Top to bottom; bottom to top
Need I remind you: the human race now sends robots to Mars. That is a thing we do! We derive that absurd capability from generations of previous epistemological revolutions which we compound and extend to create new marvels.
I like to picture the tower of abstractions that enable this kind of progress. If you looked closely, you’d see its struts form an intimidating (and not altogether reassuring) lattice. Rocketry builds on chemical engineering and relativity but not pharmacology; flu vaccines build on chemical engineering and pharmacology but not (directly) relativity.
Now, we could examine the same data or questions from any position on this tower. One fun example: why is your hair the color that it is? Because you have a certain gene which expresses that phenotype! Or… because of the cellular machinery which expressed those genetics? The electrochemistry which drove those interactions? Optics? Cultural tropes influencing the mating habits which favored those genes?
One approach suggests that the observer can best understand a complex system by breaking it into composite parts, until she trips upon axioms, which are (perhaps temporarily) where truth lies. This is sometimes called reductionism. This word has other—incompatible!—definitions, but this is the one I’ll reference.
Meanwhile, others suggest that for many complex systems, the constituents hold only noise: real meaning emerges at the highest levels. That interpretation seems especially appealing when considering human consciousness and free will. After all, who wants to think of their being in terms of roiling, deterministic chemistry? I’ll call this approach holism (with the same lexicological caviats).
This is a figure from the “Ant Fugue” in Douglas Hofstadter’s Gödel, Escher, Bach. It contains letters made of other letters. At the highest level we read “MU.” The ’M’ comprises “HOLISM” three times; the ‘U,’ “REDUCTIONISM.” Each letter of the former contains the latter word, and vice versa. Then, gleefully (and invisibly in this reduction), the smallest letters are themselves formed by repeated “MU"s.
Each level of abstraction in this figure encodes information (here, words) in its emergent properties. If you näively apply reductionism and interpret the smallest components, or take ten steps back as holism prescribes, you read only "MU.” In Chinese, “mu” means nothingness, often in a spiritual sense.
It’s not impossible to discern the middle layers of words from either of these positions, but it’s also not expedient! Independent access to each layer of abstraction delivers novel perspectives and understanding. I suspect the same is true for any complex inquiry.
Now then: how does one acquire that independent access to each layer? Which reduces to asking how one can build expertise in general—how can we learn the many facets of any complex field?
A key issue in education is that the student’s goals are often holistic. She wants to learn how to make a robot, for instance, and might not be interested in the complex analysis necesary to understand the physics necessary to understand the electrical engineering necessary to understand the servos and sensors necessary to make it move.
Two dangers lurk here. The first and more apparent: if the student plays only with pre-built motors and components (without understanding or considering the principles by which they’re constructed), she’ll be seriously limited in how she can compose them and will endure endless headaches diagnosing and fixing problems.
The second and more insidious: this naïvely holistic (vocational?) approach could appear to work for quite some time. A year or two later, when she abuts the edges of the epistemological landscape visible from her position, she’ll face enormous psychological barriers as she moves down the stack. She must unlearn parochialisms, summon humility in a field she thought she understood, muster patience for moving—temporarily!—more slowly relative to her ostensible goals.
But if she builds her tower of understanding from principles—brick by reductionist brick—boredom may dominate good intentions and drown her emerging motivation.
I propose a hybrid approach: derive motivation and intuition from top-down exploration while cementing understanding and that crucial universal access with a bottom-up climb. Critically, though, she can ground and inspire the lower levels by studying them in context of her experiences with their emergent behaviors.
Some schools will offer a physics class for physics majors and a physics class for engineers. But generally the latter distinguishes itself by watering down the concepts, skipping a great deal of material, and hand-waving over derivations—not by providing motivating context.
With this idea in mind, personalized education could provide enormous benefits not just in terms of keeping pace with the student’s progress and prior experience, but also in captivating him through critical foundational material which would otherwise never take root.
I’ll leave you with a personal anecdote. At Caltech, I spent two years studying not computer science (my major) but math, physics, chemistry, biology, astronomy, laboratory practice, etc—because I had to! Every student had the same core requirements, regardless of major: Caltech history majors have studied statistical thermodynamics.
At the time, I was incredibly bitter about this situation! I felt like I was wasting my time, so I didn’t take the material seriously. I was supposed to have learned linear algebra my freshman year, but I didn’t: I learned it after I bumped up against a brick wall in my studies of machine learning, which I’d practiced for a few years with misleading success.
So I went back to re-learn linear algebra in a real context. This time I cared, so this time it stuck. This time I understood how these abstractions might be applied, so they made sense. Which was nice, because years later they enabled me to (more or less on a whim) radically improve how iOS’s 3D “page curl” effect tracks a user’s finger.
I’ve had similar experiences with those physics, chemistry, and biology classes. They’re a means, not an end, but it wasn’t until the end was in sight that I could truly learn and build from their subject matter.