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.
> 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.
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.
I know what you mean, in fact I remember earlier on when I started the course, I had wanted to use these kinds of substitution techniques, etc. and thought I could finish the course in a few days. Boy was I wrong!
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.
Reusing and blackboxes do appear a lot in higher level mathematics. Indeed, the idea behind abstract algebra is to hide 'implementation' details. The concept of abstract data type in programming is similar to structures studied in algebra.
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.
Here's a criterion for testing this core understanding calculus - Can somebody given a problem (say optimization, or finding volumes) convert it into a standard type of differentiation or integral, then use symbolic software like Mathematica to do the computation and then get the right answer. Often, calculus students memorize standard recipes for problems and get confused by a problem which is not hard symbolically, but requires some thought to set up correctly.
It’s almost always algebra in the early calculus classes I think. I tutored an “into to calc for non-STEM majors” class for a couple years, and it was always algebra. If you have teaching assistants for the class, and you go to them with: I think I understand the calculus, but I’m struggling to simplify things in algebra, they might be able to help you out.
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…
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.
My experience certainly isn't representative! I just happen to build things where university level maths rarely comes up. Stats comes up more than anything and sadly only had to take one course in that area.
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
I’m a product manager and I use the concepts to read and understand new algos, research papers, etc. you’re right that you won’t be calculating (that builds problem solving) but grasping the principles will help you proceed to more advanced concepts in other fields
It was a while ago now but I remember our university mathematics required passing a small algebra module that covered essentially all of highschool algebra.
> 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.
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.
This does depend on the curriculum to some degree, and whether you’re just trying to grasp a concept firmly enough to move onto a more advanced concept or whether you’re trying to build a practical skill in solving problems. For instance, it’s entirely possible to understand higher level mathematics without having much skill at all in pencil-and-paper arithmetic. I know this because one of my best friends in college got straight A’s in upper level mathematics and EE classes but, due to his unusual background, only bothered learning arithmetic when he needed to prepare for the GRE.
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.
> 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.
That's exactly what happened to me!
This is why I'm learning about differentiation yet struggling to factor simple fractions with a surd.
Algebra is the simple part. I’d say it’s more about math maturity. At least 1/3rd of my classmates had a hard time grasping the epsilon-delta definition of limit, let alone the deeper definitions like Cauchy sequence or those used in the proof that R is dense(and we were in an elite university’s competitive program). Among the survivors of single-variable calculus, at least 1/3 could barely get by the multi-variable calculus. I saw too many of my friends struggle with different integrals, and got massacred by Green’s equation.
My guess is that most people hit a wall of abstraction at certain point.
> 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.
my view is software by comparison is like a single surface of knowledge; once you know the basics, thats it, nothings too hard to learn. maths on the other hand is more like a volume of knowledge.
