A Portrait of the Storyteller as a Young Child

When Eldest Daughter was four years old, we visited an artist friend of mine who had set up a table where the children could paint while we moms talked.

E.D. sat down at the table, selected a brush and some bright pink paint, and painted a girl.  As she painted, she talked about what the girl was doing, and changed colors to paint a hill and grass, and then the sky and sun, and the girl's house.  When the girl in the picture started picking flowers, E.D. not only painted in the flowers, she painted the girl kneeling to pick them.  Next, the girl took her flowers home, so she was painted traveling across the picture to the house.  As the girl entered, its interior was painted too.  E.D. spent at least half an hour telling everything the girl did and every place she went, and by the time the story was done, the paper was a muddy mess.

It was fascinating.

Math and Philosophy

Don't you love it when seemingly unrelated things come together?

A few years ago, I heard (or read, I don't remember which) Andrew Kern say that teachers ought to read The Meno, one of Plato's dialogues, regularly in order to become a better teacher.  I printed it out and tried reading it, but seriously.  It was heavy going and I didn't get very far.

When I noticed that Coursera was offering a class called Reason and Persuasion: Thinking Through Three Dialogues by Plato I immediately checked to see which three dialogues, and sure enough, The Meno was one of them, so I signed up.  The class started last month and it's been a huge help.  We spent two weeks on Euthyphro, and now I'm near the end of our second week of Meno.

Before starting this class, I had no idea that math would come up, but it plays an important role in The Meno, which you'll already know if you've ever read it.  The dialogue discusses the questions What is virtue? and Is virtue the sort of thing that can be taught?  But right smack in the middle of this dialogue is a geometry lesson.

While discussing this, the Coursera instructor mentions an essay by Bertrand Russell, "The Study of Mathematics," and quotes from it:

One of the chief ends served by mathematics, when rightly taught, is to awaken the learner's belief in reason, his confidence in the truth of what has been demonstrated, and in the value of demonstration. This purpose is not served by existing instruction; but it is easy to see ways in which it might be served. At present, in what concerns arithmetic, the boy or girl is given a set of rules, which present themselves as neither true nor false, but as merely the will of the teacher, the way in which, for some unfathomable reason, the teacher prefers to have the game played.

I don't know about you, but once I got to fourth grade math where they were actually teaching me new stuff (everything before that was pretty much just me getting to show off what I'd already figured out on my own in the normal course of childhood) this is exactly what math felt like to me -- arbitrary rules.

Russell goes to to critique my tenth-grade geometry class.  Do you remember geometry?  In that class they drive you insane by making your prove things that don't need to be proven -- you can tell just by looking at them.  Russell says this is a huge mistake:

[T]he learner should be allowed at first to assume the truth of everything obvious, and should be instructed in the demonstrations of theorems which are at once startling and easily verifiable by actual drawing . . . . In this way belief is generated; it is seen that reasoning may lead to startling conclusions, which nevertheless the facts will verify; and thus the instinctive distrust of whatever is abstract or rational is gradually overcome. *

This is what happens during the geometry lesson in The Meno.  Socrates calls over a slave boy, who has had no instruction in geometry, and draws a square that's two feet long on each side, and asks the boy what the square's area is.  The boy says that it is four square feet.

Next Socrates asks the boy what he would do in order to make a square whose area is twice this one.  The boy says that he would double the length of each side of the square.  They draw that out, but of course that turns out to make a square that's sixteen square feet, so the boy tries again -- three feet on a side, which still makes too big a square.

Of course, you probably know that the square root of 8 is not a rational number -- it's not something that can be expressed as a fraction.  It's less than 3, but it's more than 2 3/4.  In Socrates' time they didn't have a way of expressing that number since they weren't using the decimal system, so how in the world can this poor slave boy come up with the right answer?

Turns out it can be done and Socrates helps the boy figure out how to do it.  It's pretty clever, and I'll leave it to your imagination.**

The next passage quoted in my class was this one:

What is best in mathematics deserves not merely to be learnt as a task, but to be assimilated as a part of daily thought, and brought again and again before the mind with ever-renewed encouragement. Real life is, to most men, a long second-best, a perpetual compromise between the ideal and the possible; but the world of pure reason knows no compromise, no practical limitations, no barrier to the creative activity embodying in splendid edifices the passionate aspiration after the perfect from which all great work springs. Remote from human passions, remote even from the pitiful facts of nature, the generations have gradually created an ordered cosmos, where pure thought can dwell as in its natural home, and where one, at least, of our nobler impulses can escape from the dreary exile of the actual world.

A lot of my reading the last couple of years about what math is has been hinting at this kind of thing.  Math is ennobling.  This was so exciting that I decided to find and read the entire essay.  Even if I hadn't had this introduction Russell would have won me over in his first paragraph:

In regard to every form of human activity it is necessary that the question should be asked from time to time, What is its purpose and ideal? In what way does it contribute to the beauty of human existence? As respects those pursuits which contribute only remotely, by providing the mechanism of life, it is well to be reminded that not the mere fact of living is to be desired, but the art of living in the contemplation of great things. [Emphasis added]

If you've read the John Gould Fletcher quote in my sidebar you'll know why that phrase resonates with me.

Our imaginations are meant to live in a kind of temple, and originally it was this idea that gave shape to education, or what we would call a Classical Education.  But in recent generations why we're teaching what we're teaching has been forgotten, or, as Russell puts it:

Dry pedants possess themselves of the privilege of instilling this knowledge: they forget that it is to serve but as a key to open the doors of the temple; though they spend their lives on the steps leading up to those sacred doors, they turn their backs upon the temple so resolutely that its very existence is forgotten, and the eager youth, who would press forward to be initiated to its domes and arches, is bidden to turn back and count the steps.

I haven't finished The Meno yet, so I've only got an inkling of what geometry has to do with virtue, but dear Lord, please save me from being a dry pedant!

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*Here, Russell is discussing teaching geometry with Euclid.  Euclid starts off with a long list of definitions and postulates, which is a fine way to organize the information, but a lousy way to teach it -- not that I know what the best way is.  I'm just reporting what I've learned so far.  :-p

**Hint:  It has to do with triangles.

The Year of Math

I've mentioned before that math is my weakest point, but I thought I'd say it again just in case you missed it.  ;-)

In January I noticed that several bloggers were choosing a word of the year, and after thinking about it I realized that I really need to focus on math, so MATH is my word for 2014. 

The first thing I've done is to begin collecting and rereading the books and articles that I've found helpful, beginning with the sections on math in Harvey and Laurie Bluedorn's book Teaching the Trivium, especially the appendix on the history of teaching math.  What surprised me most was how it explained my own experience with math as a student.  But that's another story.

I'm also trying to organize all the teaching ideas I've found in the last couple of years.  It's all scattered -- internet links, articles I've printed out, books, notes . . .  It's a mess, and since I'm very much an out-of-sight-out-of-mind person, I need to put the information I want to remember in one place.

I've been thinking about how to do math with my youngest two for a few years now, but it all feels so scattershot.  Writing it all down will force me think through and mentally organize everything I've been reading over the last few years, so this series (if it materializes at all!) will be me thinking out loud and writing some about what we're doing, not me telling you what you ought to do.

While I think some more, I'll leave you with this article, which you really must read if you're having the kind of trouble I am.  I first read it three years ago, and let me tell you, it really made me angry.  It's what finally spurred me to start doing real math with my children instead of relying on textbook math. 

It's a 25-page long essay, but you'll want to print it out and read and reread it.

A Mathematician's Lament 

Lent

by George Herbert

Welcome deare feast of Lent: who loves not thee,
He loves not Temperance, or Authoritie,
                But is compos’d of passion.
The Scriptures bid us fast; the Church sayes, now:
Give to thy Mother, what thou wouldst allow
                To ev’ry Corporation.

The humble soul compos’d of love and fear
Begins at home, and layes the burden there,
                When doctrines disagree.
He sayes, in things which use hath justly got,
I am a scandall to the Church, and not
                The Church is so to me.

True Christians should be glad of an occasion
To use their temperance, seeking no evasion,
                When good is seasonable;
Unlesse Authoritie, which should increase
The obligation in us, make it lesse,
                And Power it self disable.

Besides the cleannesse of sweet abstinence,
Quick thoughts and motions at a small expense,
                A face not fearing light:
Whereas in fulnesse there are sluttish fumes,
Sowre exhalations, and dishonest rheumes,
                Revenging the delight.

Then those same pendant profits, which the spring
And Easter intimate, enlarge the thing,
                And goodnesse of the deed.
Neither ought other mens abuse of Lent
Spoil the good use; lest by that argument
                We forfeit all our Creed.

It ’s true, we cannot reach Christ’s fortieth day;
Yet to go part of that religious way,
                Is better than to rest:
We cannot reach our Savior’s purity;
Yet are bid, Be holy ev’n as he.
                In both let ’s do our best.

Who goeth in the way which Christ hath gone,
Is much more sure to meet with him, than one
                That travelleth by-ways:
Perhaps my God, though he be far before,
May turn, and take me by the hand, and more
                May strengthen my decays.

Yet Lord instruct us to improve our fast
By starving sin and taking such repast
                As may our faults control:
That ev’ry man may revel at his door,
Not in his parlor; banqueting the poor,
                And among those his soul.

“What do you DO with your kids if they don’t learn to read before age 10 or so?” Part 2

(Here is Part 1:  What I did with my older kids, and the post that sparked this series, Different children learn to read at different ages, and that's okay.)

I'm sorry it took me so long to get to part 2 of this series.  I pretty much hibernate in the winter, and my brain is just starting to shake off the winter fuzziness.  Has it actually been five months!?!  Golly, I'm sorry!

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Part 2: What I'm doing with my younger children

Last time I wrote about these basic elements that made up my older children's school year:

• plenty of outside play time
• Nature Study
• Bible, history, and literature
• talking about what we're reading
• math
• classical music
• hymns

I'll run through them in the same order so you can see how things have changed over the years.

Probably the biggest change involves their time spent out of doors and Nature Study.  In 2005, my husband retired from the military and got a job in rural Virginia.  We bought a house on a little over three mostly-wooded acres.  Over the years we've accumulated goats, chickens, geese, ducks, guinea fowl, cats, and a dog, so my younger children have outside chores that my older children never had -- we never even had a pet while Mike was active duty (except for a tankful of fish for a few years) because of moving so often, and having babies regularly, and my husband being out of town for a week or more at a time on occasion, including one time when he was gone for a year, and I just didn't want the added responsibility of looking after an animal.

Having the animals has been a good thing in many ways, besides the obvious benefit of having fresh, raw goat milk, fresh eggs, and meat we've raised ourselves.  The biggest benefit is giving meaningful work to my second son, who is nineteen years old and profoundly delayed in many ways.  He'll never be able to read, to live on his own, to drive, or to do most of the things that an adult needs to do in order to fit into our society.  I doubt he'd be able to hold down any kind of a job, and anyway I wouldn't really want him working away from home unless it were with a family member since his ability to communicate is so limited -- if he were ever mistreated he'd have a hard time letting us know about it, and he wouldn't be able to defend himself.

Second Son has faithfully milked the goats every day for two or three years.  We bred the does last fall and they're expecting in another few weeks and we dried them off (that is, quit milking them) last month, so he's getting a break right now, but he loves this work.  He also collects the eggs each day, makes sure all the animals have food and water, and just generally keeps an eye on them.  If they get out of the pasture the other children have to round them up for him because he doesn't have the speed or dexterity to handle them when they're on the loose.  Mike and two of the other younger children keep the goats' hooves trimmed, and doctor them if they need it, but Second Son is the one who looks after the animals' daily needs.

Whenever we butcher animals (which hasn't been very often this past year) most of the children help their daddy with the processing, and he's good about letting them do a lot of the work themselves, and teaching them the names of the organs and pointing out other interesting things.

Hopefully, learning to pay attention to the animals and anticipate their various needs will help them grow into the righteous man of Proverbs 12:10, who "regardeth the life of his beast."

I still occasionally ask them to describe something they've seen, and we've been in the habit of watching the song birds that visit the birdbath and the feeders ever since Ambleside Year 1.  We've even added Nature notebooks, though checking them just now I see that they've only got one entry each.  One August day I put a pretty little bird's nest that my youngest daughter had found that morning on a dish and sat it in the middle of the table and had them sketch it.  A pleasant activity, and I don't know why we haven't done anything more like that.  Always room for improvement.  :-D

Bible, history, and literature have been very much influenced by our use of AmblesideOnline.  I used Years 1 through 3, and by then had gotten over the rough patch I was going through with my health, so I'm mostly branching out on my own, while bringing with me what I learned from AO and still using some of the suggestions.

Our Bible readings are included in our Morning Prayers, which I've blogged about here and here.  In a nutshell, we start off with the greeting and response:  "The Lord be with you."  "And with thy spirit." "Let us pray."  This is followed by a selection from Psalm 51 ("Open my lips, O Lord / And my mouth shall proclaim your praise. . ." ).  Then we read the passages for the day.  Some years we follow the lectionary, but lately we've being going straight through the Psalms, reading a passage from Luke, and a chapter of Galatians.  Then we say the Apostles' Creed, the Lord's Prayer, and finish with a benediction, followed by "Go in peace to love and serve the Lord."  "Thanks be to God."  We say this even though they're going to be sitting down and starting school, not actually going anywhere.  When the older kids are home, they join us for prayers and stay if they have time, or leave if they need to.  Eldest Daughter particularly likes to stay when it's a Plutarch day.

Next on the agenda is poetry.  I read one selection and then we talk about it.  The conversation is always better when Eldest Daughter joins us.

These two things we do each school day.  The next three items we do once or twice a week, depending on how much time we have.  My third daughter's violin lessons have been moved from Tuesday afternoon to Monday morning and I haven't really adjusted to the change yet.

Plutarch -- we're using Anne White's study guide from AmblesideOnline as we read the Life of Nicias.  I have to stop often to let someone narrate.  Generally I ask, "What's he talking about here?" or "What just happened?"  Stopping often while reading Plutarch is necessary because the sentences are long and complex.

Then we move to Homer.  We're about half way through The Wanderings of Odysseus.  At the end of the chapter I ask, "Who would like to tell this part of this part of the story?"  Usually someone volunteers, but if needed I'll just tell someone to do it.  After the narration I'll usually ask whether anyone else wants to add anything.

I don't really know how to categorize this next item -- history, music, art. . .  It's all that.  We're studying through Professor Carol's Early Sacred Music course.  If #1 Son is home, this is what he's most interested in.  Much of the early part was filmed in Jerusalem and describes the worship at the Temple during the time of Christ.

Next is math.  I think I will need to write a post just on this subject since we're doing a lot of different things.  I've been learning a lot about teaching math and I'm pretty excited about it.  But one of the things I'm doing with my two youngest and with Second Son, is reading through the Life of Fred books.  There are math games that I play with the two youngest, together or one on one.  Also, the two youngest do a lesson from the Teaching Textbooks most days.

Something new we're doing is oral composition, using James Selby's Classical Composition series, which isn't meant to be done orally, but I don't see why delayed reading and handwriting skills should stop a child from thinking about stories and composing variations in his head.  So far I'm following his order fairly closely -- I'll write a separate post on how it works if anyone's interested.  When it's time to "write" the assignment, my eleven year old daughter wants to type it on the computer, and my thirteen year old son wants to recite the story into an audio file which I then transcribe. So far I like the way it's working.

What we're missing -- we haven't been singing hymns much lately and that's a shame.

Also, I've been using the cursive handwriting program from The Logic of English, but we haven't picked it back up yet since Christmas break.  This doubles as reading lessons because phonics instruction is built in to and I can easily add more phonics practice to this time if needed.  Obviously, as we progress with the program, it also serves as spelling lessons.

I mentioned in the "Different children read at different times" post that my then-twelve year old son had suddenly started reading, and over Christmas break, my daughter turned eleven, and started reading too.  Youngest Son has been reading several books for school on his own, but Youngest Daughter isn't to that point yet.

Something really bizarre happened last month.

It’s taken me a few weeks to get my head wrapped around it, but I think I’ve finally managed it.

My fifteen year old daughter came to me complaining about her math.

Okay, that’s par for the course. My kids and I are all math dunces, and, knowing that, I prefer to take a low-stress approach. They have to learn a certain amount of math to function in this world, just like they have to learn to drive (that’s not a frivolous analogy – I have two children who would never have learned to drive if I hadn’t made them!) and that being the case it’s only counter-productive to let them learn to hate math. So I let them move along at an exceptionally slow pace, which would freak out any parent who was worried about the kids being at grade level, but I really don’t care about that sort of thing at all.

The bizarre thing was that her complaint was that the lessons were so boring and repetitive. She understood the point and wanted to move on, already. Here is a child who has managed, on her own, to get further along than any of her older siblings ever managed to get before the age of eighteen – about half way through Saxon 87. In other words, she was using a book that was only one year behind “grade level” and finding it too easy.

Wait. I have a child who possibly has a grade-level ability in math?

I finally came out my daze enough to figure out what to do for her. I gave her the pre-algebra placement test at TeachingTextbooks.com and she passed with flying colors so I ordered the curriculum for her. I’m switching from Saxon because she’ll be getting beyond my ability to help if she needs it, and I love the way this curriculum explains every single problem, if the student needs it.

So we’ll see how it goes.