Although many think of Common Core Mathematics as a standard that makes learning math more difficult and too disconnected from the mathematics that people use in their everyday lives, there is one type of mathematics focused on by Common Core that is both intuitive and useful for both the day to day and advanced scientific research.
This kind of math is called modeling.
Learning how to model mathematical problems is important because it is both something that we often do without realizing and a process by which very complicated problems can be made simpler. Formally teaching modeling to young kids makes it so they are ready for difficult STEM problems.
Some may balk at the idea that they use modeling in their day to day life, but it’s true. Professor of Education at the University of Chicago gives the example of traveling by plane. It’s a stressful experience. You have to leave your home with enough time to arrive at the airport, get your boarding pass, drop off your luggage, go through security, and arrive at your gate.
Any time you’ve anxiously stared at your watch while in the security line and silently cursed everyone in front of you for not taking their shoes off fast enough, and then sprinted to your gate, carry-on bag swinging wildly, to just barely made it onto your plane, you have been unsuccessful in modeling.
However, if you have ever serenely strolled throughout the airport, gone through security in as carefree and breezy way as possible, leisurely wandered to your gate and arrived with enough time to wait briefly to be called to board, you have effectively implemented a model.
Figuring out how long it takes to get from your front doorstep to the airplane is a complicated issue: you have no way of knowing what stop lights you will hit, how many other cars will be on the road, how many people will be in line ahead of you for boarding passes, or ahead of you in line to go through security, etc.
Modeling asks the question: how can we simplify this to individual parts that we can reasonably estimate?
In this case, the simpler parts we need are variables that we can later estimate to the best of our ability. The variables we decide we need could be the following: T, B, D, P, S, and G where T is the time you need to leave your house, B is the time your flight boards, D is the time to drive to the airport, P is the time it takes to get your boarding pass, L is the time it takes to drop off your luggage, S is the time it takes to go through security, and G is the time to get to your gate.
We then write down the relationship between the variables in a way that helps us solve our problem. The kind of relationship we write down is dependent on what is appropriate for our problem. In this case, an equation directly relating the variables is what we need to find. We can then come up this equation, or one similar, that represents the relationship between the variables: T = B – P – L – S – G.
We can come up with a reasonable estimate for each of these values based on information we do have. For example: considering the size of the airport, the time of day of your flight, whether there is a major holiday that people often fly home for, etc. will all give you an idea of how long to expect the security line to take.
The more accurate your estimates for the values of D, P, L, S, and G, the better your estimate of T (the time you need to leave your home) is. So if you regularly find yourself running through the airport, maybe you should go through this model and evaluate which of those values you’re underestimating.
The process we just used is a simple version of that described by Director of the Secondary Mathematics Program at the University of Arizona at Tuscan, Cynthia Oropesa Anhalt and Professor of Mathematics at Tulane University, Ricardo Cortez in their paper about mathematical modeling for the National Council of Teachers of Mathematics.
Anhalt and Cortez extend our process by assessing the solutions given by a particular model, and deciding whether additions need to be made to make the model produce better solutions. For instance, if in our airport problem, you find that you always wish you had time to buy a snack before getting on the plane, perhaps you should add a snack variable to your model.
It might seem like this is something that could be taught to adults or older students, but that it might be too conceptual for younger students, but there has been research done on middle school students’ ability to tackle modeling problems.
Associate Professor of Learning Sciences and Human Development A. Susan Jurow from the University of Colorado at Boulder has run an experiment in which she watched a group of middle school boys as they collaborated on a modeling problem. In her conclusions, Jurow talks about how how she believe students need to be taught modeling, in order to be able to model successfully. She emphasizes the use of discussions and reflections on modeling within the classroom.
So now we see how this method is intuitive for adults and kids alike, but how can this method be useful in fields like computer science and biology?
Modeling is valuable because it takes complicated problems and turns them into something with much simpler parts.
Researchers John F. Sanford and Jandeep T. Naidu from Philadelphia University believe being able to model problems is essential for being able to write computer programs. They translate deciding the variables we needed for the airport example into finding the “parameters and features” (for our purposes we can think of “parameter” as being synonymous to “variable”) that impact the solution to their programming problem.
What we then did was come up with an equation that relates our variables together. Sanford and Naidu also look at the relationship between their parameters, except instead of writing an equation, they begin to write their program, which they will use to solve their problem.
Sanford and Naidu advocate for teaching modeling to younger students because they see these parallels and recognize that a student who gets good at solving problems like our airport example will find it easier to solve those programming problems, because that thought process will be more natural for them.
Biologists like Milsee Mol, Milind S Patole, and Shailza Singh from the National Centre for Cell Science in Pune, India, also use modeling, but in a way that is even more different from our airport example. These researchers study a specific parasite (the leishmanial parasite) and look specifically at the way this parasite affects the afflicted cell’s signaling pathways to other cells. In order to find out more about this interruption, the Mol, Patole, and Singh modeled the network of cells around the afflicted one. Modeling this network let them track the way cells signaled to one another when one is afflicted by the parasite.
In this example, the individual parts of the problem are the cells, and their relationships are the connections between them that allow for immune system signaling between them. These researchers were able to take something as complex as cell signaling, and develop a simpler model that allowed them to learn more about the problem they are looking at.
Of course, with any problem we are trying to answer through modeling, there isn’t necessarily a “right” answer. In our airport problem, maybe you grouped going through security and getting to your gate into one variable, or added finding a parking place as its own variable, or maybe you purposefully over estimated the time needed for each variable because you wanted extra time to buy a snack before your flight. What matters is that the model you create is able to give an answer that is acceptable for you and your purposes.
We’ve seen why modeling is applicable to many different fields, which is why it can be such a great thing for people to begin to learn while still young. Giving people the tools they may need early on in their education allows for their mastery to grow, and for those skills to become more intuitive.
by: M. Knostman
References:
Anhalt, Cynthia and Cortez, Ricardo. “Mathematical Modeling: A Structured Process.” The Mathematics Teacher, 108.6 (2015): 446–452. JSTOR, www.jstor.org/stable/10.5951/mathteacher.108.6.0446.
Zalman Usiskin. “Mathematical Modeling and Pure Mathematics.” Mathematics Teaching in the Middle School, 20.8 (2015): 476–482, JSTOR, www.jstor.org/stable/10.5951/mathteacmiddscho.20.8.0476.
Jurow, A. Susan. “Generalizing in Interaction: Middle School Mathematics Students Making Mathematical Generalizations in a Population-Modeling Project.” Mind, Culture, and Activity, 11.4 (2004): 279–300, doi:10.1207/s15327884mca1104_4.
Sanford, John F., and Jaideep T. Naidu. “Mathematical Modeling and Computational Thinking.” Contemporary Issues in Education Research (Online), 10.2 (2017): 158-168, ProQuest Central, http://libproxy.lib.unc.edu/login?url=https://search-proquest- com.libproxy.lib.unc.edu/docview/1911759823?accountid=14244, doi:http://dx.doi.org.libproxy.lib.unc.edu/10.19030/cier.v10i2.9925.
Mol, Milsee, et al. “Signaling Networks in Leishmania Macrophages Deciphered through Integrated Systems Biology: a Mathematical Modeling Approach.” Systems and Synthetic Biology, 7.4, (2013): 185–195, doi:10.1007/s11693-013-9111-9.