[graboid]

Say you have concepts/items/cards A, B and C, with

A -> B -> C (C encompasses B, B encompasses A, keeping the notation from the article).

As I understand it, the article advocates for showing C first, then you can assume that you also know B and A to at least some part, and save yourself the repetitions for these.

Intuitively, I would have guessed the opposite approach to be the best: Show A first, suspend B until A is learned (by some measure), then show B, etc.

That means no repetitions to skip, but also you get less failures (and thus additional repetitions) that occur as follows: you are shown C, but don't know B anymore, and thus cannot answer and have to repeat C.

If you are shown C before B, you kinda make C less atomic (you might have to actively recall both, B and C to answer it), showing B before C makes C more atomic, as you will have B more mentally present/internalized and can focus on what C adds to B.

[JustinSkycak]

Hey! Good questions. I'll try to clarify:

1. First want to clarify that the learner is first introduced to the topics through mastery learning (i.e., not given a topic until they've seen and mastered the prereqs). So, they would explicitly learn A before learning B, and explicitly learn B before learning C. It's only in the review phase when we do all this stuff with "knocking out" repetitions implicitly.

2. When you say "then you can assume that you also know B and A to at least some part," I want to emphasize that if C encompasses B and B encompasses A in the sense of a full encompassing that would account for a full repetition, then doing C fully exercises B and A as component skills. Not just exercises them "to some part." For instance, topic C might be solving equations of the form "ax+b=cx+d," topic B might be solving equations "ax+b=c," and topic A might be solving equations "ax=b."

3. This scenario should never happen: "you are shown C, but don't know B anymore, and thus cannot answer and have to repeat C." There are both theoretical and practical safeguards.

3a-- Theoretical: if you are at risk of forgetting B in the near future, then you'll have a repetition due on B right now, which means you're going to review it right now (by "knocking it out" with some more advanced topic if possible, but if that's not possible, we're going to give you an explicit review of B itself. In general, if a repetition is due, we're not going to wait for an "implicit knock-out" opportunity to open up and let you forget it while we wait. We'll just say "okay, guess we can't knock this one out implicitly, so we'll give it to you explicitly."

3b-- Practical: suppose that for whatever reason, the review timing is a little miscalibrated and a student ends up having forgotten more of B than we'd like when they're shown C. Even then, they haven't forgotten B completely, and they can refresh on B pretty easily. Often, that refresher is within C itself: for instance, if you're learning to solve equations of the form "ax+b=cx+d," then the explanation is going to include a thorough reminder of how to solve "ax+b=c." And even in other cases where that reminder might not be as thorough, if you're too fuzzy on B to follow the explanation in C, then you can just refer back to the content where you learned B and freshen up: "Huh, that thing in C is familiar but it involves B and I forgot how you do some part of B... okay, look back at B's lesson... ah yeah, that's right, that's how you do it. Okay, back to C." And then the act of solving problems in C solidifies your refreshed memory on B.

Anyway, I think I've clarified all your questions? But please do let me know if you have any follow-up questions or I've misinterpreted anything about what you're asking. Happy to discuss further.

[graboid]

Thank you for clarifying!

I guess math is uniquely suited for this kind of strategy, but would you say it translates to learning concepts in other domains too?

I was thinking about whether something like "what is X?" -> "What field is X used in?", which seems to form a hierarchy for me, would benefit of this technique? Personally, I found that for something like the preceding example, I could answer the second question without thinking about what X is at all, just by rote memorization of the wording. Happened to me quite a lot when I was using Anki. And actually, I guess this is even acceptable in some way, since the question is not about activating "what X is", but "what X is used in". What I am trying to express: I feel like I would not necessarily activate a parent concept by answering a child concept, and I think that might be true for a lot of questions outside math problems, although they form a hierarchy. So I am wondering what you think about the general applicability of this technique...

Please don't take all of this questioning wrong, I think you are doing pretty cool stuff, and I am grateful for everyone trying to push the boundaries of current SRS approaches :-)!

[JustinSkycak]

Yeah, you're right that the power of this strategy comes from leveraging the hierarchical / highly-encompassed nature of the structure of mathematical knowledge. If you have a knowledge domain that lacks a serious density of encompassings, there's just a hard limit to how much review you can "knock out" implicitly.

> I feel like I would not necessarily activate a parent concept by answering a child concept, and I think that might be true for a lot of questions outside math problems, although they form a hierarchy.

This is where it's really important to distinguish between "prerequisite" vs "encompassing." Admittedly I probably should have explained this better in the article, but you are right, prerequisites are not necessarily activated. If you do FIRe on a prerequisite graph, pretending prerequisites are the same as encompassings, then you're going to get a lot of incorrect repetition credit trickling down.

We actually faced that issue early on, and the solution was that I just had to go through and manually construct an "encompassing graph" by encoding my domain-expert knowledge, which was a ton of work, just like manually constructing the prerequisite graph. You can kind of think of the prerequisite graph as a "forwards" graph, showing what you're ready to learn next, and the encompassing graph as a "backwards" graph, showing you how your work on later topics should trickle back to award credit to earlier topics.

Manually constructing the encompassing graph was a real pain in the butt and I spent lots of time just looking at topics asking myself "if a student solves problems in the 'post'-requisite topic, does that mean we can be reasonably sure they truly know the prerequisite topic? Like, sure, it makes sense that a student needs to learn the prerequisite beforehand in order for the learning experience to be smooth, but is the prerequisite really a component skill here that we're sure the student is practicing?" Turns out there are many cases where the answer is "no" -- but there are also many cases where the answer is "yes," and there are enough of those cases to make a huge impact on learning efficiency if you leverage them.

I still have to make updates to the encompassing graph every time we roll out a new topic, or tweak an existing topic. Having domain expertise about the knowledge represented in the graph is absolutely vital to pull this off. (In general, our curriculum director manages the prerequisite graph, and I manage the encompassing graph.)

Happy to answer any more questions if you've got any! :)