Teaching Systems Dynamics

“Stock and flow diagrams” are a nice graphical tool for modeling systems, part of a tradition called System Dynamics. Our new software based on category theory lets teams build models using these diagrams:

• John Baez, Xiaoyan Li, Sophie Libkind, Nathaniel D. Osgood and Eric Redekopp, A categorical framework for modeling with stock and flow diagrams, to appear in Mathematics for Public Health, Springer.

But here’s something cool Nathaniel just told me: people have had success teaching stock and flow diagrams to students starting at a young age! You can use these diagrams to teach math, economics, ecology, and other subjects in a unified way.

When you include functions describing the flows—shown as faucets above—you can turn these diagrams into differential equations. But you don’t need to do that for young kids: there’s a lot you can learn from these models in a purely qualitative way. Basic concepts like feedback, etc.

And once you introduce the flow functions, you can let software solve the resulting differential equations and graph their solutions even before the kids know anything like the definition of derivative! This is a good way to gently get them interested in calculus.

For example, below you can see a model of reindeer population on an island created by middle school students. The population soared and then crashed:

Students built System Dynamics models to study human population dynamics, non-renewable and renewable resource utilization, economic influences, etc. In these lessons students were asked to build the model, anticipate model behavior, explain discrepancies between anticipated model behavior and actual model output, analyze feedback, then test policies on the model to determine leverage points.

For more details try this:

• Diana M. Fisher, Systems thinking activities used in K-12 for up to two decades, Frontiers in Education 8 (2023).

You can even introduce various kinds of mathematical functions—linear, quadratic, exponential, etc.—by discussing stock-flow diagrams that give dynamics described by these functions:

• If water flows into a lake at a constant rate, the amount of water in the lake grows linearly.

• If money flows into your bank account at a rate proportional to the amount of money in your account, the amount in your account grows exponentially.

• If the velocity of an object grows linearly, its position changes quadratically.

And so on!

You can teach kids these patterns before they have any explicit knowledge of calculus, differential equations or even things like polynomials or exponentials. You can use these patterns to get them interested in these math concepts. Math is not just a bunch of symbols written on a page. It expresses the patterns our universe is made of—for example, patterns of how things move and change.