A Math Fable

Notwithstanding its lighthearted style, perhaps the best tip for students for their first college mathematics course(s) is the following, quoted from "To the student: A Fable" in Karl Smith's textbook, The Nature of Mathematics:

"Once upon a time, two young women, Shelly and Cindy, came to a town called Mathematics. People had warned them that this is a particularly confusing town. Many people who arrived in Mathematics were very enthusiastic, but could not find their way around, became frustrated, gave up, and left town.

"Shelly was strongly determined to succeed. She was going to learn her way through the town. For example, in order to learn how to go from her dorm to class, she concentrated on memorizing this clearly essential information: she had to walk 325 steps south, then 253 steps west, then 129 steps in a diagonal (southwest), and finally 86 steps north. It was not easy to remember all of that, but fortunately she had a very good instructor who helped her to walk this same path 50 times. In order to stick to the strictly necessary information, she ignored much of the beauty along the route, such as the color of the adjacent buildings or the existence of the trees, bushes, and nearby flowers. She always walked blindfolded. After repeated exercising, she succeeded in learning her way to class and also to the cafeteria. But she could not learn the way to the grocery store, the bus station, or a nice restaurant; there were just too many routes to memorize. It was so overwhelming! Finally she gave up and left town; Mathematics was too complicated for her.

"Cindy, on the other hand, was of a much less serious nature. To the dismay of her instructor, she did not even intend to memorize the number of steps of her walks. Neither did she use the standard blindfold that students [usually used] for learning. She was always curious, looking at the different buildings, trees, bushes, and nearby flowers or anything else not necessarily related to her walk. Sometimes she walked down dead-end alleys in order to find out where they were leading, even if this was obviously superfluous. Curiously, Cindy succeeded in learning how to walk from one place to another. She even found it easy and enjoyed the scenery. She eventually built a building on a vacant lot in the city of Mathematics."

What do you suppose the author of this fable (attributed originally to Prof. Emilio Roxin at the University of RI) meant?

Hopefully it comes through that a math class strategy centered mostly on memorization (and thinking of "learning" in math as memorizing facts rather than as active exploration and engaging in mental reasoning processes, relationships, etc) is in the final analysis not helpful, and is a "life-saver" made of lead..?

Likewise, a professor who merely "shows you" the same path "50 times" until you memorize it isn't really doing you a favor, as when memorization did not ultimately serve Shelley very well.

The Fable's lesson is: Focus on active exploration of concepts and ideas, creating your own examples, your own pictures, your own analysis, investigating by trial-and-error -- your instructor as well as undergraduate tutors will then be glad to help you with any difficulties you encounter during such efforts on your part to build a rich internal mental map of the mathematics, as Cindy did.

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