Your inability to effectively communicate the kind of scale you had in mind might be because we don't actually know what the prerequisite tree looks like at that scale, and just finding out would be a major step forward for educational science.
It's also not clear that any unique dependency tree even exists at such a fine-grained scale, because there exist multiple different ways of accomplishing a goal. Feynman talked about this, when recounting the episode where he learned that he counted time by mental speech, whereas his friend counted time by watching a visualization of a tape with flowing numbers:
By that experience Tukey and I discovered that what goes on in different people's heads when they think they're doing the same thing - something as simple as counting - is different for different people. And we discovered that you can externally and objectively test how the brain works: you don't have to ask a person how he counts and rely on his own observations of himself; instead, you observe what he can and can't do while he counts. The test is absolute. There's no way to beat it; no way to fake it.
It's natural to explain an idea in terms of what you already have in your head. Concepts are piled on top of each other; this idea is taught in terms of that idea, and that idea is taught in terms of another idea, which comes from counting, which can be so different for different people! I often think about that, especially when I'm teaching some esoteric technique such as integrating Bessel functions. When I see equations, I see the letters in colors - I don't know why. As I'm talking, I see vague pictures of Bessel functions from Jahnke and Emde's book, with light-tan j's, slightly violet-bluish n's, and dark brown x's flying around. And I wonder what the hell it must look like to the students.
Similarly, komponisto had an explanation of how he understood speed, and how that made it hard for him to understand some kinds of problems, because his mental concept of speed was a nonstandard one.
So there's unlikely to exist any single dependency tree, because at such a fine-grained level, it's likely that different people will have multiple different ways of comprehending any single concept. If my comprehension of "counting" is some specific variant X, then that might make some follow-up concept so obvious as to not even be worth explaining, while making some other follow-up concept hard to understand even with explanation. Whereas if my comprehension of counting was based on variant Y instead, then the pattern might be reversed.
I think you can see this assumption in constructivist teaching methods, which start out from the assumption that everyone filters their learning through and bases it on their existing knowledge, which will be different for everyone. Thus, rather than there being a pre-existing curriculum with a clear dependency tree that the teacher tries to transfer into the heads of the learners, the methods instead
rely on some form of guided discovery where the teacher avoids most direct instruction and attempts to lead the student through questions and activities to discover, discuss, appreciate, and verbalize the new knowledge.
Thus, instead of the teacher trying to chart out the exact fine-grained dependency tree, the students are encouraged to work on the problems in such a way that they themselves find the ways of learning the topic that are suitable for their pre-existing knowledge and ways of conceptualizing the world. You do still have some dependency tree, of course, but it's at a higher level than the one you seem to be talking about.
Problems
Problems have bottlenecks. To solve problems, you need to overcome each bottleneck. If you fail to overcome just one bottleneck, the problem will go unsolved, and your effort will have been fruitless.
In reality, it’s a little bit more complicated than that. Some bottlenecks are tighter than others, and some progress might leak through, but it usually isn’t anything notable.
Education
There is a lot wrong with education. Attempts are being made to improve it, but they’re glossing over important bottlenecks. Consequently, progress is slowly dripping through. I think that it’d be a better use of our time to take the time to think through each bottleneck, and how it can be addressed.
I have a theory of how we can overcome enough bottlenecks such that progress will fall through, instead of drip through.
Consider how we learn. Say that you want to learn parent concept A. To do this, it’ll require you to understand a bunch of other things first
My groundbreaking idea: make sure that students know A1…An before teaching them A.
The bottlenecks to understanding A are A1…An. Some of these bottlenecks are tighter than others, and in reality, there are constraints on our ability to teach, so it’s probably best to focus on the tighter bottlenecks. Regardless, this is the approach we’ll need to take if we want to truly change education.
How would this work?
1) Create a dependency tree.
2) Explain each cell in the tree.
3) Devise a test of understanding for each cell in the tree.
4) Teach accordingly.
Where does our system fail us?
My proposal
I propose that we pool all of our resources and make a perfect educational web app. It would have the dependency trees, have explanations for each cell in each tree, and have a test of understanding for each cell in each tree. It would test the user to establish what it is that he does and doesn’t know, and would proceed with lessons accordingly.
In other words, usage of this web app would be mastery-based: you’d only proceed to a parent concept when you’ve mastered the child concepts.
Motivation
Motivation would be another thing to optimize.
One way to do this would be to teach things to students at the right times. Lack of interest is often due to lack of understanding of child concepts, and thus lack of appreciation for the beauty and significance of a parent concept. By teaching things to students when they’re able to appreciate them, we could increase students’ motivation.
Another way to optimize motivation would be to do a better job of teaching students things that are useful to them (or things that are likely to be useful to them). In todays system, students are often times forced to memorize lots of details that are unlikely to ever be useful to them.
By making teaching more effective, I think motivation will naturally increase as well (it’ll eliminate the lack of motivation that comes with the frustration of bad teaching).
Pooling of resources
The pooling of resources to create this web app is analogous to how resources were pooled for Christopher Nolan to make a really cool movie. When you pool resources, a lot more becomes possible. When you don’t pool resources, the product often sucks. Imagine what would happen if you tried to reproduce Batman at a local high school. This is analogous to what we’re trying to do with education now.
How would this look?
I’m not quite sure. Technically, kids could just sit at home on their computers and work through the lessons that the web app gives them… but I sense that that wouldn’t be such a good idea. It’d probably be best to require kids to go to a “school-like institution”. Kids could work through the lessons by themselves, ask each other for help, work together on projects, compete with each other on projects etc.
Certificates
I envision that credentials would be certificate-based. You’d get smaller certificates that indicate that you have mastered a certain subject. Today, the credentials you get are for passing a grade, or passing a class, or getting a degree. They’re too big and inflexible. For example, maybe the plant unit in intro to biology isn’t necessary for you. Smaller certificates allow for more flexibility.
Deadlines
Deadlines are a tough issue. If they exist, there’s a possibility that you have to cram to meet the deadline, and cramming isn’t optimal for learning. However, if they don’t exist, students probably won’t have the incentive to learn. For this reason, I think that they probably do have to exist.
My first thought is that deadlines should be personalized. For example, if I moved 50 steps and the deadline was at 100 steps, the next deadline should be based on where I am now (step 50), not where the deadline was (step 100).
My second thought is that deadlines should be rather loose, because I think that flexibility and personalization are important, and that deadlines sacrifice those things.
My third thought, is that students should be given credit for going faster. In our one-size-fits-all system now, you can’t get credit for moving faster than your class. I think that if you want to work harder and make faster progress, you should be able to and you should be given credentials for the knowledge that you’ve acquired. Given the chance, I think that many students would do this. I think this would allow students to really thrive and pursue their interests.
Tutoring
I think that it’d be a good idea to require tutoring. Say, in order to get a certificate, after passing the tests, you’d have to tutor for x hours.
Tutoring helps you to master the concept, because having to explain something will expose the holes in your understanding. See The Feynman Technique.
Tutoring allows for social interaction, which is important.
Social Atmosphere
The social atmosphere in these “schools” would also be something to optimize. It's not something that people think too much about, but it has a huge impact on how people develop, and thus on how society develops.
I’m not sure exactly what would be best, but I have a few thoughts:
The idea of social value is horrible. In schools today, you grow up caring way too much about how you look, who you’re friends with, how athletic you are, how smart you are, how much success you have with the opposite sex… how “good” you are. This bleeds into our society, and does a lot to cause unhappiness. It should be avoided, if possible.
Relationships are based largely on repeated, unplanned interactions + an environment that encourages you to let your guard down. I think that schools should actively provide these situations to students, and should allow you to experience these situations with a variety of types of people (right now you only get these repeated, unplanned interactions with the cohort of students you happen to be with, which limits you in a lot of ways).
Rationality
I propose that rationality be a core part of the curriculum (the benefits of making people better at reasoning would trickle down into many aspects of life). I think that this should be done in two ways: the first is by teaching the ideas of rationality, and the second is by using them.
The ideas of rationality can be found right here. Some examples:
After the ideas are taught, they should be practiced. The best way that I could think of to do this is to have kids write and critique essays (writing is just thought on paper, and it’s often easier to argue in writing than it is in verbal conversation). Students could pick a topic that they want to talk about, make claims, and argue for them. And then they could read each others’ essays, and point out what they think are mistakes in each others’ reasoning (this should all be supervised by a teacher, who should probably be more of a benevolent dictator, and who should also contribute points to the discussions).
I think that some competition and social pressure could be useful too; maybe it’d be a good idea to divide students into classes, where the most insightful points are voted upon, and the number of mistakes committed would be tallied and posted.
Writing
Right now, essays in schools are a joke. No one takes them seriously. Students b.s. them, and teachers barely read them, and hardly give any feedback. And they’re also always on english literature, which sends a bad message to kids about what an essay really is. Good writing isn’t taught or practiced, and it should be.
Levels of Action
Certain levels of action have impacts that are orders of magnitude bigger than others. I think that improving education this much would be a high level action, and have many positive effects that’ll trickle down into many aspects of society. I’ll let you speculate on what they are.