Toy Models

Using Math in Physics:
4. Toy Models, by Joe Redish. 

In Intermediate Physics for Medicine and Biology, Russ Hobbie and I introduce many “toy models.” These are simplified models that strip away detail to expose fundamental processes. Why use toy models in biology, which is notoriously complex? To explore this question, I want to focus on an article by the late Joe Redish published in The Physics Teacher: “Using Math in Physics: 4. Toy Models” (Volume 59, Pages 683–688, 2021). The paper is one in a series of articles that Redish wrote about using math in physics, several of which will be cited in the 6th edition of IPMB.

In his introduction, Redish writes

As physicists, we consider our highly simplified models an obvious and natural way to approach physics. Mathematical models of complicated systems can be tricky, so the best way to understand the math is to take the simplest possible example that illustrates a phenomenon, then take it apart and put it back together again, matching the math with physical intuitions and building a mental blend of what the math means physically.

He goes on to say

Simple systems help build understanding: Learning to use this resource effectively to build new understanding is an important step in learning to be an effective scientist.

Toy models help students to learn

to blend physical concepts, knowledge, and intuition with mathematical symbols and processing.

I couldn’t agree more.

One important skill when using a toy model is deciding what to include and what to ignore. Redish addresses this issue:

In choosing a model, we have to decide what phenomena we are trying to describe, how to quantify the quantities involved, and, perhaps most important, what matters and what doesn’t. The world is too complex for us to include everything that’s going on. Deciding what matters and what can be ignored (at least at first) is an essential scientific skill, one that is, unfortunately, rarely taught explicitly even to our physics majors.

Toy models are useful for teaching students how to go back and forth from physics (and biology and medicine) to mathematics. When I was teaching, I noticed that many students understood the physics qualitatively and had good math skills, but had trouble translating between the two. They tend to think of these skill as being separate. Redish says

Once we’ve mapped our physical quantities onto math, we inherit processing tools from mathematics that let us solve problems that we might have difficulty solving. But once we have completed our calculation, we have to interpret the result back in the physics. What did the solution tell us about the physical world? Finally, we have to evaluate that interpretation. Is our model good enough for what we needed to do? Or are there refinements that we have to make, additional factors or effects that we really need to include?

Redish is explicit about why toy models are useful and important. 

We use toy models widely in introductory physics because they support multiple pedagogically valuable developments. 

• Toy models help students build the blend by focusing on the math-physics connection. 
• Toy models are built into most of our problems and can help build physical intuition. 
• Some toy models work way better than we might expect.

I consider the second bullet point to be particularly critical. Students need to gain intuition into how systems behave. They need insight. If they use no math, any insight is totally qualitative. If they use math, they risk missing the insight because they are focused entirely on manipulating the mathematical symbols. In graduate school, I took a course on general relativity. I learned how to do the math well enough to get an A, but I never felt I understood what was happening physically. I would have benefited from some toy models.

Some biologists and medical doctors like to put all possible details into a complicated “black box” computer model. While such an approach has its uses, such as for making numerical predictions to compare to experiments, it provides no insight. (Perhaps the researcher who writes the computer program gains some insight, but the user does not.) Redish says

Many real-world phenomena include lots of competing effects. Making sense of them, figuring out what matters most, and how to approach them can be challenging. Toy models are not just a way of learning to build the blend; they are an analytical tool for approaching a complicated system.

The sixth edition of Intermediate Physics for Medicine and Biology relies even more heavily on toy models than previous editions. If students can gain the intuition from these toy models and can practice building models and analyzing them mathematically, they will be ready to examine even more complicated and diverse biological and medical systems quantitatively.