Rethinking the two dimensional abstract

Graphing paper is slowly going the way of the abacus.  It still works but why use it when there are more efficient and accurate models to use?  I find myself stuck in the 2 dimensional vortex when tutoring students in math.  Their first method of action is to refer to a graphing calculator or a software application that instantly will graph a function.  Technology can do functions that were something we could only think of in a past generation.  As educators, we can take this level of teaching to...well, another level.   

Prior to working in IT, I always thought of mathematical functions in terms of plotting ordered pairs on graphing paper and running a line through them to represent a straight line.  But what does that really mean? 

 I try to think in terms of the modern day thinker.  To simplify, a pair of variables can only have one function.  That applies in computer programming.  If you give a set of variables more that one function, you have essentially confused your computer.  It can not carry out a command that gives them two options.  Explaining algebra in terms of computer programming or language may be a way to bridge the gap with kids that do not see the relevance in graphing lines and plotting functions on paper.  

There is also an art to math and its usually rigid rules.  If a pair of variables can only have one function, then how do you explain programming for video games?  Are the functions algorithmic?  Do variables have an implied irrationality to them?  The art of math allows for variance or unpredictability.  When I was an artillery officer, I performed calculations for accurate targeting.  The science of math allows for accuracy in a world with no external variables.  No air, no terrain, no humidity, no human error.  In the real world those all exist which require an art to mathematics.  The Earth is not flat.  It rains.  There are mountains.  And humans are...human.  With modern technology, it is possible to add the art of mathematics in virtual graphing and conceptualization.  Death to 2 dimensions!