The money shot of math. Good explanation.
I was assured that mathematics would not be involved.
Everything is math. Sometimes it's just too hard to figure out the equation.
Heebie, these posts are great. I'm learning so much! (Or at least, I feel like I am, even though I don't remember enough algebra to confidently do the math myself.)
I kind of can't believe I (mostly?) understand this! You're an amazing teacher. Either that, or my high school calculus teacher was an asshole. Both could be true!
Love how you relate the slope concept back to the difference quotient, which I first saw in the mid-1970s and didn't understand for years and years.
Thanks, jms and bill! I'm really enjoying putting this down on paper and it's gratifying to hear that it's communicating what I'm hoping to communicate.
When you're done, have you thought about polishing the collection of posts up a little and turning them into a little book? Not for your own students -- this probably has an awful lot of overlap with how you teach the same concepts in class -- but it seems like it might be useful for other people.
I think I understand this in a conversational way, but when I got to this
And s(3+h)=(3+h)3-8(3+h)2+15(3+h),
=h3+h2-6h
(exponents don't paste correctly)
I realized I would need a refresher on the binomial theorem(is that right?), to actually solve any problems.
8: I like the idea, but the overcoming the fear and procrastination to navigate that kind of process seems daunting!
9: The algebra bogs students down so much more than the actual calculus. Or rather, they're often so bogged down in the algebra that they don't have mental space leftover to see the big picture of calculus.
We do such a shitty job of teaching algebra in this country, but I haven't taught it myself a zillion times, so I don't have a great idea of how I'd reinvent it - ie what I'd count as the key goals, and how I'd work backwards to create a curriculum.
But it's such a good example of why do algebra. It's a cheat code to the universe.
I think back to how I came to understand algebra. What it boils down to is that any time I came to a crossroads of possible ways to simplify or whatever, I went off to the side of my paper and tested numbers until I could reason out which way was the correct way to proceed.
That makes me think that we need to go far slower through the material, and teach methods of reasoning your way out of ambiguous situations. But in practice, that would displace all sorts of things we find valuable in the curriculum. Or displace things that a few people find extremely valuable.
I'll add my voice to the chorus of praise for this series. I'm loving it and learning a lot. Please keep it going!
I really regret not learning math in high school. I had a terrible attitude about academic work that didn't come easy, so it wasn't ALL my teachers' fault, but I was also a student who wanted explanations. My teachers did not provide them. This does! It's so great!
13. The content monster. Can't stop, got to get through the material (what, 70% of my class isn't understanding what I'm 'covering?'). Can't stop.
I like this series, too. I'm lazy so I generally skip doing any actual work. So I'll probably get out of it what I put in. But I'm liking it.
It's pronounced "cun TENT". I've been doing a lot of work on myself and I have learned how to step and accept what I've been given. Thank you.
"stop" not "step". But I'm O.K. with the idea that I don't type so well all the time.
16: Are you really still "Opinionated" then? Maybe it's time to listen to Megan on this.
I've learned to embrace that not everyone will like everything I do.
As soon as I finally read 16 correctly, I thought it was funny.