Working my way up to calculus for the first time in my life, at nearly 40 years old. I've always hated math :(
When I was a kid, they always told me math would be super useful, especially if I liked computers. Well, 20+ years of a dev career later, I still have never used anything more than basic arithmetic and rudimentary algebra (to calculate responsive component sizes). But with web dev jobs going the way of the horse-drawn wagon, I figured it was time for a career change. Hoping to get into (civil/environmental) engineering instead, but I guess that field actually does use math, lol. We'll see how it goes...
In the meantime, also taking singing classes at the community college, and enjoying THAT way more. We performed at a nursing home a few weeks ago, and that brought SO much joy to the audience there, even though we're just a bunch of amateurs. It's just such a different reception than anything I've ever seen as a dev. Tech rarely inspires such joy.
If I could start all over again, I wish I would've pursued music over computer stuff. Much harder life though!
It sounds like you might enjoy my book on MACH+CALC, which comes with prerequisites included (a summary of the essential parts of high school math requited to understand calculus).
Last but not least, I highly recommend you check out the computer algebra system SymPy, which provide functionality for doing all kinds of basic algebra (`solve`, `simplify`, `expand`, etc.) You can use it to invent practice problems for yourslef, check answers, etc. It also has useful functions for calculus which you will need when you get to that. Here is a little tutorial on it:
https://minireference.com/static/tutorials/sympy_tutorial.pd...
Always just saw it as the handful of operations and a handful of generalized objects (sets, variables, etc) that represent data. So knowing those parts is all there is. Contrived compositions used in school to prove ourselves to some teacher just seems like patronizing helicopter parenting type vibes.
Went at most of my math classes with that in mind and didn’t really worry about relating it to past things
I don’t really buy into a notion like “more abstract” or “less” abstract. Everyone is its own abstract.
I dunno. That philosophy has worked for me so far shrug
> Does math have many prerequisites? [...] handful of operations and a handful of generalized objects (sets, variables, etc)
This handful of MATH is precisely what I was referring to as the prerequisites: notation, numbers, sets, equations (and the general procedure for solving them), functions, and a bit of geometry. I agree that with these concepts in place, you can tackle A LOT of topics, possibly filling in other knowledge in a just-in-time manner. This is what I find very fascinating (and a big part of my startup's mission): the fact that (re)learning a handful of math topics opens so many doors, that currently a vast number of adults are keeping firmly shut because they believe they are not "math people."
* Sorry for the confusion: there is a typo in my comment, I meant to write my book is on MECH+CALC, and the prerequisite are the handful of MATH topics you listed.
Have you heard about stokes theorem? Or Fourier Transforms? Those aren't particularly high level math concepts, but they have years worth of math prerequisites, without prerequisites there is no way to understand such things properly, and there are things that are way deeper with more prerequisites than those.
And those aren't some arbitrary equations, they are core concepts that are very useful in so many situations. There is a reason those are taught to most engineers.
I highly recommend the online book "Calculus at Moravian University", it does a pretty good job in building a fundamental understanding of calculus from the basics to multiple integration and vector calculus. I also used Joplin to take notes while reading, it's a notebook app where you can also use LateX (admittedly, many people won't like reading and taking notes on the phone, but I like it since it gives me a chance to learn even when you can't easily use books and laptop)
I'm taking a class at the local community college. It's OK, just some class notes and an e-textbook that I never bother to read. The pedagogy is much as I remember it, the teacher going through the problems on a whiteboard and expecting everyone to rote-memorize things.
But mostly, I'm learning from ChatGPT. You can enter (or take a picture of) any problem and ask it to break it down step by step and it does that very well, and explains it better than most resources I've found. There are some OK YouTube or KhanAcademy videos too, but overall I prefer ChatGPT for its higher signal to noise ratio.
At home I'll usually ask ChatGPT to explain the first one of a problem type, then try to do it again on my own and double-check it against the posted answers. For subsequent problems, I'll do it myself first (pen and paper or iPad), check it for correctness, and then ask ChatGPT for a breakdown if I screw up. I can usually tell it the mistake I made (i.e. how come I got X in step Y) and it can often correctly guess and explain where I went wrong.
Some examples (keep in mind that I'm still working up to calc, so still in pre-calc right now!):
Overall, I find this method of learning math (by rote memorization and parroting) very unsatisfying, and I'm unable to retain most of it in long-term memory. A few days after I learn anything I already forget how to do it. I ended up with an A in the class mostly just cramming the night before + morning of, using ChatGPT and class notes to refresh myself before the tests.
But IMO it's a terrible way to learn (at least for me) and part of why I hate math. I never really learn the whys and wherefores of anything, it's just a bunch of magic shortcuts and black-box algorithms that I have to memorize and re-use without any actual understanding. It's the educational equivalent of solving every problem with someone else's function/library :( I have no idea why anything works the way it does, only that I must remember it and re-use it exactly.
If anyone has a better approach to learning and retaining this stuff, I'm all ears!
Memorization can be a useful part of mathematics, but that's more true of higher level courses where you might expend some effort in memorizing definitions to have them at hand to use in proofs.
But most of the time memorization is a side-effect of just solving lots of problems, which is the most effective way to learn mathematics in the same way that writing programs is the most effective way to learn programming.
I'd highly recommend Cal Newport's books on studying [0] and his older blog posts on effective study habits [1]. Barbara Oakley's Learning How to Learn course on Coursera is also excellent [2].
In your current context, I'd probably just focus on time management and consistently grinding problems in areas you don't find easy with things like Schaum's Problem books and Outlines. You could use software like Anki to schedule review of problems and concepts that you've solved or understood.
If you aren't constrained by time, you could also use the Art of Problem Solving books to rebuild your math foundation at a much deeper level [3].
There's a popular math book called Infinite Powers by Steven Strogatz which is about the story of calculus. It's really good and may help bring the subject to life as opposed to rote learning problems.
I’m in the exact same boat as solardev (non-traditional student, taking precalculus, relying on LLMs, getting good grades but not having any true understanding) and I came here to recommend this same book.
It’s a great supplement because it gives you a chance to understand the ‘why’ of things and not just the ‘how’. The writing style is neither too dry nor too watered-down. It feels like the piece that was missing from K-12 education.
On a side note, I’ve found the LLMs to be terrible at math, but insanely good at writing LaTeX. I’ve been using GitHub Copilot to speed up the rewriting of my class notes and I’m just gobsmacked at how accurately it can print out the steps to some calculation after feeding in the original problem.
Pre-calc mathematics can be a bit boring because it consists mostly of a lot of technicalities that aren't themselves super interesting. A lot of those technicalities are connected in ways that unfortunately become only apparent in much higher-level maths. So I can understand that it's a bit of a pain, but I also think that it gets better afterwards (at least on the conceptual level).
I'm a bit skeptical about the use of ChatGPT because it can be very off about maths. If it's something it has seen exactly like this in its training materials, it will get it right (e.g. it will spit out the correct quadratic formula). But if you ask it to solve a specific problem that it hasn't seen before with those exact parameters, it might be bogus.
It could still help, but proceed with caution. For example, there's one part where you ask it why it multiplied by 4. Your question is wrong, because it multiplied by -4, but ChatGPT, always eager to please, doesn't correct you on that, instead it says:
"Multiplying by 4 is equivalent to dividing by −0.25 because 1/-0.25 = -4" - which is self-contradictory.
There are also other services you could use for this, such as symbolab (which is rule-based), but I think it needs a subscription to see all the steps.
I would recommend actually reading the textbook. Or, if it's a boring textbook, try out other textbooks. People find different kinds of explanations intuitive / different styles of exposition engaging, so you can experiment. I think it would make it easier for you to retain the material because you would learn some of the why.
In mathematics, it's often a better strategy to understand something and be able to know what it's true, than to just memorise the result.
The LLMs completely changed education for me. Whether it's math or music theory or programming, I'd rank ChatGPT amongst the best teachers I've ever known. (There are amazing real-human ones too, but it's really hit and miss!). For $20/mo it's totally worth it, way cheaper than tutoring or buying more textbooks. But for infrequent use, probably the free plan is enough...?
I tried MIT OCW but found the videos too long and tedious... it's hard for me to just sit still and listen to videos like that for hours =/ I chose an in-person math class on purpose just to have that real community feel and a live teacher, but YMMV... probably many would prefer the online or async versions instead.
Hi! I'm probably not your target audience here, unless you're specifically developing a product that targets math education for midlife washouts, lol. But if you want some rando dude's opinion, sure, I'd be happy to jump into a call with ya. Just let me know and I'll find a slot on your Calendly.
But I think you'd get better feedback from someone who's actually in your target audience/user demographic :) I'm just an annoying grumpy gramps who hates math.
I'm actually building a tool in this space. Would you be open to jump on a 15 mins call with me and help shape the product? https://calendly.com/vel-yan/15min
There is a book, “how to ace calculus, the streetwise guide” that sounds a little gimmicky but is actually a bit entertaining to read. Looks like it can be had on thrift books or eBay for about $5. It helped me understand some of the concepts at a deeper level than math text books. The problem with math text books are they’re mostly written by mathematicians who seem to think different than the rest of us, so when they try to explain things it’s in a mathematician way.
IMO taking Calculus later in life is probably easier for most (study of 1) people (assuming you have seen some of it before). I took Calculus at 19 years old and passed but I did not have any idea what it was really used for. Taking it again during Grad School (mid 30s) Calculus made way more sense. Ideas would click (mostly not CS related) but more in the general world view (think washers for nuts and bolts – area under a curve). I will be the first to admit that making front ends and tuning a DB will almost never need any calculus (but it does happen) and admit I did horrible in high school but turned it around in college (I was behind when I got there the first time for sure). How much do I remember right this second (not much), but I do feel its worth doing as if only a right of passage to say we deserve that 6-figure salary and maybe not feel too bad about calling our self’s engineers.
Good luck with the singing lessons and new job, but if your married (or have children) do NOT quit that day job just yet If not, well hell go for it my friend!!!
Lol, I hope I get to feel that way in hindsight! Right now it's just the struggle and the grind... math feels like a bad Asian MMO where you just run around crunching numbers to make other numbers go up slowly. And if you die/miss an assignment, you have to go back a few levels and do it all over again. Sigh.
> that 6-figure salary and maybe not feel too bad about calling our self’s engineers
Hah! I've only ever briefly made a 6-figure salary, and I quit that job because of its bureaucracy. I'm just some rando dev and I would NEVER call myself an engineer (even though my job title sometimes says that). It's an insult to all the REAL ones who actually survived the math, lol.
(Frontend) web dev has always been more similar to me to graphic design than anything I would call "engineering": It's pretty-looking shiny stuff backed by spaghetti code, not anything I'd ever trust to life-and-death scenarios like dams or bridges.
Anyway, that's not the point :)
> well hell go for it my friend!!!
Hah. This is the upside of a minimal DINK life. We don't have much money (at all), but we have fun!
chatgpt is proficient enough in writing web dev CRUD app code, which leads some to believe it'll lead to obsolescence of that particular skill set, and collapse the job market and pay for that, much like the skills of a horse-drawn wagon driver, or a buggy-whip manufacturer.
When I was a kid, they always told me math would be super useful, especially if I liked computers. Well, 20+ years of a dev career later, I still have never used anything more than basic arithmetic and rudimentary algebra (to calculate responsive component sizes). But with web dev jobs going the way of the horse-drawn wagon, I figured it was time for a career change. Hoping to get into (civil/environmental) engineering instead, but I guess that field actually does use math, lol. We'll see how it goes...
In the meantime, also taking singing classes at the community college, and enjoying THAT way more. We performed at a nursing home a few weeks ago, and that brought SO much joy to the audience there, even though we're just a bunch of amateurs. It's just such a different reception than anything I've ever seen as a dev. Tech rarely inspires such joy.
If I could start all over again, I wish I would've pursued music over computer stuff. Much harder life though!