In high school students are often told to “define your terms.” This is horrible advice. What they should be told is to “label your concepts.”

The best essays know the rules, and then break them. (You can count the number of five-paragraph essays on this blog on no hands.) If an inexperienced writer defines his terms, he is likely to pick out interesting terms, come up with a definitions that don’t mean anything even to him, drop them in the first paragraph, and never refer back to them again. When I write essays, especially essays for school, I brainstorm ideas until they take on a life of their own. They evolve into roaring uncaged beasts prowling through my mind, with features of startling clarity, demanding one thing: to be named. Once a moniker has been bestowed, the animal gracefully enters the cage of categorization and becomes part of my menagerie of clear and communicable ideas.

In the course of my daily life, I often find myself thinking about advanced concepts in STEM fields. The geometry and loading of skyscrapers, the acceleration of trains and elevators, the paths and patterns of falling snow and rising steam. How are patterns (of parentheses (that are nested)) related to (isomorphic to) bit strings (of 1 and 0 (only) (or is there a more elegant choice of numbers (there is)))?

This is the point where my sister would complain, “why does he take the fun out of *everything*?” But I think that analyzing these sorts of thing *is* fun. There are patterns all around us, left either by those who came before us or by the physical universe and mathematics. When I see these patterns emerge, the greatest gift is to find out that it already has a name. With a name, I can find out more about the pattern, bypassing the process of reinventing to wheel, and talk about it with others. I can learn about the pattern’s details as clean accurate facts, but first I must explore and understand the core concept myself.

Too many people don’t see STEM knowledge in this light. But that’s just because STEM (math in particular) is often taught by defining terms (term, then definition) instead of labeling concepts (definition, then term). “Today we’re going to learn about *x,*” where *x* equals any high school math or science topic, is as common as it is boring. “Where am I going to use this?” the students complain. A lot of math is either unexpectedly useful or mindblowingly cool (and if you’re an engineer, both) but unless the students discover that first hand, it’s just material to memorize for the test and then discard.

Instead, we need to have our students engage the material, explore it and see it in all its glory. Then they will ask not “when will I use this?” but “what can I call this?”.

Posted by Roy Wright on November 27, 2011 at 9:53 pm

If I understand you correctly, you’re basically saying that it’s best for students not to have a name for a concept until they actually understand and appreciate the concept? If so, I agree wholeheartedly. In fact, in my “teaching statement” (an essay that potential faculty members usually have to submit when applying for teaching jobs), I wrote the following:

“So the ideal in my teaching is that a formal definition should only organize what students already know. Perhaps, from the point of view of a modern mathematician, this approach reverses the logical structure of mathematics, in which every new concept is introduced by way of a definition before it is ever applied. On the other hand, my approach is faithful to the way in which much of mathematics originally came to light…

“As Richard R. Skemp points out in The Psychology of Learning Mathematics, when we introduce an entirely new mathematical idea to a student, the concept ‘cannot be communicated to him by a definition, but only by arranging for him to encounter a suitable collection of examples.’ Once this is done adequately, the statement of a formal definition is a welcome and natural organizational tool, rather than a sterile fact to be memorized.”

Posted by Max Goldstein on November 27, 2011 at 11:08 pm

Exactly. We put this into practice with derangements. Indeed, the I now see that my reply to you there was the genesis of this post. It’s also what makes walking the solar system very effective.