I’ve been reading the book “Metazoa” by Peter Goddfrey-Smith, and I was struck by his description of the “molecular storm,” the setting for cellular life:
“Inside a cell, events occur on the nanoscale, the scale at which objects are measured in millionths of a millimeter, and the medium in which things happen is one of water. Matter in this environment behaves differently from anything in our midsize, dry-land world. At this scale, activity arises spontaneously, without having to be made to happen. In a phrase due to the biophysicist Peter Hoffmann, within any cell is a “molecular storm,” a ceaseless turmoil of collisions, attractions, and repulsions.”
Peter Hoffmann wrote a book that looks just as fascinating, titled “Life’s Ratchet: How Molecular Machines Extract Order from Chaos.” More on the molecular storm, from the description of his book:
“The secret of life, he says, is not some “vital force,” but the unique operations of the second law of thermodynamics at the nanoscale, where molecular machines from kinesins to DNA synthase, fueled by ATP, can harness the energy of the “molecular storm”—the random bombardment of water molecules at jet-plane speeds—to move and work. Hoffman convincingly demonstrates how such “motors” could have evolved from simpler self-assembling structures, but admits that how all these cellular components came to regulate one another so precisely is still a mystery.”
Later, Goddfrey-Smith points to the storm as a crucial difference between the functioning of a biological system like a cell and a computer:
“In the days of ubiquitous computers and AI, it is natural, almost inevitable, to ask about the relationships between living systems and these artifacts. Do organisms and computers do essentially the same thing with different materials? Similarities between the two do arise, often unexpectedly, but it’s also important to recognize dissimilarities. One difference is that much of what a cell does, its main business, is something a computer never has to do. A great deal of the activity in a cell is concerned with maintaining itself, keeping energy coming in, keeping a pattern of activity going despite decay and turnover in materials. Within living systems, the activities that look like the things computers also do — electrical switching and “information processing” — are always embedded within a sea, a mini-ecology, of other chemical processes. In cells, everything that happens takes place in a liquid medium, subject to the vicissitudes of the molecular storm and all the chemical digressions that living systems engage in. When we build a computer, we build something whose operation is more regular and uniform; we build something that will be distracted as little as possible by the undirected ruminations of its chemistry.”
All of this has me thinking about knowledge and its absence. Just as cells are constantly bombarded with molecules, we exist in a storm of images, sounds, smells, and thoughts. Of course we don’t experience it this way — thank you brains! — and instead experience it in a reasonably orderly manner. This order is thanks to two things: the biological structure of our minds, and the structure provided by what we learn.
But the chaos is still lurking beneath the surface. I think you can see this when people make mistakes, especially when they are experiencing cognitive overload. In those moments the structure of knowledge seems to bend under the stress of the challenge, and we get to see some of the random dynamic churning of association.
You can see this at times when students are pushed to apply numerical skills to a new context. You could maybe, maybe, find a way to explain how a student multiplied 0.8 by 1.6 and got 8.0. But to me, that misses the point. The point is that there is no explanation. It’s just the intellectual storm of experience and association bubbling across the mind.
Mistakes in math are usually like this: the result of having insufficient structure, not of having the wrong structure. Whatever structures are present are not strong enough to withstand the storm of experience.
This is why I bristle when people talk about student misconceptions. Not because I’m sheepish about judging student ideas as right or wrong. (I’m not into “alternate conceptions.”) And I’m not totally skeptical that misconceptions matter for learning, particularly in science or history. But with math, I think the situation is different. There aren’t a lot of ways to have conceptions about some of these abstract topics. There aren’t a lot of “folk theories” in math, I don’t think.
And when I personally don’t understand something in mathematics, it usually feels like the storm to me. Undifferentiated thoughts, slipping and sliding between different intellectual currents. I’m searching for something to hold on to! And I think our students often feel similarly.
A lot of math teachers feel that every mistake needs to be addressed and explained. If every mistake were a misconception — a faulty or limited mental structure — then I think this would make sense. But mistakes aren’t usually like this. They are chaotic, a snapshot of the storm. And because of this, they can only be addressed is by learning, which is to say building something new.