Learning Strategies

This week’s New York Times Book Review contains a fascinating Literary Map of Manhattan, preceded by an explanatory article written by Ethicist Randy Cohen. Quoting Meg Wolitzer, Cohen defines his (their?) “cartographic motto”:

a strong sense of specificity, even though everything is made up.

I was immediately reminded of two very different connections. The first, of course, was Marianne Moore’s description of poetry as “imaginary gardens with real toads in them.” The fictional characters on the map live in imaginary gardens, but the convincing details about Manhattan (whether genuine or imagined) are the real toads.

The second connection was with the problems we assign in school mathematics. All too often these imaginary gardens contain concrete or cardboard toads, certainly not real ones. I try to populate my problems with details that are specific enough that they can become real in the reader’s imagination, even when the occasional student takes them all too literally. Here was a problem on my Algebra II test:

Before founding Macy’s department store, John and Mary Macy had five children. Unfortunately they named the first three Casey, Stacy, and Tracy. Then they stopped the rhyming; the last two children were Lee and Chris. You’ve already noticed that it’s impossible to tell from the names which children were boys and which were girls.

1. Find the probability that all five were boys.


2. Find the probability that both Casey and Lee were boys.


3. I don’t know what it’s like in your house, but the Macys were never able to get all five children to show up for breakfast at the same time. Oddly enough, it always turned out that exactly four of them turned up (but not always the same four). Their parents made them line up for breakfast in single file. How many different arrangements of children were possible under these conditions?


4. Eventually, when they retired, John and Mary wanted to pick two of the children to run the store as co-presidents. From how many different pairs could they choose?

I don’t know how real those toads were, but that’s definitely what I was aiming at in my imaginary garden. Similarly, I wrote this problem, inspired by my colleague Josh, on today’s precalculus test:

In the famous movie, “Honey, I Kept Shrinking the Kids,”” Rick Moranis played a biologist who cloned his daughter, Chloe, and then cloned the clone, and then cloned the clone of the clone, and so forth. Chloe was 64 inches tall, but the clone was only 48 inches, and the next clone was only 36 inches, and so forth (each clone was three quarters of the height of the previous incarnation). Their cheerleading coach thought of a unique routine, in which each girl would stand on the shoulders of the next larger girl, and so on down until Chloe was at the bottom. The coach pointed out to Moranis that they could fit an infinite number of clones on top of Chloe within the gym!

1. Find the combined height of Chloe plus the first four clones. (Ignore the fact that each clone is actually standing on her predecessor's shoulders, not head.)


2. What is the minimum height for the ceiling of the gym in order to fit Chloe plus the infinite number of clones?

Math as fiction?

Labels: math, teaching and learning