People who're engaging in learning partly or wholly for the explicit purpose of human capital need to be strategic about their learning choices. Only some subjects develop human capital useful to the person's goals. Within each subject, only some subtopics develop useful human capital. Even within a particular course, the material covered in some weeks could be highly relevant, and the material covered in other weeks need to be relevant. Therefore, learners need to be discerning in figuring out what material to focus their learning effort on and what material to just skim, or even ignore.

Such discernment is most relevant for self-learners who are unconstrained by formal mastery requirements of courses. Self-learners may of course be motivated by many concerns other than human capital acquisition. In particular, they may be learning for pure consumptive reasons, or to signal their smarts to friends. But at any rate, they have more flexibility than people in courses and therefore they can gain more from better discernment.

Those who're taking courses primarily for signaling purposes need to acquire sufficient mastery to attain their desired grade, but even here, they have considerable flexibility:

• People who're already able to get the top grade without stretching themselves too much have flexibility in how to allocate additional time. Should they try to acquire some more mastery of the entire curriculum, or delve deeper into one topic?
• People who're far from getting a top grade may have the same grade-per-unit-effort payoff from delving deep into one subtopic or acquiring a shallow understanding of many topics. Considerations regarding long-term human capital acquisition can then help them decide what path to pursue among paths that confer roughly similar signaling benefits.

What self-learners and people with some flexibility in a formal learning situation need is what I call utilitarian discernment: the ability to figure out what stuff to concentrate on. Ideally, they should be able to figure this out relatively easily:

1. Sequencing within the course: Important topics are often foundational and therefore done early on.
2. Relative time and emphasis placed on topics should give an indicator of their relative importance.
3. Important topics should be explicitly marked as important by course texts, videos, and syllabi.
4. Important topics should receive more emphasis in end-of-course assessments.
5. Important topics should be more frequently listed as prerequisites in follow-on courses covering the sort of material the learner wants to do next.
6. The learner can consult friends and websites: This includes more advanced students and subject matter experts, as well as online sources such as Quora and Less Wrong.

The above work better than nothing, but I think they still leave a lot to be desired. Some obvious pitfalls:

1. Sequencing within the course: While important topics are often done early on because they are foundational, they are sometimes done later because they rely on a synthesis of other knowledge.
2. Relative time and emphasis: Often, courses place more time and emphasis on more difficult topics than more important ones. There's also the element of time and emphasis being placed on topics that subject matter experts find interesting or relevant, rather than topics that are relevant to somebody who does not intend to pursue a lifetime of research in the subject but is learning it to apply it in other subjects. Note also that the signaling story would suggest that more time and emphasis would be given to topics that do a better job at sorting and ranking students' relevant general abilities than to topics that teach relevant knowledge and skills.
3. Important topics marked as important: This is often the case, but it too fails, because what is important to teachers may differ from what is important to students.
4. Emphasis in end-of-course assessments: The relative weight to topics in end-of-course assessments is often in proportion to the time spent on the topics than their relative importance, bringing us back to (1).
5. Important topics should be more frequently listed as prerequisites: This would work well if somebody actually compiled and combined prerequisites for all follow-on courses, but this is a labor-intensive exercise that few people have engaged in.
6. The learner can consult friends and websites: Friends who lack strong subject matter knowledge may simply be guessing or giving too much weight to their personal beliefs and experiences. Many of them may not even remember the material enough to offer an informed judgment. Those who have subject matter knowledge may be too focused on academic relevance within the subject rather than real-world relevance outside it. People may also be biased (in either direction) about how a particular topic taught them general analytical skills because they fail to consider other counterfactual topics that could have achieved a similar effect.

In light of these pitfalls, I'm interested in developing general guidelines for improving one's utilitarian discernment. For this purpose, I list some example head-to-head contest questions. I'd like it if commenters indicated a clear choice of winner for each head-to-head contest (you don't have to indicate a choice of winner for every one, but I would prefer a clear choice rather than lots of branch cases within each contest), then explained their reasoning and how somebody without an inside view or relevant expertise could have come to the same conclusion. For some of the choices I've listed, I think the winner should be clear, whereas for others, the contest is closer. Note that the numbering in this list is independent of the preceding numbering.

1. Middle school and high school mathematics: Manipulating fractions (basic arithmetic operations on fractions) versus solving quadratic equations (you may assume that the treatment of quadratic equations does not require detailed knowledge of fractions)
2. High school physics: Classical mechanics versus geometrical optics
3. Precalculus/functions: Logarithmic and exponential functions versus trigonometric functions
4. Differential calculus: Conceptual definition of derivative as a limit of a difference quotient versus differentiation of trigonometric functions
5. Integral calculus and applications: Integration of rational functions versus solution strategy for separable differential equations
6. Physical chemistry: Stoichiometry versus chemical kinetics
7. Basic biology: Cell biology versus plant taxonomy
8. Micreconomics: Supply and demand curves versus adverse selection

PS: The examples chosen here are all standard topics in the sciences and social sciences ranging from middle school to early college, but my question is more general. I didn't have enough domain knowledge to come up with quick examples of self-learning head-to-head contests for other domains or for learning at other stages of life, but feel free to discuss these in the comments.