Wow! Let's see if I'm as dumb at this as I was the first time around.
1. hopefully not!
2. yay! thanks Teo.
That's nicely presented Heebie. The physics adjacent parts of calculus always seemed to make the most sense... maybe because they've been around longest, so have the most examples and are the best described?
That's nicely presented Heebie. The physics adjacent parts of calculus always seemed to make the most sense... maybe because they've been around longest, so have the most examples and are the best described?
But what was my instantaneous velocity between those two posts?
I don't get it. You can easily divide by zero. You get #DIV/0!.
This _is_ a great idea! The previous discussion of calculus instruction reminded me of all the annoying parts of calculus that I didn't enjoy, but this reminds me why calculus can be fun -- it's a great way to answer to questions that are easy to understand but difficult to answer (without using calculus).
9: Dividing by 0! isn't a problem, because 0! =1.
Maybe I'm too steeped in the traditional pedagogy of this, but saying that the point of limits is to figure out dividing by zero rubs me the wrong way. Saying it's about dividing zero by zero is only slightly better. Verbally, I'd want to emphasize the "and the estimate keeps getting closer" part rather than the zero part.
The concept has never made sense to me but you can plug stuff into the formula and it works.
Is the problem that you can't figure out velocity by just looking at a single instant? You need to have an instant and the instant after that to get motion.
We were presented this as the New York State Thruway Police theorem (aka mean value theorem). Our math teacher said the the police were allowed to give you a ticket for your average speed between entry and exit tolls even if they never observed you going that speed, because your instantaneous velocity must equal your average velocity at some point on your trip. With the caveat that your position was a continuous function but maybe teleportation is also illegal.
That's why you always stop at the plaza.
You should also tell the story of the crazy academic who tried to claim she invented the trapezoid method of integration and name it after herself.
Thanks so much for this, heebie! I will try to be worthy of your generosity.
Lesson 1 is very clear, probably helpful to whatever portion of students read and understand it. But LIMITS SUCK! They are gobblegook until you actually understand what derivatives are. Teaching limits right at the start of a calculus course is a guaranteed way to confuse and turn off a large part of the class. Teach people how to do a derivative and how that's useful, and then maybe go back in the middle of Calc 2, after you've introduced integration, and use all those stupid little getting thinner all the time rectangles under the line to explain how integration relates to calculating the area under the line.
I took calc 3 times: one semester version in high school which was pretty simple. First semester of college where I went to class for about 2 weeks, decided "oh yeah we covered all this in my high school course" so skipped the next two months, and when I went back found that I was totally fucked, not least because the instructor expected us to know how to do proofs. And finally grad school 15 years later where I did 4 semesters (through partial dif equations) plus both an undergrad and a grad version of math for economists.
The final time worked because I just focused on the mechanics of how to do the calculations and gave zero attention to proofs, theoretical justification, or the crazy brilliant shit that Newton and/or Leibniz had to do to figure out how we get to the point where the mechanical computations come in because they were twice in a century geniuses and I was just a schmuck struggling to get through grad school.
Starting calculus instruction with limits is like starting an English and American literature course for non-native English speakers with Finnegan's Wake
Maybe I'm too steeped in the traditional pedagogy of this, but saying that the point of limits is to figure out dividing by zero rubs me the wrong way. Saying it's about dividing zero by zero is only slightly better. Verbally, I'd want to emphasize the "and the estimate keeps getting closer" part rather than the zero part.
I was being a little flip, but it'll get developed in the next section. This is just intended to be an accessible situation that motivates why we might want to develop the idea of limits.
Also "5/0" vs "0/0" are both just shorthand notation to talk about limits, right? 5/0 usually means the limit of f/g, where f -> 5 and g -> 0, whereas 0/0 means f -> 0 and g -> 0.
Is the problem that you can't figure out velocity by just looking at a single instant? You need to have an instant and the instant after that to get motion.
Yes! Except that derivatives will provide a more exact method. They're still fundamentally using nearby points, though.
Oh man peep you better learn calc now or this thread is going to be a waste of time.
But LIMITS SUCK! They are gobblegook until you actually understand what derivatives are. Teaching limits right at the start of a calculus course is a guaranteed way to confuse and turn off a large part of the class.
Just you wait! It will be great! (I am not planning on doing epsilon-delta limits though. More just descriptive - limits give us some language to describe different situations at a point.)
Shoot, I forgot to say no pressure. Now it's all on me.
I like the conversational frame. Reminds me a bit of Feynman.
I would simply teach calculus in the manly way of the Ancient Greek geometers.
That's why calculus should be taught in college instead of high school.
You should also tell the story of the crazy academic who tried to claim she invented the trapezoid method of integration and name it after herself.
Is that the biologist who published a paper on the total population of something, if you only knew how many babies were born each year?
The one I know of was a 1994 diabetes journal. But maybe calculus has been independently discovered multiple times throughout history!
https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.831.4725&rep=rep1&type=pdf
I think that's the one I'm thinking of. I just filled in a plausible other context in my head.
I would like to try teaching calculus with "dx is a made-up infinitely small quantity" Leibniz-style rather than using "limits" as such.
I introduce derivatives immediately after reviewing function, but using the notion of a limit. And try to emphasize throughout that the point of limits is to take derivatives. Lots of graphing x^2/x.
I also want to try teaching it where you skip limits and instead use "slope of the best linear approximation" as the definition of derivative. I think that viewpoint makes the sum rule, product rule, chain rule a lot easier.
This is great.
I failed Calculus. When finals came around, I was starting to understand the fourth week and it could be good. I hope you continue.
This is wonderfully clear!
I took both Calculus I and II in high school, and it really says something about my school that I got As both years, despite having not the slightest clue what was going on. I remember feeling totally lost and just memorizing a mess of steps and formulae without any idea why or what they meant.
I reread 38 and realized that it sounds like a humblebrag, but let me emphasize, those two years of math went totally over my head. My teacher was this lovely but obviously exhausted and burnt-out guy who, when asked for clarification, would literally scream I DON'T UNDERSTAND WHAT YOU DON'T UNDERSTAND at us until stopped asking questions. He liked me because I kept my mouth shut.
Anyway, looking forward to Lesson 2!
would literally scream I DON'T UNDERSTAND WHAT YOU DON'T UNDERSTAND
Fuck. That doesn't work?
"Harvard-calculus" was the fad when I learned it (lots of talking about whether the line is going up or down). So I'm curious about Heebie's grandmother's calculus text from the 30s and how different it is.
34. Related to infinitesimals, I saw somewhere calculus done with little-o notation.
I recall making more and thinner rectangles until suddenly they turned into lines and you not only knew the area under the curve, you knew why you knew it. Except I forgot the second part.
In the criminal justice system, the area under the curve is represented by two separate yet equal methods, the one that makes sense and is impossible to actually calculate and the one that I don't get how it works but is right. These are their stories.
40: I mean, we did stop asking questions.
My section of AB calculus had to learn Riemann sums first. The other section just memorized derivative formulas from the beginning.
Excited to re- learn some calc. If anyone wants to do conversational statistics, I'd like that too.
I like the lesson but I've always been That Student*: The graph drawn doesn't match your 48mph example, maybe you meant to flip the curve so it's fastest around t(1/2)?
*I once got a question thrown out for the entire country on an SAT subject exam because I asked the proctor about something I thought was unclear, they of course couldn't help but have to write down and submit anything a student asks, it turns out the SAT masters agreed it could be interpreted unclearly, and removed the entire question from grading. They sent me a follow up letter explaining this.
But did you get a certificate for asking the question? That would be 'way cool.
38: Oh good! After 1, I was starting to think you'd clicked through and then slunk away out of the opposite.
46 is actually an excellent point, and maybe I'll go back and fix that.
Somewhere along the way I confused Riemann and Neumann. Looking up both, I'm not sure I ever learned enough matrix math to understand a Neumann series. I definitely don't remember any of that.
I like explaining things to people, but I don't like honest feedback if they didn't understand what I said. That's their problem.
Oh wow I had actually wanted to revisit Calculus which I took a year of in high school and totally don't remember. My brain is a little not used to doing anything remotely like this thirty years later.