I wrote a bit about my journey of saying goodbye to U-Substitution on the One Good Thing blog (here and here). As I said in that blog, I will probably never go back to teaching U-Substitution. When I see u-sub, I see it as a way to simplify the integrand so I can easily reverse Chain Rule. But, no matter how many colored highlighters I got out or how slowly I went, my students seemed to only ever see it as a set of abstract steps that they would get tangled up in. Mathematics should be revealing. For my kids, u-sub was muddling. The whole idea of u-sub is to simplify. If it's not simplifying things for your kids, it may be time to leave the u-sub train.
Some have asked for more details, so here we go!
I spent about four days on this:
Review of Chain Rule and begin to reverse Chain Rule only when the inside function is linear
Anti-Chain Rule Part I: What happens when the inside function is not linear but when our integrand is in the perfect form of f'(g(x))g'(x)?
Anti-Chain Rule Part II: What if there's a coefficient left over?
Anti-Chain Rule Part III: Can we apply what we've done to definite integrals?
I taught these lessons live (typically I flip) so these videos are my first draft, but will give an idea for what we spent time on in class.
What else do we insist on teaching in muddling ways? What do we need to reinvent?