*The Having of Wonderful Ideas and Other Essays on Teaching and Learning*by Eleanor Duckworth (Teachers College Press, 1986), a book that's been on my shelf for more than 25 years and one that I return to time and again for inspiration and guidance.

## Uncovering Pi

^{2}), introduced Π as the symbol for pi, explained that we could use either 3.14 or 3 1/7 for the value of Π, and then had students apply the formulas to solve problems. In other words, I covered the subject, but I didn't uncover it. I taught the formulas for area and circumference and how to apply them, but I didn't help students understand why those formulas made sense.

*notice*focuses them on looking for patterns, structure, and regularity, all important for making sense of mathematical ideas and procedures. Asking students what they

*wonder*focuses them on extending what they've noticed to make conjectures. This kind of thinking is fundamental to doing mathematics.

## Procedures Versus Understanding

*pi*and the symbol Π that we use to name the ratio of the circumference to the diameter of circles. No amount of thinking and reasoning alone will reveal this knowledge to students. This is content that we as teachers need to cover. In such a case, teaching by telling is appropriate and necessary. But the actual ratio of the circumference to the diameter is a mathematical constant that exists in the physical world for all circles. Students can uncover this for themselves through firsthand learning experiences. And they should.

## Exploring the "Why"

## 1. Why is it OK to add a zero when multiplying whole numbers by 10 but not when multiplying decimals by 10?

## 2. Why is the sum of two odd numbers always even?

*two*odd numbers of things in pairs, each of them will have one extra without a partner. These two extras can always be paired, so there won't be an extra anymore.

## 3. Why is zero an even number?

*divisible*by another when the result is a whole number without a remainder.) You can also use multiplication to explain this instead of division. An integer is even if you can write it as "2 times something;" for example, 26 = 2 × 13, so 26 is even. Or you can use addition: Even numbers can be represented as a number plus itself (13 + 13 = 26, so 26 is even.) Zero passes all three tests: It's divisible by 2 (0 ÷ 2 = 0); you can write it as 2 times something (2 × 0 = 0); and it can be represented as a number plus itself (0 + 0 = 0).

## 4. Why does canceling zeros produce an equivalent fraction in the fraction 10/20, but not in the fraction 101/201?

10/20 = 1/220/30 = 2/320/40 = 2/4

Does 101/201 = 11/21?

11/21, 22/42, 33/63, 44/84, 55/105, 66/126, 77/147, 88/168, 99/189, 110/210

## Those Who Understand, Teach

*ask*and when to

*tell*. Even more important, I've had to learn

*what*to ask and

*what*to tell, which calls for thoroughly understanding the mathematical content I'm teaching.

*Those who can, do. Those who understand, teach.*I agree with this message. Even with elementary math topics that seem fairly uncomplicated and easy to understand, unexpected twists and turns can emerge during classroom teaching. But if our math knowledge as teachers is robust enough, we can treat these surprises not as difficulties but as opportunities to guide students in uncovering their understanding of mathematics.