Mathematical Modeling in Physics: More than “Plug & Chug”
This is based on a presentation that I gave at the Wisconsin Society of Science Teacher’s Annual Conference in March 2017.
What is your experience when you see an equation? Fear of the unknown? Or the comfort of a relatively concise, simple explanation of a very complex phenomenon? (In this case, the Schrodinger’s equation for the Hydrogen atom.) One of the features that defines physics as a discipline is the obsession with writing out ideas in mathematical forms. Physicists favor mathematical models above all else for their precision and ability to make quantitative, testable predictions. Unfortunately, our students often miss the point of using equations. They tend to see equations as formulas, and their task is simply to figure out what numbers to plug in where.
Fortunately, the Next Generation Science Standards provide an excellent excuse for us to spend valuable class time helping our students to start thinking about physics as a way to mathematically model the world. We need to teach students to build, test, and revise mathematical models, and hopefully in the process gain some deeper understanding of the discipline of physics and how physicist use mathematical models to understand the world.
The Physicist’s Perspective
To a physicist, an equation is a representation of a scientific model, a conceptual explanation describing how or why a system behaves the way it does. Part of what defines physics as a discipline is this preference for mathematical representations of ideas. A mathematical description of a system is given higher status than a verbal description. A mathematical model can make very precise, quantitative predictions that can be tested experimentally. These mathematical models can play many roles in physics:
Identify causal links. In many physical systems, there is an interaction between two objects that causes a change in the system. Many mathematical models show how these phenomena are linked together. For example: A net force acting on an object causes an acceleration (Newton’s 2nd Law). A current moving through a wire causes a magnetic field to be generated (Ampere’s Law).
Description of the behavior of a system over time. The fundamental relationships such as Newton’s Laws can them be used to predict the behavior of a system over time. The most common example of this is called the “equation of motion” (a fancy way to say position as a function of time). We can use this equation to predict where the object will be at a given point in time in the future, or where it was at some point in the past. This one equation tells us everything about the motion of the object from the initial conditions to its position ten years from now. At the introductory level, the kinematics equations are a good example of how we use mathematical models to make predictions (e.g. with what speed will the ball hit the ground? Where will it land?).
To make sense of complex systems. Another way we use models is to make complicated problems easier to understand. To do this, scientists have invented various abstract concepts that help us to articulate the changes we observe in a system. Energy conservation is an example of this. Energy is not a real thing that I can hold in my hand; it is an abstract concept that was developed to understand changes in a system. Electric and magnetic fields are another abstract concept that help us to conceptualize how charged particles behave.
Mathematical Models & Scientific Practices
The Next Generation Science Standards (http://www.nextgenscience.org/) define a set of Scientific Practices, which recognize that there is no one right way to do science, but rather a collection of practices that constitute scientific inquiry. These practices vary slightly (or significantly) between scientific disciplines. Here we will explore how a physicist uses mathematical modeling as a tool when engaging in scientific inquiry.
Asking questions and defining problems. The goal of any scientific endeavor is to answer questions about how the world works. In physics, these questions are often framed in terms of mathematical models. Does a particular theoretical, mathematical model hold for a given system? To a physicist, the “hypothesis” of an experiment is a mathematical model.
Developing and using models. Developing a mathematical model goes hand in hand with asking the question. A scientific model is an explanation for how or why a system behaves the way it does. Scientific models can be represented in a variety of ways (verbal, visual, graphical, etc.) but the physicist’s preference is for the mathematical model because of their ability to make precise, testable predictions about the behavior of a system.
There are two types of models that we could build. Theoretical models are based on fundamental principles, such as energy conservation or Newton’s Laws. If we start with a theoretical model, we are usually trying to determine whether or not it holds for a particular circumstance. This is a deductive approach to physics. On the other hand, empirical models are based on data. For example, a best fit line through a set of data is an example of an empirical model. This represents an inductive approach to scientific inquiry. One could also argue for a third type of model: a computational model or computer simulation. Typically, computational models are based on theory or sometimes what is called a “semi-empirical” model that is based on a combination of theory and data.
We must keep in mind that a model is a simplification of a system. There is an old joke about a physicist being asked to figure out how much milk a cow will produce. The physicist responds “Imagine a spherical cow…” The joke is funny because the physicist has clearly taken the idea of simplifying a system too far. These simplifications are often the bane of introductory physics students who are forced to solve unrealistic problems about massless pulleys and frictionless inclines. There is a balance that one must strike when building a model. The system must be simplified to the extent that it can still be modeled by fundamental physics principles, but not simplified so much that we lose all the interesting information and are unable to solve the problem. (This is essentially what separates physics from the other fields of science. Biologists and chemists are solving problems that are too complex for physicists to simplify into a solvable equation.)
Planning and carrying out investigations. Once we have determined what the model is that is going to be tested, we have to design the experiment. While a clever student might have enough intuition to make a guess at what to measure in an experiment, we really need a more systematic approach. The model provides a clue as to what kind of data we want to collect. One experiment that I have my students do is to test the work-energy theorem. The setup is an Atwood machine (a cart on a track pulled by a hanging mass that falls off the end of the table). Essentially, they have to determine whether or not all of the work done by gravity is converted into kinetic energy (in which case the work-energy theorem is verified). If they set up the equations, they get something that looks like this: .
Looking at this expression gives them several possibilities for setting up the experiment. They can fix the masses, change the height, and then measure the resulting velocity of the cart. Alternatively, they can keep the height constant and vary the masses, and still measure the resulting velocity. Without the equation (mathematical model) as a prompt, students will try to measure the position of the cart, or the acceleration of the mass. They can measure these parameters, but they aren’t going to help them answer the question at hand.
Analyzing and interpreting data. Now that the data has been collected, we have to analyze it. This can happen in many different ways, depending on the experiment. One common type of analysis is to develop an empirical model based on the data. The best example of this is when you have students find the best-fit line (or curve) for their data. That best-fit line is often based on what we expect to find as the results of the experiment. Thomas Kuhn called this “theory-laden” data analysis. For example, if I drop a ball and plot the velocity vs. time data on a graph, I expect the result to be linear because I expect that the ball is undergoing constant acceleration. Even if the data isn’t quite looking linear, I might still choose that as my best-fit curve. From that best-fit line, I have an experimental value for the slope, which in this case I interpret to be the acceleration of the ball. I’m using my mathematical model to give physical meaning to the best-fit line.
Constructing explanations. At this point, I’ve developed a theoretical, mathematical model based on fundamental principles (say, kinematics). I’ve also analyzed my data so that I have an empirical model based on my data. The question is: does the empirical model agree with the theoretical model? If the answer is “yes”, then you can say that you have gathered evidence in support of your model and move on to the next step.
However, more often the answer is some version of “no” or “sort of,” which brings up the question of how close is “close enough”? Maybe I find that the slope is not as I expect. This is a great opportunity to talk about error analysis. How carefully we did we make our measurements? What is the inherent error in the measuring device? If you made multiple measurements, then you can have students put error bars on their data. As a general rule, if the error bars overlap with your best fit line, then we can say “close enough.” If a significant number of your error bars don’t intersect your best fit line, then it’s harder to build an argument that your data and your model are in agreement. (Depending on the level of the course, this is could be a good time to introduce statistical methods such as chi-squared or the least-squares fit.)
If the data and the model still don’t agree, there are two possibilities: either there is a problem with our data, or there is a problem with the model. Problems with the data could include measuring errors, as described above. Maybe the tools that you are using are just not sophisticated enough to get the level of precision you need to say whether or not your model holds for this circumstance. The other possibility is that we need to revise our model. What assumptions and approximations did we make? What are the limits of the model?
This where physics gets interesting! Have I discovered that there is an anomaly in the gravitational field? Or have I simply made an assumption in my model that doesn’t hold (e.g. drag can be neglected)? Pushing the limits of current models for understanding the physical world is what physicists do! They don’t simply want to reconfirm old models, they want to determine what the limits are to these old models, and if a model breaks down, to think about how they can be revised or replaced. For example, the dual discoveries of special relativity and quantum mechanics placed some limits on Newton’s Laws; they don’t work if objects are moving really fast or are really small. This doesn’t mean that Newton’s Laws are wrong; it means that they only work for certain cases. Understanding where the models break down is the exciting part of physics.
Engaging in argument from evidence. Although I always find it surprising, students need reminding that the evidence they are using to build their arguments is the data. Lately, I have been using the Claim-Evidence-Reason framework to help students structure their arguments. First, make a claim. Then support your claim with evidence (i.e. data). Finally, explain your reasoning. Two examples are below:
Claim: I am the tallest person in the world. (Side note: I am in fact 5' 1".)
Evidence: I am the tallest person in the room.
Reasoning: The room contains a good sample of the overall population, so I think if I’m the tallest person in the room I am the tallest person in the world.
Claim: The motion of coffee filters dropped off the balcony cannot be modeled using the kinematics equations.
Evidence: The velocity graph is not linear.
Reasoning: The kinematics equations require the object to undergo constant acceleration. The velocity graph is not linear, which indicates the acceleration is not constant. Therefore the kinematics equations cannot be used.
Then it’s up to the reader to determine whether or not they buy the argument. For the first example, you can probably find a flaw in my reasoning. You know that I am short, so you wonder what room am I in? Maybe it’s a room of 2nd graders, which means that my reasoning is not very good. The evidence has to support the argument, but the reasoning also has to be logical.
To a physicist, the argument should be based on the agreement (or not) of the theoretical and empirical models. If the models don’t agree, then there should be discussion about how both models could be improved or revised. If they do agree, then a similar analysis could be done speculating on where the models could break down, essentially articulating the limits of the current model.
Obtaining, evaluating, and communicating information. This scientific practice is used throughout the process described above, but often culminated in some kind of written paper, poster, or oral presentation. What I am looking for is that students are fluent in the various representations of whatever concept we are studying. I want them to be able to see the relationship and connections between graphs, equations, vector diagrams, and verbal descriptions, and to be able to articulate these ideas either in writing or orally.
Using mathematics and computational thinking. It seems silly to even address this one of the scientific practices, as the argument I’m trying to make is that mathematical models are important every step along the way. So to conclude, in physics students use mathematics and computational thinking in many ways:
Building & revising mathematical models
Using models to make predictions
Data analysis (Statistics, error analysis)
And the list could go on…
Implications for Instruction
What does this mean for teaching physics? To get students to engage in this sort of mathematical thinking, where equations are treated as mathematical models that are representations of scientific ideas, requires reframing how we talk about equations in the classroom. One way to engage students in this mindset is to have them building, testing, and revising mathematical models. This can take a lot of time, but the payoff could be huge.
One change that we made a few years ago to our first-semester physics courses (both calculus and algebra-based) was to reframe the first unit as “Modeling Motion.” We still include all of the standard parts of the curriculum (graphing, vectors, kinematics equations), but we now present it as a model building exercise. The students make a series of videos throughout the unit, and use video analysis software to track the motion. They develop increasingly sophisticated empirical models for motion based on the data and graphs, and then we introduce the kinematics equations after they have “discovered” free fall. The final project for this unit is for the students to do a video analysis of some two-dimensional motion and determine whether or not the kinematics equations can be applied to this system. The final report requires them to build an argument using multiple representations (vectors, diagrams, graphs, mathematical models).
On a smaller scale, an individual lab can be turned into a modeling exercise. For example, the standard projectile motion lab asks students to fire a projectile and hit a target. This can be re-framed in terms of modeling by asking the students to build a mathematical model that can predict where the ball will land. This requires students to find the initial conditions of the projectile (typically height of the launcher and initial velocity), and then solve for the position where the ball will land. In my class, I have them also make a computational model in Excel, where they can easily change the initial conditions and see how their prediction changes. Finally, they conduct the experiment and check to see if all three models (in this case theoretical, computational, and empirical) agree.
On an even smaller scale, we can think about rewriting the standard problems as modeling exercises. For example, take this standard problem:
A coin rolls along the top of a 1.33 m high desk with a constant velocity. It reaches the edge of the desk and hits the ground 1.00 m from the edge of the desk. (a) What was the velocity of the coin as it rolled across the desk? (b) What is the final velocity of the coin (magnitude and direction)?
We can take the same prompt, and step the students through a modeling approach to solving the problem:
A coin rolls along the top of a 1.33 m high desk with a constant velocity. It reaches the edge of the desk and hits the ground 1.00 m from the edge of the desk. (a) What assumptions can we make in modeling this motion? (b)What are the known initial conditions of the coin? (c) What questions can we answer about the motion of the coin? (d) Find the answer to one of your questions.
All of these curricular changes do take time and effort. In reality, the simplest thing that you can do as a physics teacher is to emphasize modeling in the way that you present the material. Just by introducing the concept of mathematical models as a form of scientific models, or using the word “model” instead of “formula” or “equation”, can make a difference in how students view the process of solving problems.