Monks, Distracted

We’ve always been easily distracted. For survival’s sake. Didn’t notice the large predator rustling in the bushes? Then you’re dead meat. But what if you didn’t have to worry about hungry lions, and you could cloister yourself away from distractions? Turns out that’s not so easy either. The subject of Jamie Kreiner’s book, The Wandering Mind, is what medieval monks and the desert fathers thought about distractions and how to avoid them. Their goal was to focus on God; but it’s hard to beat evolution.

 

What did monks from the 300-900 C.E. in Europe and around the Mediterranean try to do? Some tried to get away from it all. They lived as hermits and ascetics, away from the distractions of civilizations and other people. But as the fame of their devotion grew, they would get more and more visitors wanting to learn from the holy hermit. It’s difficult to set yourself apart when hordes are knocking at the makeshift door of your cave or lean-to, out there in the wilderness. Also, when you’re alone, you can lose your anchor without the reference state of other human beings.

 

Thus, living communally became a popular option. Mutual support towards a common goal, iron sharpening iron, accountability, encouragement, and necessary rebuke – these might be an aid more so than a hindrance. Rules helped. But no one could agree which ones were best, thus the variety of monastic communities with their different codes of conduct. Should silence be enforced most of the day? Should there be a rigid schedule of work, prayer, mealtimes, alongside personal study and meditation? By following rules, decision-making can be reduced and delegated. But who decides who does what? What if there is a breach of discipline? Historical records suggest no agreement between different sects as to what was best.

 

Kreimer’s chapter on the body was eye-opening. I had expected to read about fasting and abstinence from sex, and the challenges of these “disciplines” and strategies that different groups employed. But I didn’t realize that personal hygiene was a big issue. Some groups believed in “high and dry”, depriving themselves of baths and clean apparel. Others wondered if such practices were “on oxymoronic performance of humility, that it fostered feelings of self-obsession rather than obliterating them.” They even argued about haircuts and shaving. I was also surprised to read about folks who worried about sleep. Some thought that reducing sleep “had the beneficial effect of restricting the mind’s susceptibility to distracting dreams and demons.” Praying and keeping vigil in the middle of the night could be considered a virtue.

 

The second half of Kreimer’s book discusses how reading, memory, and mental exercises played a role in the life of monks. Monasteries were book repositories – the local libraries of the medieval world. Reading was thought to help focus the mind, but it could also be distracting especially if it was too interesting and lodged itself too firmly in one’s mind thereby becoming a distraction. Active reading was encouraged, by taking notes in the margin or rewriting what you had read in a way that clarified the subject matter. That’s likely how commentaries got started. Monks were also encouraged to memorize portions of scripture or important sayings and stories. The memory palace was one among several strategies, as were visual mnemonics. Having memorized something, one could then meditate on it, twisting it around in one’s mind to get at the meat. Also highly prized was the art and discipline of metacognition – thinking about thinking – so you could recognize distractions and consciously teach yourself to refocus your concentration.

 

Back before it was popularized in psychology today, the concept of flow was something monks thought about and, in some cases, tried to achieve. When in that state of flow, it was if time no longer mattered, distractions ceased, and one medieval practitioner observed “the feeling of having his mind grow so calm and his heart expand so much that it felt as if heaven and earth were contained within him”. A transcendence of sorts, except difficult to describe experientially. Even metacognition has dissolved. The popularity of meditation techniques and ideas of achieving a state of Zen-like calm is still sought after today, as it was centuries ago.

 

Did I learn anything about my own wandering mind from the monks’ experience? While I have become more hermit-like since the pandemic, I’m not sure it improved my focus. I’m certainly not going to stop taking a shower and I enjoy my beauty sleep and comfortable bed. I read a lot, but perhaps I don’t read actively enough. Maybe that’s why I retain less, but I suspect that an aging brain has different priorities when processing. I memorize less and less, but I’ve built significant knowledge in chemistry that I can recall at will. I do mull over such concepts and try to push myself towards greater depth of knowledge. I think my metacognition has improved ever since I started trying to help students improve in this area. All this seemed natural to do as part of my job and my intersecting interests.

 

The medieval monks recognized that distraction was systemic, and not just an individual personal failing. They actively combatted distraction because for them it was a moral issue. The devil can whisper distractions in your mind’s ear. But the divine also speaks and breaks into our conscious thoughts. Therefore, discernment was highly prized, and could only be improved through discipline and study. You can only think “higher” thoughts if you keep practicing. I’m reminded of the advice of Sertillanges that I read not too long ago. I don’t think I have actively developed the discipline of focused thought, but for some reason I’m not bothered by it. Maybe I should be. And maybe there’s something I can learn from the medieval monks.

Experimental Details

I’ve been guilty of glossing over details. I suspect, but can’t remember, that I had educated myself on the experimental details that went into the discovery of the structure of the atom. Thomson, Rutherford and Millikan led teams that performed painstaking experiments that I now summarize in less than an hour. Worse still, over time I have conflated multiple experiments to make the conclusions easier to digest for my students. I’m reminded of my shortcomings while reading The Matter of Everything, written by the experimental physicist Suzie Sheehy. As a theorist, I acknowledge my blinders and lack of experience when it comes to appreciating the experimental details.

 

Sheehy focuses on those details! In the first three chapters, she covers three stories: the cathode ray tube, the gold foil experiment, and the photoelectric effect. I cover all three in my General Chemistry course. Today’s post is to remind myself where I have over-glossed the details, and do a better job when I resume teaching in the fall semester. I’d like to blame the G-Chem textbook which also glosses over the details, but it’s my own fault for not remembering. As the expert in the classroom, I should know better.

 

The Cathode Ray Tube. Sheehy begins her story with the discovery of X-rays and the use of the Crookes tube. I skip this story and go straight to J. J. Thomson’s experiments. I had forgotten how crucial vacuum is to this story. If most of the gas particles had not been removed (which was no easy task back then), they would interfere with the cathode ray, and it would be difficult to observe the crucial bending of the ray that led Thomson to conclude that this ray of “light” was made up of negatively charged particles (electrons!). The observable green glow was because stray electrons would hit gas particles or the wall, so-called “braking radiation” because electrons stopped by the glass would emit light (i.e., electromagnetic radiation). While Thomson tried different gases, the results were the same. I had conflated this by telling the students that this allowed him to conclude that no matter what the element, what all atoms had in common was the electrons, and where they differed was in the nature of the positive particle (the nucleus with different numbers of protons). Thomson did change the identity of the metal electrodes, which would support what I said, but I’ve been failing to mention this to the students. I resolve to do better.

 

The Gold-Foil Experiment. Thomson proposed a plum-pudding model for the atom. I tell the students that Rutherford wanted to test the model. What I fail to also say is that Rutherford had previously observed that shooting alpha-particles through a thin metal piece produced a fuzzy image (on a photographic plate), i.e., the particles were scattered in some way but no one knew why. His assistants Geiger and Marsden performed a series of experiments where they measured the deflection of alpha-particles against a metal slab, and then decreased the thickness of the metals so that more alpha-particles could pass through. Thus, they would need detectors for both the wide-angle reflection and for those that passed through. G-Chem textbooks show a detector “ring”. I have been erroneously telling the students that having detectors that picked up wide-angle deflections was fortuitous – but that’s wrong. They were there from the beginning because of how the experiment was carried out. I will get this right next time.

 

The Photoelectric Effect. I present the full set of experimental details before providing Einstein’s explanation; so does the G-Chem textbook. That’s backwards. Lenard’s early experiments puzzlingly showed that changing the light intensity had no correlation with the speed of ejected electrons. I fail to mention Millikan’s early experiments showing that the photoelectric effect was independent of temperature, adding to the puzzle. Einstein then came up with his strange “quantum explanation”, making the crucial predictions that the speed of ejected electrons would vary linearly with the frequency of light, that there would be a threshold frequency, and that changing the intensity would change the number of electrons emitted (but not their speeds). It took Millikan a good ten years to verify all this, and it took so long because he was trying to disprove Einstein’s weird hypothesis. Einstein deservedly won the Nobel prize for his insight; and I need to tell the story in the correct sequence to underscore this point.

 

In G-Chem, I don’t discuss the “cloud chamber” experiments of Wilson that supported the discoveries of the Thomson and Rutherford teams. How does one take a picture or a snapshot of a tiny particle too small to be seen by a light microscope? By seeing their tell-tale effects as they whiz as a wisp through the vapor – akin to the trails we see behind a jetplane. I don’t foresee myself covering this in G-Chem. I do spend all of five minutes on Millikan’s Oil Drop experiment, briefly going over the setup, and then quickly telling the students the result. I used to do more, but I think my truncated version is about right given the goals of the course. But I would like to incorporate changes to the three stories as recounted above, and I’m reminded that it is important to sweat the experimental details.

Mostly Moseley

“It has been the fate of some men to accomplish in their youth a work of surpassing importance, and then to have their career suddenly cut short by a great catastrophe.” Thus writes Bernard Jaffe in his book Crucibles of Chemistry in his chapter on Henry Moseley. How do we distinguish the elements in the periodic table from one another? Students learn this early on in chemistry class: By atomic number – the number of protons that an atom of an element possesses. It was Moseley who figured this out.

 

Shortly after earning his undergraduate degree, Moseley joined the research lab of Ernest Rutherford (of gold-foil experiment fame). At that time, anyone who joined the lab “went through a period of intensive training and experimentation in electricity, magnetism, optics, and radioactivity.” Thus, Moseley was very well trained as an experimentalist. His first project to determine the half-life of an actinium isotope led to his first published paper in 1911. That same year, Geiger and Marsden working in Rutherford’s lab determined that the positive charge of the atomic nucleus was roughly half its relative atomic mass. Moseley’s task was to figure out how exactly charge and mass were correlated.

 

To do this, Moseley fired electrons at a metal plate. The metal emanated X-rays and the characteristic lines of these X-ray spectra were photographed. Putting together these spectra led to the generated pattern known as “Moseley’s staircase” (shown below). The work was not easy. Building a device that allowed him to make these measurements was painstaking, but Moseley had the ingenuity, skills, and most importantly patience, to overcome one problem after another. In 1912, after a solid six months of work, he published his “Law of Atomic Numbers”. It was integrated into Mendeleev’s periodic table, helping to confirm Mendeleev’s intuition but also correct some of the discrepant ordering. Moseley was greatly aided by his discussions with Bohr who went on to publish his famous model of the atom in 1913.

 

Moseley’s work also helped disprove many claims that yet another new element had been discovering: “More than seventy elements had been announced during the [previous] generation to fill sixteen gaps in Mendeleev’s Table.” Moseley whittled the seventy down to nine, leaving only seven remaining gaps. Moseley even made predictions of what their X-ray spectra would look like, and that they will soon be found. The story of the discovery of these seven is now well known. Moseley’s law of atomic numbers also revived Prout’s Hypothesis, proposed a century before, but without gaining traction. Prout proposed that hydrogen was the basis of everything, and that all other elements were multiples of hydrogen. Where nuclei of atoms are considered, that’s not far from the truth.

 

Moseley enthusiastically signed up for active duty and went to the front lines in World War I, despite the pleading of Rutherford that he should work at a military research lab as his service. Moseley was killed at Gallipoli. He was twenty-six years old. His colleagues and fellow officers described him as a fearless man full of good cheer. He might well have won the Nobel prize, or played a key role in finding isotopes, and we can only speculate on what else he would have accomplished scientifically. Jaffe suggests we call atomic numbers Moseley Numbers; I like that idea!

How to Write

I started blogging for two reasons: In the short-term, I wanted to foster more thoughtful engagement from my students on what they found interesting or confusing in class. If I want students to do something, I should also do it myself. The long-term reason came from the realization that my writing wasn’t very good. I could churn out a jargon-laden stultifying academic research paper that checked all the necessary boxes, but I didn’t enjoy reading my own papers. And if I didn’t enjoy them, chances are few others would.

 

While I feel that my writing improved over the first five years or so, being more flexible and experimenting with different styles, lately I feel like I’m in a bit of a rut. In earlier days, I tried to think about what other readers might be interested in and how to keep them engaged. But I feel I’ve turned inward, writing mostly for myself and using my blog as an extended memory bank. I’m not sure this is a good thing, although it functionally serves its purpose.

 

This morning I felt a small jolt to improve my writing once again. The impetus is a refreshing article in the Chronicle of Higher Education by Rachel Toor. It is titled “How to Be Yourself on the Page” (linked here, may be behind a paywall). Her catchy subtitle: “You may understand the power of first-person writing, but can you wield it gracefully?” Lately, I’d rate my writing as rushed and awkward. Certainly not graceful. Toor’s key principle is that you need to tell a story, it needs characters, and the one you know best is yourself. That’s what first-person writing is all about.

 

Toor provides the following list of suggestions. I found each of them a helpful reminder. Each of her suggestions is bold-faced. Brief first-person commentary from me follows.

 

1. Think of a specific person as your reader. Lately, that person is me. But I’m not sure that’s the best choice. Sometimes I aimed my writing at my students or my fellow teachers; and these are probably the more interesting (and useful) articles. Other times, it was a stream of consciousness data dump to no one in particular. Toor suggests thinking about a specific individual when writing – I find this imagery helpful. For today’s article, that individual is me. I’m writing an advice column to myself.

 

2. Write a draft as an email. The idea behind this is to draw out why you’re writing on the particular topic. I don’t know how this works, but I will have to experiment. Today’s post is not being written as an email.

 

3. Write the way you speak. This is one suggestion I’ve incorporated early on. I think it has helped. However, I should combine it with #1 and think of someone I’m speaking to. At present, I’m speaking to myself. In my own head. And I can be quite verbose in my own head.

 

4. Make sure personal details in your writing serve a purpose. I haven’t thought about this point carefully. Perhaps I’ve been doing so implicitly; my approach is to be minimalistic about personal details but occasionally use them as anecdotal examples. Toor asks the question: “[Do] those details illuminate something important” that you’re trying to get across in your writing? I should stop to think about this when writing.

 

5. Don’t make yourself the hero. This seems like good advice. I’m self-deprecating but that’s culturally built-in and I haven’t been thoughtful about how I self-present. Echoing #4 above, I’m doubly reminded that I could be more thoughtful in the writing process. I need to up the meta-cognitive quotient.

 

6. Write long drafts and then cut ruthlessly. I don’t do this. I’m too lazy. That’s probably why some of my blog posts are over-long and bloated. Maybe a part of me secretly thinks that most of my writing is good and worth preserving. Maybe I subconsciously excuse myself by thinking I’m following #3. Except that I can be verbose in my own head. I don’t know when I’ll actually try this. But I should. Though not in today’s post. Too lazy.

 

7. But don’t fail to elaborate. I run into lapses where I get stuck and don’t know how to continue. So does Toor, and she makes a great suggestion: “After you write a declarative statement, add the phrase ‘by that I mean’ or ‘I think this because’ and see what you have to say.” This dovetails with #6. Toor explains: “Writing should be an exercise in discovery. Often we need to write our way into our subjects. The first three paragraphs… are throat-clearing. In the heat of banging out a draft, we use plenty of needless words. I find pleasure in reducing flab.” I like her philosophy. If only I had her patience and discipline as a writer.

 

Okay. It remains to be seen if I will apply any of these principles in my next few posts. I’m glad for the reminders.  

Naïve Analogies

Chemistry is an esoteric science. It’s all about trying to explain why nature behaves the way it does from the point of view of tiny particles that you and I cannot see. These invisible “electrons” are at the heart of chemistry, and their behavior leads to chemical bonds being made and broken, with atoms being exchanged between reacting molecules. How do we even being to imagine this invisible world, governed by mathematical rules, and couched in the language of symbols? We do it via naïve analogies.

 

Today’s post is on Chapter 7 of Surfaces and Essences by Douglas Hofstadter and Emmanuel Sander, continuing my previous post on how we move from novice to expert by abstraction and by using analogies to make reasonable arguments. But we have to start somewhere. And that somewhere must be in the world of the concrete, visible, and visually imaginable. Without this, it’s impossible to conceive the fleeting, invisible, and being able to “see” abstractly with the aid of mathematics. Naturally, we begin (as children) with naïve analogies.

 

The authors begin with the following scenario: A four-year old boy watches his father shaving. Shaving cream is applied. Then a metal object, that to him looks like a scraper (and not a scissors of a knife) is used to wipe away the cream. The cream must therefore dissolve the hair. He’s seen things dissolve in water before so that’s not so strange. This seems like a very reasonable conclusion given his life-experience and observations thus far. According to the authors naïve analogies in general “have a certain limited domain in which they are correct, and which justifies their existence and their likelihood of survival over years or possibly even decades.”

 

We constantly make naïve analogies and use them regularly when encountering something new. Such analogies are easily activated in our memories in the encounter – as we’re trying to make sense of our novel observations. The problem, however, is that while the naïve analogy may be true in some contexts, it becomes utterly misleading in others. Thus, the authors provide three key ideas about learning and education. First, in the classroom, all ideas are “understood via naïve analogies… children unconsciously make analogies to simple and familiar events… these unconscious analogies will control how they will incorporate new concepts.”

 

Second, and sometimes exasperating to teachers even though we should expect it: “naïve analogies are in general not eliminated by schooling. When teaching has an effect on a student, it usually just fine-tunes the set of contexts in which the student is inclined to apply a naïve analogy. The naïve notion does not displace the new concept being taught, but coexists with it. Both types of knowledge can then be exploited by a learner, but they will be useful in different contexts. And this is fortunate, since banishing naïve analogies from people’s minds would be extremely harmful.” (The authors provide examples.) This is why, we need to use multiple examples to help students learn a new concept: to help them figure out the contexts in which it is applicable and those in which it is not. Electronegativity comes to mind here!

 

Third, and this is something I wrestle with as a theorist: “a formal description of a given subject matter does not reflect the type of knowledge that allows one to feel comfortable in thinking about the domain. Humans do not generally feel comfortable manipulating formal structures; when faced with a new situation, they favor non-formal approaches. Learning is thus the building-up not of logical structures bot of well-organized repertoires of categories that themselves are under continual refinement.” The curse of the expert is that it’s so much more efficient and useful to think in terms of abstract formal models, be they mathematical or conceptual. To move students from novice to expert, we want them to acquire the ability to “see” in this way. But it turns out that the journey is hampered if you begin with the formal and abstract. The student is lost, at sea, with no touchpoints. And since they’re going to automatically apply a naïve analogy, it’s likely to way off base.

 

Hence, in education, familiarity is a crucial stepping stone. As the authors whimsically state: “for most of us, rockets are less familiar than cars”. And thus, we have the idiom that something “straightforward” is “not rocket-science”. And when we introduce terms such as electronegativity and electron affinity in chemistry class, we need to get students familiar with them through examples. Students get easily confused between the two and I’ve had years of observing this as students grapple with learning them. That’s also why I ask them to memorize the definitions. Being able to draw on the definition helps them sort out the confusion, even though it takes multiple attempts and examples to draw this out.

 

When I introduce atoms to students on the first day of class, we picture them as balls: hard spheres with a boundary. That’s an analogy they’re used to. Why not cubes? Or spiky tetrahedra? We talk about this with reference to Platonic Solids. We also discuss representation and draw pictures together. The “sizes” of different atoms is drawn on a flat surface as smaller and larger circles. The “identities” of different atoms constituting different “elements” is represented by coloring them or labeling them with a symbol (as Dalton did in his description of Atomic Theory). But atoms are not hard spheres. Rather, they have a tiny nucleus and a bunch of empty space where the electrons can be found. The analogy often used is an electron “cloud”. Orbitals are technically probability distributions, but students have trouble grasping this abstract idea, and so “cloud” sort of works – except when it doesn’t. That’s the challenge with naïve analogies. It doesn’t help that orbits and orbitals are conceptually quite different, but the words seems so closely related that students conflate the two.

 

When we get to chemical bonds, things become trickier. Students learn about ionic bonds, metallic bonds, and covalent bonds. I’d say that most of them think of electrostatics as some sort of occult force. No, they’d never use the word occult or magical to describe their thinking, but that’s likely the naïve analogy they’re making. Balls with plus signs attract balls with minus signs. If the signs are the same, they repel each other. Metallic bonds are an electron cloud that acts as a glue between positive ions. Covalent bonds are like hard sticks that connect atoms. It’s what the pictures (the ball-and-stick models) look like in textbooks and on the internet! We talk about different representations of the chemical bond in class. Sometimes we represent them as springs rather than sticks. Sometimes we represent them as sponge balls that merge into each other. And when orbitals get involved, students find it more confusing. Hybridization makes the beast stranger still.

 

In their book, Hofstadter and Sander focus on computer and technology-related analogies. Our computer is a desktop with folders and files. We can manipulate these with a mouse – a device that allows us to touch the virtual (what a concept!) and move objects around. Drag something to the Trash to delete it. Windows are opened and closed. You have an e-mail address, not a physical location but a symbolic one. I found these interesting, especially the examples of reverse analogies where computer lingo slides back into real-world operations. This made me think of the disconnect students sometimes have when working on the learning management system and the online homework system. If the interface isn’t smooth or intuitive, it adds to the student’s cognitive load. In terms of user experience, you want to have an interface that is unobtrusive, so much so that you don’t notice it’s there. And despite the many claims that students can learn “just as well” online (and there’s no doubt these systems are getting better), I think the learning happens differently and perhaps more directly face-to-face rather than through an interface. Conversation with a human (expert) interlocutor is a difference that makes a difference.

 

The authors also discuss mathematical equations and how physicists arrange their equations in a particular form to represent conceptual links. I’d never realized that Maxwell’s equations are written so that the physical “cause” is on the right-hand side and the “effect” is on the left-hand side. This is opposite to the mathematical “cause” which runs from left to right (“effect”). Bizarre. I started to recount all the equations my students use in chemistry class. While there are some in G-Chem, there are many more in P-Chem. For me, the equation encapsulates conceptual knowledge in a chunk making it a stepping stone for something more complex. For students, the equation is an operational manipulation to get to the “right” answer, and which one to use depends on the variables given in the question. Many students don’t actually try to understand what the question is asking conceptually. Which is why they don’t really understand chemistry, and they use poor study strategies. Trying to shift their way of thinking and approaching chemistry “problems” takes effort and practice. It’s why I repeatedly ask them to explain their answers in words. The answers can be surprising and revealing.

 

There is no Matrix plug-in to upload expertise. We humans learn through naïve analogies. As a teacher, I maintain a stock of analogies for different concepts in chemistry. But all analogies have their limitations, and it is important for students to learn to recognize such context-limitations. We can only do this through providing more examples. The road to expertise is paved with analogies. And hopefully they become less naïve as we climb the mountain into the stratosphere of abstraction.

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.