Which competency in calculus do you find hard to maintain, exactly? My job does require math, but only occasionally calculus (and even then we don't integrate anything by hand, there's Wolfram Alpha for this stuff).
If you can't remember the core intuitions about calculus (what does integrating a function mean, how to use the derivative or gradient to find local minima or maxima, which functions are continuous, which of those are derivable everywhere) then if you're like me and most people I know, you need to spend more time applying calculus.
You can read stuff in fields that benefit from application of calculus to get a better feel for it (geometry if you're doing anything with curves or curved surfaces; computational geometry is also quite fun, also in most engineering fields there is some calculus to model the application of the forces, if you like building things learning to simulate real-world structures and systems can be super motivating). Build a (very small) neural network without a framework, that'll make you work at grasping the concepts of measuring small variations in the output of a function in a way that is very goal-directed.
(I used your example of calculus because I think you're talking about being an adult with a day job who can't find a use for high school math. If you did math in university and are worried about losing your skill at writing proofs, you should find a job writing proofs. There is about zero overlap between proof writing and a developer career, and proof writing is tedious and difficult to gain the focus to do on the side.)
If you're good enough then volunteer or hire yourself as a tutor or night college teacher.
Also, there are a number of people that produce calculus tutoring youtube videos. Maybe you can try that.
Unfortunately, a large percent of what we learn will eventually be forgotten due to lack of use. Most of what we learn will not be used at work. I remember spending tons of time studying calculus yet I've yet to use any of it. If ever I need to use it I will need to review it to refresh my mind but most likely I won't remember most of it.
I am not very good at classical mathematics. I am quite confident that there are better more intuitive ways of describing the same ideas, so until one such system emerges I tend to learn the concepts that are being expressed in mathematics and convert them away from mathematics to just a logical description of the operation. I haven’t run across anything in calculus for instance that I can’t describe in python with loops and arrays. One might argue that it is all math ultimately but that’s fine so long as I can deal with the raw idea and not the idea expressed in terms of math operations.
You might be more comfortable with Python syntax than with standard mathematical syntax, but I wouldn't think of it as being closer to "just a logical description" or closer to being the "raw idea." You're just choosing to use different notation.
I would guess that most people comfortable with both representations would feel that standard mathematical notation is lighter and conveys the "raw idea" more directly.
I believe the place of calc (just like trig) is overinflated in pre-real- analysis education. As far as I am concerned, sin is just a function defined a certain way and I ain't worried about SOH-CAH-TOA or anything to that effect. I don't think one requires 1000 page book to learn a few core calc concepts either. Derivative is a slope. That's it. It's the same value along the length of a line. The only problem is non-linear functions where slope changes along every point. In which case we bring in the machinery of tangent/secant lines and take a limit for a very intuitive reason, but the underlying concept remains the same. This informs the definition of derivative. From this point on derivative is just a function defined a certain way and in this regard is no different from any other function like indicator function or whatever. The rest of the talk about derivatives is often just a padding and general interest. Have you ever seen a 500 page book written about the step function?
When you take a bit more advanced class, the symbols for partial derivatives and integrals will be interspersed with other symbology and flash before your eyes like elements in a humongous matrix. You won't have time to think about stuff you learned from 1000+ page doorstop. You have to learn to think more nimbly and abstractly like a mathematician. To that end, check out "intro to math proofs" textbooks.
If by calculus you meant modern math analysis, then simply disregard the stuff above.
It might depend on the person, but I've retaught myself some math stuff, and I think you relearn it a lot faster the second time. The real problem is not getting bored. It was a bit slow-going at first because I was out of practice, but after a little bit of settling in, it was alright. YMMV, it depends how well your long term memory works.
For deeper mathematics, I feel like that's hard. I come back to my notes from Grad School once in a while, or try to follow proofs for topics of interests. But to be honest, most of them are a little beyond me at times. Maybe Fermat's library (https://fermatslibrary.com/) can offer some annotated reading which would be helpful?
Regardless, I wish you good luck. Please do share if you have any more good ideas.
Simple answer is past-times. For example, I have a side interest in ODEs and PDEs, which I basically never get to apply for work.
So I'm working on 3 problems from orbital mechanics, control theory and investment science respectively, each which is interesting in itself but each which flexes my ODE and PDE muscles.
In short, just invent problems you find engaging and then scratch away at them on planes and trains, and in all those in-between times.
I struggle with this so much and am glad you asked it. I reopen the books and things come back to me slowly, but it is very hard to retain without daily use.
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[ 3.1 ms ] story [ 46.3 ms ] threadThere's also Schaums outline for Calculus
You can also try going through Art of Problem Solving Calculus (though it's much more difficult than the typical calculus text)
That seems to favour rote learning instead of actually developing or keeping core competences.
In my experience, it's the mechanical stuff that you forget most quickly without use, as opposed to the conceptual stuff.
If you can't remember the core intuitions about calculus (what does integrating a function mean, how to use the derivative or gradient to find local minima or maxima, which functions are continuous, which of those are derivable everywhere) then if you're like me and most people I know, you need to spend more time applying calculus.
You can read stuff in fields that benefit from application of calculus to get a better feel for it (geometry if you're doing anything with curves or curved surfaces; computational geometry is also quite fun, also in most engineering fields there is some calculus to model the application of the forces, if you like building things learning to simulate real-world structures and systems can be super motivating). Build a (very small) neural network without a framework, that'll make you work at grasping the concepts of measuring small variations in the output of a function in a way that is very goal-directed.
(I used your example of calculus because I think you're talking about being an adult with a day job who can't find a use for high school math. If you did math in university and are worried about losing your skill at writing proofs, you should find a job writing proofs. There is about zero overlap between proof writing and a developer career, and proof writing is tedious and difficult to gain the focus to do on the side.)
Also, there are a number of people that produce calculus tutoring youtube videos. Maybe you can try that.
Unfortunately, a large percent of what we learn will eventually be forgotten due to lack of use. Most of what we learn will not be used at work. I remember spending tons of time studying calculus yet I've yet to use any of it. If ever I need to use it I will need to review it to refresh my mind but most likely I won't remember most of it.
I would guess that most people comfortable with both representations would feel that standard mathematical notation is lighter and conveys the "raw idea" more directly.
When you take a bit more advanced class, the symbols for partial derivatives and integrals will be interspersed with other symbology and flash before your eyes like elements in a humongous matrix. You won't have time to think about stuff you learned from 1000+ page doorstop. You have to learn to think more nimbly and abstractly like a mathematician. To that end, check out "intro to math proofs" textbooks.
If by calculus you meant modern math analysis, then simply disregard the stuff above.
If you mean the ability to solve and think critically about problems, I like a combination of "How to solve it" by Pólya (https://en.wikipedia.org/wiki/How_to_Solve_It) for more general ideas and regularly finding puzzles onm different topics such as the ones proposed here (https://fivethirtyeight.com/tag/the-riddler/).
For deeper mathematics, I feel like that's hard. I come back to my notes from Grad School once in a while, or try to follow proofs for topics of interests. But to be honest, most of them are a little beyond me at times. Maybe Fermat's library (https://fermatslibrary.com/) can offer some annotated reading which would be helpful?
Regardless, I wish you good luck. Please do share if you have any more good ideas.
So I'm working on 3 problems from orbital mechanics, control theory and investment science respectively, each which is interesting in itself but each which flexes my ODE and PDE muscles.
In short, just invent problems you find engaging and then scratch away at them on planes and trains, and in all those in-between times.
2. Youtube (see: Professor Leonard, 3blue1brown, Gilbert Strang, etc.)
3. Books like:
https://www.amazon.com/Humongous-Book-Calculus-Problems-Book...
https://www.amazon.com/Calculus-Practice-Problems-Dummies-On...
https://www.amazon.com/Schaums-Solved-Problems-Calculus-Outl...
https://www.amazon.com/gp/slredirect/picassoRedirect.html/re...
4. And maybe not as calculus specific, but doing Project Euler problems might also be useful.