Most of your first point is … algebra? Yes if your algebra is weak you will not be able to cope with solving calc equations. The solution to that problem is not to be found in a calculus made easy. It would be found in algebra made easy.
Math isn't like programming. In programming you can often solve a problem using a library, framework, language facility, etc. without entirely understanding why it works all the way down to the binary level.
In math you can't often solve a more advanced problem such as Calculus problem without understanding the more foundational math such as algebra, fractions, etc.
If "information hiding" / layers of abstraction was possible in math, I would have completed my university entrance course months ago, but here I am still struggling.
Sure, we could have Algebra made easy and also Trigonometry made easy, Fractions made easy, Functions made easy, etc. etc.
I just find it personally irritating that all this foundational knowledge is brushed aside when it's really core to someone's actual competence dealing with actual math problems.
Maybe it's just assumed that people went to a good high school or had a private math tutor and already learned the foundations very well, but I think at least that assumption would be coming from a place of privilege.
It's similar to telling someone to take a Bootcamp in React and that will be enough for them to succeed as a software engineer. But to solve the kind of problems they are going to face in reality they will eventually have to learn at least some foundational Javascript and maybe a little about algorithms and data structures.
Certainly not everyone is in the same place in their learning journey as you. Material on calc, at a university level, is typically going to focus on calc. Yes it is assumed that you have learned the fundamentals before taking that course.
I was in a similar situation as you. If you really want to learn it there's no substitute for skipping over the fundamentals. I did that and did fairly well but it's all long forgotten. Never use the stuff :)
So many people tell me this that it's become cliche at this point.
I find it demotivating, but unfortunately I have to press through, as there is literally no other way I'm going to gain entry to my university's bachelors program.
A part of me wonders if this kind of fundamental knowledge could be actually useful, similar to being able to cook your own food instead of takeaway.
Kind of like how "first principles" thinking can apparently lead to new discoveries because you're not just mimicking / re-using the same structures that were already built.
Good luck you’ll get there.
Since you bring up food. As a former professional baker it would also take me some time to make croissants professionally at the level I used to. At least for me personally, if I don't use it I lose it. But I can certainly pick up faster than someone seeing it for the first time if I needed to.
Along the way you'll pick up some intuition that you can use elsewhere that's hard to quantify. Outside of the loans I don't regret taking any of the maths required for my CS degree.
Personally, I found the calculus lifesaver by Adrian Baker to be helpful in my studies as someone that was missing some fundamentals
Yep, this is the number one reason people think they aren’t suited for math. Everything is built on everything else, and if you missed anything you’re screwed. It takes a while to realise you are screwed, you can get by on rote for a surprising distance.
Ultimately, “there is no royal road”, but a good tutor will help you find those gaps and build out the missing bricks.
That's exactly what happened to me!
This is why I'm learning about differentiation yet struggling to factor simple fractions with a surd.
It's similar to the "expert beginner" problem described by Erick Dietrich (https://daedtech.com/how-developers-stop-learning-rise-of-th...).
I didn’t enjoy math as a child, and I used to be a lot more bitter about this when I first started to grasp what mathematics actually was. As a child, mathematics seemed like a small amount of “learn and understand a new abstract concept” (which I was pretty good at) bogged down with a huge amount of “okay now you have to solve a a bunch of problems based on that concept over and over again before we’ll trust you with another concept”. Eventually I figured out that mathematics itself really is the concepts, and that the concepts eventually build up to a level of complexity where it was increasingly challenging and fun to grasp them.
Maybe the reason it’s taught this way is because the vast majority of people aren’t mathematicians and aren’t really attracted to mathematics out of an abstract intellectual appreciation for the beauty of mathematical concepts; they just want to solve problems. And this is perfectly reasonable. But if I had it to do over again, I probably would have put more effort into mathematics and study more of it, at much higher levels, if I knew it would eventually get a lot more interesting.
And eventually things do start to branch out a bit. The standard K-12 curriculum up through calculus mostly builds up like a single tower where everything is built on everything else, but there are parts of mathematics where you can just sort of go in a different direction for awhile.
This is true to a good degree, but maybe a bit less so than you believe. Trigonometry is a topic that only clicked for me after finishing my uni calculus curriculum, I didn't get a great grade, but got by with a technique similar to how we handle complex numbers: Instead of giving up after being unable to solve an eg. weird chain of sin, arccos etc. functions, just declare it to be u(x) and do the calculus bits around it. In the last step substitute the actual function back in and you have an incomplete, yet technically correct solution.
These techniques definitely won't work in a tough online multiple-choice test (of the kind I'm getting) where they deliberately sprinkle in subtle quirks to deceive you, which would require very disciplined algebra, fractions, powers, etc. to identify.
It is common for mathematicians to rely on theorems as black boxes(ex: classification of surfaces) even without knowing the proof. Secondly, people can even write research papers without knowing how to work with some object covered in the paper, by working with collaborators who are experts on a different topic.
It would be helpful to isolate the essence of calculus itself from the symbolic techniques, for ex to actually calculate integrals(especially magical seeming substitutions and nontrivial factorizations) as many of these symbolic techniques will appear in different topics even outside calclus.
Math classes build up, and at some point unfortunately they do have to start assuming that your previous classes were solid. Calculus is where algebra and trigonometry gets some of that treatment. It is extremely common for a calculus class to reveal some shaky algebra foundations though, so I’d hope your school has some help there…
My guess is that most people hit a wall of abstraction at certain point.
I don't think it's a limit to their abstraction, I think it's that they didn't work properly on the fundamentals, so they had a superficial understanding of the abstractions.
To give a fitness analogy it's like trying to do heavy barbell presses before you can even do 10 pushups in a row.
My experience with programming is that once you get really really good with fundamentals you suddenly leap ahead and pick up new languages, paradigms, etc. incredibly fast.
Maybe this partly explains the 10x phenomenon - it's because they worked very hard on the fundamentals.