There was the struggle between classroom management and engaging instruction. I invested myself equally in both until the depressing day I realized that my investment in engaging instruction also paid off certain dividends in easier classroom management. I spent that day re-evaluating my assumptions about teaching and re-balancing my investment portfolio.
Then there is the more current struggle between teaching skills (multiplying two exponentials) and teaching concepts (proportional reasoning). I figured, until recently, that in a 120-minute classroom, any time we spent on goofy conceptual digressions was time away from skill instruction we'd have to make up later.
So it's strange, then, that after a semester of frequent digression, my classes are still on pace with every other Algebra 1 class and my kids set the curve for the semester final exam.
We spent thirty minutes on Friday, for example, investigating Jessica Hagy's infographic work. I cherrypicked some interesting relationships, covering up Hagy's graph in each, and asked the students to draw the relationship, also labeling each "direct" or "indirect" variation because, why not.
By the end, we were disputing Hagy's graphs on technicalities, altering the intercepts ever so slightly to reflect the fact that (eg.) plastic surgeries could, at first, make someone less frightening to children.
After digressing for fully 25% of the period, we got down to the new business of adding and subtracting polynomials. And it struck me as I put an example up on a slide and asked them "what can you do with this?" how little time I spend "teaching" anymore, how these goofy conceptual digressions have trained my kids to look for connections, not just between "plastic surgery" and "frightened children," but between "old skills" and "new skills."
I realized, Friday, why we lost dozens of hours in the first semester to goofy conceptual digressions but still outpaced the school.
We didn't need those hours anymore.