Pedagogy 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.
You can develop muscle memory without rote memorization by doing exercises. There, you end up memorizing facts by solving problems and therefore building intuition. I agree that you need some level of memorization when you first "perform things," but if you're performing, it's not rote memorization.
Rote is memorizing integrals with flash cards. Building intuition is doing exercises. Might seem pedantic, but the difference is huge pedagogically.
I switched schools in tenth grade (1994). Till then, I was mostly interested in the science subjects. We had Physics, Chemistry and Biology as separate subjects. I had a pretty good chemistry set (compared to the crap you get these days) at home, I had a small microscope, I could create slides, do basic chemical analysis etc. That was sort of my domain. Math always seemed too abstract for me and while I liked some of the topics, I was never too good at it and treated it as "below me". Science subjects, I could reason from first principles and didn't have to study many things by rote. I rationalised this saying that it was because I was too intelligent to waste my time memorizing stuff and considered the ability to reason way more superior.
After switching schools, we had a mathematics teacher (who since passed away). His style of teaching was to make us practice with harder and harder problems. He had a tower of books with example problems in all the topics that were relevant to us and he made us do all of them. No calculators. It was tedious work and doing large integrals and complicated equations can be a chore. However, because of the sheer effort, I broke through some kind of wall and saw it was that I was really doing. I suppose it's analogus to an inductive rather than deductive way of learning. I was able to derive several formulae using basic Calculus which I had previously learned by rote in my physics classes. All the hidden patterns started to make sense and I learned to appreciate the subject very much and started seeing places where I could use it.
I've forgotten some of the topics which I don't use on a daily basis but the basic idea of doing grunt work to master a subject stuck with me. I also emphasise it when I teach my kids math. Lots and lots of problems. Till it becomes second nature. It's how I learnt how to code. It's how I learnt calligraphy. It came full circle when I started studying martial arts. The repetition of forms and drills to "learn" a technique is quite different from "learning the theory". It helps to have a teacher (coach) who pushes you but the basic idea that "doing is the best way to learn" is foundational.
I'm one of those people who find maths really easy. I've also had extensive experience working in mathematics education, from primary to tertiary levels. I've even written policy for our state education department.
You are spot on the money. I've seen "progressive" educators who believe that technology, including CAS is the answer to teaching calculus to high school and tertiary students. I think the tech is mostly useful _pedagogically_ after the student has a very firm grip on the content. Using it as a shortcut to "real world problems" leaves the student with no real understanding of the dynamics/functions involved, leading in turn to believing the machines are magic and always right, regardless of whether you've hit all the right buttons in the right order.
The most extreme example of this I've seen was in a classroom I was auditing. The students were learning about harmonic motion and trig functions by using a mass on a spring and an ultrasonic transducer. The students were told to "keep all the digits, because the machine gave them". Students were doing sums with 14 significant digits. When I pointed out that most of them were meaningless, being measures in the Angstrom range, the "teacher" said "the machine gives those figures".
Even when I pointed out that measuring 10^-14 m required a very high frequency, which meant high energy, all supposedly from AA batteries that last for months, he agreed that it did seem strange, but after all, the "machine gave the figures".
The man teaching (he was head of department AND author of the state mathematics syllabus) was unaware that only fractions with denominators with only 2 and 5 as factors can be expressed exactly in binary without using custom datatypes and arithmetic logic.
Now as I said, for me maths is easy. It's not for most people. Most people won't grasp it all as "but that's obvious", so for these people having the skills to check the result the machine gives is vital if they are not to cause all sorts of fun as bridges collapse, cars explode and mortgages go into default. Those skills are developed by doing it by hand, and learning what numbers change in what ways.
TL;DR Modern trend to use CAS in mathematics education is well intentioned but misled, in this retired maths educators very informed opinion.
> only fractions with denominators with only 2 and 5 as factors can be expressed exactly in binary without using custom datatypes and arithmetic logic.
You mean decimal, not binary. For binary, it’s only 2.
This is getting offtopic, but back when I was in grad school I TAed some physics labs and the number one thing that annoyed me to no end was students didn’t understand or care about significant digits at all. They had no idea that 2.1cm they measured was in fact (2.1 \pm 0.05)cm, they just put 2.1 in the calculator and wrote down 9.7831m/s^2 the calculator spit out. I would point it out, and next week, same thing. Thankfully I never TAed an electronics lab to see what they read from digital meters. Btw that was in Princeton, the students weren’t dumb.
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.
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.