Abstraction and Expertise

I’m slowly reading my way through Surfaces and Essences by Douglas Hofstadter and Emmanuel Sander. It’s a tome; 500+ pages not including references. I’m about two-thirds through. The book is subtitled “Analogy as the Fuel and Fire of Thinking”. Essentially, the authors argue, learning and thinking consists of categorizing what we observe by making analogies. Most of the book draws its analogies through examples in language. Comparing idioms in two languages provides a stark example of how analogies differ in different cultures, yet baseline abstract categories may coincide at a “deeper” level. Observing the language “errors” of young children reveals how those categories fluidly change as they make sense of the world around them. I’m finding it very fascinating!

 

While I’m waiting to get to the final chapter on the power of analogy in scientific thought and discovery, which is why I started reading this book in the first place, I wanted to post some thoughts on the role of abstraction in learning. As a teacher, my role is to help move students from novice to expert. What is the road to expertise? How do experts gain rich interconnected knowledge that they retrieve from long-term memory? Hofstadter and Sander will argue that it comes from abstraction – the ability to strip away surface-level particularities and get to the deeper essence. Hence, the title of the book.

 

As a physical chemist, I’ve been thinking about the essence of P-Chem. More so because I will be participating in a midsummer workshop asking this essential question! What is P-Chem? Well, it’s physical chemistry, so a surface-level answer would be the intersection of physics and chemistry. (This is what a student might tell me before the first day of P-Chem class.) In my G-Chem class, I will occasionally tell my students that if they take P-Chem, they will get to explore the source of the equations we’re using. This prospect excites some students (the curious ones!), and makes others blanch. Nowadays, I think the essence of P-Chem is the construction of models, buttressed by mathematics, that allow us to both understand chemical phenomena and make subsequent useful predictions.

 

In Chapter 4, Hofstadter and Sander discuss an activity where students label a bunch of four-sided objects with names: parallelogram, quadrilateral, rectangle, rhombus, square. I’ve ordered them alphabetically here, but the more interesting question is how students think about these categories and how a mathematician considers what these names mean. The authors argue that expertise comes from vertical hierarchical ordering (with horizontal cases also present). Think about the image of how we classify organisms in a tangled tree, albeit one that is nested hierarchically: Kingdom, Phylum, Class, etc. For a non-expert like me, I can identify a dog and distinguish it from a cat. I can name several breeds of dogs, and while I know there are multiple breeds of small white dogs, I couldn’t tell you one from the other. An expert could, but there might not be one “correct” way to categorize them. A dog show judge and a biologist might well make different groupings when constructing their nested hierarchy.

 

The big question: Why is abstraction key to gaining expertise? The authors argue that experts are able to reason their way to an answer in their area of expertise, even when asked a question they don’t initially know the answer to right off the bat. A true novice can only throw up their hands and say “I don’t know” or provide random shot-in-the-dark answers that don’t get them any closer. The authors have this to say: “… true experts have knowledge not just about many specific cases in their domain, but also, through analogical links, a set of expectations about cases that are far less familiar to them… experts never have access to all categories, but a genuine expert has a dense enough mesh of categories that specific gaps at various levels can be gracefully sidestepped by the process of analogy-making, and this helps to fill in missing knowledge in any specific area of the domain…”

 

The examples and quotes above are from Chapter 4, titled “Abstraction and Inter-category sliding”. This ability to slide turns out to be crucial. The authors continue: “Whenever one changes one’s categorization of some aspect of a situation, one changes one’s perspective on the situation. Experts have so many potential perspectives that even in an unfamiliar situation, they can often find a highly pertinent one. Specific, concrete categories are precious to experts because, as they are all genuinely different from one another, they furnish the most precise insights that have been gained over a lifetime. On the other hand, general, abstract categories are also useful to experts because they summarize many cases at once, and also because they are closer to the ‘essence’, the ‘conceptual skeleton’ of concrete situations… down-to-earth categories allow one to be precise, while highly abstract categories allow one to be deep.”

 

Precision and depth. This is what it means to be an expert, rich in knowledge and its interconnections. Clearly, one needs to know a lot to be an expert. It’s why a curriculum needs to be knowledge-rich. But the interconnections need to be made and explicitly modeled. Analogy lends a hand here. Our brains are wired to make analogies, and that’s how we build that rich, interconnected knowledge. The expert sees beyond the surface to the essence. To the novice, only the surface stands out. And how does one get from the surface to the essence? By making abstractions via analogies. I’m reminded of the physicist joke that begins an answer to a real-world question: “Consider a spherical cow…”

 

I close this post with the three-disc Tower of Hanoi problem [PIC] which the authors use in Chapter 5. The goal is to move the three-disc stack on the left pole to the right pole. Only one disc can be moved at a time and it must be at the top of its “stack” to be moved. You can’t place a larger disc on top of a smaller disc. Children often take ~30 moves to “solve” it while an adult would take far less. I was able to picture in my mind’s eye the minimal 7-move solution. It’s not that I’m particularly smart. The authors have an intriguing explanation: The children self-impose an “extra rule” that a disc must be moved from left to middle, before it can be moved from middle to right. The children conceive motion as having to pass through “intermediate” states “along the way”. Adults see the source and the destination, i.e., they conceive the motion as a state change.

 

I hadn’t thought of it this way before, and it rings a bell. In thermodynamics, students have trouble with the idea of energy as a state function. The path doesn’t matter. In fact, teleportation is just as viable of a “path”. I hadn’t realized I hit on state change to solve the Tower of Hanoi problem. It was easy, even automatic. This reinforces the authors’ many language examples where native speakers don’t realize the analogies and abstractions they make in every day speech, most starkly in the use of idioms. They bypass the surface-level meaning of the idiom, and cut to the chase automatically. Did you think of old-school motion pictures when I used the phrase “cut to the chase”? Probably not, if you’re fluent in the English language. I went straight to the abstract essence and so did you. That’s what fluency does. That’s what expertise is.