Mathematical ModelingMy purpose here is to try to clarify what it means to "model with mathematics," and more broadly to deal with related phrases...
Recently, 45 states adopted what have come to be called the Common Core State Standards in Mathematics (CCSS-M). This document deals with a variety of content and pedagogical aspects of mathematics education for grades K-12. One section of the document deals with Mathematical Practices, and one of the mathematical practices, the fourth, uses language that is somewhat different from prior standards documents like the NCTM Standards. (NCTM refers to the National Council of Teachers of Mathematics which is the largest mathematics professional society concerned with K-12 mathematics education.)
For example, the Nobel Memorial Prize winning economist Robert Solow developed a mathematical model using a differential equation approach for understanding economic growth. This differential equation is now known as the Solow Equation. Kenneth Arrow won the Nobel Memorial Prize for work having to do with understanding issues related to elections, rankings, and voting.
For example, if you are a parent with a sick 8-year-old arriving at an emergency room that is staffed with two doctors, it may occur to you that there is a mathematical approach to designing a system as to how to schedule arrivals for treatment - some people in the waiting room may have arrived earlier than others but the level of danger in not being treated quickly for later arrivals may be much higher than for people who have been waiting for some time. One can imagine a system where one of the doctors treats people independent of arrival time in order of how dangerous it is for them not to be treated quickly, while the other doctor might treat people more with regard to their arrival time. However, one might still need to take into account what to do if two very ill people arrived at the same time. Also, are there circumstances under which one or more doctors would stop working on the patient they are treating to treat a new arrival because the rapidity of the treatment for the new arrival mattered so much? With more training in mathematics students/citizens can be taught to look for other situations where similar kinds of decisions arise. Thus, scheduling treatment of patients in an emergency room has features in common with scheduling operating rooms at a hospital. There are also analogies with scheduling the use of runways at an airport, checking out customers at a supermarket, or collecting tolls at a tunnel or bridge. All of these problems have some features of "scheduling" and also features of what happens when one must wait on line. Today, mathematics has subdisciplines that study scheduling problems and queuing problems. For scheduling, some of the mathematics is deterministic (doesn't take into account randomness issues), and some has stochastic (random) aspects. Problems that involve queues also have deterministic and stochastic aspects.
The word model
As used in common parlance the word model has many meanings. There are many phrases in which model appears in conjunction with other words. We hear of model homes, model railroads, and model planes. We hear of runway models - the glamorous women or handsome men who model clothing, including lingerie and underwear. Making model planes can be a hobby and they can be used in advertisements or museums to show people historic or interesting kinds of planes - a model of the Wright Brothers first aircraft or the new (no longer in service) supersonic plane, the Concorde. However, an aircraft company can make a model of a plane under development and test it in a wind tunnel to guarantee that its behavior is what was predicted in theory. It is much cheaper to see the pros and cons of a new design from a model than to build a prototype and then discover that its characteristics are not what one anticipated. Similarly, car manufactures will test model cars to check issues ranging from stability when making turns to gas mileage. In these contexts the word model is being used for a physical object which represents some other physical object. In many cases where industry has used physical models in the past, computer simulations or computer "models" are being substituted. A more prosaic physical model is a doll. Dolls allow children to interact with "representations" of people and thus improve social skills while having fun at the same time.
Mathematical modeling is now viewed as a branch of mathematics just as algebra, geometry, number theory and topology are viewed as branches of mathematics. Because it is a relatively new designation within mathematics, the way it is described or "defined" will vary a bit from place to place. However, it is useful to note that even an established branch of mathematics such as geometry is not easy to define because the domain of what is considered geometry has changed and is changing. Mathematical modeling took off as a subject when the term started to be used in some textbook titles. The courses that these texts were used for started to proliferate. While at one time the number of students who took advantage of higher education was quite small, the number of high school graduates going on to college has grown tremendously. As the pool of people who could major in mathematics grew, there was the need for courses that provided these students with skills to get jobs which used their mathematics in some way. Mathematical modeling courses played a role here. Mathematical modeling courses also help promote the dual aspects of mathematics - theory and putting theory to work.
Figure 1 (Modeling diagram)
Here is a brief account of the process as indicated by the diagram. One nearly always starts with a situation (upper left box, above) or phenomenon one wants to understand better. Because this situation involves lots of messy details, one wants to simplify it using mathematical (analytical) ideas to create something more tractable. The way this is done is through stating assumptions and purposefully disregarding a lot of the details but being explicit about those details. One can think of simplification in terms of a "control valve." When the control valve (see Figure 2) is turned a lot, one simplifies a large amount, while one can turn the valve a small amount which yields a model which is "closer" to the messy real world and that is used to get the mathematical model of the real world situation.
Figure 2 (A control valve)
(Courtesy of Wiki)
Usually, an underlying real world problem that one wants to address is tied to the real world situation, though sometimes the reason for constructing a mathematical model involves little more than intellectual curiosity to understand the world better. Once one has the mathematical model for the original situation, one tries to use the mathematics to obtain understanding, which typically means trying to use the model to solve the associated real world problem. Now comes the most critical step in applying mathematics in the real world. This is represented by the arrow on the bottom of Figure 1. One is interested in seeing how much insight one gets by applying the mathematical model. Does solving the problem expressed within the model help to better understand what is going on in the real world? If the results are not "helpful" or don't give you a way to deal with the situation that concerns you, you may want to adjust the original mathematical model by altering the way you performed the simplifications in order to get a mathematical model which is more helpful.
This model will predict short- and long-term exponential growth for the population. This may be true in the short run but cannot continue in the long run. Thus, one might want to modify one's population growth model. One wants to use a less simple assumption than that the growth rate for the population never changes because this model gives unrealistic predictions for the future.
The solution to this differential equation will be a function (the logistic function) which does not grow forever, but approaches a fixed level with time. This new model, while in many respects an improvement may still not fulfill the use for which it was designed and might have to be improved to achieve the purposes that were hoped for.
An operations research example of modeling
Operations research is the branch of mathematics concerned with helping governments and business operate more efficiently and smoothly. Imagine a section of urban roads consists of two-way streets (Figure 3). Once a year a line is painted down the center of each street. Design an efficient route for the truck that paints this center line, for the layout of streets shown below. The location of the garage for the painting truck is indicated at G.
Figure 3 (A schematic diagram of a section of streets)
A certain amount of modeling, simplifying, has gone on in drawing this diagram. The diagram is a schematic - it does not show stop signs, traffic lights or accurate widths for the streets, and it only hints at what the intersections of the streets are like, to name but a few of the "messy" real world details that are simplified.
Figure 4 (A graph model for the streets in Figure 3)
The diagram in Figure 4 illustrates issues in combinatorial or discrete geometry. Some of the vertices (dots) have two line segments which meet at them and others have 3 or 5 line segments at them. This geometrical property of the diagram reflects the complexity of the intersections in the original street network. The line segments In Figure 4 have physically different lengths but these lengths don't have any meaning for this model. If one wants to indicate that sections of the street have different lengths or driving times, that would be done by placing a number near each edge which represents a weight or a cost or by using a metrical diagram rather than the one we have used which does not represent distance information correctly. A diagram with weights would provide more information than the diagram above. For the moment, though, we will not think of the edges as having different lengths or weights. The tradeoff in doing simplifications in the construction of a model is the amount of time it might take to analyze the model (typically more complex models take more time to understand) as well as what progress can be made in "solving" the problem one is interested in. Often (but not always) simpler models are more amenable to easier solution or insight than more complex models.
Figure 5 (Portrait of Leonard Euler)
Euler was responsible for solving what is now called the Koenigsberg Bridge Problem.
Figure 6 (A schematic of the bridge pattern in Koenigsberg in Euler's day)
Today a tour of a connected graph (one in a single piece) that starts and ends at the same vertex and visits each edge of the graph once and only once is known as an Eulerian circuit.
Here is a proof of this theorem. Whatever vertex of the graph G one picks, if G has a tour that visits each edge of the graph once and only once, then at any vertex w the number of edges that one leaves from, must equal the number of edges that one enters w on. If one leaves w on an edge, there must be a different edge (unused edge) that one reenters w on. Thus, the edges at w can be counted in pairs, and the valence of w must be even.
While adding the edge from vertex 0 to vertex 7 turns the graph into a new graph which has an Eulerian circuit, the edge 07 does not duplicate an existing edge. To eulerize the graph one needs to repeat every edge, even though the number of odd-valent vertices is only 2. The reason we care about duplicating existing edges to obtain a graph with an Eulerian circuit rather than just adding any edges to get a new Eulerian graph is that the original graph is a model for an actual applied problem in painting lines down edges of existing roads. The edge 07 probably does not exist as a road in this situation! Only by duplicating existing edges do we get further insight into the the applied problem which concerns us. For many actual graphs with odd-valent vertices one can find a trial and error solution which is optimal. However, if one needs to find an optimal solution one can use the algorithm developed by Edmonds and Johnson. From a theoretician's point of view one might not only be interested in finding one optimal duplication pattern but all duplication patterns.
The theory and practice of modeling
A pioneer in describing and promoting the teaching of mathematical modeling has been Henry O. Pollak. Pollak was an undergraduate at Yale but got his doctorate degree in mathematics at Harvard. His thesis advisor was Lars Ahlfors, who along with Jesse Douglas, won the first Fields Medals.
Figure 9 (Photo of Henry Pollak, courtesy of Henry Pollak)
However, unlike many mathematicians trained to do research in mathematics, Pollak got involved with mathematics in the world outside of mathematics. In 1951, after finishing his doctorate he took a job at Bell Laboratories and eventually became director of its Mathematics and Statistics Research Center. After retiring from industry he has pursued a teaching career at Teachers College at Columbia. During his long and distinguished career he wrote joint research papers with a number of distinguished theoretical and applied mathematicians: Ronald Graham, David Slepian, Edmund Gilbert, Edward Coffman, Lawrence Shepp, David Johnson and Henry Landau as well as having been issued patents. During his long involvement with "industry" Pollak was involved with mathematics education. He was a member of CUPM (the Committee on the Undergraduate Program in Mathematics) from 1959 to 1965 as well as being involved with the School Mathematics Study Group (SMSG) and served as president of the MAA, the first person to hold that position from outside of academia, from 1975 to 1976. More important he has written wisely and extensively about the nature of mathematics and mathematical modeling.
The AMS encourages your comments, and hopes you will join the discussions. We review comments before they're posted, and those that are offensive, abusive, off-topic or promoting a commercial product, person or website will not be posted. Expressing disagreement is fine, but mutual respect is required.
Andrews, J. and R. McLone, Mathematical Modelling, Butterworths, London, l976.
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