Vertical Learning
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Vertical Learning

Powerful Ideas and Statistical Mechanics

I had the great fortune to attend the Thinking about Thinking about Seymour symposium hosted by the on January 26, 2017. The title of the symposium refers to the idea that the best way to learn about thinking is to observe how we think as we do.

One of the things I do is curriculum development. Now, in his seminal book Mindstorms: Children, Computers, and Powerful Ideas, Seymour Papert was more than a little critical of the concept of curriculum. But just as he drew a clear distinction between math and “school math”, I see a similar distinction between curriculum and “school curriculum”.

In an article based on his talk, Rethinking Learning, Mitch Resnick writes:

Seymour had a more organic view of teaching and learning — and a more organic view on how ideas spread. The process is not like an engineer building a structure according to specifications; it’s more like a farmer or gardener tending to plants, creating an environment in which the plants will flourish.

Seymour was frustrated when schools simply taught children to draw squares and triangles, and stopped there, never truly engaging with the powerful ideas of programming. As Logo spread around the world, he worried that the core ideas were becoming diluted; he even coined the term “epistemological dilution” to describe the process.

I’m sure that Seymour would have similar concerns about many of the activities around making and coding and computational thinking today. And I’d agree with him. Even as new technologies proliferate, and more children are making and coding, Seymour’s powerful ideas are often missing.

And in his talk, Rethinking Ideas, Alan Kay states:

“Real forms” of many other powerful ideas can be found to allow children to get started understanding them earlier. […] If we want to improve education for children, let’s not try to teach them watered-down adult ideas. What we want is the full power of the adult ideas, but put in a form that the children’s experience can deal with.

As I develop curriculum, I am constantly expanding my thinking by asking myself: Is this “teaching” or providing the conditions for learning? Is there a powerful idea and is the full power of that idea in a form a child can engage with, reason and have intuitions about, and work on?

The atomic hypothesis

According to Alan Kay, powerful ideas take us from knowledge to insight, and insight creates “new contexts for thinking”. I experienced an epiphany in my junior year of college while taking physical chemistry, which grounds macroscopic phenomena in the basic interactions between particles. My entire perception of science suddenly shifted as I saw old ideas in new ways. I remember questioning why we hadn’t studied physical chemistry from the very beginning—for me, science would have made much more sense. That initial questioning is what led me to eventually leave chemical engineering to pursue a career in education.

Then, a few years later, I read Richard Feynman’s introduction to chapter 1 in The Feynman Lectures on Physics:

If, in some cataclysm, all of scientific knowledge were to be destroyed, and only one sentence passed on to the next generation of creatures, what statement would contain the most information in the fewest words? I believe it is the atomic hypothesis (or the atomic fact, or whatever you wish to call it) that all things are made of atoms—little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another. In that one sentence, you will see, there is an enormous amount of information about the world, if just a little imagination and thinking are applied.

The first formal lesson plan I developed with three classmates in my Master of Arts in Teaching program at Boston University was an attempt to put the full power of the atomic hypothesis in a form for children—and I’ve taken at least two other major whacks at it since. It’s incomprehensible to me that we can expect students to study and attempt to make sense of science for years without understanding and applying this organizing idea.

Insight and self-discovery

There is a reason why scientists often describe themselves as “standing on the shoulders of giants”. As a species, it took dozens of generations for us to develop the insights needed to formulate the atomic hypothesis. Each generation built on the discoveries made by earlier generations to reach the understanding and perspective we have now.

How can a child hope to replicate the same set of discoveries within a single lifetime? It seems impossible—which is why I spent years studying the distilled thinking of generations of scientists until my teachers thought I was prepared to study and apply the atomic hypothesis in physical chemistry as a college junior majoring in chemical engineering.

But what if the only reason why it took generations for us to arrive at the atomic hypothesis is because we grow up in a landscape where atoms are in our blind spot? We don’t see or experience them directly, so it takes a series of insights before we can even begin to think in those terms. What would happen if we grew up in a world where atoms were as real to us as trees or rocks—and we could think about and relate to them through our own every day experiences? That is the question Seymour Papert poses in Mindstorms.

Simulating random particle motion

Imagine a world where children play a game with pennies. Sitting in a row, they flip, pass, and count pennies to observe how the distribution of pennies changes over time. The mechanics of the game are simple:

  1. Flip your pennies.
  2. Pass the pennies that come up heads to the person on your right and the pennies that come up tails to the person on your left.
  3. In return, collect and count the pennies passed to you by your neighbors. Repeat these steps until the game ends.

The children vary the game by changing the initial conditions (how many children are in the row and how many pennies does each child have at the start of the game) and the boundary conditions (what happens at the ends of the row).

Now imagine that, as a rite of passage, most children invent a mechanical system that plays the game on its own. That way, a child can set the initial and boundary conditions for a game and come back hours later to analyze the outcome.

Run this simulation to see how the distribution of pennies changes over time. Edit the simulation to change its initial conditions. Pennies that come up tails at position 1 remain at position 1. Pennies that come up heads at position 6 remain at position 6.

This game is the hands-on activity my classmates and I created for our first lesson as prospective teachers at Boston University. It was (and is) the simplest and most concrete model of random particle motion I could think of which still holds onto the full power of the ideas and insights that I developed in physical chemistry and that Richard Feynman wrote about in his lectures on physics.

We model the system’s randomness using coin flips (statistical mechanics) and model particle motion as steps between discrete points (finite element analysis) along a single dimension.

Why is this important?

It’s important to note that I’m not saying a child would necessarily find this game fun or engaging. I might, but I’m someone who had an epiphany while studying physical chemistry. What I’m trying to point out is that a child can construct an adult understanding of the atomic hypothesis without needing to learn any of the insights made by earlier generations first.

Alan Kay talks about our three main brains for thinking. We use our “body brain” to flip, pass, and count the pennies. Our body brain knows on a deep instinctual level that the pennies are moving randomly — neither we nor the pennies are directing their motion. Yet, when we look up and scan the length of the row, our “eye brain” sees that, somehow, the pennies end up in predictable non-random distributions over time. How do the pennies know where to go? How do they organize themselves?

Run this simulation to see how the distribution of pennies changes over time. Edit the simulation to change its initial conditions. Pennies are added to or removed from positions 1 and 6 to keep the number of pennies at those two positions constant.

Just by playing the game, a child would discover how complex macroscopic behaviors arise from easy-to-understand local mechanisms and how dynamic systems reach equilibrium. We don’t introduce these ideas in school because we don’t believe children are ready for them. But that’s not true. While our “language brains” may not be ready for an idea because we lack the insight and context to understand it, we can understand the idea if we use our body and eye brains first, and then use our language brain to express and generalize what we think as we are doing and seeing. In Mindstorms, Papert describes this as providing raw materials for building intellectual structures.

Run this simulation to see how the distribution of pennies changes over time. Edit the simulation to change its initial conditions. Pennies are added to or removed from position 1 to keep the number of pennies at that position constant. Pennies that come up heads at position 6 remain at position 6.

Does it matter whether we understand the atomic hypothesis as children or adults? Personally, I think it does. As Alan Kay states:

If you learn a common sense version of reality when you’re young and try to tack science on it when you’re older, when you get in trouble, you tend to think in these terms that are not going to give you good intuitions; where if you learn the powerful ideas early, you will retreat to intuitions that are likely to help you a lot more.

This is why I still tend to retreat to anthropomorphisms even when thinking and talking about science—my instincts about people run much deeper than my instincts about atoms and particles. But my goal isn’t enabling children to discover a single powerful idea or even a set of powerful ideas. At the end of his talk, Alan Kay challenges us with the following question:

What kind of culture of powerful ideas must we create?

That’s something I’m always thinking about as I do curriculum development.



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