Remember when you first learned how to write paragraphs? For me I think it was seventh grade, and the approach we were given was to craft five sentence long paragraphs: the first sentence set out the topic (or thesis); the next three sentences gave further detail supporting the thesis; and the last sentence summarized the content of the paragraph.
By the end of seventh grade, we were starting to write essays. An essay consisted of five paragraphs: the introduction, three supporting paragraphs, and the conclusion. Each of those paragraphs were themselves five sentences long, following the same structure.
Egads, a fractal!
[It is fun to imagine how to extend this scale symmetry; chapters of five essays, books of five chapters, etc...; alternately, a grammar with five word sentences; five letter words; five stroke letters; etc....]
The other day in real analysis we were talking about writing proofs, doing a fill-in-the-blanks activity where the general outline of a proof was given, and the students only had to deduce (in effect) the appropriate nouns to complete the argument. The pedagogical point was to get them actively reading and anticipating steps in a formal argument, while exposing them to yet another example of a proof that two sets are equal by proving two containments.
I felt the activity to be just a little silly, especially for my students, all of whom have had at least two proof-based courses prior to this semester. But I am also painfully aware of just how difficult it is to write a proof, and I was pleased that my students had a relatively easy time with the fill-in-the-blank activity. We noted in class that they would have had an infinitely harder time crafting a proof of the same result if they had been challenged to do so but had faced a blank page. Why is writing a proof from scratch so much harder? Part of it is structure: where do I start, and where does that lead? By having the first step of the proof (partially) written out, they could dispense with thinking about how to articulate the beginning of the proof, and focus on its structure. “We begin by assuming that x is a member of [blah], and thus we know that x satisfies ____; that in turn implies that either x is ____ or _____, and thus….”
This led to some discussion of how the rigid structure made the actual writing significantly easier — we can focus on just the ideas, not the format for communicating them. Hence the flashback to 7th grade essay writing. I don’t know any fluent writers who still write in 5 sentence paragraphs, or 5 paragraph essays. But as a teaching tool, as a crutch, it works. The long-range goal, of learning to write fluently, is achieved by means of practice and repetition: after crafting dozens upon dozens of rigidly structured five sentence paragraphs, eventually our inner voice declares “ENOUGH!”, and we begin to have faith in our own ability to navigate the rules of grammar and syntax while passionately advocating for the truths we seek to demonstrate.
Some day, some of my students will write proofs with passion and verve. I think that rocks.
But in the meanwhile, we all get to relive seventh grade. And that’s okay too.