Explaining Mathematical Modelling to a nonmathematician 
Due to the interdisciplinary nature of my project, I find myself collaborating a lot with nonmathematicians especially biologists, medical doctors, etc. I work mostly on mathematical models as applied to biological systems and one constant challenge I come across is explaining what mathematical modelling is to a nonmathematician.
  I've tried various approaches but I was wondering if anyone is willing to share very intuitive ways of explaining mathematical modelling to someone who hasn't studied mathematics beyond secondary school. It doesn't have to be restricted to biological systems, hence I would like to hear from people working on models in other fields.

Thanks   
 A: As a general principle, I'd say that mathematical modeling was the translation of intuition (derived from observation, experiment, and experience) into mathematical formalism. The goal is not so much to "solve the problem", that is frequently unrealistic.  Rather the goal is to get a good basis for making testable predictions (which will either tend to confirm the model or point to areas where the model must be modified).  Thus, if you tell me that "Y tends to grow proportionately with X if everything else stays the same" I can fit the line Y = CX to the data and we can see if we believe it or, alternatively, if it has to be modified in some ways (maybe it works only in a range, for instance).  If it fits, we can then take a look at how C might depend on some of the other parameters in our system, and so on.
Looking at mathematical models in Biology specifically, I have always greatly admired the little book by J. Maynard Smith Mathematical Ideas in Biology.  In the introduction, the author points out that it in Biology the problem is generally that the insights biologists tend to have does not readily lend itself to mathematical formalism.  This is very different from, say, finance in which the insights tend to be highly mathematical from the start.  In the latter sort of field, the problem tends to be working with complex formulas whereas in the former the math, once you get to the math, tends not to be that bad.  
Smith works many solid examples:  for example, he deduces from very basic mechanical and biological principles that for the most part creatures should all be able to jump about the same vertical distance (surprisingly accurate). That suggests that an interesting thing is to look at dramatic counterexamples (some wild cats, for example, can jump 12 feet vertically)...we have to imagine that there is interesting biology in the outliers.  Likewise basic modeling of simple pumps tells us, say, that giraffes have to have some unusual biology (else there's no way blood could make it to their brains).
Hope this is of some help!
A: First of all, I really like the question, I asked myself a lot of times how to explain and translate mathematical ideas into some "real world" applications. I am also currently working in an interdisciplinary field and the first thing one has to realize is


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*be especially aware of the slang you are usually using and drop it
Mathematical language has already become a part of your thinking and therefore is part of how you express ideas - this no way to talk to someone who might never has heard of continuity, compactness, limits you certainly could also speak in Latin, someone might understand what you mean, but it is very unlikely.


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*use the power of visualization if possible, keep it simple and clean and be guided by your own intuition you developed


If you want to transport the basic idea, then I would suggest using some graphs and visualization - of course this depends on what exactly you want to present, but in my experience you can always find a way to visualize. Keep the graphics simple and clean and people will follow you and start to develop own ideas based on their own imagination, which you already slightly influenced.
An example
I studied for a couple of months Interacting Particle Systems (IPS), which is a special branch of Markov processes - the underlying theory is quite complex. I was supposed to present my work within a seminar (it was a mathematical one, but people had a very different background). So instead of explaining the whole mathematical tool set (generators, semigroups etc) I focused on a very special IPS - the contact process, which is one of the first IPS you'll discover while working in this field. 
The important thing is now, that there exists a very intuitive way to explain what is going on and what the process is actually describing (I followed the great notes of Jan Swart).
The process can be visualized on a $\mathbb{Z}\times\mathbb{Z} $-grid, the whole construction can be done in a graphical way. The process translates then into a process of infection spreading.  
You have spontaneous recoveries, infection between closest neighbors and you can draw lines on the grid according to the propagation of the infection. Even the generator can be described in very simple words.
Well, it worked great, people were following and we were discussing problems which went way beyond of what I could have ever explained by just using mathematical slang. Of course there was a lost of precision, although the graphical representation can be done precise, but we gained much more generality. 
