# Combinatorial explosion of questions and errors

(there might be a more standard term for this)

Non-interactive explanations (e.g. textbooks, blog posts, and YouTube videos, in contrast to interactive explanations like classrooms and tutoring) face the problem of combinatorial explosion of questions and errors. As the length of the explanation increases, there will be more and more potential questions a learner could ask, as well as more and more potential errors in reasoning the learner could make (or misconceptions they could have). As the explanation is non-interactive, the explainer must anticipate in advance which questions and errors are most likely for the intended audience, and must decide how extensive the explanation will be.

Consider the experience of reading a mathematical proof. Each step in the proof is an opportunity for the reader to become confused, as they might not understand a calculation or reasoning that is being done.

Here is an example of this sort of thing, as related by Nate Soares:[1]

The problem is, most of the time that I get stuck, I get stuck on something incredibly stupid. I’ve either misread something somewhere or misremembered a concept from earlier in the book. Usually, someone looking over my shoulder could correct me in ten seconds with three words.

“Dude. Disjunction. Disjunction.”

These are the things that eat my days.

## Ways to mitigate

For the explainer:

• test the explanation on many readers to catch potential errors/questions, so that these can be cached
• leave the explanation for a while to make it fresher in your mind
• move to a more interactive format
• allow comments (if it's a blog post)
• explain more of the background material to "uniformize" the audience, to make the audience more predictable
• stipulate conditions on the audience (e.g. say that X and Y are required background reading, or that this is intended for intermediate students)

For the learner:

## Related phenomena

• the thing where course instructors reuse old exam/homework problems it's hard to come up with new problems (this problem also arises from the static/non-interactive nature of learning material)
• the problem with storing static example problems in spaced repetition software, because the user will just memorize the answer instead of treating it like a new problem (this problem also arises from the static/non-interactive nature of learning material)
• inferential distance/lack of uniform background of learners (this problem also arises from the difficulty of anticipating the identity of the learner or how they will react); see e.g. "“There are 10 pre-requisites for understanding concept X. Most people have 6 or seven, and then I write a blog post for each of the 10. Most people, most of the time, feel like they’re reading a thing they already know, yet I did have to write all 10 to be able to get everyone to take the step forward together.”"[2]
• there is a kind of "reverse" or "ironic" problem that happens where the explainer did correctly anticipate a question, but the reader fails to anticipate that the explainer anticipates this, so the reader stops reading as soon as they become confused, when in fact in the next paragraph (or next section or whatever) the question is answered. Some authors try to prevent this by saying things like "more on this soon" or "see section 4.5.3 for details".
• in arguments/debates (which are a special kind of explanation), there is a combinatorial explosion of potential arguments each side could make, even if the actual path of arguments is just a single path through this tree. See https://arxiv.org/abs/1805.00899 for more on this.