Dear awesome teacher,

Thank you for everything you put into good teaching. I know that’s it’s rare to see solid evidence that your extra efforts are really working. So here are two tiny examples of ways that teachers helped me, decades ago, probably dramatically, but probably in ways that would not be detected by any study on teaching effectiveness. I intersperse these examples with two equally tiny suggestions for mathematics teaching. My hope is that these suggestions may also be helpful.

- How does one read out loud? How does one become prepared for calculus without learning calculus?

I might have been 6 years old. My classmates and I were challenged to read passages from a book. It sounded like we would read a word, then, eventually, read the next word. One day there was a fleeting comment from the teacher—telling us to have our eyes a little bit ahead of the word we were saying. Was the start of great reading? It probably took years to get good enough at it for it all to pay off. I don’t believe my teacher got better reviews for this transformational suggestion, and I don’t even remember her name. But the idea works everywhere. Sight reading on the violin?—easier. Preparing for your future?—it’s just more natural to anticipate the next step.

I would love to see more of these fleeting comments in grade school mathematics classes! Here is a possible example involving the formulas for the area of a triangle and the area of a trapezoid. To make it easier, we’ll use a right angle triangle. One commonly used way to approach a trapezoid, is to view it as a difference of two triangles. (See the green trapezoid in figure 1).

*The green area represents the difference in area between two triangles—both triangles have a vertex at the origin.*

Now take it further—imagine a growing right angle triangle, one side formed by part of the x-axis, another side formed from part of the line y=cx, shown in figure 1. (The third side is parallel to the y-axis). Ask about the last bit of growth in the last second—measure the change. Calculus is all about measuring change, anticipating growth, looking slightly ahead, and having a measurement for it. I believe that middle school students will be inspired by gaining a sense of what calculus is, even without having to learn any of the details beyond the ones they are already learning anyway. I have no proof that this line of discussion would prepare any student for calculus courses 10 years later; but surely it can’t hurt; my sense is that many university students pass courses deprived of really knowing or caring why they’re doing what they’re doing. So your role in this process, and therefore your potential influence on your young students, may be much bigger than it appears. Much of this piece, especially “Example 1, Section 3” is accessible without too much prior knowledge of mathematics. While it was originally intended for calculus teachers, I now believe all math teachers could use a small part of it to help their super young students prepare for *the idea of calculus*.

- How do we memorize? Why do we memorize? How do we generalize?

“Repeat it here, repeat it there, repeat it everywhere. Write it down, compare it with the correct version and ask yourself if the differences are important. Eventually you will discover what it all means.”

I don’t actually know if anyone told me exactly this, or where I learned to memorize this way; but almost for sure I was given advice like this, from somewhere, when I was very young. Don’t just assume that your students will simply pick this up. Tell them, show them, and it will help (although maybe not soon enough to help your teacher evaluations—sorry!). Here’s an example of how memorization in mathematics led to a possibly new, definitely elegant proof of an old theorem (stated near the end of the post). It’s another triangle fact—simple, yet sophisticated.

**Memorize this fact**: The distance from the vertex of an equilateral triangle to the opposite side does not depend on the choice of vertex.

What does this mean? Look at triangle ABC in figure 2.

*Figure 2. An equilateral triangle.*

It means the (drawn) distance from B to AC is equal to the (not drawn) distance from C to AB.

“Repeat it here, repeat it there, repeat it everywhere…” What does our fact mean in figure 3?

It could mean that the distance from P to QB is the same as the distance from B to QP, provided that QBP is an equilateral triangle (so let’s assume that QP is parallel to AC).

*Figure 3. Two equilateral triangles.*

But equilateral triangle ABC is still in this picture, and guess what else is equal to the distance from B to AC?—the sum of distances from P to AB and from P to AC (see figure 4—can you see why?)

*Figure 4. The sum of the lengths of the two dotted blue lines is equal to the altitude of triangle ABC.*

By repeating our old fact in a different place, and with a bit of thinking, we now have a new, **second fact:** If P is on a side of an equilateral triangle (in this case side BC on triangle ABC), the sum of the two distances to the other two sides is the altitude of the equilateral triangle.

Repeat this second fact, looking at figure 5. “Eventually you will discover what it means.”

*Figure 5. Viviani’s theorem.*

This second fact—applied to equilateral triangle MBN and with P lying on MN—means that the sum of two distances from P to MB and P to BN is equal to the altitude of MBN. (As you guessed, MN is parallel to AC).

Now switch to triangle ABC—the sum of the three distances—from P to AB, from P to BC and from P to AC gives the altitude of ABC (see figure 5). Beautiful; we have:

**Third fact:** If P is any point in an equilateral triangle, then the sum of the three distances, from P to each of the three sides is equal to the altitude of the equilateral triangle.

This is known as **Viviani’s theorem**. For a more detailed discussion regarding this proof, please see this preprint.

Memorization leads to generalization, which is essential for understanding how modern mathematics works. The importance of generalization, and therefore of memorization, can be communicated at very early stages of mathematical development.

Notes:

- I first saw Viviani’s theorem in “A Friendly Mathematical Competition: 35 years of teamwork in Indiana,” a problem book edited by Rick Gillman, published by the MAA. Our proof of Viviani’s theorem and the resulting preprint mentioned above was a result of discussions with Dr. Armstrong after Putnam training sessions, using this book.

- I’ve suggested “Example 1” from my paper (coauthored by Darlene Olsen). However, example 2 from the same paper addresses two issues. The first is that it provides a simple calculus-based explanation of the fact that the ratio of area of a circle to the square of its radius is the same as the ratio of its circumference to its diameter. However it also prepares students for the method of cylindrical shells—a common difficult part of a standard calculus course. In fact, an informal preview, 10 years before a calculus class, of the intuitive idea of example 2 could help (we have absolutely no statistical evidence of this). It does seem to me though, that this is one of the possible benefits of society asking that the people who teach math know something about where their students may be headed. I sincerely hope that our society asks for even more mathematics knowledge and passion from our mathematics teachers.

- I recommend that you continue to follow your own thoughtful judgment regarding pedagogy. Research on pedagogical techniques (just like research on which foods are good for you) may take a lot of time to settle. Please keep in mind that research suggesting that teaching via memorization is not effective, may subtly assume that students already know how to memorize; if one experimental class focuses on something students aren’t getting elsewhere then it may appear to be far more valuable than it would be if, say, all classes were focusing on that same thing. Perhaps memorization will be considered far more valuable in the future when fewer teachers are focusing on it! My suggestion is that you continue to try to do everything well and with a purpose, whether it is memorization, inquiry based learning or some combination of many techniques. Thanks!