You might know I got obssessed with Jordan Ellenberg's book, "How Not to Be Wrong." It's not written for teachers, specifically, but it has lots of lessons and inspiration for us.
On Mathematical Knowing
I've spent lots of time thinking and wondering about why we spend so much time fighting about math. I jokingly talk about the #MathWars a lot, but in truth it's time to lay down our arms. One can stake out a position on the "back to basics" side, or the "discovery" side, but the truth is, and always will be, somewhere in the middle.
@PeelSchools, my employer, recognizes this-we have a balanced math instruction document now (as do several other boards). There are many voices of moderation out there. Practice needs to be balanced with problem-solving. Number facts are the scaffolding upon which strong mathematical buildings are made, so they must be known. Yes, students can "discover" a whole lot of math, but they usually need a lot of guidance to see what they found means.
One thing both "sides" agree on is- actually, forget that, there are no "sides". We ALL stand for student understanding, and being able to use math skills and concepts. (One common debate is how and when we "know" a math fact, versus how and when we "understand" a math fact- i'll leave that one to the researchers and cognitive scientists)
Perhaps the best strategy, in any given situation, to borrow from Mr. Ellenberg's quote, is the one that helps our students know the mathematical big idea or concept under study "all the way to the bottom." If we are working with circles, that means the deep pleasure of finding pi using circles and string, AND developing, using, and applying the circumference formula. Seeing how grade 3s can move beyond repeated adding as their schema of multiplication develops, AND help them start to know their facts with practice and games. Watching junior students apply their sense of what proportionality means, and watch their toolbox full of strategies grow.
What we are not fighting about is the beauty and utility of mathematics-we all agree about that. Our methods and our means should help our students follow their thoughts about math deep down, all the way to the bottom. Let them find the spark (with our guidance, of course)...