Goodbye, U-Substitution

EPSILON-DELTA · one year ago · 101 views

 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:Day 1Review of Chain Rule and begin to reverse Chain Rule only when the inside function is linearNotes |  Problem SetMy kids worked the problem set together around the room on vertical whiteboards (absent students were asked to work questions on the PDF).*****Day 2Anti-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)?Video lesson | Blank Notes*****Day 3Anti-Chain Rule Part II: What if there's a coefficient left over?Video lesson | Blank Notes*****Day 4 Anti-Chain Rule Part III: Can we apply what we've done to definite integrals? Video lesson | Blank NotesI 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? 

 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:

Day 1
Review of Chain Rule and begin to reverse Chain Rule only when the inside function is linear

Notes |  Problem Set

My kids worked the problem set together around the room on vertical whiteboards (absent students were asked to work questions on the PDF).

*****

Day 2
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)?

Video lesson | Blank Notes

*****

Day 3
Anti-Chain Rule Part II: What if there's a coefficient left over?

Video lesson | Blank Notes

*****

Day 4 
Anti-Chain Rule Part III: Can we apply what we've done to definite integrals? 

Video lesson | Blank Notes

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?