Ask HN: Can calculus be taught without differentiating or integrating by hand?

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.

Repetitio est mater studiorum as they say.

Maybe a good tool does not need to be understood on a deeper level to be used.

If we gave students more exposure to maths as a tool to be used, rather than arcane formulae and symbolic manipulations, they could build an intuition and appreciation for it that allow them to use it even if they cannot perform the mechanical transformations of equations on their own.

I'm not saying x+1=7 is a good example of that -- my son, who is not even literate, could answer that in the blink of an eye. But recently I needed to get the implied marginal probability out of a partition of conditional probabilities, i.e. solve for p in something like e = ap + b(1-p). Could I have done it manually? Sure. Did I plug it into a symbolic solver? Absolutely.

For me the insight lay in (a) knowing that was the shape of the equation I knew, and (b) that I needed to rearrange it to get p as a dependent variable. Actually going through the motions is not that important, in my mind.

I don't need to master any of the dozens symbolic integration techniques.

I am not sacred of any equation looking too complex to solve.

I just write less than 30 lines of python and have the solution to the problem via numerical methods.

At the end of the day what I care about is that answers, not symbolic techniques.