I maintain a database of questions. For each question, I assign it two metrics that describe its difficulty: "[Unnamed concept]" and "complexity".
"[Unnamed concept]" is how much you have to know about the topic; how much expertise is required to answer it competently. "Complexity" is how much "stuff" there is going on in the question.
Questions that are high in "[unnamed concept]" and low in "complexity" are elegant and deep. Questions that are low in "[unnamed concept]" and high in "complexity" are closer to "busywork". Maybe still challenging to get through everything without making a mistake, but not in an interesting way.
For example, in math, here's a question that's high in "[Unnamed concept]" and low in "complexity":
Two circles of the same radius overlap by a distance of that radius. What's the area of the overlapping section?
And here's a question that's low in "[unnamed concept]" and high in "complexity":
What is (2 + (4 / 5)) - (65 * ((3 / 5) + 20))?
When demonstrating a new concept, you want a question that's low in "complexity", since high "complexity" questions will distract/overwhelm the learner with irrelevant other things to think about.
High "complexity" questions are good repetative work for helping someone internalize/memorize a process and become good at executing it quickly and reliably.
In a not-quite-synonymous-but-kinda-close-in-many-cases definition, "[unnamed concept]" refers to the difficulty of figuring out the answer to the question, while "complexity" refers to the difficulty of figuring out what's being asked in the first place. (Excluding domain-specific knowledge, which falls under "[unnamed concept]".)
I want a name for "[unnamed concept]". The first word that comes to mind is "difficulty", but that's not accurate because my concept of "complexity" also contributes to a question's overall difficulty.
Ideally I'd like it to be a single word, or maybe two short words. (My use-case requires it to fit in a small amount of space, just like "complexity" can.)
Resolves NO if I haven't found a good name for this in a week.
If this doesn't seem like a coherent concept to you, or you think there's a better way to subdivide the different aspects of "difficulty", I'd be interested in that too.
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Maybe you could call these two different types of complexity - the complexity you already named could be something like (1) procedural complexity (or procedural detail), and the unnamed thing could be (2) conceptual complexity (or conceptual sophistication, similar to what Duncan said)?
If I understand what you're looking for correctly, "complexity" refers to complexity in the statement of the problem itself, while [unnamed concept] refers to complexity in the solution. So what you are getting at is the idea that a problem can be high in [unnamed concept] but low in "complexity", making it very simple to state, but extremely hard to prove, like Fermat's Last Theorem.
I suppose, based on that, you could follow the long tradition of turning famous mathematicians' names into adjectives and call it Fermatness, but I think that Duncan's ideas below of calling it either "depth" or "sophistication" are better names.
@JosephNoonan Another potential names could be "opacity", since that would mean that a transparent problem is one that is easy to solve by elementary means, while an opaque problem doesn't have an obvious path to a solution if you don't know the methods required to solve it.
@Duncan I am tempted to add 'finesse' or 'nuance'. If any particular shade of meaning calls out to you, let us know.
@Duncan I have a paper on depth and transparency as virtues of mathematical proofs, and this is what I immediately thought too! It specifically talks about, e.g., how depth isn’t the same thing as mere difficulty or complexity.
@Duncan These are the closest anyone has come so far to words I'd find satisfying. "Depth" and "sophistication" are the best matches I think.
(I don't like "expertise" because I think that usually refers to people, not inanimate conceptual objects. "Expertise required" would be a more fitting phrase, but I don't want to use multiple words like that.)
Not sure whether I find them satisfying enough to resolve the market to YES, let me turn them over.
(I was also thinking of scrapping my categories entirely and just using "difficulty".)
I maintain a database of questions. For each question, I assign it two metrics that describe its difficulty: "[Unnamed concept]" and "complexity".
@KatjaGrace I'm curious whether you were asking for more context on my question, or suggesting "context" as the term I'm looking for.
it's probably not what you are looking for given that you see it as an aspect of difficulty, but I think difficulty is a good enough description of that concept.
What is (2 + (4 / 5)) - (65 * ((3 / 5) + 20))? is a question that is not very difficult but is complex
Two circles of the same radius overlap by a distance of that radius. What's the area of the overlapping section? is a question that is difficult but not too complex
@bluerat At least it’s as accurate a description as complexity is for the other concept in my opinion. I think if you’re gonna say it’s not good enough you should also change complexity to something better.
@bluerat Difficulty is defined as 'needing much effort or skill to accomplish, deal with, or understand.' Both are difficult, as the former requires effort to calculate, and the latter requires more thought to understand and think through. The former is mechanical, arithmetic, while the latter is logical, analytical.
I think if you’re gonna say it’s not good enough you should also change complexity to something better.
What would you suggest?
@IsaacKing so I don't like complex because it is often used synonymously with difficult, like you could say "will joe biden be reelected president?" is a complex problem even though I don't think it matches the concept you are going for. instead perhaps something like 'stacked' or 'multiplex'. In fact I think multiplex should be perfect because it means "consisting of many elements in a complex relationship" which seems like exactly the description you are going for, similar to complexes definition of "consisting of many different and connected parts." but without the second definition of complex: "not easy to analyze or understand; complicated or intricate.".
@XComhghall no, i'm not proposing multiplex for the unnamed concept I'm proposing changing complex to multiplex
@bluerat The former is the unnamed concept. I propose that it be called complex.
The latter is what Isaac called complex. I agree with your naming of it as multiplex.
nearby concepts: Chaos. Emergence. Intricacy. Interdependence. Nuance. Conceptual Richness. Depth.
unrelatedly, https://en.wikipedia.org/wiki/Self-organized_criticality is cool.
I don't have the word you're looking for, but here are some words: solving a math problem might require "conceptual understanding" and "technical proficiency".
"Technical proficiency" is the ability to perform routine calculations -- multiplying many-digit numbers, or differentiating functions...
"Conceptual understanding" is... I don't really know how to explain it, it's what you need when you're confronted with a problem that you haven't been taught the algorithm for.