What is the difference between word problems and 'math modeling' and 'computational' problems?

In math and science classes, students are all too familiar with 'word problems' that need to be solved on a homework set. These are problems that ask, in words, what a particular answer should be to an everyday type application of the topic being studied. In algebra, it may be something like "the sides of a rectangular plot of land has two sides three times as long as the short sides, and the sides add up to 120 meters. What is the length of a long side of the rectangle?"

You would need to think of an equation to solve this. These can be difficult for many students, but these types of problems at least show a practical side to the material being studied in a math class. It is also nice that there is a definite numerical answer, and only one correct answer.

Then there are the problems professional mathematicians, scientists, and engineers deal with on a daily basis: Math Modeling and Computational problems. These are generally much more complex, do not necessarily have a single correct answer, and can be open-ended, meaning you must go out and find information that can help you develop a possible solution. These types of problems have many possible ways of thinking about the problem, to the point where you must start with a series of assumptions that a solution could be built upon.

For instance, what if your problem was: For a specific city, develop an optimal recycling plan that will maximize participation by the city's inhabitants.

And that's it! No information given to you that hints at an answer, no data provided upfront. You must go out and find information, you might need to obtain information that does not yet exist, such as what is the current level of participation. There are different ways of collecting recyclables, such as given each household a recycling bin or having a central depository. There are cost restraints in the city budget. You will need a staff to do all this - how many do you hire to make this process effective, but also keep costs down so the city can afford it. You need facilities to process and separate the materials, transportation to then take those materials to plants that actually do recycling, and so on. Wow!!

These complex, open-ended problems need assumptions to be made, justified, and then used with real data to develop potential, viable solutions. When doing this type of problem, you actually need to develop equations, i.e. the mathematical model, that results from those assumptions and best-fit functions from the data, which can then be used to make predictions and find results to initial conditions that you have for the city.

Got all that?!

Check out the Moody's Mega Math Challenge Handbook for more details about mathematical modeling. There is a video with a brief explanation of math modeling from professional applied mathematicians. Check out archives of nationally winning papers from high school students, so you can see the level of work that is possible! There is a matrix of all the problem statements given for the COMAP math modeling contests, including the high school contest (HiMCM) for many more examples of what open-ended problems look like!

For an example of computational work, which typically involves writing computer simulations for complex systems, check this out. It is an example of how computational research is done for climate science.