Check out the extended preview here: https://minireference.com/static/excerpts/noBSmathphys_v5_pr...
Here is the concept map from the book which might be useful to you as general orientation in the "concept space", now matter which book you're learning from: https://minireference.com/static/conceptmaps/math_and_physic...
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...
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.
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!):
* Breaking down the quadratic formula and explaining imaginary numbers: https://chat.openai.com/share/b267e954-fa0b-46ea-8fd6-11f71a...
* Explaining properties of logs: https://chat.openai.com/share/01d79b4f-81b4-46b3-9767-be93b7...
* e and natural logs: https://chat.openai.com/share/fbb3f957-ed6f-4c1a-af95-917d0e...
* Word problems: https://chat.openai.com/share/502f7a91-9c2c-4c74-834a-5c791a...
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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!
In terms of courses I am looking at Khan Academy but also https://ocw.mit.edu/courses/18-01sc-single-variable-calculus... (haven't started so I cant speak for quality)
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.
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.
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].
[0] https://www.amazon.com/gp/product/0767922719?linkId=98a11bfd...
[1] https://calnewport.com/case-study-how-i-got-the-highest-grad...
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.
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.
> 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!
Could you elaborate please ?
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!