When scientists talk about developing models as part of their research, they can refer to quite different things. Even within industrial and applied mathematics, modeling can have a variety of meanings. To avoid confusion and clearly communicate the relevance of mathematical modeling, it's important to understand what are those different meanings.
To me, a mathematical model is a simplified description of a
phenomenon that allows us to gain quantitative
insight. In the best cases the model prunes out extraneous detail, leaving
only the key factors that illuminate some previously puzzling behavior.
Perhaps the best example comes from Isaac Newton's
masterpiece, the law of universal gravitation. Kepler had earlier proposed three laws of planetary motion motivated by
data alone: e.g., he had no a-priori reason to expect orbits to be ellipses
(his first law), but observations from Tycho Brahe led him to propose the
shape. Newton took an idea that was debatably
"in the air" at the time---the possibility of inverse square
gravitational attraction---and worked carefully through the implications to
show that Kepler's "curve fit" laws were actually inevitable
consequences of this. (Newton also
showed that much other astronomical data could all be understood via the new
model.) The simplicity of the model and
the depth of insight that came with it caused a revolution in our understanding
of the universe.
Mathematical models can lead to new insights in many fields, not
just physical systems. Alan Turing's bifurcation model for morphogenesis 
helps us understand how body plans can be encoded in chemical
interactions. Watts and Strogatz's
"small world" network model  explains why information can travel
fast on social networks even while they might remain clustered and cliquish. John Nash's "game theory" models for
human economic behavior (e.g., ) show why and how cooperation could have
Not every model leads to revolutionary insight, but the
aim of mathematical modeling (in my humble opinion) should be the same as
Newton's goal: to elucidate a natural (or other) phenomenon by connecting it to
ideas that are fundamental, understandable, and quantitatively testable.
This opinion puts me at odds with some others who describe
their work as modeling of natural phenomena. For example: are machine learning methods that generate
"nonparametric" models directly from data doing something
similar? Research into machine learning
is unquestionably valuable, but it does not appear to be modeling in the same
sense of the word.
There are many other
cases where "modeling" can take on differ meanings:
- Many biological, medical, and social scientific
models propose fundamental and understandable ideas for phenomena in question,
but lack the ability to make quantitative predictions that can be tested with
data.These qualitative models may contain the seeds of a quantitative mathematical
- In some econometric, statistical, and health modeling,
researchers aim to discover previously unknown correlations between variables in
data. This can be extremely significant, but can still lack the connection to a
fundamental quantitative mechanism (the insight)
that I consider key to mathematical modeling.
- A distinction should be made between
mathematical and computational
modeling, which is important in a wide range of fields and goes hand in hand
with mathematical modeling. Numerical
experiments consisting of high accuracy simulations can offer tests of
simplified intuitive models, and can provide useful predictions for engineering
applications, but complicated models that throw in "everything and the
kitchen sink" can also obscure key ideas rather than deliver insight.
- In medicine, a "model" may refer to a
living creature used to test treatments or conduct experiments (e.g., mouse
model of pancreatic cancer).
- The most popular sense of the word
"model" is (I surmise from internet search) the fashion sense, with
which there is not a lot of overlap or danger of confusion. Still, I can see at least two points of
intersection: (1) Danica McKellar, the actress who has also worked on
mathematical models for magnetic phenomena ; and (2) the study of draping
, which can help us understand the folding of dresses that models like
Danica wear on the red carpet.
 Turing, A. M. (1952). The
chemical basis of morphogenesis. Philosophical Transactions of the Royal
Society of London B: Biological Sciences, 237(641), 37-72.
 Watts, D. J., & Strogatz,
S. H. (1998). Collective dynamics of 'small-world' networks. Nature, 393(6684),
 Nash, J. (1951).
Non-cooperative games. Annals of mathematics, 286-295.
 Chayes, L., McKellar, D., &
Winn, B. (1998). Percolation and Gibbs states multiplicity for ferromagnetic
Ashkin-Teller models on. Journal of Physics A: Mathematical and General,
 Cerda, E., Mahadevan, L., & Pasini, J. M. (2004). The
elements of draping. Proceedings of the National Academy of Sciences, 101(7),
Photo caption: Mathematical
model? Actress and mathematical modeler Danica McKellar at the 2015
American Music Awards on November 22, 2015.
Dress shows folds of a characteristic wavelength that can be understood
via mathematical modeling. Image
obtained from https://www.flickr.com/photos/disneyabc/23225679855 and used
under Creative Commons license.