I wrote a half dozen entries here in October, November and December last year. I'd like to pick it up again.
In the second half of 2020 I felt that I had caught on to something exciting. When Maria was younger, and even before she was born, I had a dream of homeschool that seemed unrealistic. Over the course of teaching her to read a realistic picture of it emerged for me, not all at once. I did some reading and some experiments, and in the fall I could feel the urgency to write down the good ideas I had got and the good advice I had found. I was going to forget them.
I have forgotten them. Writing is slow and growing up is fast. But the curriculum from July 2020 still seems fresh and relevant in July 2021. From the beginning of July until the end of November, I presented the math lessons in Give Your Child a Superior Mind pretty faithfully. Some of the headings from that book tell a story, a cryptic one, about what this material was and what order Maria learned it:
Showing the Relationship between Counting and Adding — Reading Addition Problems — Word Problems Involving Adding 1 — Number pairs — Algebra Problems — Algebra Problems Involving +1 —Algebra Problems Involving Number Pairs — If-Then Propositions — Rule for +2 — Continuity Game — Subtraction —
I've copied the section on "Algebra Problems" below. I enjoyed explaining it to a kid who could work out only a few sums (6+1 and 6+6 by rote, 6+4 on her fingers — from the earlier sections copied in another post) and who didn't yet know the minus sign.
—Review —Multiplication — Ones — Tens — Twos — Fives — Nines — Eights — Fours — Sixes — Sevens — Threes — Multiplication Problems — Multiplication Involving Algebra —Area of Rectangles —Area Problems Involving an Unknown — Telling Time — Fractions —Subtracting Fractions —Adding Parts to Make a Whole —Rule for Fractions That Are Equal to One Whole — Rule For Fractions That Are More Than One Whole —Algebra Problems With Fractions — Division — Money—Column Addition — Column Subtraction — More About the Relationship Between the Terms of a Problem
Those are the section titles between pages 193 and 296, and there are also many untitled sections. I was pleased with how they worked. If there are things I now suspect I should have done differently, they weren't in this section:
Algebra problems
No, algebra is not necessarily a high-flown, abstract subject. In fact, the child already knows algebra. He simply doesn't know that he does.
1. Write this problem on the board:
3 + 1 = B
It is an algebra problem with one unknown—the B.
2. Explain that we read the problem as, "3+1 is B." Then give him a rule for interpreting an unknown. "Whenever you have a letter in a problem, it asks you, 'How many?' This problem asks, '3 plus 1 is how many?' It asks, 'How many do you end up with?'"
3. Work it as you would any other +1 problem. "We're adding 1, so we say, 'What comes after 3?' Good: 3 plus 1 is 4."
4. Explain that you know what B equals when you answer the question, "How many?" "When you add 3 plus 1 you end up with how many? You end up with 4. So we can say B equals 4." Write:
3 + 1 = B
B = 4
Give the child six to eight algebra problems of this form during each session for about a week. Use familiar +1 and number-pair problems. Use a variety of uppercase letters for unknowns, but avoid I, O, Q, S, and X. Here's a typical exercise:
Problems Solutions
3 + 3 = K K = 6
8 + 1 = R R = 9
1 + 13 = D D = 14
6 + 6 = M M = 12
4 + 4 = U U = 8
4 + 1 = E E = 5
Algebra Problems Involving +1: The unknown in an algebra problem asks the question "How many?" regardless of where it is in the problem. The equation 3 + 1 = F tells you that if you start out with 3 and add 1 more, you'll end up with—how many? You'll end up with 4. So, F = 4.
The problem 3 + F = 4 tells you that you start out with 3 and add how many to get to 4. So, 3 plus how many is 4? If you answer the question "how many?" you solve the problem: 3 plus 1 is 4. F = 1.
1. Present the problem
6 + C = 7
2. Explain, "This is a new kind of algebra problem. See? The C is here in the middle. But it still asks 'How many?' This problem says, '6 plus how many is 7?' You say it. Good."
3. Show the child how to use a variation of the rule for adding 1 to solve the problem. "Look. We're adding something to 6. How many? We're adding something to 6 and we're ending up with 7; 7 is the next number after 6. We're adding something and we're ending up with the next number. We must be adding 1. Because what's our rule? When you add 1 to a number, you end up with the next number,"
The pattern of logic involved in this kind of deduction is basic, so basic in fact that we, who are used to it, have trouble reducing it to words. The child is not used to it. He has to learn it. And he'll learn it more rapidly and more thoroughly if you show him the steps you're taking. Study the sample explanation we just gave. Practice it.
4. Summarize the problem so that the child can see the relationship between the familiar +1 fact and the present problem: "6 plus how many is 7? 6 plus 1 is 7. Say it. Good."
5. Write the answer,
6 + C = 7
C = 1
Give the child six to eight problems of this kind during each session for the next week. Put the unknown first in some problems and second in others. Here would be a typical exercise.
A + 2 = 3
9 + C = 10
4 + R = 5
V + 6 = 7
F + 6 = 7
1 + G = 2
The explanation is the same whether the unknown comes first or second. For instance, to explain A + 2 = 3, you point out that you're adding something to 2 and ending up with 3. How much are you adding? Well, you're ending up with the number that comes after 2, so you must be adding 1. How many plus 2 is 3? 1 plus 2 is 3. A = 1.
The child will soon spot a shortcut for solving these problems. He'll learn that all he has to say is "C equals 1," or "A equals 1," without really knowing why. The best way to handle this is to say, "That's right. And how do we know that A equals 1. We know it because we're adding something to 3 and ending up with the number that comes after 3. Right? Let's hear you say it. How do we know that we're adding 1?" Help him through the explanation.
Algebra Problems Involving Number Pairs: Introduce these problems only after the child is used to relating algebra problems to addition facts. "3 plus how many equals 4? 3 plus 1 equals 4."
1. Present the problem, 4 + H = 8.
2. Show that the rule for adding 1 does not apply. "Look. We're adding something to 4 and we're ending up with 8. Are we ending up with the next number after 4? No. So we're not adding 1."
3. See if the child can answer the question "How many?" "Think hard: 4 plus how many is 8? ... 4 plus ..."
4. If he doesn't know, show him how you can figure it out. Represent the problem this way:
◦ ◦ ◦ ◦ + _________ =◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦
Explain, "We start out with 4 and we add. We don't know how many we add, but we must end up with 8. The problem tells us that we started out with 4 of those 8. Let's draws lines to show which ones we started out with."
"These are the four we know about. The ones that are left are the ones we must have added. How many are left? Let's count and see . . . 4." Fill in the diagram so that it looks like this:
"The circles that are underlined are the circles we must have added. 4 plus how many equals 8? 4 plus 4 equals 8."
5. Refer to the original problem.
4 + H = 8
H = 8
The sample problem above shows the relationship between the answer term and the unknown in a way the child can appreciate. Use it liberally during the first three or four number-pair sessions. Then discontinue it, and encourage the child to solve the problem by remembering the addition fact. "Come on. You can remember that if you think hard: 3 plus how many is 6? 3 plus . . . " If he doesn't remember, tell him.
Include a few +1 problems in with the number pairs. Also, introduce the number-pair problems in which the unknown comes first, C + 6 = 12. The child should catch on to these algebra problems rapidly. He should be able to work all +1 and all number pairs involving an unknown a month after you introduce the first algebra problem.
"The child should catch on rapidly," I think Maria did.
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