Thursday, July 17, 2014

Priorities in Homeschooling

 [First posted on this date in 2003, meant to be the beginning of a series, but I'm not sure I ever finished it.]

The obvious priority, because it applies to all parents, is to "bring them up in the nurture and admonition of the Lord." Since "man's chief end is to glorify God and fully to enjoy him forever," everything we do should be with the ultimate goal of glorifying God. Beyond this obvious priority, I've had quite an *ahem* adventure figuring out how to work academics into the equation.

My first year was an academic disaster. The curriculum I had chosen was so overwhelming! Every day I had to do Bible, reading, math, calendar, penmanship, history, science, health and safety, music, poetry, and physical education. The curriculum said it could be done in two hours a day, but I found out that whoever gave that estimate obviously did not have, in addition to the first-grader, a preschooler, a toddler, and a nursing baby.

The first day, I got through Bible, reading, math, calendar, and history (plus two loads of laundry, cooking and cleaning up after three meals, and countless dirty diapers). On the second day, I did Bible, reading, math, calendar, and science (plus two loads of laundry, cooking and cleaning up after three meals - you know the rest). The third day I did Bible and calendar, and got caught up in history. I was beginning to realize that I might never get to some of the other lessons!

On the fourth day, Hurricane Opal interrupted us, so we did Disaster Preparedness, that is, we went to the BX and bought a camp stove and a lot of bottled water, then came home and taped up the windows – plus two loads of laundry.... That night and the next day we had a family with us that had had to evacuate a military base in Florida because of Opal.

Because of that storm and the chance we had to minister to another military family, I realized that Real Life should be as much a part of homeschooling as learning math.

By the next Monday, I had decided that the two really important things were to read through the Bible with my children so that they would become familiar with God's Word and love it, and to teach my children to read! History lessons were so much fun we did them in the afternoon during our regular story time.

Wednesday, July 9, 2014

Wednesday with words: Isaac Asimov is so funny

The crucial point in man’s mathematical history came when more than patterns were required; when more was needed than to look inside the cave to assure himself that both children were present, or a glance at his rack of stone axes to convince himself that all four spares were in place.

At some point, man found it necessary to communicate numbers. He had to go to a neighbor and say, “Listen, old man, you didn’t lift one of my stone axes last time you were in my cave, did you?” Then, if the neighbor were to say, “Good heavens, what makes you think that?” it would be convenient to be able to say, “You see, friend, I had four spares before you came to visit and only three after you left.”

~ Isaac Asimov, Realm of Numbers, pp. 1-2

Friday, July 4, 2014

Happy Independence Day!

[The following is a repost from 2011]

In her post Video Games, Home Education, and the U.S. Supreme Court, Brandy discusses what is being called the third wave of home school persecution, and has encouraging things to say.

In the course of her post she mentions the earlier days when “socialization” was the big issue that home schoolers had to deal with, but then explains that this newer form of persecution is based on the fact that when a parent educates his own children at home, he is passing his own ideas down to his children, and those ideas might be dangerous or unacceptable.

But I’ve come to believe that this is what the whole socialization argument was about – not that home schooled children won’t know how to interact politely with other people on an individual basis, but that they won’t know how to fit into Society at large, meaning, they won’t grow up to be good contributors to the national economy.

The other night we were at Home Depot looking at new flooring for our kitchen and the young woman who was helping us, mentioned the installation fee a couple of times. After a while, when we’d finished picking out what we wanted, she said something about calling to schedule installation, but I said, “Oh, I have a son – he does all my installation.”

She responded with mock horror at the idea of us not paying someone to do our work, and I said, laughing, “I know – our family is so bad for the economy.”

And this is the point: As soon as you figure out that you can raise your own kids from infancy to adulthood without needing a paid professional to do it for you, you figure out that there are scads of things you can do yourself, and those kids grow up assuming that doing things as frugally and as independently as possible is the way Normal people function. They pay for fewer and fewer services, and in a service economy, if everybody did that, where would we be? This, I believe, is what so many fear about parents educating their own children at home.

One of the first times we visited George Washington’s birthplace, one of the blacksmiths was telling visitors about how economically independent from Britain the Virginians strove to be, refining their own iron ore, for example, and forging it into the necessary items, instead of sending the ore the England to be refined and forged there, as Parliament wished. In fact, Parliament wanted all raw materials to be sent to England for processing, and then bought back (as value-added products, in today’s speech) by the colonists. So the colonists were supposed to raise sheep and harvest the wool, but send it straight to England for carding, spinning, and weaving into cloth which would then be purchased by the colonists to make their clothes from. The same with timber, which the colonists were expected to harvest and ship to England, to be turned into the lumber and shingles they would buy to build their houses and barns with.

But at the Pope’s Creek Plantation, where George Washington was born, all of the family’s basic needs were provided by the farm. The plantation functioned like a village, with a blacksmith shop, a spin shop (for spinning, dying, and weaving wool and flax). Cobblers and carpenters had their shops, too. Most of the Virginia plantations worked this way, and allowed their craftsmen, who were nearly all indentured servants and slaves, to hire themselves out to locals who needed their labor. In this way, local communities provided for all of their basic needs. Wealthy families bought luxuries from Britain when they shipped their tobacco harvest to London, but not the daily necessities Parliament wanted them to buy, such as cloth for everyday clothes, lumber, and hardware.

Well, this blacksmith, in giving us this history lesson, remarked that, “When a people have gained economic freedom, political freedom won’t be far behind.”

That’s something to keep in mind this weekend, as we celebrate our political independence from Great Britain.

Wednesday, July 2, 2014

Maria Comic update -- chapter one is complete

At a page and week and 30-something pages per chapter it's going to take a while for the whole thing to be done, but the girls finished the first chapter last week, and this week published a bonus page -- "In case you ever wondered what our heroine is thinking at any given time."

Read the whole thing here.

Wednesday, June 25, 2014

Wednesday with Words: The Quadrivium is all about Math

[T]he quadrivium is essential to a liberal education in the traditional sense. And sense we can normally only advance from sense-perception to intellectual intuition by way of intellectual argumentation, the quadrivium necessarily involved the study of number and its relationship to physical space or time, preparatory to the study of philosophy (in the higher sense of that word) and theology: arithmetic being pure number, geometry number in space, music number in time, and astronomy number in both space and time.
(Stratford Caldecott, Beauty for Truth’s Sake, pp. 23-24)

Elaienar made a graphic for me.
Isn't it pretty?

Friday, June 20, 2014

Squaring the Circle, part 2 -- the Mesopotamians

Plimpton 322
Image via Wikipedia

Reading about mathematics in Mesopotamia has been pretty interesting.  They seem to have done a lot of math just for the fun of it.  They were fond of tables comparing numbers and their relationships to one another – multiplication tables, tables of reciprocals, squares, cubes, and square and cube roots.  Algebra was their specialty. 

They had a base 60 system, apparently because sixty can be divided so nicely in so many different ways – into halves, thirds, fourths, fifths, sixths, tenths, twelfths, fifteenths, twentieths, and thirtieths (did I miss any?).  We still use base-60 for telling time and for measuring angles, which is interesting because it doesn’t seem that the Babylonians had the concept of measuring an angle.

For numbers above 59 they used a place value system which worked the way ours does.  The first place to the right is “ones,” the numbers from 1 to 59 in their case.  The place to the left of that is for the base units – 10 in our case, 60 in theirs.  To the left of that is the base squared – 100 for us, 3600 for them.  (This looks like x(60)2 + x(60) + x in Math-ese, but I prefer English.)  They eventually developed a zero which they used as a place holder in the middle of long numbers, but they didn’t use it in the far right position if it was needed there, so 1, 60, and 3600 look like the same thing and have to be guessed from context.

This place value system was also used for fractions and is the reason for the unfamiliar notation I mentioned last week.  In the example quoted, the first number is the whole number.  The semi-colon serves the way a decimal point does for us – to separate the whole number from the “less than one” part to the right.  The commas are needed to distinguish the places since each place might have anything from a 1 to a 59 in it.

Those are some of the basics of math in Mesopotamia, but this is supposed to be about finding the area of a circle, right?

Remember all those tables and lists I mentioned in the first paragraph?  The clay tablet pictured at the top of this post is a table that describes the relationship of the squares drawn on the sides of a triangle, sort of like this:

This is such a handy way of understanding shapes that the Mesopotamian mathematicians noted this kind of relationship with all the regular (equilateral) polygons with three to seven sides, comparing the area of each shape to the square that could be made along its side.  One tablet lists all these relationships and gives very accurate ratios describing the relationships.

The ratio of the area of the pentagon, for example, to the square on the side of the pentagon, is given as 1;40, a value that is correct to two significant figures. For the hexagon and the heptagon, the ratios are expressed as 2;37,30 and 3;41, respectively.  In the same table, the scribe gives 0;57,36 as the ratio of the perimeter of the regular hexagon to the circumference of the circumscribed circle, and from this, we can readily conclude that the Babylonian scribe had adopted 3;7,30, or 3 1/8, as an approximation for π.
[Merzbach and Boyer, p. 35]


Me, trying to learn Math.
And yes, that does mean a hexagon with a circle drawn around it.
I checked.  :-)

So it turns out that there’s a good reason for comparing various shapes to squares, and circles to various regular polygons – it helps us understand the relationships between different shapes, which helps us understand the shapes themselves.  And noting these relationships gave the Babylonians, like the Egyptians before them, a fairly accurate way of figuring out the area of any circle.

But one thing that confused me in A History of Mathematics’ chapters on the Ancient Greeks and Mesopotamians was the phrase “an approximation for π,” which the authors use several times.  They make it sound like π is a thing that everyone knows about and only needs to be clearly defined or standardized, the way a “foot” was eventually standardized as a unit of measure.  It was pretty exciting when I finally realized why they keep saying this, but that will be in the next post.

~*~ ~*~ ~*~

Click here for Part 1, which is about the Egyptians

Thursday, June 12, 2014

Incidental learning is so fun

While reading the section on the area of a circle in a chapter on mathematics in Mesopotamia, I kept coming across things like this:

The ratio of the [blah blah blah] is given as 1;40 . . .. For the hexagon and the heptagon, the ratios are expressed 2;37,39 and 3;41 respectively. In the same tablet, the scribe gives 0;57, 36 as the ratio of the [blah blah blah] . . ..
[A History of Mathematics, Merzbach and Boyer, p. 35]

I've never seen notation like that before and don't know how to read it, so I decided to read the beginning of the chapter, which I had skipped over, it not being concerned with circles, and found this interesting sentence:

In modern characters, this number can be written as 1;24,51,10, where a semicolon is used to separate the integral and fractional parts, and a comma is used as a separatrix for the sexagesimal positions.
[op. cit. p. 25]

Did you catch that word separatrix?  Do you know what that means?  It means that commas are feminine.

The rest of the afternoon was spent browsing the dictionary and an online Greek grammar.

Friday, June 6, 2014

Squaring the Circle, part 1

Fragment of the Ahmes Papyrus
Image via Wikipedia

Have you heard that phrase before?  I have and I didn’t know it referred to anything other than trying to do something that’s impossible, and also rather silly, right?  I mean, why would anyone try to turn a circle into in a square?

Turns out there was actually a good reason for it.  Back in the days when the best tools mathematicians had were the compass and straightedge, they needed a fairly simple way of determining the area of a circle, which is devilishly hard to do.  It’s very easy to find the area of a square, so one solution they thought of was to figure out a simple way to draw a square that had the same area as a given circle. 

This endeavor went on for millennia and turned out to be futile.

In the mean time they did a pretty good job of estimating the area of a circle.  Here’s what they did, and how we know about it.

Back around 450 B.C. the Greek historian Herodotus traveled around Egypt interviewing priests and observing all the great work that went on in Egypt, from building monuments to farming along the Nile.  He said that geometry was invented in Egypt in part because the land had to be resurveyed every year after the Nile’s flood waters receded.  (Aristotle disagreed with him – he said that it was the priestly leisure class who did all the cool math in Egypt.  I don’t see why they can’t both be right – I guess it’s the age-old rivalry between engineers, who have practical problems to solve, and mathematicians, who are frequently philosophers and deal in the purely abstract realm.)

A piece of evidence on Herodotus’s side of the story, is the Ahmes Papyrus, which resides in the British Museum.  This papyrus was named for the scribe who around 1650 B.C. wrote it, using material from about 2000 to 1800 B.C.  In the papyrus, Ahmes says that the area of a circular field with a diameter of 9 units is the same as the area of square that’s 8 units on each side.

A little earlier on he had worked a problem that shows how this relationship was discovered.

First, you draw a circle with a diameter of 9 units.  Then you draw a square around it.  This square is 9 units on each side.

Then you cut off an isosceles triangle from each corner . . .  

. . . which gives you an octagon with an area of 63 units that has roughly the same area as the circle, though it’s not clear whether Ahmes thought it was exactly the same, or “close enough for practical purposes.”

From this they came up with the rule that “the ratio of the area of a circle to the circumference is the same as the ratio of the area of the circumscribed square to its perimeter [A History of Mathematics, Mertzbach and Boyer, page 15].”  Apparently this works out to a mathematical constant, 4(8/9)2, which they used the way we use π, but I don’t really understand that part of the story.  Mertzbach and Boyer say that it’s “a commendably close approximation” of π and I’m taking their word for it.

The Ancient Babylonians also had a method, but I’ll have to tell y’all about that another time.

Tuesday, June 3, 2014

Math update... sort of.

Since I've decided to devote myself to studying math this year I've had the overwhelming desire to pick up my French studies where I left off a few years ago.  Ain't that always the way?