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.