Maybe the focus could be on solving calculus problems with the help of a symbolic algebra system instead?
Ask HN: Can calculus be taught without differentiating or integrating by hand?
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Re: Ask HN: Can calculus be taught without differentiating or integrating by hand?
#2Pedagogy has at least two aspects. The rote component, which people usually deprecate, and the inductive/reasoned component, where you learn "how it is" not "what to do"
The thing is, the rote component is (in my personal experience) much more important than people give credit to. You need to learn some axiom applied rules to be able to use "mental muscle memory" to perform things, and then once you can operate as a bit of a black box, you can get the uplift.
Sure, in practice the "why" comes as the stories, the instances of calculating acelleration over time, or distance travelled under non-constant speed, or whatever the analogy-instance is you're using to say why being able to compute derivitives and integrals matter, but if you can't do recall on 2+2 = 5 then this is a bit of a mixed bag.
I think maths is like cookery. You need basic knife skills and a discipline about order and time and sequence. For veggie stew it's less important but for baking its everything or the cake doesn't rise. Well, maths is the same but with more egg on your face and less sugar coating.
TL;DR you need to do things by hand. Thats what learning is, sometimes. I don't personally think teaching without doing some hand examples works as well. BTW I consider myself functionally illiterate in maths, but reasonably competent in arithmetic. Calculus is the dividing line. The massive mountain range which I climb on, but never cross.
Re: Ask HN: Can calculus be taught without differentiating or integrating by hand?
#3Calculus isn’t solely about learning how to derive and integrate equations using algebraic rules. There are underlying principles behind differentiation and integration that are missed if one only focuses on learning symbolic manipulations. Sure, many could get by with just a computer algebra system, but eventually someone will run across a problem that a CAS won’t handle. That person will need to have an understanding of the deeper mathematical principles to solve the problem.
Re: Ask HN: Can calculus be taught without differentiating or integrating by hand?
#4I think it could be, but I found realizing the correlation between differentiation and integration to be just beautiful. I think offloading those calculations to a computer system, at least while initially learning, would be a disservice.
Re: Ask HN: Can calculus be taught without differentiating or integrating by hand?
#5Pedagogy has at least two aspects. The rote component, which people usually deprecate, and the inductive/reasoned component, where you learn "how it is" not "what to do" The thing is, the rote component is (in my personal experience) much more important than people give credit to. You need to learn some axiom applied rules to be able to use "mental muscle memory" to perform things, and then once you can operate as a…
A certain kind of student who has a lot of programming experience could write an automatic differentiator in the process of learning how to differentiate.
Re: Ask HN: Can calculus be taught without differentiating or integrating by hand?
#6Yes, it can. The biggest failure in math education is that we spend an inordinate amount of time solving equations without understanding how to apply them to solve real-world problems. Thanks to the calculator we can now spend less time figuring out how to solve x+1=7 and put that time on how to understand when the equation is needed in the real world or how to create an equation that will help solve a real-world problem.
The same can be said about calculus. Given the abilities that computers give us, it's much better to teach applied calculus than to teach students how to solve equations by hand. Calculus is a tool, and it should be available to as many people as possible.
For most students, Calculus should be seen in the same light as drills. We understand when we need a drill so we get one and use it but we never have to go out and learn how to build one just because we have a need to use it. Calculus should be similar. We need to learn when we need it and then use a computer to get an answer that helps solve a problem that advances our needs.
Re: Ask HN: Can calculus be taught without differentiating or integrating by hand?
#7Yeah, would have been more useful to spend relatively more time on numerical approximations.
Re: Ask HN: Can calculus be taught without differentiating or integrating by hand?
#8In SICP (Structure and Interpretation of Computer Programs) an early distinction is made between "declarative" and "imperative" ways of knowing. If you teach the declarative aspect in mathematics you are teaching what I would consider the "language" of mathematics. It is in fact a language and has value on its own. Now, later we can put on the programmer's hat and work with algorithmic aspects (the imperative part). This seems very valuable to me.
Re: Ask HN: Can calculus be taught without differentiating or integrating by hand?
#9Pedagogy has at least two aspects. The rote component, which people usually deprecate, and the inductive/reasoned component, where you learn "how it is" not "what to do" The thing is, the rote component is (in my personal experience) much more important than people give credit to. You need to learn some axiom applied rules to be able to use "mental muscle memory" to perform things, and then once you can operate as a…
A certain kind of student who has a lot of programming experience could write an automatic differentiator in the process of learning how to differentiate.
Sure, for somebody who is already on board with inductive reasoning and abstraction the best path in might be code. Personally I suspect "hand" has merits but I won't say there aren't other approaches.
I remain unconvinced theory-first, and "theory, the machine does it" would work out ok.
Reminded of my pascal lecturer at uni in 79 "this compiled cleanly but I haven't run it" for every coding example. He really did behave like syntactic and basic semantic checks were all you needed to prove correctness. I am sure he also knew runtime is everything.
Re: Ask HN: Can calculus be taught without differentiating or integrating by hand?
#10I think a course like that would need to be called something like “Mathematics Appreciation,” in analogy to “Music Appreciation.” I would never hire someone whose only music education was the latter to actually play in a band or write a movie score, and I would never hire someone who didn’t know how to do differentiation or elementary integration by hand to do any work that required an understanding or use of calculus.
Now, there is such a thing as overkill in assigning calculus homework; integration problems can be made arbitrarily difficult, and past a certain point I don’t think there’s much educational payoff.
But learning advanced math is hard and requires hard work. Sorry about that. Math students often start off in elementary school finding everything easy and intuitive, but then at some point everyone hits the wall. Same thing can happen when learning a language or developing a physical skill. At that point there is no alternative to working hard in order to make progress past that wall.