I am a software engineer wanting to learn math. I also do a little bit of drawing.

What I am wondering is, how do you formulate a vague notion of something you're trying to model into a mathematical expression? Here is some background to give you more context of what exactly I mean...

When I write code, I have a vague notion of something I want to build. Say I want to build an [ORM](http://en.wikipedia.org/wiki/Object-relational_mapping) (a database basically), which is something I am currently doing as an open source project (it is building on ideas already discovered about database modeling, but missing from all the current solutions and the next evolution of a database). I often then look through some open source code for reference, to see how other's have done similar things. Then I *sketch* out the code a few different ways in a text file, and see if I can write pseudocode that will be simple and accomplish the objective. Then once I like the general code architecture, I write the real code.

When I draw, I have a vague notion of something I want to draw. Then I find some reference work perhaps, or look at something. Then I sketch out different ways of drawing that thing. Then once I like a sketch, I draw the real picture carefully based on that sketch.

So, it seems there should be a similar process for math, but I have been unsuccessful so far in figuring that out. I tried asking [this](https://math.stackexchange.com/questions/895417/how-to-model-a-real-world-graphical-structure-into-a-mathematical-formulation) and [that](https://math.stackexchange.com/questions/895596/how-do-you-express-a-tree-is-composed-of-leaves-branches-and-roots-using-math) question as a specific problem to solve, but the questions don't seem to resonate with the way mathematicians approach problems, so this is me asking a different way :)

When I first was learning to code, I had an idea in mind of what I wanted to build, and so stumbled through tutorials and such that showed me how to model things. After a few months, I could easily take an abstract concept and formulate it into code (functions, classes, variables, etc.).

When I was first learning to draw (still kinda in that phase), everyone says "just draw or just practice". But that doesn't work for me, I have to have a sensible method behind it, and motivation. Motivation comes from having a specific example of something to draw. For me that was nature drawing, I wanted to draw a leaf. And it took a while to find someone (a nature illustrator) who could show me how they think about taking an actual leaf from nature, and abstracting it into a drawing.

Now I'm wondering the same thing for math. How do you take an abstract, vague notion of something, and convert it into a mathematical expression? The two questions I linked to were specific examples I wanted to use to try and understand, but maybe they aren't good examples. I would try asking another specific question but I am a little disheartened.

I have read most of [How to Prove It](http://rads.stackoverflow.com/amzn/click/0521675995), which has been very helpful. But, it doesn't provide much guidance on the process of turning vague notions (like the ones linked) into mathematical expressions.

The main thing is, I am not learning anything by doing textbook practice problems that say "Given you have a set A = {1, 2, 3} and B = {4, 5, 6}, then the union A∪B = {1, 2, 3, 4, 5, 6}" type thing. They are very demotivating. I need a real-life example to learn from, and those two questions were a problem I thought could be formulated into a mathematical expression. Since those were bad questions, the general question is, how do you formulate some vague notion into a mathematical expression?

I am mainly looking for a specific example of how you take some problem/pattern/model, and start sketching out variables and mathematical expressions (using the symbols like $\sum$ and such) to try to describe the pattern/model. It doesn't have to be something you completely solve (or that is even solvable), just the process of how you model things out using mathematical symbols.