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Our undergraduate mathematics program has recently seen a large drop-off in majors (suspected reason: our growing (but separate) undergraduate statistics program is seen as being a more employable degree). In an attempt to appeal to students looking for a more employable math degree, we've discussed adding more modeling to our program.

However, I have to admit that (as a pure theory person) I'm not clear what's modeling versus application versus problems in context. Consider the following standard problems from Calculus:

1) Farmer Brown has some fence and wants to put it in a rectangle that encloses the largest area. Determine the dimensions.

2) Find the work required to build an Egyptian pyramid given the dimensions and the density of stone.

3) Pure water is flowing into a salt-water tank at a given rate. At the same time, salty water is leaving the tank at a given rate. Determine the salinity at time $t$.

4) What's the velocity of a car that's traveling down a straight highway with given position at time $t$?

Problems 2 & 3 seem to me like applications. Problems 1 & 4 seem to me like problems with context. Are any of them modeling? If not (or if so), then what makes modeling modeling?

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What is Mathematical Modeling?

You might think this to be a simple, straightforward answer, but unfortunately we have no such luck. The definition of mathematical modeling varies depending on the author unlike other more clearly defined mathematical terms like prime or group.

Pollak Definition

Arguably, one leading mathematician in the field of modeling is Pollak, former Bell Labs director who has also been an educator for the past 30 years or so. In his chapter A History of the Teaching of Modeling in Stanic and Kilpatrick's A History of School Mathematics, he opens with providing a clean definition for mathematical modeling. To quote from here:

What distinguishes modeling from other forms of applications of mathematics are (1) explicit attention at the beginning to the process of getting from the problem outside of mathematics to its mathematical formulation and (2) an explicit reconciliation between the mathematics and the real-world situation at the end. Throughout the modeling process, consideration is given to both the external world and the mathematics, and the results have to be both mathematically correct and reasonable in the real-world context. (p. 649)

Pollak goes on to describe an 8-step process that form mathematical modeling. These 8 steps can be split apart into different groupings to form other types of mathematics [e.g., "'Applied mathematics'" is traditionally a name for a collection of fields of mathematics that arise frequently in step 5." (p. 650)].

An abridged version of these 8 steps are below. [For the full version see H. Pollak, Solving problems in the real world, in Why Numbers Count: Quantitative Literacy for Tomorrow's America, (ed.) L. Steen, College Board, New York, 1997, pp. 91-105.]:

  1. Identify the real-world problem;
  2. Identify important factors on which to focus;
  3. Select which factors and which interrelations to keep and which to toss away;
  4. Translate the real-world problem into a mathematical one;
  5. Identify the mathematics required;
  6. Use mathematics to determine results;
  7. Translate results back into the real world; and
  8. Determine if your answer is reasonable. If it is, report on this. If not, repeat the process.

Common Core Definition

More recently, the authors of the Common Core have brought mathematical modeling to the forefront by placing it prominently within their standards document. It appears in two distinct locations: First, as one of the 8 Standards for Mathematical Practice and then again as a High School Standard unto itself. What is unique about this second appearance is that it does not provide bite-sized content pieces (e.g., CCSSM.HSG.C.A.4) as do all of the other sections (e.g., Geometry or Grade 4). The authors do define mathematical modeling in the High School section as:

"the process of choosing and using appropriate mathematics and statistics to analyze empirical situations, to understand them better, and to improve decisions."

Note that while there are similarities here to Pollak's definition above, the emphasis on the movement from the real world to the mathematical one, and back again, does not appear.

The Common Core authors do choose to include what I call "visual maps" of mathematical modeling. (Note, it was originally included in the PDF versions of the Standards but has become difficult to find on the online version.)

http://2.bp.blogspot.com/-yrpvvVk3uPg/T7gUWqbLeVI/AAAAAAAAAaw/djAmW9xor7I/s1600/MathModelingModel.png

This image does call to Pollak's definition of re-cycling through the steps if needed.

Are any of your Calculus questions modeling?

Based on Pollak's definition, I would say as stand alone problems, that no these are not modeling problems. Being given specific problems like these takes away from the earlier steps in the modeling process that Pollak indicates. It is certainly possible to use these problems as parts of the modeling process, and even to adjust them to include many or all of Pollak's requirements.

Based on the Common Core definition, I might claim that yes all of these could be considered modeling problems, as long as these are based on empirical and not theoretical situations (which could be argued in and of itself).

As for my personal opinion, I think that the Common Core definition is (understandably) lacking in some of the areas that Pollak's definition seems to detail. On the other hand, Pollak's definition might make it difficult to ever deem any textbook or teacher provided problem as modeling. The definition of mathematical modeling is very much in the eye of the beholder.

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  • $\begingroup$ @MichaelE2 is mathematics not a subset of science? In my mind, it always has been. What differentiates Pollak's method from other sciences would be steps 4-6 that specify work in the mathematical world. As a note, the visual map presented in the answer is one of many different ones you can find on modeling, and not all of them are mathematical in nature. $\endgroup$ – Andrew Sanfratello Jan 16 '16 at 17:59
  • $\begingroup$ The visual map presentation, a mention of Pollak, and some other relevant info around modifying pre-existing textbook word problems for mathematical modeling can be found in the Dec 2015 / Jan 2016 issue of the Mathematics Teacher: Wendt and Murphy's "Integrating Modeling Steps into the High School Curriculum." (Note also its excerpt: "Modeling activities such as those found in Gould, Murray, and Sanfratello (2012) ... are great starting points to get ideas of the types of modeling activities that teachers use in their classrooms" [p. 376].) $\endgroup$ – Benjamin Dickman Jan 19 '16 at 16:28
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These questions could technically be called modelling, in the same sort of way that questions like 'simplify $2x-4+3(x-1)$' can be called algebra. They are not modelling in the sense of being useful or interesting to students as examples of modelling.

Mathematical modelling is the process of turning a (physical) problem into a mathematical question you can analyse. You first work out what sort of objects are involved, and attempt to establish relationships between them. You then see what outcomes you get, compare with the situation you started with, and make adjustments accordingly, in a cycle until your model matches reality to an accuracy you're happy with. Writing this, it occurs to me that modelling might be said to be the mathematical version of the scientific method.

Typically modelling questions in maths classes lead to differential equations, and so often it is paired with that, but it would be possible to model other types of systems, although I'd have to think about how.

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In my opinion, what differentiates modeling from applications or contexts is data. To be a modeling problem, you need data. Either the students should be collecting data, or the data can be given to them. One possible "flow" for a modling problem is: data -> patterns/relationships -> creating mathematical model -> using the model to predict -> going back to the "physical world" to determine accuracy/results

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