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I've got a problem with regards to grasping and conceptualising mathematics.
I've been out of education for around 15 years and was OK with maths in school. However, I missed a lot of high school due to health problems and didn't study a lot of algebra, calc, etc.

Fast forward to now, I recently took up maths again and for some reason, I am really struggling to truly understand and conceptualise maths as an adult. I don't have dyscalculia or a learning issue, but rather I can't visualise the concepts or see the purpose of them, and they don't really sink in. I forget them very easily.

I really enjoy it when I manage to solve questions correctly but I feel as though I'm simply rote learning. I can't understand when people say "maths is everywhere, it's creative, it's the universal language, etc" because I'm lacking in the core concepts or theory behind it. Trigonometry for example, and cosine, sine, etc, completely goes over my head.

So I wondered whether anybody could please recommend any learning resources where I could learn the idea and theory behind it all, and how it's applicable in the real world.

Thanks so much.

Sometimes, such question have much more meaningful interest than just exercises. The process of learning mathematics, studying in the right way and getting to properly understand the true meaning of them is something that not only bothers lower-level or standard level students (of any age) but also anyone who is involved with mathematics. The process of learning never ends !

Now, after my philosophical entry, let's get to the facts. An adult may find it harder to adapt to studying mathematics and understanding new ideas compared to its younger self, because a younger brain is keen on learning things easier, while being fresher and more relieved of obligations. But, it is never late to learn new things !

Getting down to studying now, it really depends on the level of mathematics you want to comprehend. It also depends on the subject you're trying to study. For example, if you are interested in Geometry, visual representations, drawing sketches, seeing thorough examples will be of very big help. On the other hand, a more algebraic-theoritic or applied related mathematics course, needs first of all very good understanding of the theory and the ideas behind the tools used. What defines a good book for every student, while primarily opinion based, would always come down to the fact that it boasts a healthy balance of both theory and applications, while also some real life examples.

Understanding (fully and correctly) the theory behind the tools and the ideas of every chapter is very, very important. True understanding and grasping of things as mathematical substances is a really strong weapon into developing a strong and rigorous mathematical thought and approach to problems. I would suggest that your first priority should be the proper understanding of the theory, regardless the subject, before moving on to studying examples and exercises.

Important : A lot of students and studiers, make one very common but big mistake. Studying solved exercises by just reading through them and watching them does not make you truly better in mathematics. You may grasp some stuff, but the most important thing is practice. They say that practice makes perfect and while no-one can be perfect in mathematics, it definitely makes up for a lot of struggles. At start, it may seem a grind or it may make you feel like you have gaps in your knowledge, but continuous practicing (which means picking up a pen and trying to solve exercises) is very important. Even if you cannot solve an exercise completely, the process of thinking and trying also makes up for a very good level of future understanding.

Finally, for visualising concepts behind mathematics but also getting to see more stuff, internet is your friend. You can find a lot of applications, videos, textbooks of any kind and even online courses (most of them free).

A note : Do not get frustrated and never feel sad of not understanding something. Everybody goes through the process of failing to solve something or understanding something, but we shall never give up on our attempts to make ourselves better and try to find solutions to our problems. Seeking help by a professional mathematician, buying books or studying yours, searching online or practicing by yourself until you can solve the exercises that bother you, will lead you to getting rid of all the initial frustration.

A note (2): If any more information about why you want to learn mathematics or what courses and subject exactly you want to learn, are very welcome and you may carry on by commenting !

Hint: A great introduction into the concepts of mathematics is

• 
  What Is Mathematics? An Elementary Approach to Ideas and Methods by R. Courant and H. Robbins.

This classic is mathematically profound and good to grasp. It is from my point of view a wonderful example how mathematical thoughts can be provided in a pedagogically valuable manner to the interested audience.

From the referenced review of the book:

• It is ironic really. One of the ten lessons Gian-Carlo
  Rota wished he had been taught is that mathematicians are more likely to be remembered for their expository work than for their original research. Courant and Robbins may become the
  most convincing examples of this lesson.

I have a similar situation and found Khan Academy to do a great job explaining concepts and they have exercises so you can test what you have learned. It's free and they also have you do a placement test so you can start right where you need :)

https://www.khanacademy.org/math

Also, if you have questions the community is very helpful.

To the advice of existing comments and answers I just want to add something on one particular part of the original question:

I really enjoy it when I manage to solve questions correctly but I
feel as though I'm simply rote learning. I can't understand when
people say "maths is everywhere, it's creative, it's the universal
language, etc" because I'm lacking in the core concepts or theory
behind it.

Unfortunately the maths lessons in school focus a lot on rote learning, and can give the impression the whole subject is a limited series of repetitive problem types, when it's nothing of the kind. I don't think, however, that the solution to that in your case is to look at more advanced material. What I would recommend is material that gets across the true spirit of the subject, which is to follow the implications of what we know (or assume). This is what mathematicians mean when they talk about "proof". That's the real most important concept of the subject.

If I had to advise on how to encounter this at your level, I'd suggest dipping in at out of "proofs from the book". This is a concept due to Paul Erdős, who imagined a book in Heaven containing the most elegant proof of each theorem. There's even a book by that title that collects a few such proofs, but you needn't shell out money for that resource in particular. The internet is awash with examples. Some will be a little beyond you for the moment, but that's OK; skip them for now. A few at your own level will, at a minimum, fire up your appreciation for how creative the subject can look. Quite a few of these proofs are (i) accessible to newcomers and (ii) quite elegant.

"Proofs without words" (again, it's a book title where the book is less important than the concept) are another cute insight into what mathematical reasoning is all about. The idea is to prove a result with a diagram alone, whereby if you stare at it you can see what it proves. For example, here is a proof of the Pythagorean theorem, wherein the same four triangles can be positioned in the same large square in two ways. One causes the remaining area to be obviously one side of the equation we wish to prove; the other arrangement obviously gets the other side as the remaining area.

Donald Duck in Mathmagic Land is a classic exploration of both the creativity and real-world applicability (almost exclusively to geometry) of mathematics, at what I hope is a suitable level, and it's likely to inspire your choices for which topics you'll read about next. The only downside is there's an early misstatement of the first few decimal places of $pi$.

What does it mean to say maths is everywhere, or that it's the universal language? The above video partly addresses this; it even ends with a quotation from Galileo, who said "mathematics is the alphabet with which God has written the universe". Mathematics isn't really about numbers or trigonometry or quadratic equations; it's about what our assumptions imply. A small number of "axioms" can imply a stunning number of theorems, many of them unexpected. Just about every system in the real world obeys (exactly or approximately) a small list of assumptions we can state in quantifiable terms, and from these much follows, including surprising similarities. And that's why we find not only that maths is broadly applicable, but that the same few usual suspects show up a lot. For example, physics is mostly a story of a calculus topic called second-order differential equations.

I hope this answer helps. If you're really only interested in learning the specific school topics and what their syllabuses cover in it, maybe I've not helped much at all. But I must congratulate anyone voluntarily returning to the subject as an adult for knowing they're missing out on something fun, and many of them - including, with any luck, future readers of this page - will find this is just what they need.

I'm a programmer who started studying math in my spare time after more than a decade away from high school, and now I do discrete math for fun.

There are many advantages to studying as an adult. Adults are able to focus more, and have more discipline than kids, and can think better in many ways, so don't worry about being older!

I think one of the most helpful courses I did was this course about How To Learn Math. It has a lot of great information about how best to learn math.

It's been a while since I've done the course, but I think that they said it's important to:

• 
  Develop procedural fluency - knowing how to multiply, or factorize


• 
  Understand the big idea - like how division is about relating two quantities to each other


• 
  Connect concepts like how division and fractions and rates are all related, or the ways geometry and algebra are related.


• 
  Think in many ways about problems and concepts, like how you can solve some problems with pictures, or with numbers, or with geometry, or how you can calculate the same sum in many different ways: 3*5 = 5 + 5 + 5 = 30/2 = 3*3 + 3*2.

After doing that course, and understanding how I should be learning, I found the AlgebraX and GeometryX courses extremely valuable.

After those, I signed up on Khan Academy, and started doing all the math there. The important thing is to do it every day.

It pays off eventually. At a coding competition, I solved one of the problems with basic algebra, rather than using code. The other participants with university degrees were a little stunned that they didn't think of it, and had to resort to using programs to solve it.

I feel like the way mathematics is taught in schools doesn't focus on what it really is, instead focusing on the particular sub-fields that were useful for the industrial era economy. Algebra/Calculus/Geometry are of course brilliant, incredibly important, and have had a massive effect on the world and on every other part of mathematics. But we're taught those subjects in school as if they are what math is. But it's much larger than that.

For me, taking a Discrete Mathematics course in college is what finally woke my mind up and got me excited about what it was truly capable of. Discrete Math essentially gets down underneath the rote formulas people are taught in school. It focuses on logic, reasoning, and proofs, which is what math really is. It's a general system of reasoning about the world and proving things about it, which can apply to areas that don't even include numbers.

Discrete Mathematics: Mathematical Reasoning and Proof with Puzzles, Patterns, and Games is the book we used, and I found it incredibly accessible and well-written. I bet if you hunt around you could find it for cheaper than even the used listings on that Amazon link, but it's a book I enjoyed so much I decided to keep it rather than sell it like I did with most of my textbooks. It's still sitting on my shelf.