant tomorrow.”

:: Beyond Newspaper Chewing: Why it Matters What is Read in High School (Part II of II) (A proposed “humane letters” book list for high school students. I enjoyed this nugget.):

“Fiction is truer than fact: I mean that in great fiction we obtain the dis­tilled judgments of writers of remarkable perceptions—views of human nature and society which we could get, if unaided by books, only at the end of life, if then.”

::  On Teaching From a State of Rest @ Amongst Lovely Things:

“We homeschooling mothers worry over everything. We worry that we aren't doing a good enough job, that our kids aren't progressing, that they aren't reading as well as they should be. We worry that they won't develop the study skills they need, won't do enough science experiments, won't read enough Great Books, won't get good SAT scores, won't won't wont. We worry about who they are hanging out with and how well they are eating and if they are getting enough sleep. We worry that the curriculum we've chosen isn't a good fit, that it isn't challenging enough, or maybe that it's too challenging. We worry that we aren't teaching Latin and everybody else is teaching Latin, that our child is two math books behind the other homeschooled kids we know, that we haven't done enough poetry or nature study or grammar.”

::  A Game-Changing Education Book from England by E.D. Hirsch, Jr. @ Core Knowledge (Read the excellent preview of her book at Amazon using the “Look inside” feature: Seven Myths about Education):

“The author gives evidence from her own experience of the ways in which potentially effective teachers have been made ineffective because they are dutifully following the ideas instilled in them by their training institutes. These colleges of education have not only perpetuated wrong ideas about skills and knowledge, but in their scorn for “mere facts” have also deprived these potentially good teachers of the knowledge they need to be effective teachers of subject matter. Teachers who are only moderately talented teacher can be highly effective if they follow sound teaching principles and a sound curriculum within a school environment where knowledge builds cumulatively from year to year.”

::  The End of the Matter! @ Odoro Amoris (Clearly coming from a place of mother-anguish.)

::  Nine Throw-Away Ideas With Which to Think by Andrew Kern @ CiRCE Institute

“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.”

::  English Metrical Law by Stratford Caldecott @ The Imaginative Conservative:

“[W]hat good prose has in common with good poetry is music, “harmonious numbers,” and specifically rhythm. (Flaubert is famously said to have worked out a rhythm for the final pages of Madame Bovary before coming up with the words.) Rhythm or metre is a mathematical structure, a structure of repetition and variation. It creates a shape in time, a dynamic flowing movement that carries the mind along with it. If prose lacks rhythm, it leaves us behind. Our attention is too easily diverted from the direction the author intends us to move.”

And

“[H]e also argues that the best poetry does not follow the rules tamely and as if mechanically, but will convey feeling by constant little tensions with the underlying structure, little departures from the standard pattern. (The same is true in music. It must constantly surprise us in little ways; which it can only do if the form to which it basically conforms creates a framework of expectation.) Thus “there seems to be a perpetual conflict between the law of the verse and freedom of the language, and each is incessantly, though insignificantly, violated for the purpose of giving effect to the other.”

::  The Three Rules of Work by Lisa Bailey @ Classical Conversations:

“However, when I read that Albert Einstein once posited that there were three rules of work, I had to pause to consider his position. As I pondered, I realized that the genius was right and that these rules for work might well be rules for homeschooling, too. The rules are elementary and surprisingly easy to remember: out of clutter find simplicity, from discord find harmony, and in the middle of difficulty lies opportunity.”

::  5 things I hate about homeschooling (& how I deal with them) by Jamie Martin @ Simple Homeschool

“Fear of not enough time keeps many from considering home education. After all, how could someone possibly keep a clean home, educate multiple kids, work part-time, have a strong marriage, and stay sane at the same time?”

::  Coming Soon to Theaters near You! Saxon Advanced Math by Jennifer Greenholt @ Classical Conversations (so funny)

“Our Mother Tongue would be a heartwarming production co-written by Nicholas Sparks and Alex Kendrick, and starring Rachel McAdams and Kirk Cameron. The Smiths are just your average American family: Dad (a newspaper editor who works a bit too hard), Mom (a high school English teacher who worries about the future of her profession), and their children (tech-savvy teenagers who could not care less about the written word). Then, a devastating accident brought on by texting while driving threatens to destroy the family. Enter an inspiring therapist (Sean Connery), who uses sentence diagramming to restore the son’s fine motor and cognitive skills. As the Smith family works through classes of words and kinds of sentences, they rediscover their love for one other as well as the power of the English language.”

::  My Summer with Percy Jackson by Kathy Sheppard @ Classical Conversations:

“The fake “little g” gods of Percy Jackson provide a springboard to discussion concerning our God. Riordan does a wonderful job representing the “little g” gods in the way that the ancients portrayed them. They are always intervening and scheming, just like in the Iliad, the Odyssey, and the Aeneid. They are anthropomorphic (i.e., they act exactly like humans) and have human imperfections such as greed, lying, selfishness, and extreme hatred. They are bumbling, always fighting, and have no foresight into the future of people. They were created instead of being a creator.”

:: 6 Homeschooling Misconceptions Erased @ Simple Homeschool (excellent article!):

“Like any other parent, I’m driven to provide my children with the essential ingredients that lead to lifelong happiness and success. Late at night, unable to sleep, I’ve entertained my share of doubts. What if homeschooling will limit their chances? I finally realized I was looking at it from too narrow a perspective.”

:: From Brave Writer on Facebook. (If you haven’t liked her page, do it. Her thoughts and posts are intelligent and encouraging.):

"We are the lucky ones. We have work, love in our lives, and the personalities to not settle. We look for ways to grow, improve, and become people who make contributions that last and matter.

"We don't give in to the status quo. We aren't content to do 'what we've always done' just because we've always done it.

"We aren't afraid to take risks, even though sometimes we feel afraid. We aren't worried that others 'won't understand' so much so, that we stay in situations that aren't healthy or good for us, or our loved ones.

"We are willing to try hard and harder, and we are willing to rest when we need a rest. We make changes when they need to be made.

"We have friendships and optimism and hope for the future and our good health. We are the lucky ones."

::  Stop Penalizing Boys for Not Being Able to Sit Still at School @ The Atlantic:

“A study released last year in the Journal of Human Resources confirms my suspicions. It seems that behavior plays a significant role in teachers' grading practices, and consequently, boys receive lower grades from their teachers than testing would have predicted. The authors of this study conclude that teacher bias regarding behavior, rather than academic performance, penalizes boys as early as kindergarten. On average, boys receive lower behavioral assessment scores from teachers, and those scores affect teachers' overall perceptions of boys' intelligence and achievement.”

::  Pleasure and Practice in the Literature Classroom @ The Art of Poetry:

“I start every class having students recite either a poem or a prose passage-they do this 3-4 times in a semester.  This gives them a direct intimacy with the language and it gives the listeners a certain relation to the language as well-receiving it as a gift from another human being.  IT also starts our class on the right foot-a liturgy of beginning, if you will, we we are reminding ourself of the importance of the words of these beloved writers, taking them in, making them part of our meanings.”

::  Old-Fashioned Play Builds Serious Skills @ NPR:

“It turns out that all that time spent playing make-believe actually helped children develop a critical cognitive skill called executive function. Executive function has a number of different elements, but a central one is the ability to self-regulate. Kids with good self-regulation are able to control their emotions and behavior, resist impulses, and exert self-control and discipline.”

::  Meaning Is Healthier Than Happiness @ The Atlantic:

“Cole and Fredrickson found that people who are happy but have little to no sense of meaning in their lives — proverbially, simply here for the party — have the same gene expression patterns as people who are responding to and enduring chronic adversity. That is, the bodies of these happy people are preparing them for bacterial threats by activating the pro-inflammatory response. Chronic inflammation is, of course, associated with major illnesses like heart disease and various cancers.”

::  The Right Kind of Happy @ The Economist (same song, different hymnal as my friend Pam says)

“THE Greek founders of philosophy constantly debated how best to live the good life. Some contended that personal pleasure is the key. Others pointed out that serving society and finding purpose is vital. Socrates was in the latter camp, fiercely arguing that an unvirtuous person could not be happy, and that a virtuous person could not fail to be happy.”

::  16 Fancy Literary Techniques Explained By Disney @ BuzzFeed (Actually quite helpful!!)

5. FOIL
Definition: A character who illuminates the qualities of another character by means of contrast.
Example: Gaston’s combination of good looks and terrible personality emphasizes Beast’s tragic situation. The former is a monster trapped inside a man; the latter a man trapped inside a monster.

 

Lists and Lessons

It was a busy month, and I tried my best to keep track of our reading lists at least. (The boys did a little math here and there, as well.)

Language Arts: 
All About Spelling
Penpal Letters

History/Literature/Historical Fiction:
The Story of the World Volume 2: The Middle Ages (Ch 3-10) 
The Kingfisher History Encyclopedia (Levi, assigned pages)
The Usborne Encyclopedia of World History (Luke, assigned pages)
Hostage Lands by Douglas Bond (historical fiction, Roman Britain 3rd century A.D., 228 pp, Levi-IR)
Augustine, the Farmers Boy of Tagaste (Augustine of Hippo, 354-430 A.D.)
Against the World: The Odyssey of Athanasius (373 A.D.) 
Beowulf: Dragonslayer retold by Rosemary Sutcliff
Across a Dark and Wild Sea (Ireland in 521 A.D., Columcille, writing books by hand) (library) 
The Book of Kells: An Illustrated Introduction to the Manuscript in Trinity College, Dublin
The Secret of Kells (DVD, Netflix and Amazon streaming)
The Ink Garden of Brother Theophane (historical fiction, monasteries, making colored ink) 
Patrick: Patron Saint of Ireland by Tomie dePaola (library)
The Story of Saint Patrick (400s)
Patrick: Saint of Ireland
Saint Patrick: Pioneer Missionary to Ireland
The Life of Saint Brigid: Abbess of Kildare
Saint Ciaran: The Tale of a Saint of Ireland 
Augustine Came to Kent (historical fiction, Augustine of Canterbury, 597 A.D., 179 pp, Levi-IR) 
Fin M'coul: The Giant of Knockmany Hill (Irish legend, literature) 
Scottish Fairy Tales
The Holy Twins: Benedict and Scholastica (480-547 A.D.)
Who in the World Was The Acrobatic Empress?: The Story of Theodora
How the Monastery Came to Be on the Top of the Mountain (Romanian oral tradition)
Byzantine Empire (Explore Ancient Worlds) (library)
The Byzantine Empire (Exploring the Ancient World) (library)
The Real Santa Claus by Marianna Mayer
One Thousand and One Arabian Nights retold by Geraldine McCaughrean (literature)
The Story of Sinbad the Sailor and Other Tales
Science in Early Islamic Cultures
Empress of China, Wu Ze Tian: Written by Jiang Cheng an ; Illustrated by Xu De Yuan
The Silk Route: 7,000 Miles of History 
Ancient China (See Through History) (library)
Through Time: Beijing (library)
Maples in the Mist: Poems for Children from the Tang Dynasty (library)
Ms. Frizzle's Adventures: Imperial China (Magic School Bus) (library)
We're Riding on a Caravan: An Adventure on the Silk Road (library)
The Crane Wife (Japanese Folktale) (library)
Little Oh (Japanese Folktale) (library)
Cool Melons-Turn to Frogs!: The Life and Poems of Issa (Japan) (library) 
Chieko and the Pine - A Japanese Folktale (DVD) (library)
The Boy Who Drew Cats adapted by Margaret Hodges (Japanese Folktale) (library)
The Pumpkin Runner (Australia) (library)
The Story of Rosy Dock (Australia) (library)
Through Time: London (library)

Literature Study:
Book Detectives literary analysis book club: The Raft
The Golden Ass of Lucius Apuleius (adapted from the Latin original and appropriately retold for younger audiences)

Levi’s Free Reading:
Re-read the Fablehaven series by Brandon Mull (4 books) (library)
A World Without Heroes (Beyonders) (library)
Seeds of Rebellion (Beyonders) (library)
100 Cupboards (100 Cupboards, Bk 1) (library)
Dandelion Fire: Book 2 of the 100 Cupboards (library)
The Chestnut King: Book 3 of the 100 Cupboards (library)
The Dragon's Tooth (Ashtown Burials #1) (library)
The Drowned Vault (Ashtown Burials #2) (library)
(Here is an interesting article about 100 Cupboards and N.D. Wilson at The Rabbit Room. I’ve used many of their book selections for Levi this month.)
Auralia's Colors (The Auralia Thread Series #1) (library)
The Martian Chronicles by Ray Bradbury (library)
Reread first three books in the Temeraire series by Naomi Novik
(Note: If anyone uses Levi’s book list as a reference, Temeraire is not a children’s/young adult series. I am merely listing the books he reads, not wholeheartedly endorsing them for all children/teens.)
Empire of Ivory (Temeraire, Book 4) (library)
Victory of Eagles (Temeraire, Book 5) (library)
Tongues of Serpents (Temeraire, Book 6) (library)
Crucible of Gold (Temeraire, Book 7) (library)
The Bark Of The Bog Owl (The Wilderking Trilogy) (library)
The Miraculous Journey of Edward Tulane (library)
The Outcasts: Brotherband Chronicles, Book 1 by John Flanagan (author of Ranger’s Apprentice series) (library)
The Invaders: Brotherband Chronicles, Book 2 (library)
Pages of History, Volume Two: Blazing New Trails (historical fiction from Veritas Press, 1500s to the present)

Luke’s Free Reading:
100 Cupboards (100 Cupboards, Bk 1) (library)
The Miraculous Journey of Edward Tulane (library)
Pages of History, Volume Two: Blazing New Trails (historical fiction from Veritas Press, 1500s to the present)
The Candy Shop War by Brandon Mull (library)

Leif’s Free Reading:
More progress!!
Pippi Longstocking by Astrid Lindgren
Pippi in the South Seas
Pippi on the Run
Dominic by William Steig
The Miraculous Journey of Edward Tulane (library)
(Diary of a Wimpy Kid, Magic Tree House…

My Reading!:
His Majesty's Dragon (Temeraire, Book 1)
The Picture of Dorian Gray by Oscar Wilde (library)
We Are All Completely Beside Ourselves (library)
Birds by Aristophanes
100 Cupboards by N.D. Wilson (library)
The Duel by Anton Chekhov
The Little Prince by Antoine de Saint Exupery

Miscellaneous Lovely Picture Books:
Non-fiction:
Diogenes' Lantern (The Greek philosopher)
Diogenes (The Greek philosopher…as a dog)
Wise Guy: The Life and Philosophy of Socrates (library)
On a Beam of Light: A Story of Albert Einstein (library)
Fiction:
The Treasure by Uri Shulevitz (library)
Picture books by William Steig

Extras:
A full day of science camp (including shark dissection) (Luke and Levi)
Three days of logic camp using The Fallacy Detective: Thirty-Eight Lessons on How to Recognize Bad Reasoning (Levi-3 days, Luke-1) 
Star-gazing with Great-Grandpa (and time spend with both Great-Grandpa and Great-Grandma)
Weekly+ get-togethers (potluck, volleyball) in the garden with family and friends
Monday evening concerts in the park
A week of VBS for the boys (Levi volunteering)
(Russ out of state for two weeks!)
A day in the mountains, a day on the water
Play dates with friends, birthday party, swim practice
All sorts of good stuff…

Intermission

But probably not nearly as entertaining for all of you as it is for me.

At least you’ll get a little taste of her personality.

Notes, Quotes, Links, and More ~ Part 3

Part 1 (Basic Overview: Classical Education and Mathematics)
Part 2 (Day 1 Notes: Cosmos!)

 

Fibonacci

In the West, the Fibonacci sequence first appears in the book Liber Abaci (1202) by Leonardo of Pisa, known as Fibonacci. Fibonacci considers the growth of an idealized (biologically unrealistic) rabbit population, assuming that:

A newly born pair of rabbits, one male, one female, are put in a field; rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits; rabbits never die and a mating pair always produces one new pair (one male, one female) every month from the second month on. How many pairs will there be in one year?

(The photo above is an example of how we went about looking at the problem.)

At the end of the first month, they mate, but there is still only 1 pair.

At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field.

At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field.

At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs.

At the end of the nth month, the number of pairs of rabbits is equal to the number of new pairs (which is the number of pairs in month n − 2) plus the number of pairs alive last month (n − 1). This is the nth Fibonacci number.

 

The Fibonacci sequence is named after Leonardo Fibonacci. His 1202 book Liber Abaci introduced the sequence to Western European mathematics. It is a number pattern found in nature—such as in branching in trees, the arrangement of leaves on a stem, the fruit sprouts of a pineapple, the flowering of artichoke, an uncurling fern and the arrangement of a pine cone—even galaxies and the curing of waves. It also has many practical applications—the Fibonacci sequence is also the foundation of how apparel is sized (called "grading"), and it’s used in knitting. There is so much more to say about it, but for now I’ll just tell you that the sequence starts with the numbers 0 and 1. Then every subsequent number is the sum of the previous two numbers. So you have 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, and so on.

 

Blockhead: The Life of Fibonacci is a beautiful (and entertaining) picture book introduction to the life of Fibonacci for elementary students. 

“It is a blueprint that describes how living things such as flowers grow in an orderly, harmonious way. The numbers even pop up in works of human imagination—buildings, music, art, and poetry.” Blockhead: The Life of Fibonacci

The Fibonacci sequence is related to the “golden ratio,” or phi. Take any two neighboring Fibonacci numbers. Divide the larger by the smaller and you get roughly 1.618. (Phi is an irrational number, which means that it is a non-terminating, non-repeating decimal, 1.6180339887….) (Phi shows up in art and architecture, body proportions, the heart, DNA, spiral in snail shells, etc…)

 

Notes, Quotes, Links, and More ~ Part 2

In no way can I replicate the richness and depth of the conversations (I am immensely thankful for all of the participation of attendees!) and atmosphere from the two practicums where I spoke, but I pray that—whether you attended the Eugene or Albany (Oregon) practicums, attended a CC practicum in another city, or were unable to attend—these abbreviated notes are helpful or inspiring. (I am sharing my personal speaking notes, so they will not necessarily correspond with other practicums.)

I posted about the purpose of the practicums at this link, the idea of Copiousness at the end of this link, and the main themes of classical education and teaching mathematics within that framework in Part 1 of this series, so we’ll jump straight to Cosmos in this post. (If you are interested in the basics of Classical Conversations, check out this link in which I share details of our own experience.) (I’d share my opening joke, but no one laughed. Ha!)

Day 1

Theme: Cosmos (Words matter.)

In all my reading (in the spirit of copiousness) leading up to the practicum, I continually came across the word cosmos. I had a general idea of what cosmos meant, but as I personally have been on a “words matter” focus lately, I decided to look up the definition.

A cosmos is an orderly or harmonious system. The word derives from the Greek term κόσμος (kosmos), meaning literally "order" or "ornament" and metaphorically "world,” and is diametrically opposed to the concept of chaos.

While we’re at it, let’s look up the definition of ornament: (Merriam-Webster)
2a. something that lends grace or beauty
3: one whose virtues or graces add luster to a place or society

Order. (Form. Structure. Truth.) Ornament. (Beauty. Harmony. Grace. Virtue.)

Order + Beauty (literally) = World (metaphorically)

(We’re really starting at the very beginning, here.)

Genesis 1:1-2 In the beginning God created the heavens and the earth. Now the earth was formless and empty, darkness was over the surface of the deep, and the Spirit of God was hovering over the waters.

Formless. And what did God do? Created form: separated light and darkness, waters and sky, land and seas.

Empty. And once the form established, he filled the place with beauty: plants, stars, birds, sea creatures, animals, man.

Genesis 2:1 Thus the heavens and the earth were completed in all their vast array.

(Words matter!)

Array: verb (used with object):
1. to place in proper or desired order
2. to clothe with garments, especially of an ornamental kind; dress up; deck out.

And, as Leigh Bortins says, that’s how you teach everything to everybody. Figure out what the form is, and then you have all the content in the world to make it creative, beautiful!

Sentence forms
Latin ending forms
Math formulas
The structure of story

You can put in whatever content you wish once you know the form. The content is what makes it unique and interesting.

This includes the form of classical education.

Day 1: Focus on Grammar

Grammar is the first art or tool of the trivium. This is the stage of input. Grammar is not just the study of the structure of language, it is defined as the “science of vocabulary” (Leigh Bortins, The Core: Teaching Your Child the Foundations of Classical Education, page 48). The student (of any age) must first internalize (by exposure and mastery) the vocabulary, definitions, facts, stories, ideas, names, dates, and rules of a subject or skill.

“No matter what your children’s strengths and weaknesses are, or their likes and dislikes, or their gifts and talents—their brains want to gather, sort, and store, and retrieve information.” (The Core, page 52)

“It is not surprising that, for the Greek mind, the Muses—of epic, history, astronomy, music, dance, tragedy, comedy, lyric poetry, and sacred poetry—should be daughters of Memory.” (Anthony Esolen, Ten Ways to Destroy the Imagination of Your Child, page 9)

“One simple and immutable fact about the human brain is that you can’t get something out of it that isn’t there to start with. Supernatural inspiration notwithstanding, human beings in general—and children in particular—really can’t produce... thoughts or concepts that they haven’t first experienced and stored. In other words, we cannot think a thought we don’t have to begin with. Even the most unique, creative, and extraordinary ideas can only exist as a combination and permutation of previously learned bits of information.” (Andrew Pudewa, 1 Myth, 2 Truths)

“There are times when memorization is out of favor in education. Some might say that “rote memorization” is not appropriate as a teaching strategy. “Rote memorization,” however, is loaded language, biased against the discipline and effort required to learn things permanently. There is nothing wrong with challenge. We must remember that the alternative to remembering is forgetting, and when we teach something as important as grammar, that will be needed for one’s entire life, the ban on memorization makes little sense. There are areas of knowledge that should be memorized, and in the past, there was a better term for it: to learn by heart.” (Michael Clay Thompson)

“But more than that, we would desire to bring children into the garden of created being, and thought, and expression. Caldecott reminds us that for the medieval schoolmen, as for Plato, education was essentially musical, an education in the cosmos or lovely order that surrounds us and bears us up. Thus when we teach our youngest children by means of rhymes and songs, we do so not merely because rhymes and songs are actually effective mnemonic devices. We do so because we wish to form their souls by memory: we wish to bring them up as rememberers, as persons, born, as Caldecott points out, in certain localities, among certain people, who bear a certain history, and who claim our love and loyalty.” (Anthony Esolen, author of Ten Ways to Destroy the Imagination of Your Child, in the Foreword from Beauty in the Word: Rethinking the Foundations of Education by Stratford Caldecott)

Mathematics

 

“The sheer amount of information available in every discipline is far too great to be mastered by one person even in a lifetime. 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...understood in terms of number and cosmic harmony.” (Stratford Caldecott, Beauty for Truth's Sake: On the Re-enchantment of Education, page 28)

Michael S. Schneider writes, “Numbers are a map of the beautiful order of the universe, the plan by which the divine Architect transformed undifferentiated Chaos into orderly Cosmos.” (Beauty for Truth’s Sake, page 54)

God spoke in words (His very name “I AM” is a subject and a predicate, the form of language, Exodus 3:14), but he also spoke in math.

“Mathematics is the language of science, but it is also the hidden structure behind art…, and its basis is the invisible Logos of God.” (Beauty for Truth’s Sake, page 30)

Galileo Galilei (1564-1642) said, "The universe cannot be read until we have learned the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth."

“The laws of nature are but the mathematical thoughts of God.” ~Euclid

(Words matter!) The word mathematics comes from the Greek μάθημα (máthēma), which, in the ancient Greek language, means "that which is learnt", "what one gets to know," hence also "study" and "science", and in modern Greek just "lesson."

From Merriam-Webster, the definition of MATHEMATICS: the science of numbers and their operations, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations. (Whew!!)

Another source defined math as the study of relationships using numbers.

Mathematics is the abstract study of quantity, structure, space, change, and many other topics. It has no generally accepted definition.

Let’s talk about a brief history of math.

Why?

“An integrated curriculum must teach subjects, and it must teach the right subjects, 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.” (Beauty for Truth’s Sake, page 28)

From Mathematics: The Science of Patterns by Keith Devlin:

Up until c. 500 B.C. mathematics was indeed the study of number. During the period of Egyptian and Babylonian mathematics, it consisted almost solely of utilitarian arithmetic or counting.

500 B.C. to 300 A.D., the era of Greek mathematics was primarily concerned with measurement and geometry—number and shape. Not just utilitarian, mathematics was regarded as an intellectual pursuit having both aesthetic (beauty!) and religious elements. From the Greeks we have Euclid’s Elements.

In the middle of the seventeenth century, Newton and Leibniz independently invented calculus—essentially the study of motion and change—making it possible to study physics and the motion of planets, among other things. So mathematics became the study of number, shape, motion, change, and space.

In the year 1900, all the world’s mathematical knowledge would have fit into about 80 books. Today it would take maybe 100,000 volumes to contain all known mathematics.

As you can imagine, many quite new branches of mathematics have sprung up.

[Mark Twain said “We could use up two Eternities in learning all that is to be learned about our own world and the thousands of nations that have arisen and flourished and vanished from it. Mathematics alone would occupy me eight million years.”]

Many mathematicians agree that mathematics can be considered the science of patterns. Mathematicians examine abstract patterns using symbolic notation—mathematical language.

The History of Counting is a fantastic picture book introduction to the history of counting across cultures.

The quadrivium consists of arithmetic (pure number), geometry (number in space), music (number in time), and astronomy (number in space and time) (Beauty for Truth’s Sake, page 24)). Our focus is arithmetic.

There are only are 3 (three!) basic things to learn in arithmetic…everything else is just more complex combinations of these three categories:

Numbers (8), operations (6), and laws (4). That’s it!

This is our FORM!

Math in a nutshell: “There are digits, you do things with them, and they follow laws.” Leigh Bortins

So let’s begin with the definitions. You cannot have a conversation about something if you don’t have words to use—not “thingy”! Kids soak up vocabulary. If they can say Tyrannosaurus Rex, they can say denominator. It’s adults who are intimidated!

(Understanding Mathematics: From Counting to Calculus is a great all-in-one resource for adults who are in the process of redeeming their education or as a teaching reference.)

A Number is the IDEA that I have three of something.

A Numeral is the SYMBOL used to express the idea. (For example, we use the heart symbol to express the idea of love.)

We are talking about real numbers today. Real numbers can be written as a fraction (as opposed to irrational #s which do not have a ratio—cannot be written as a fraction—such as pi).

Natural Numbers are the counting numbers. They are positive and exclude 0. 1, 2, 3, 4, 5… and so on.

Whole Numbers are the counting numbers and 0. 0, 1, 2, 3, 4… This is easy to remember if you look at the word Wh0le and think of the o as a 0.

Integers are the whole numbers and their opposites (including 0). …-4, -3, -2, -1, 0, 1, 2, 3, 4…

A number located in between integers on the number line is called a Decimal. Add a decimal point and add numbers to the right to indicate a decimal. This is part of our base 10 system. A natural number can be expressed as a decimal by adding .0

Fractions. A fraction is a number in the form of a over b. the line represents division. A fraction is simply an integer or decimal number, prior to completing the division. The top number is called the numerator. The bottom number is called the denominator. Also called a ratio. (A fraction may also be defined as a ratio of numbers, where a ratio is just the division relationship a/b.)

A proper fraction has a value less than one (the numerator is less than the denominator).

An improper fraction has a value greater than one (the numerator is greater than the denominator) such as 6/1 or six divided by 1 equals 6.

Mixed numbers include an integer and a ratio. Could 6 be expressed as a mixed number in different ways? Yes. The integer of a mixed number will be a 0 if the value is less than one.

Percent. % symbols mean the number was multiplied by 100, so divide by 100 to return to a non-percent number.

Scientific Notation is used when working with very large or very small numbers and is a very specific form. In scientific notation a number is rewritten as a value between 1 and (less than) 10 (a digit in the one’s place) expressed as a decimal, multiplied by a factor of 10.

10 to a positive power means that you move the decimal that many places to the right (used for large numbers). 10 to a negative power means that you move the decimal that many places to the left (used for small numbers).

Anything to the 0 power = 1, so the number three expressed in scientific notation would be 3.0 x 10 to the 0 power or 10^0.

Why? Here is one example of why it works:

I challenge you to express a number in as many different ways as possible. This is a fun activity to do with kids. The boys and I filled a page with symbols expressing the value of the number six, including Roman numerals, percent, tally marks, stick figure people, fraction “pies,” dots on a die, and various numerical expressions (numbers and operations).

“When I was a boy, we had to memorize the multiplication tables in the second grade, up to 12 x 12 = 144. Let’s set aside the fact that it takes a deal of intelligence and some ingenuity to accomplish that task. Forget that you would have to learn that anything multiplied by 5 ends in 5 or 0, alternately. Forget that if you were sharp you’d see that odd times odd is odd, and everything else is even. Forget the patterns showing up among the 2s, 4s, and 8s. Forget the nice progression in the 9s, with the tens digit gaining one and the ones digit dropping it: 09, 18, 27, and so forth. What that memorization did was to free you up to become comfortable with numbers themselves, and with the structure of arithmetic. Once you had done that, you could play with numbers creatively, long before you’d ever suspected the existence of algebra or calculus, with their toboggan curves and their infinite series and their radio waves, their transcendental numbers and the mysterious i, the square root of -1, whose existence we must leave to philosophers to determine.” (Anthony Esolen, Ten Ways to Destroy the Imagination of Your Child, page 20)

And another interesting dive into the meaning of words:

“In English, integrity is a word related to a number of other familiar ones. It is built from the root word, integer. Now, as most of us learned in early math, an integer is a whole number, as in 1,2,3,3- that is, whole numbers as opposed to fractions. They are whole or complete numbers, not parts of a whole.”

Integer
2. A complete entity.

[1500-10; < Latin: untouched, hence, undivided, whole]

In·teg·ri·ty
1. Steadfast adherence to a strict moral or ethical code.
2. The state of being unimpaired; soundness.
3. The quality or condition of being whole or undivided; completeness.

[Middle English integrite, from Old French, from Latin integrits, soundness, from integer, whole, complete]

“In relating the word integrity to our lives, it describes an uncompromised character, an unjaded soul, an unsullied heart, an undivided mind. It requires the maintenance of our hearts in entirety before the Lord. David said: "Unite my heart to fear Your name." Those words say, "God, draw the strands of my heart so firmly tight and in such reverence before Your throne, that I will be kept wholly and entirely aligned with You."