If-Then Propositions

Mathematics is a deductive science. That means that you can go from a simpler form to a more complicated form if you play according to the rules of the game. To make the correct move, however, you have to make the right kind of statement. Almost always, the statement can be expressed as an if-then proposition: "If 8 plus 1 is 9, 80 plus 10 must be 90." "If 8 plus 8 is 16, 8 plus 7 must be 15." "If 8 times 5 is 40, 8 times 6 must be 48." "If 8 equal the fractions 8 x 1/1, 8 x 2/2, 8 x 6/6, it also equals the fraction 8 x Dg/Dg."

Lewis Carroll wrote a funny dialog in 1895, between the runners in a Zeno paradox. The tortoise teases Achilles that he understands an assertion A, and he understands that A implies B, but he still does not believe B. The Engelmanns' instructions for training a 4-year old to follow these inch-long deductions reminded me of it. I felt a little demented discussing it with 5-year-old Maria: "If this is a book, then this can't be a book." I sometimes had to keep from laughing. But ignorance of logic is a normal state of mind, and I don't mind having relied on something pre-logical (not demented at all) to cure that ignorance.

Present if-thens this way:

1. Make a green mark on a piece of paper with a crayon. Next to it, make another green mark. Explain, "We're going to figure things out a new way. We're going to if-then." Point to the first mark. "If this is green . . . [point to the second mark] this is green. We can say that because they are the same. Let's do it once more. If this is green, this must be green." Repeat with two blue marks and two red marks.

2. Introduce something a little more complicated. Take two books. Don't tell the child what they are called. Just show that the first book has pages with printing on them. Show that the book has a cover. Now show that the second book also has page, printing, and a cover. Conclude. "They're the same, just as the colors were the same. So I can say, 'If this is a book, this other one must be a book.' Your turn . . ."

3. Put a green mark on your paper. Next to it make a red mark. "Here's a new kind of if-then. If this one is green, this one can't be green. It can't be green because it's not the same. We can't figure out what it is, but we know that it can't be green." Repeat with different color combinations. Then use the book again. Place it next to a block, and make the line of if-then reasoning more specific. "If this is a book, this can't be a book."

"When we if-then, we always start out with what we know. We know that a book has pages, has printing, has a cover. Does a block have pages, printing, and a cover? No. It can't be the same."

4. Repeat with various examples, comparing spoons with dishes, lamps with rugs, and so on.

5. Demonstrate some basic if-thens with number-type concepts. Put a row of 4 circles on your chalkboard. Beneath it put another row of 3 circles.

◦  ◦  ◦  ◦

◦  ◦  ◦

Have the child count the circles in the top row. Then say, "If there are 4 up here, there can't be 4 down here  because these rows aren't the same." Show why they're not the same by drawing a line from every circle in the top row.

◦  ◦  ◦  ◦

|  |   |  |

◦  ◦  ◦

"If these lines were the same, there would be a circle below for every one above. But there's no circle for the last circle."

Give similar examples in which the line below is either the same or different from the line above.

6. Introduce if-then with addition. Start with the familiar problem, 3 + 1 = 4. Represent it with circles.

◦  ◦  ◦ + ◦ = 

◦  ◦  ◦    ◦

Next to the original problem present 3 + 2 = . Modify the 3 + 1 diagram.

◦  ◦  ◦ + ◦┃◦ = 

◦  ◦  ◦    ◦┃◦

The difference between 3 + 1 and 3 + 2 is now evident. A circle has been added to the right of the heavy vertical line, changing the 1 into a 2. The top row is no longer equal to the bottom row. To make it equal we have to add 1 circle.

◦  ◦  ◦ + ◦┃◦ = 

◦  ◦  ◦    ◦┃

We can translate the diagram into an if-then proposition by covering up the portion of the problem to the right of the vertical line and saying, "If 3 plus 1 is 4 . . . "

◦  ◦  ◦ + ◦

◦  ◦  ◦    ◦

Now uncover the top row to the right of the vertical line, ". . . 3 plus 2 can't be 4. It has to be 1 more than 4." Uncover the bottom row to the right of the vertical line. "It has to be 5." Repeat, "If 3 plus 1 is 4, 3 plus 2 is 5."

◦  ◦  ◦ + ◦┃◦ = 

◦  ◦  ◦    ◦┃◦

By using the diagram properly, you can show the child just exactly why 3 plus 2 can't be 4, and why it has to be 1 more than 4 (because the circles in the top row must correspond to the circles in the bottom row).

Present six to eight if-then problems during each of the next fifteen sessions. Base these on familiar +1 and number-pair facts: 6 + 1 =, 3 + 3 =, 1 + 4 =, 5 + 5 =, 5 + 1 =, 2 + 2 =. Always change the second number, never the first. If 6 + 1 = 7, 6 + 2 = ? Take the child through the steps in reasoning. Don't expect him to pick up the patter overnight. He won't.

7. Use diagrams during the first four or five sessions. Then drop them and work from the problems, using a modified approach.

2 + 2 = 4

2 + 3 =

"Look. We start out with 2 plus 2. In the second problem we don't have 2 plus 2; we have 2 plus 3. We know that 3 is bigger than 2—1 bigger." Draw a line from the 2 down to the 3, and put a +1 in the middle of it. "We added 1 . . .

2 + 2    =  4

   + 1 ) 

2 + 3    = 

"Here's a rule. If we add 1 on this side of the equal sign, we have to add 1 on the other side of the equal sign.

2 + 2    =    4

   + 1 )     ( + 1

2 + 3    =    5

What's 4 plus 1? Sure, 5." Write it, completing the bottom equation. "Let's go through the whole thing. We start with 2 plus 2. If 2 plus 2 is 4, 2 plus 3 can't be 4. It must be five." It helps to phrase the if-then the way we suggest here. The child finds the line of reasoning easier to follow if he knows that 2 plus 3 can't be 4. He can then see that it has to be 1 more than 4. It has to be 5.

Keep the child responding. As he becomes handier with if-then reasoning, let him do more and more of the talking. "What are we starting out with? How are we changing it? What's the rule? If we add 1 here, we must . . . Good. So give me the whole if-then. Fine."

8. After the first two-weeks, introduce if-thens in which the first term changes.

3 + 3 = 6

4 + 3 =

Show that they diagram the same way as the original problems and that they follow the same rules.

◦┃◦  ◦  ◦ + ◦  ◦  ◦ = 

  ┃◦  ◦  ◦    ◦  ◦  ◦

They should present no real stumbling blocks.

By the time the child has been working if-thens for two weeks, he should be able to phrase an if-then conclusion and he should have reached the point where the words are starting to mean something. But it will be a while before the full meaning comes through.

Dorothy Sayers

Dorothy Sayers gave a speech in 1947:

"I want to inquire whether, amid all the multitudinous subjects which figure in the syllabuses, we are really teaching the right things in the right way; and whether, by teaching fewer things, differently, we might not succeed in “shedding the load” (as the fashionable phrase goes) and, at the same time, producing a better result."

She thought her coevals were not educated as well as literate people of the middle ages. The early part of the speech gives examples of sloppy speaking and writing and thinking that disappointed her. They don't disappoint me in 2021, actually I don't find them easy to react to at all. But farther into the speech, her description of what school could be like is interesting:

Let us now look at the mediaeval scheme of education—the syllabus of the Schools. It does not matter, for the moment, whether it was devised for small children or for older students, or how long people were supposed to take over it. What matters is the light it throws upon what the men of the Middle Ages supposed to be the object and the right order of the educative process. The syllabus was divided into two parts: the Trivium and Quadrivium. The second part—the Quadrivium—consisted of "subjects," and need not for the moment concern us. The interesting thing for us is the composition of the Trivium, which preceded the Quadrivium and was the preliminary discipline for it. It consisted of three parts: Grammar, Dialectic, and Rhetoric, in that order.

Was it really like that? Anyway if you are responsible for educating a young kid it is food for thought as a prescription. Decades later it was influential for America's homeschoolers. When you search google for "homeschool" you find some of those influences. Two of them were the authors of The Well-Trained Mind, quoted at the end of this post.

Maria, now six years old, has 12 grades ahead of her. Sayers paints a picture of them: 4 grades of Grammar, 4 grades of Dialectic, and 4 grades of Rhetoric. I can't anticipate even 12 months ahead, but I can recognize some of my six-year-old in this passage:

My views about child-psychology are, I admit, neither orthodox nor enlightened. Looking back upon myself (since I am the child I know best and the only child I can pretend to know from inside) I recognise in myself three states of development. These, in a rough-and-ready fashion, I will call the Poll-parrot, the Pert, and the Poetic—the latter coinciding, approximately, with the onset of puberty. The Poll-parrot stage is the one in which learning by heart is easy and, on the whole, pleasurable; whereas reasoning is difficult and, on the whole, little relished. At this age one readily memorises the shapes and appearances of things; one likes to recite the number-plates of cars; one rejoices in the chanting of rhymes and the rumble and thunder of unintelligible polysyllables; one enjoys the mere accumulation of things.

If a young kid has some enthusiasm for repetition, how can you make the most of it as a tutor? More on that later. The speech describes kids drilling the times tables, Latin vocabulary, and the names of constellations and of backyard plants and insects. I haven't tried to do that. This part, just as tendentious, had more appeal:

The grammar of History should consist, I think, of dates, events, anecdotes, and personalities. A set of dates to which one can peg all later historical knowledge is of enormous help later on in establishing the perspective of history. It does not greatly matter which dates: those of the Kings of England will do very nicely, provided they are accompanied by pictures of costume, architecture, and all “every-day things,” so that the mere mention of a date calls up a strong visual presentment of the whole period.

Geography will similarly be presented in its factual aspect, with maps, natural features and visual presentment of customs, costumes, flora, fauna, and so on; and I believe myself that the discredited and old-fashioned memorising of a few capital cities, rivers, mountain ranges, etc., does no harm. Stamp-collecting may be encouraged.

I found this idea expanded and made practical in The Well-Trained Mind, by Jessie Wise and Susan Wise-Bauer (mother and daughter):

Over the four years of the grammar stage, you'll progress from 5000 BC to the present, accumulating facts the whole way. These four years will be an exploration of the stories of history: great men and women of all kinds; battles and wars; important inventions; world religions; details of daily life and culture; and great books. ...

World history is divided into four segments, one segment per year of study. In the first through fourth grades, the child will study history from 5000 BC though the present day. In fifth through eight grades (the logic stage), he'll study it again, concentrating on cause-and-effect and chronological relationships. In grades 9 through 12, he'll repeat it again, this time studying original sources and writing thoughtful essays about them.

Planning for the shorter term, they recommend covering "ancient history" in one year, "medieval history" the next year, and "the renaissance" and "the modern period" in years three and four. We've learned a lot taking an even shorter-term view. In history (we still call it "reading lesson") each day it is easy to solve the problem of what to teach: review yesterday's reading, and then inch forward in time.

It has not been as easy to decide what to do each day in math lesson.

Zig and Therese Engelmann on Algebra for 4-year olds, 1966

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.