The Case For Discontinuous Curricula
why following a textbook is not the best strategy + autodidact apologism
My favourite thing about auto-didactism is you get to cheat.
One of the ways that happens is when I skip an important part of a subject because I either:
Couldn't be bothered to learn it at the time because I had other goals.
Took too long to wrap my head around it and decided to come back to it later.
To be fair, this sometimes ends with me doing perfectly alright without learning them. In which case it isn’t cheating as much as it’s efficient optimisation. But often, I simply can’t absorb a particular chunk of information into my web of understanding. At least, not at that moment.
Except this phenomenon isn’t specific to auto-didactism. It’s a part of any learning experience that follows a fixed course. Any ordered curriculum that has a sequence of modules, creates the possibility for such gaps to appear. But it's usually only self-learning where you're allowed to get away with it indefinitely.
And that's what got me thinking, why don't we have more variation in curriculum designs?
It may sound terribly inconvenient, but it's really just a matter of introducing a bit of randomness into the standard syllabus.
Instead of following the order of topics provided by a textbook, or even their own decades-old idiosyncratic methods, teachers would be encouraged to use a different ordering of sub-topics each year. Or, if you’re really feeling adventurous, teaching entire courses before their “official” prerequisites.
Sure, some things can't be taught the wrong way around- or can they?
The idea of objects was only one example. In the past, I’ve also skipped past complex numbers, certain econ equations, even trignometry!
Returning to them weeks later to find them much more easily comprehensible. Far from being the easy way out, I believe this is how learning *should* be. Relatively effortless comprehension, instead of needless head-against-wall-banging. Go off and solve some other problems, this one will be easier when you come back. Learn to love technical debt.
the greatest whitepill of all is that local minima are rare in high dimensional spaces - Roon
Sure, you could call this lazy instead of efficient. Maybe if I stuck with it, I’d make it through with the necessary knowledge earned sooner than I would have otherwise. Like real students do it.
But another view is that because auto-didacts aren’t usually trying to prove their knowledge to anybody but themselves, I’m simply being more honest. I know when I don’t understand things and have no incentive to pretend otherwise. This virtuous framing makes it easier for me to justify skipping past stuff I’d normally remain stuck on. Knowing I’ll return because I *want* to.
As more extreme examples, how bad would it really be if we didn't teach kids multiplication but instead jumped straight to division? Or went through linear algebra without touching matrices? Essay writing without caring about spelling?
My answer is "not that bad". Heck, we’d probably see a lot of fun stuff come out of it. You’d have a class with a bunch of different ways of understanding and doing multiplication, independently discovered and tailored to their individual intuitions.
And if my experience of how kids operate is any hint, the essays would contain whole new dialects and word-forming. If you really wanted to, you could always teach them the “right way” later on.
Granted, learners don't need to reinvent the wheel. But figuring out what the spokes are for isn't particularly difficult, and possibly a useful exercise. They’ll only get the chance to do that if they haven't already been told the reason.
Some fields lend themselves particularly well to this re-ordering process. The humanities (primarily history and philosophy) come to mind. But my argument is that even the more sequential fields can be learned the “wrong” way. Math, physiology or chemistry, they’re still worth exploring in a roundabout fashion.
As an example, if you went through a physics course without having heard of Newton’s laws, you’d probably end up developing a vague intuition of what they should be. And when you learnt them properly at a later date, they’d just *click*.
And those are only a few of the possible benefits.
Wanna teach problem solving and critical thinking? Why not add implicit problems into the learning process by making stuff ever-so-slightly-harder? Not “solve this triple integral in under 3 minutes” harder, but the challenge of “go explore and see what you can dig up”.
School is mostly about "this is right" whereas work is mostly "nobody knows, figure it out" and the delta is jarring. - Rohit Krishnan
If you accept the premise that self-education is the only kind of education that makes a difference in the long-run, then such a teaching process would be the ideal prepearation.
As any book-child knows, the way to ascribe meaning to unfamiliar words is not through the opening of a dictionary. That would be a cop-out, and one that (paradoxically?) takes too much effort in the long run.
Instead, the way it’s actually done is by seeing the word in a dozen different places, and guessing at it’s meaning by identifying the common context in which it appears. This “earned knowledge” proves to be far stickier than simply looking it up.
Sure, you may guess wrong once in a while, but you’ll be corrected pretty soon if you encounter the word often enough. Filling in the blanks like this is a useful exercise, and we should do more of it.
In fact, the correct place to do this is probably in the early, elementary part of an education. And not the later specialised stages, where things get really complex.
Just like reading, where it’s the rare-but-not-super-rare words that lend themselves best to self-discovery. It isn’t worth waiting to encounter “eschaton” again, I would rather google it instead.
Inventing your own definition for multiplication works because it’s both trivial and used often enough that you’d get corrected if it were wrong. Reinventing calculus on the other hand, is probably not the best of plans.
But you can still teach it in a weirder order. Skip details and let students fill them in. Count on the sheer randomness to generate rarer (and hopefully better) ideas and perspectives.
I think the downsides of such an approach are negligible in the long-run.
If you’re worried about high schoolers never using soh-cah-toa, engineers forgetting the gravitational constant, surgeons operating without knowing the Latin name for the elbow joint- What’s that? Doesn’t sound that scary, you say?
Well, of course not. I don’t intend to do away with tests or qualification standards. And we can still count on the usual selection methods to test the essential stuff and pick reliable experts.
Sure, there’s a certain class of people who take pride in not knowing stuff that ought to be essential to their lives. Like professional programmers claiming it’s okay to not remember how Big-O notation works because they never use it anyway. And they had perfectly-ordered syllabi!
It’s easy to see that the problem is, and will always be, willful ignorance. If you were ever truly interested in X, you'd have learnt it without being forced to.
Once upon a time, it may have been the case that bad teaching methods were inexcusable. Because teachers were the sole sources of knowledge. But
that was in the past
it happened anyway
Now we have search engines, and a multitude of explanations for any given topic. If you can’t be bothered to fill in the details yourself, the least you can do is look it up.
Of course, such a suggestion will never make it past the parent boards or the media. I can see the headlines already (Teachers willfully delude students! Systemic misleading rampant in schools!).
Because there's cultural forces at play here. We simply do not have systems that encourage, or even tolerate, the possibility of being wrong. We pay lip service to it, we even ask parents to encourage "imagination and fearlessness". And then we turn right around and mock mistakes and frown at anything that comes with short-term costs and uncertain benefits.
Something something bad Nash equilibriums.
An amusing side-effect of this was me developing a programming style that was pretty close to the functional programming paradigm without having heard of it before.
Great post, basically all of this fits with my inuitions/experiences from teaching high school biology for the last 5 years. I’ll share a few scattered thoughts
The distinction of “just in time” vs. “just in case” learning seems relevant here. Discontinuous learning is all “just in time” and like you say this creates a totally different mode of learning and teaches very different attitudes/skills vs. continuous, progressive, just in case learning.
There are two really valuable lessons that you will only get from self-directed discontinuous learning – you learn how to identify your weaknesses/blind spots and then fill them – which (as you say) is easy because you then have this natural motivation to find that missing piece of your personal knowledge puzzle. You also learn an intuition for identifying what is not essential to the larger picture, which in our age of information overload is also an essential skill.
I’ve written about the lack of curriculum diversity in science before – it’s asinine that continue to insist that we need non-conformists who think outside the box and bring new perspectives yet we still teach science (and everything) in a one-size-fits-all, lowest-common-denominator paradigm
"Education can be reformulated for scientific discovery if we conceive of each student as a unique experiment in the recombination of knowledge and opportunity. From this perspective, the lack of diversity in scientific education is a significant limitation on our ability to innovate - it is as if we are running the same experiment over and over again and hoping to get different results (the definition of insanity). I wrote about this in “Exploring the Landscape of Scientific Minds”; modern science education is highly homogenous - the vast majority of students are funneled through educational systems in which the same subjects (biology, chemistry, physics) are taught in the same sequences (either biology first or physics first) and students take the same assessments (AP, IB, SAT, etc.) with the same goal of getting into a top university - and we would do well to create educational systems that allow for students to develop more idiosyncratic sets of knowledge and skills. One thing that has always struck me is our insistence on teaching the natural sciences before psychology and the social sciences - might flipping the sequence lead to some unique experiments in the landscape of scientific minds? Why do we even teach science in classes in the first place? Why even teach scientific content (e.g. facts about biology or psychology) anymore? Why don’t we just teach students experimental design, data science, a little history of science, a little philosophy of science, and then tell them to go learn about the scientific disciplines that most interest them? (sorry rant over)"
There are many reasons why we can’t reform for curriculum for more open-ended, discontinuous learning is that we are painfully risk averse when it comes to education (except as you say, there isn’t really a risk). We are for some reason scared of the thought that a kid might have not taken history classes or never taken chemistry/physics by the time they graduate high school, which is stupid because it ignores the fact that (1) you forget most of what you learn anyways and re-learn what you actually care about or need to know, (2) if you realize that you are weak in a subject then you will be motivated to learn it on your own and will do it better/more efficiently because you are motivated.
The problem with discontinuous learning is that you actually might develop some gaps in your knowledge – these gaps will eventually be filled or will be inconsequential in the real world, but they might hurt your SAT score and then you don’t get into a top college and then your life is ruined forever (/s).