I was asked to lead a local Classical Conversations Essentials Academic Orientation this past month. I have spent three years in Essentials class as a parent and another three years as a tutor (and parent), but this was my first opportunity to lead Tutor Orientation at a CC Practicum.
As I was preparing to lead the orientation and then spending time in discussion with the tutors and directors during the orientation, I was reminded (again) why I love Essentials.
It is the class in which students are beginning to play with Cosmos. They are learning FORM.
I’ve written about some of these ideas before, after speaking at the math practicum and then as I was preparing to tutor Essentials the first year, but I re-organized my notes to correspond with the three elements of an Essentials class: math, English grammar, and writing. I’ll be sharing these thoughts in a 4-part series, beginning with this introduction.
“Cosmos” is the thread that ran through the three days of training and connects all three class elements together.
Let’s begin here.
A cosmos is an orderly, harmonious system or “world.” The word derives from the Greek word “kosmos,” meaning “order” or “ornament.” Cosmos 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
(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.
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In Classical Conversations communities, Essentials students are learning the FORM of three arts.
Math: Learning the Form of Numbers, Operations, Laws
Grammar: Learning the Form of Sentences
Writing: Learning the Form of Paragraphs (Reports, Stories, Essays, and Critiques)
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Orient and Invite
As tutor trainers, tutors, parents, and fellow students, we have the opporunity to “orient and invite.”
Orient Our Tutors, Parents, and Students to Essentials and the Arts of Math, Grammar, and Writing
Review Past Concepts
Introduce New Grammar
Invite Our Tutors, Parents, and Students to the Conversation
Begin Dialectic Discussion in Class
Point to Available Resources
Continue Grammar and Dialectic Discussion at Home
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May I invite you along on my learning journey?
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“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’s Omnibus III Textbook.
[I received a free copy of this book for review purposes; opinions are my own.]
While I do love a charming fiction picture book, I often gravitate toward beautiful biographical picture books or picture books that help explain a variety of ideas in ways that help kids engage with the concepts. My shelves are abundently loaded with picture books about science, art, music, history, geography, bible, and math. I’m always on the lookout for new titles to add to my collection.
Granddaddy Parallelogram is a picture book introduction to parallelograms, rhombuses, rectangles, right angles, and perpendicular diagonals. The illustrations are bright and simple. Geometry was not my strong suit in high school, and I learned a thing or two from this one! The story is silly and appeals to my six year old. I found her reading it in bed, way past her bedtime.
The author invented a clever way to show that rhomuses have perpendicular diagonals which make right angles.
Parents and teachers will appreciate the full page of tips and interactive ideas at the end of the book, including discussion questions, activities, and helpful explanations. The book also includes a page of math vocabulary and definitions.
Reading the book aloud, I found some of the conversation awkward. The main character has an eight syllable name. Paired with all the synonyms for “said” (interjected, suggested, grumbled, asserted, assured, pronounced), this was quite a mouthful. Also, some parents may not appreciate the sibling squabbling or school atmosphere (friends laughing and pointing when the rhombus trips and falls), though kids will likely relate!
I’m looking forward to trying some of the math activity suggestions with Lola.
Math. Honestly, it has never been my favorite subject to teach.
I started out using RightStart Math with Levi. It’s an incredible program, but incredibly teacher-intensive. It was difficult to teach Levi (my distractable non-math kid) with two younger brothers running around getting into mischief. It was even more difficult to consider teaching two boys at different levels. RightStart Math was only going to be great if I actually used it, and it started to sit on my shelf much more often than it was off the shelf in use. [I’m considering pulling it back off the shelf to teach Lola early math, however. We’ll see.]
After much math frustration with Levi and then a long break to regain sanity (around 2nd or 3rd grade), I purchased Teaching Textbooks and ended up using it for all three boys (my two younger boys were advanced in reading and math) for a few years. Honestly, it was a God-send. Math was much more enjoyable for everyone. I loved that the boys could do it independently, that it gave them instant feedback, and that it was self-grading.
Last year, Levi’s first year in the Classical Conversations Challenge program, we switched him to Saxon Math. I can see how Saxon Math is a thorough, rigorous program. But it almost killed us. Even doing only half the problems.
What I really wanted was an interactive, inspiring, engaging, self-teaching (with excellent visual/audio instruction), instant-feedback, self-grading, mastery-based, challenging, attractive, comprehensive math program. Similar to Teaching Textbooks, but better.
I had used Khan Academy occasionally in the past for a video here and there, and I loved Sal Khan’s teaching style. What I hadn’t realized is just how much they’ve added to Khan Academy recently. It is now a complete math program.
So we’ve been using Khan Academy as our main math “spine” since September and I adore it.
It is an online math (and so much more!!) program, and it’s free. Let me repeat that in case you didn’t read it correctly the first time:
It’s FREE.
It blows my mind.
Students can work online on a computer or on mobile devices with the Khan app.
Parents sign up for an account and then their students sign up for their own account under the parent.
Students choose a grade level (K-8th) or a subject (pre-algebra and up through college math). They complete a Mission Warm-Up to assess their current knowledge.
When a student logs in, they can go to their mission page (the grade level or subject they are working through).
This is what Luke’s mission page looks like:
On the left it tells him what percent of the mission (grade level or subject) he has completed. It also tells him which skills he has practiced, which skills he has mastered, and which skills he has yet to complete in each topic. He can click “show all skills” to see all the little boxes, or “hide skill breakdown” to minimize it. He can click on any one of the little squares if he wants to choose his next skill to practice. When he hovers over the square, it will tell him what the skill is and give a preview.
On the right he is given suggested next tasks.
When he clicks on a skill to practice, his screen looks like this (I think this is a screen-shot of a 5th grade skill):
The program is mastery-based. In the upper right-hand corner, students can see exactly how many problems they need to complete correctly and independently to successfully practice the skill. For this particular skill, they must get the first two correct or five in a row if they miss one.
If they need instruction, each problem gives them a direct link to the video with instruction for that particular skill. The video pops up on their screen. They can watch it and then return directly to the practice. If they need help working through the problem, they can click on “show me how.” Each time they click the button, they are shown one step of the problem. (This screen shot shows one hint.) They can watch every problem worked through and explained step by step! If they ask for a hint, that problem does not count as correct. Students then work through the problems until they can get the designated number in a row correct.
Students can use a scratchpad on the screen when needed (with a mouse on the computer or finger with the app), but my boys usually use scratch paper and a pencil. A calculator function pops up on the screen when they are allowed to use it for the skill.
After a student has successfully practiced a skill, they are given a mastery challenge after a specified amount of time has passed (often 16 hours). Previous skills are randomly tested in mastery challenges to determined whether the skill or concept is still mastered. If not, it gets bumped back down to “practiced” status rather than “mastered” status.
The levels are connected and build on each other. Some skills are covered in multiple levels. If a student masters a skill in 4th grade that is also covered in the 5th grade level, it will already show as mastered when they move up a level so they do not have to repeat concepts (unless they show up briefly in mastery challenges).
Students work at their own pace. They work on skills and concepts until they are mastered. They level up as soon as they are ready.
I’m not even touching the surface of the program. Students earn “badges” and avatars. They can see graphs of their activity. You can add “coaches” to their account so other adults can encourage or challenge them.
One of the best aspects of the program is the parent page. Parents have access to detailed, customizable reports for each of their students.
I can see with an easy glance at his activity summary, for any specified period of time (including daily), just how much time my child has actually spent working on Khan, what videos he watched, what skills he practiced, what skills he is struggling with, and more. Or I can click on “full progress report” (below). I can expand or minimize each category.
So here are the cons:
Students have to have internet access. (But they can log in from anywhere at any time!)
Students are not given a specified day’s lesson. I usually give my boys a set amount of time, and I can verify the time they spend and their activity from my parent account. I’ve found this helpful because the boys can work on math even if we have varied amounts of time available depending on the day.
Some kids may struggle with deciding what to do next. They are given suggestions, but they may feel it is too open-ended. Some kids may need more parental direction.
A student must be able to read and follow directions or have parental assistance. (Teaching Textbooks, on the other hand, has a narrator reading the problems aloud, so that is helpful for struggling readers.)
Khan is constantly upgrading and improving the program, as well, so look for more features in the future!
Well, there you have it. I didn’t even mention the computer programing or the science or many other subjects that Khan offers. You’ll have to check it out yourself. [grin]
I’ll end this post with one of Sal Khan’s instructional videos, just to give you a taste.
:: 15 Poverty Habits Parents Teach Their Children @ Rich Habits. This post could generate some interesting conversation. His statistics are fascinating. I think we have to remember that correlation does not imply causation, though, and I think that some of the statistics may be effect rather than cause. It’s important to note that this post does not address systemic concerns surrounding poverty nor should we consider monetary wealth as the single indicator of a rich life. His “success habits,” however, aren’t all directly related to money and could help lead a person to a rich life, regardless of income. Most of them involve taking care of what you do have: physical health, time, relationships, character, and mind.
“Wealthy people do certain things every single day that sets them apart from everyone else in life. Wealthy people have good daily success habits that they learned from their parents.”
“Taking home small souvenirs of the woods is just the beginning of things kids can’t do in nature. In many parks and other public lands, kids are told by rangers, parents, or teachers not to leave the trail, not to climb rocks or trees, not to whack trees with sticks, not to build forts or lean-tos, not to dig holes, not to move rocks from one place to another within the park, not to yell or even talk too loudly. Are we having fun yet?”
On trying to recruit Mindy Kaling, who wound up voicing Disgust:
"I turned around, because I was pitching kind of some visuals on the computer, and she's crying ... she really responded emotionally, and she said, 'Sorry, I just think it's really beautiful that you guys are making a story that tells kids that it's difficult to grow up and it's OK to be sad about it.' We were like, 'Quick! Write that down.' Because that was really what we were trying to say."
Around the World and Close to Home
:: #BringBackOurGirls: Meet Some of the Survivors From the Boko Haram Chibok Kidnapping @ Cosmopolitan [I can’t believe I’m linking to a Cosmo article.] These girls are finishing their high school education at the Christian boarding school in a little town in Oregon that my great grandfather founded. My grandmother taught there for years and years. My Dad (and siblings) attended the school. A couple uncles have taught there, and my Aunt Judi and Uncle Phil are there now. You can see my grandmother’s house behind the girls in the last picture. I spent some beautiful days of my childhood wandering the town and the grounds of the school. I am so happy for these girls that they are able to be there.
“Kids are by nature mean. Smart phones make them meaner. Why? They can’t see the faces and experience the reactions of those they diss. Their “humor” is more cruelly fun than it might otherwise be, because it’s unchastened by empathy. Smart phones work against the emotion that evolutionary psychologists say we need to moderate our selfish struggle for status.
“…And an insightful comedian today reminds us that nobody with eyes to see really believes that kids or the rest of us are getting less mean. These might be the toughest times ever not to be smart and pretty.”
“There are many forms of online shaming: The angry blog, the critical tweet, the vicious comment on Facebook. Whatever the method—people try to hurt people. Sometimes the shaming escalates into a mob, a faux-community that latches on to the negative verdict and piles on. Under the pretense of righteous indignation, the mob licks its chops as it goes about demonizing, diminishing and destroying its target.”
"The Internet has enabled the schoolyard bully to crash a family dinner, the parental tyrant to stalk his child through the school halls, and the school administrator to punish a girl for the things she does when she leaves the campus... Digital villagers are no longer relegated to the sidelines; online, everybody gets a gavel."
A friend asked if I ever just wanted to quit the internet.
I feel like quitting the internet as often as I feel like quitting everything else involving humanity, including parenting. [wink] But I've come to the conclusion that I need to be the best human I can be wherever I am, and that includes the internet. I've seen so much encouragement, intelligence, and kindness on the internet (FB in particular) as well, and I want to contribute to that if I can (even though I am far from perfect).
You have the power to make the world a better, kinder place, friends. Wherever you are--work, school, community, internet--be the best human you can be.
"My father was very charismatic and a very good storyteller but he couldn't invent anything so he would tell me stories about things that had just happened. And these stories would be amazing and there was sometimes violence in them, many extreme things, but at the same time, they were full of love for mankind and even the people who would do those extreme things, you would still understand them and like them. The protagonist in those stories, they would always be prostitutes and mafia guys and drunk people.
“…Those stories, for me, were always the model for the function of stories and storytelling in our lives — the idea is that you kind of look reality straight in the face, it doesn't matter how ugly it is, and you try to find humanity in it, you try to find beauty in it, you try to find hope in it. So you can't beautify it, but at the same time, you should find these tiny things that you know that would make sometimes very violent and unhappy occasions still human and emotional.”
“Joyce’s novel offers a great opportunity to talk about the purpose and nature of literature, as well as the project of the early 20th century modernists. At CenterForLit we believe that all great literature is worth reading, even when we disagree with the worldview of the author. It is through reading opposing viewpoints that we come to have compassion for other worldviews, while being strengthened in our own. And there is always the slightest chance that the author we don’t agree with has noticed something true about the world, which can then be magnified and deepened with real Truth.”
Maps and Globes is one of my most favorite picture books to introduce kids to basic geography (What are maps? What do maps tell us? How is the earth measured? What are the marking on maps?), and now I have a favorite picture book for the features of the earth! In alphabetical order, Geography from A to Z: A Picture Glossary defines more than sixty geographical features and weaves in another 60+ definitions and synonyms. For example, the definition of oasis is given within the definition of desert; the definitions of knob and knoll are included in the definition of hill. The added terms are in bold, so they are easy to spot.
Challenge A students are instructed to define these terms and create their own picture glossary. This is one of the projects that Levi fell behind on, and I wish we had this book at the beginning of the year. My other kids will enjoy reading and learning the terms before they hit Challenge.
We recently enjoyed a couple picture books about small things, Tiny Creatures: The World of Microbes and What's Smaller Than a Pygmy Shrew? so it was fun to go in the opposite direction and think about something big. If… is a book full of visual representations to help kids (and adults) understand mind-boggling big numbers, from our galaxy to a timeline of inventions to water and continents to energy to population. This book makes a variety numbers come to life! [Yes, there are also two pages of the evolutionary history of earth.]
My boys love fairy tales and we have several beautiful picture books of fairy tales by Hans Christian Andersen. I try to find biographical picture books of authors whenever possible, and this picture book by Jane Yolen is lovely. The pictures are soft and muted. I also love The Young Hans Christian Andersen, the story of his childhood (with darling illustrations) by Karen Hesse.
We have borrowed this one from the library before, but it was time to revisit. My boys love Appalachian Spring (because they love the original Shaker hymn, Simple Gifts), so this story is a favorite. The illustrations are wonderful! Even more exciting is that the original ballet can be viewed on YouTube!
[I apologize for the long list of links. I missed last week and only posted two the week before. I had to catch up! Remember, you can always follow me on Facebook—link in the sidebar—if you are interested in reading the links in “real time” as I discover them.]
“A man should hear a little music, read a little poetry, and see a fine picture every day of his life, in order that worldly cares may not obliterate the sense of the beautiful which God has implanted in the human soul.” ~Goethe (HT: Homegrown Learners)
“The truth is that the modern world has committed itself to two totally different and inconsistent conceptions about education. It is always trying to expand the scope of education; and always trying to exclude from it all religion and philosophy. But this is sheer nonsense. You can have an education that teaches atheism because atheism is true, and it can be, from its own point of view, a complete education. But you cannot have an education claiming to teach all truth, and then refusing to discuss whether atheism is true.” ~Chesterton
Literature and the arts in general create pathways to discovering personal vision—to imagine a world that values one’s creativity. Imagination informs innovation.
“In this course, you are going to encounter words and images and ideas that are going to change your life. We’ve got Hamlet and Lear, Achilles and David, Frederick Douglass and Elizabeth Bennett, Augustine’s pears and Van Gogh’s stars—beauty and sublimity and truth. If you miss them, you will not be the person you could be.”
My boys have enjoyed N.D. Wilson’s children’s books, and I loved his Notes From The Tilt-A-Whirl: Wide-Eyed Wonder in God's Spoken World in which he expresses many of the same ideas as he does in this video. I’ll have to disagree with his assessment of Harry Potter, though. And that sends me on a new tangent:
Stunned by his insight, I returned to cleaning the kitchen and wondered how my life would be different if I walked confidently in my knowledge of The Ending.
But listen: we’re free to enjoy the good and the beautiful, even from the most unlikely places. We’re free—and this is huge—to look for the light in people (and things!), to give them the benefit of the doubt, to laud their beauty, to outlove unloveliness—in short, to love as Christ loves us. That includes billionaire authors like J.K. Rowling. She didn’t grow up in the Bible Belt of America; she grew up in England. And yet, in defiance of a culture that tends to snub its nose at Christianity, she wrote a story that contains powerful redemptive themes, stirs a longing for life after death, piques the staunchest atheist’s suspicion that there just might be something beyond the veil, and plainly shows evil for what it is—and not just evil, but love’s triumph over it.
Many fairy tales offer hope — hope of redemption, hope that good can conquer evil, hope that our enemies will be vanquished. G.K. Chesterton said it best, “Fairy tales do not tell children the dragons exist. Children already know that dragons exist. Fairy tales tell children the dragons can be killed.”
Music cannot be separated from time nor can the timing of music be thought of as something entirely constructed by man. Because of music’s physical and temporal character, music reminds us that time belongs to the very framework of God’s creation.
If we wish to produce young men and women who are capable of thinking like Galileo and Kepler, advancing the boundaries of mathematics and science, they must be educated in all areas of study, including music as well as arithmetic, and be able to integrate the disciplines together.
There is now a battle raging over the scientific method itself, particularly between those engaged in cosmology and those pursuing the study of fundamental physics.
“A powerful and personally developed structuring of information — an active and selective memory — is as necessary for scientists as it is for poets.” [John-Steiner]
But perhaps the most potent use of memory in the creative mind is the cross-pollination of accumulated ideas and the fusing together of seemingly unrelated concepts into novel configurations — something Stephen Jay Gould, arguably the greatest science essayist of all time, captured when he said that his sole talent is “making connections.” John-Steiner quotes a similar sentiment by the Polish-born mathematician Stan Ulam:
“It seems to me that good memory — at least for mathematicians and physicists — forms a large part of their talent. And what we call talent or perhaps genius itself depends to a large extent on the ability to use one’s memory properly to find analogies, past, present and future, which [are] essential to the development of new ideas.”
“The fact of the matter is that we are all at different stages of this journey towards the true, the good, and the beautiful. I have yet to make it to Augustine and Pieper — I’m still reading Lewis, Esolen, and Caldecott and that’s okay, because I am reading and learning. My journey has begun. We all bring different backgrounds, different expectations, and different educations. We also bring different experiences of classical education.”
“Good stories involve conflict, which is just a nice word for pain. People don’t become heroes without sacrifice, and as creatures of comfort, this is the last thing we want to endure.”
“What’s a discussion about language doing in Ask The Headhunter? Poor spelling, incorrect grammar, lousy writing and poor oral presentation are all signs of illiteracy. I don’t care what field you work in, how much you earn, or whether you’re a production worker or a vice president. The way you use language reveals who you are, how you think, and how you work. And that will affect your career profoundly. You can pretend otherwise, but you can also walk around buck-naked believing you’re invisible because you’ve got your eyes closed."
"Movies and TV shows built to transfer particular abstract ideas wind up fitting the story to the ideas, instead of letting the story and characters breathe and live like real people, who are messy and inconsistent and confusing. Like you. ...Like me...
"Humans actually are pretty good at figuring out if someone is telling them a story in order to talk us into believing they're right. We hate it. But we also like seeing the results of our ideologies played out on screen in ways that are favorable to us."
“All I know for certain is that, if a limitless God can call something good and sit down and rest and enjoy his work, who are we to battle long past the end of our strength or obsess over trivialities or hover anxiously over what ought to be released and laid aside?”
“Baseball is a game. And math, for kids, is a game, too. Everything for them is a game. That’s the great thing about being a kid. In Little League, you play hard and you play to win, but it doesn’t actually matter who wins. And good coaches get this. They don’t get mad and they don’t throw you off the team. They don’t tell you that you stink at baseball, even if you do — they tell you what you need to do to get better, which everybody can do.”
If your brain doesn't have to work as hard on simple maths, it has more working memory free to process the teacher's brand-new lesson on more complex math…'So learning your addition and multiplication tables and having them in rote memory helps.'
After reading Fractals in Frozen? @ Running with Team Hogan, I had been wanting to learn a little more about fractals—just a little, mind you, as I’m still in the “grammar” stage of such complex mathematical and scientific ideas (not my forte) and I find myself easily intimidated.
Today, while at the library, I stumbled across a beautiful new picture book by Sarah and Richard Campbell. It caught my eye immediately because I remembered enjoying their book about Fibonacci Numbers, Growing Patterns: Fibonacci Numbers in Nature. Their most recent book, Mysterious Patterns: Finding Fractals in Nature, is just as lovely. The photography is beautiful, and both are great (simple) introductions to the mathematical and scientific patterns for younger children (and their parents who did not learn such things in school).
I read it in the truck while waiting for Levi at swim practice, and my fingers were itching to do some doodling when we arrived home. Mom doodling with markers at the kitchen table is apparently an irresistible sight to young boys, so Leif joined me. He excitedly read the book and started in on his own fractal trees. And then Levi and Luke wanted to know all about fractals…
Math, be not proud. Thou art mean and base. Thou hath no royal luster in thy eyes. Give me those who art tired of thy blusters and brags. Send these to me. Math, thou shalt die. Thou shalt die a death so profound that none shall remember thee, or revive thee. Thy death shall be cause of rejoicin’. All the school masters shall be merry for math was a subject none would learn. The schoolboy would no longer creep like a snail, now he would run faster than a cheetah. A cheetah would wonder why he had been so challenged. One king will decree that addition symbols will be fed to his falcons. Ah, these simple musings do no good. I must be done, gentle listeners, for even papers have ears.
Part 1 (Basic Overview: Classical Education and Mathematics) Part 2 (Day 1 Notes: Cosmos! Memory. Numbers.) Part 3 (Fibonacci) Part 4 (Day 2 Notes: Playing With Cosmos (Poetry). Operations.) Part 5 (Day 3 Notes: Worship. Attention. Rhetoric. Laws.)
(I’ve been sharing my speaking notes from the local Classical Conversations parent practicum, and I think this about wraps it up!)
The History of Counting is an excellent picture book introduction to the history of counting across many cultures. I learned a great deal from this simple resource!
The Phantom Tollbooth is a fantastically witty and hilarious romp through an imaginative world filled with words and numbers. This chapter book is worth reading whether you are 8 or 80.
Carry On, Mr. Bowditch is a deeply inspirational work of historical fiction based on the life of a self-educated, eighteenth-century nautical and mathematical wonder.
Little Britches: Father and I Were Ranchers is the first book in an autobiographical series by Ralph Moody. While not a book about mathematics, Moody’s childhood is filled with the practical application of a strong education. Excellent.
Half-a-Hundred Acre Wood (An impressive resource for anyone involved in Classical Conversations, a plethora of links, lists, planners, ideas, and more—all free.)
Khan Academy (One of the best free resources on the internet. Do. Not. Miss. Math, Science and Economics, and Humanities for all ages.)
:: It’s All About Value! by Kate Deddens @ Classical Conversations (phenomenal, lengthy article—go read it!):
"Mysterious though it may be, and precisely because mathematics does seem to delve down into the bare essences of things, whatever is inessential is removed. Lovely as they can be, all the distractions and bunny trails of other forms of expression (such as rhetorical devices in writing or flourishes in art and music) that seek to enhance reality are eliminated. Indeed, the best mathematical proofs are the ones which come to the concluding point in the fewest steps, using the most appropriate laws and principles; the same is true for proofs in Formal Logic. This could even be argued to be true in many practical areas, such as in cooking; the best cakes are those in which the ingredients come together well in perfect accord, with no extraneous, distracting flavors or textures. Anything that does not speak precisely to the end-goal—even if it may have intrinsic value and even, in fact, add value—is unnecessary to the unadulterated task at hand."
“As we have seen, the “Liberal” Arts are precisely not “Servile” Arts that can be justified in terms of their immediate practical purpose. “The ‘liberality’ or ‘freedom’ of the Liberal Arts consist in their not being disposable for purposes, that they do not need to be legitimated by a social function, by being ‘work.’” …At the heart of any culture worthy of the name is not work but leisure, schole in Greek, a word that lies at the root of the English word “school.” At its highest, leisure is contemplation. It is an activity that is its own justification, the pure expression of what it is to be human. It is what we do. The “purpose” of the quadrivium was to prepare us to contemplate God in an ordered fashion, to take delight in the source of all truth, beauty, and goodness…"
"Education is simply the soul of a society as it passes from one generation to another." G.K. Chesterton
“Classical education encourages us that we are capable of becoming an Oxford don who builds bicycles, or a plumber who reads Milton, or a business owner who spouts theology. The classically educated are not defined by their occupation so much as by their breadth of knowledge and understanding.”
And page 61:
“We need to offer children a broad, freeing education that allows them to think well and to be lifelong learners. Children need to be prepared for any challenge, even for job opportunities that may not exist until well into the future.”
“The only people who achieve much are those who want knowledge so badly that they seek it while the conditions are still unfavourable. Favourable conditions never come.” ~C.S. Lewis quoted in The Core, p 6
What is a student?
“To make the content of the curriculum relevant to the everyday life of the pupil, it is essential not to shrink the content to match the pupil’s present experience, but to expand the life of the pupil to match the proposed curriculum.” Beauty in the Word: Rethinking the Foundations of Education by Stratford Caldecott, page 35
“We do not know what or how to teach children, because we do not know what a child is, and we do not know what a child is, because we do not know what man is—and Him from whom and for whom man is. How decisive for…any educator of good will, is the revelation that man is made in the image and likeness of the three-Personed God? That is like asking what difference it will make to us if we keep in mind that a human being is made not for the processing of data, but for wisdom; not for the utilitarian satisfaction of appetite, but for love; not for the domination of nature, but for participation in it; not for the autonomy of an isolated self, but for communion.” Anthony Esolen in the foreword to Beauty in the Word (Stratford Caldecott)
Subjects tell us more about God and are connected with one another.
“To all of us who hold the Christian belief that God is truth, anything that is true is a fact about God, and mathematics is a branch of theology.” Hilda Phoebe Hudson, English mathematician in the early 1900s Integration of subjects.
“Do you often come across people for whom, all their lives, a ‘subject’ remains a ‘subject,’ divided by watertight bulkheads from all other ‘subjects,’ so that they experience very great difficulty in making an immediate mental connection between let us say, algebra and detective fiction, sewage disposal and the price of salmon--or, more generally, between such spheres of knowledge as philosophy and economics, or chemistry and art?” Dorothy Sayers, from “The Lost Tools of Learning,” an essay presented at Oxford in 1947
“I am only pointing out that every education teaches a philosophy; if not by dogma then by suggestion, by implication, by atmosphere. Every part of that education has a connection with every other part. If it does not all combine to convey some general view of life, it is not an education at all." -G.K. Chesterton
“Music, architecture, astronomy, and physics—the physical arts and their applications—demonstrate the fundamental intuition behind the Liberal Arts tradition of education, which is that the world is an ordered whole, a “cosmos,” whose beauty becomes more apparent the more carefully and deeply we study it. By preparing ourselves in this way to contemplate the higher mysteries of philosophy and theology, we become more alive, more fully human. This beautiful order can be studied at every level and in every context, from the patterns made by cloud formation or river erosion to that of the leaves around the stem of the most obnoxious weed, from the shape of the human face as it catches the light, or the way keys are ordered in a concerto by Bach, to the collision of stellar nebulae and particles in an atomic furnace.” (Beauty in the Word, pages 116-117)
Susan Wise Bauer writes: “[T]o the classical mind, all knowledge is interrelated. Astronomy, for example, isn’t studied in isolation; it’s learned along with the history of scientific discovery, which leads into the church’s relationship to science and from there to the intricacies of medieval church history. The reading of the Odyssey allows the student to consider Greek history, the nature of heroism, the development of the epic, and humankind’s understanding of the divine.”
Marva Collins states in Marva Collins' Way: “I taught my students how to add and subtract, but I also taught them that arithmetic is a Greek word meaning to count and that numbers were called digits after the Latin word digitus, meaning finger, because people used to count on their fingers. I taught them about Pythagoras, who believed that mathematics made a pupil perfect and ready to meet the gods. I told them what Socrates said about straight thinking leading to straight living.”
And Parker J. Palmer on teaching well (HT: Mental multivitamin): “Good teachers possess a capacity for connectedness. They are able to weave a complex web of connections among themselves, their subjects, and their students so that students can learn to weave a world for themselves. The methods used by these weavers vary widely: Socratic dialogues, laboratory experiments, collaborative problem solving, creative chaos. The connections made by good teachers are not in their methods but in their hearts -- meaning heart in its ancient sense, as the place where intellect and emotion and spirit and will converge in the human self.”
From Cultural Literacy: What Every American Needs To Know by E. D. Hirsch, Jr.: “In the technological age, Washington and the cherry tree, Scrooge and Christmas, the fights historical, the oceans geographical, the "beings animalculus," and all the other shared materials of literate culture have become more, not less, important. The more computers we have, the more we need shared fairy tales, Greek myths, historical images, and so on. That is not really the paradox it seems to be. The more specialized and technical our civilization becomes, the harder it is for nonspecialists to participate in the decisions that deeply affect their lives. If we do not achieve a literate society, the technicians, with their arcane specialties, will not be able to communicate with us nor we with them. That would contradict the basic principles of democracy and must not be allowed to happen.”
“An education worthy of the name would develop an awareness of the totality through art and literature, music, mathematics, physics, biology, and history. Each subject has its own autonomy, but at its heart it connects with every other.” Beauty for Truth’s Sake, page 31
Tools or Arts of Learning
Dorothy Sayers: “For the tools of learning are the same, in any and every subject; and the person who knows how to use them will, at any age, get the mastery of a new subject in half the time and with a quarter of the effort expended by the person who has not the tools at his command. To learn six subjects without remembering how they were learnt does nothing to ease the approach to a seventh; to have learnt and remembered the art of learning makes the approach to every subject an open door. “
“The key for me was to discover that the three elements of the Trivium link us directly with three basic dimensions of our humanity. No wonder they are so fundamental in classical education! ...To become fully human we need to discover who ...we are (Memory), to engage in a continual search for truth (Thought), and to communicate with others (Speech)." ~Stratford Caldecott, about Beauty in the Word
‘According to Hugh of Saint Victor [during the Middle Ages], “Grammar is the knowledge of how to speak without error; dialectic is clear-sighted argument which separates the true from the false; rhetoric is the discipline of persuading to every suitable thing.” Quoted in Beauty in the Word
Verses
Ephesians 3:17b-19 “And I pray that you, being rooted and established in love, may have power, together with all the saints, to grasp how wide and long and high and deep is the love of Christ, and to know this love that surpasses knowledge—that you may be filled to the measure of all the fullness of God.
(Roots! Math! Abundance!)
Col 1:19 “We…desire that you might be filled with the knowledge of his will in all wisdom and spiritual understanding.”
Prov 24:3-4 By wisdom a house is built, and through understanding it is established; through knowledge its rooms are filled with rare and beautiful treasures.
(Form! Beauty!)
Hebrews 1:3a The Son is the radiance of God’s glory and the exact representation of his being, sustaining all things by his powerful word.
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.
“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.”
"[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."
“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.”
“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!
“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)
(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…)
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.
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.”
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.
