H ave students choose (or suggest) a simple addition fact. 5 + 2 = 7. Have them give examples of that fact as applied to real things. Ask for full answers and write them down: 5 books + 2 books is 7 books 5 dollars + 2 dollars = 7 dollars 5 cats + 2 cats is 7 cats. Ask what math facts these answers illustrate. 5 + 2 = 7. These answers also are examples of a rule: Listen: “Books ,,, books ,,, books.” “Dollars … dollars … dollars.” “Cats … cats … cats.” This is the typical rhythm of additions and subtractions. Learn to hear it and to expect it. “We can only add (and subtract) numbers that have the SAME NAME.” It’s a fundamental rule of arithmetic with constant applications. It also makes sense: 5 + 2 = 7. But if we try to add 5 dogs + 2 cats, we don’t have 7 dogs. We don’t have 7 cats. But those cats and dogs are pets. We can decide to call them pets, and now we can add: we have 7 pets. Let’s apply the SAME NAME rule to PLACE VALUE. “What’s 1 thousand + 2 thousand?” “What’s 1 hundred + 2 hundred?” “What’s 1 thousand + 2 hundred?” 1 + 2 = 3. But we don’t get 3 of anything when we try to add 1,000 + 200. We can only say: “1 thousand 2 hundred.” Or we can change 1 thousand into 10 hundred. Now we can add: “10 hundred + 2 hundred = 12 hundred.” Now, the two numbers have the same name and we can add. The “SAME NAME” rule is a powerful rule of arithmetic. Math formulates it as a rule when it asks us to add mathematical objects: With fractions: “Add the numerators and keep the COMMON DENOMINATOR." “COMMON DENOMINATOR” is another way of saying “SAME NAME.” Check TOPICS “Fractions 1” to connect adding fractions with the rule that applies to anything we add. In algebra: “You can only add LIKE TERMS.” “LIKE TERMS” is another way of saying “SAME NAME.” Check “SAME NAME 2” for other details on the rule. Also “PLACE VALUE.” CLICK to download this article

We expect students to learn by answering questions. A version of that same approach consists in expecting students to learn from examples. We tell them, for instance, that 2/5 is: Another way of writing a division: 2 divided by 5. A quantity equal to the quotient of the division. 2 of something called a fifth. A proportion: 2 out of 5. Etc. We then take it for granted that they can generalize from these examples. “2/5 is 2 of something called a fifth.” They have no difficulty generalizing to “7/8 is 7 of something called an eighth.” Formulating the same knowledge in terms of “numerator” and “denominator” would be difficult to articulate and just as difficult to understand. Trusting students to generalize from examples is better math knowledge than a formulation of the legalistic version. Various versions of ASK don’t TELL and trusting students to GENERALIZE from EXAMPLES can be applied at all levels of mathematical instruction. CLICK to download this article