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.

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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.

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"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.

◦  ◦  ◦ + ◦ = 

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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 . . . "

◦  ◦  ◦ + ◦

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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.