Doing math on paper vs computer Generally, I do all my math with pen and paper, and only when done, type it via LaTeX.  The downsides to this are inefficiency (time taken to retype), difficulty of editing (lots of cross outs), and losing things that I end up not typing.  With so many downsides, why do I do it? Because I need to actually see the figures in mathematical notation to think about them, and waiting for LaTeX to compile them breaks up my train of thought.
I've tried repeatedly to try to "think" with LaTeX symbols, and it's failed: I can reason about $x < f(\frac \alpha {e^ \sqrt y}) \implies y \in \mathbb Q$ but not about x < f(\frac \alpha {e^ \sqrt y) \implies y \in \mathbb Q.
Likewise, I've tried "being patient" and waiting for the LaTeX to compile, and found it prohibitively disruptive to try while I'm still thinking and exploring.  Even the faster LaTeX previews make this hard - and for the slow ones, it's downright impossible.
Are there any means to do math via a computer? Are there tools or techniques designed for exploring math, as opposed to publishing or sharing it?
 A: I am a thoroughly devoted fan of Mathematica, which allows one to input formulas in "mathematical" typography (rather than confusing "computer science" typography).  (I'll teach a course on this starting next month.)
Unlike most tablets and such, Mathematica then performs the computations, thereby allowing you to think like a mathematician, without being burdened by long, error-prone hand calculation.
At the end, you can apply // TeXForm to an output to get LaTeX for pasting into documents such as scientific papers or homework assignments.

Or...

Try it... you'll never go back to pen and paper.

In a comment, the OP asks for an example showing how "mathematical" input is superior to traditional "computer science" input.  OK, here is a "computer science" (code) term as input:
Surd[x + Surd[x^2 + Surd[x^3 + Surd[x^4 + Surd[x^5, 6], 5], 4], 3], 2]

It is confusing and hides the structure.  And if you are missing a comma, or matching bracket, or have extra ones, or... it is very difficult to find such errors.
Here is the LaTeX version of that term:
\sqrt[2]{\sqrt[3]{x^2+\sqrt[4]{x^3+\sqrt[5]{\sqrt[6]{x^5}+x^4}}}+x}

Ugh!!!!!
By contrast, here is the "mathematical" (typeset) term in Mathematica as input:

If there is an error or improper syntax, you see it immediately.
Note that this is input and can be executed immediately.
SOOO much clearer.  Typesetting input this way helps you copy equations from books or (gulp) paper-and-pencil hand calculation.
Make sense?
A: I would propose using a tablet with stylus support. I am using samsung galaxy tab with the split screen of a textbook I am studying on the left side and the samsung notes on the right side. You can then modify the text on the tablet and even convert from handwriting to written text.
A bonus is that I also use a geogebra on that tablet so its really a usefull tool.
A: Most of the time, I just use a plain text file (open editor.exe on Windows, resp. TextEdit on Mac) to develop math. I know this is rather trivial, but often this is sufficient, probably at least for algebra related mathematics. For example, this recent answer of mine was completely done in a simple text file. When I want to write a fraction, I just write a/b in the file instead of \frac{a}{b}, the latter being much less intuitive to work with and longer to write. Of course, there is still some work to be done to write the result with LaTeX, but usually the end result is much shorter compared to what has been written before. Also, writing the whole thing again helps to reorganize the proofs in a better way. Believe it or not, but I have been doing this for almost 20 years now. I have been wasting quite a lot of paper in the first years (think about the trees...), but nowadays it's the exception. Of course for commutative diagrams and really complex projects nothing beats pen and paper.
A: Your way of working is correct: you are just lacking the right tool for actually doing your work: as you state correctly, you can't reason with things like x < f(\frac \alpha {e^ \sqrt y) \implies y \in \mathbb Q, but you might start using an editor for working with LaTeX instead of typing everything by hand. I just mean: I have written my thesis like this, but this was 25 years ago! Surely somebody has written a LaTeX editor by now? Maybe some people can add links towards LaTeX editors as a comment to my answer?
