Intermediate Algebra

This got long and a little ranty sorry :/ TLDR: Take a proofs course and get perspective outside the problems you are struggling on and return with the ability to ask questions about what your actually doing and how to verify you did it right.

I am not blaming you here but this mentality "The next time around I recognize the problem and basically just solve it from memory." Raises alarm bells for me. To me this says you are not engaging critically with a problem and arguable your pervious sentence tells me why. You are going over and over a problem which is fundamentally the same which is a disengaging process of learning and reasoning.

My suggestion is to focus less on you got the problem wrong and rather diagnosis why you got it wrong. Which is a very difficult thing to do alone. If you share a problem and the mistake you got I could try and help you. In general thought for me there are 2 major types of errors I run into when doing math. Yours may vary, but mine are Arithmetic errors and not understanding the theories.

Arithmetic -- i.e i said something like 1+1=3 somewhere and everything is wrong now. This happens a surprisingly a lot but the usually go back and check all your calculations almost never finds this issue for me. (if possible) I do have a lot more success with plug your answer back into the question and calculating what your input should be. This leads to the second point

Understanding the theories (and question) - If you understand what your solution is saying you should be able to find a different question to ask about your answer to be sure its right. If you can't you don't understand what your doing well enough. I have been picking on factoring because its easy to have hard problems that can be solved in many many different ways. But if you factor something essentially you found a zero so you should be able to do 2 things. One is plug your answer in and have the whole expression equal zero after (zero is always easiest to find because math is easy when you can set things to zero.) The other though is divide the the expression by (x-a). This reduce out perfectly and was a major way I checked my errors when I took a cryptography and number theory course.

Understanding something deeply is hard because there is no easy testable why to say that you do get it. But if you can ask questions like how do I know this answer is right its a good sign you have a lot more to learn what your actually doing. My final recommendation if you are trying to learn algebra is take a break from that. Do a intro to proofs class or lesson plan if you've never done that before. Those type of courses are suppose to teach systematic approaches to math that scale beyond do you know a formula that works. They also give you more intuition into what your doing because they teach you to think about the results of what you are saying and give perspective that the "algebra 2" courses you take in high school don't bother to even think about because they are focused on doing problems for problems sake which most people have enough problems they don't need more to solve for fun they want tools to handle a more diverse set better.

I think that set theory and other analytic can be instructive, but can also obscure what's happening. The exact encoding of numbers using sets is just an "implementation detail", in the sense that there's many ways you can build natural numbers (for example) using set theory and they are all equivalent.

So it's like learning data structures by coding in assembly, which is what Donald Knuth thinks is the right thing to do anyway, but some other teachers would disagree. But if you want to see some high level construction, you could look to eg. Tarski's synthetic construction of reals

https://en.wikipedia.org/wiki/Tarski%27s_axiomatization_of_t...

Which doesn't build reals using other theories as building blocks; real numbers are real numbers.

Anyway, from the analytic constructions of reals, I'm most partial to

https://en.wikipedia.org/wiki/Construction_of_the_real_numbe...

Which uses integers rather than sets as the building block, and is simpler than many constructions. And, of course integers themselves can be constructed out of sets, but they can be constructed out of lambda calculus terms as well https://en.wikipedia.org/wiki/Lambda_calculus#Encoding_datat... among many other constructions - but when we finally define integers, we can abstract away the implementation details (and that's really the crux of the question!)

Anyway there's a discussion of analytic vs synthetic mathematics in this post, https://golem.ph.utexas.edu/category/2015/02/introduction_to... (it seems that part 2 wasn't written unfortunately)

Some things are formally defined terms of art and some things aren't. There are some people here who don't merely want to learn a model for the naturals, they want some kind of spiritual essence of a number. It's useful to say that number is an informal term, and that the fuzzy boundaries you think with are adequate for communication.

Similarly, "crazy" is not a term of art for psychiatry. That doesn't mean a lack of definition, and lacking formalism does not mean inadequate for communication. By using a fuzzy term for communication, you imply a taste in curation and abstraction. It means you don't want to get into the details of what crazy means.

> That is like saying that physicists have no precise definition of "distance" because there is Euclidean distance, Geodesic distance, Hyperbolic distance, Hamming distance, Levenshtein distance etc etc etc

A metric is a term of art, and if that's the level of specificity you wanted, then it's adequate.

A student may be confused about why just these axions and not some others? It can then be explained that there in fact be alternate set of axions.

But still most teachers give us the same standard set of axions. Why? What would happen if they dropped some of them or replaced them with others?

I just wrote a TI83 program to solve the equation and then spit out a couple randomized "guesses" so I could "show my work".