TL;DR There are too many things going on, I was unable to make a summary, sorry.
One of the issues I see here and would like to address is:
How the properties of mathematics (whatever they are)
affect questions and answers on teaching mathematics?
The fact is, there might be questions posed as general-education problems, but the characteristics of mathematics may make some answers more applicable and others perhaps even inappropriate. I would deem such questions on topic here, at MESE.
I'll try next to cover (possibly in a shallow way) the following topics:
- What are the properties of math that makes it different?
- How does it affect the answers and questions?
- Does it apply to the original question?
1. What are the properties of math that makes it different?
To list a few characteristics:
- Math is a zcience. (Using 'z' to avoid name-calling. To me this is just yet another characteristic for which I could not find a better term.) What I mean is that studying a mathematical object/structure does not change it (i.e. it is fixed). Exploring the mathematical world, we discover relations, theorems, etc., that is, these connections were there before we uncovered them; even before the mammals took over the Earth, $2 + 2$ was equal $4$ (and I'm not talking about the symbols, but the relation between the abstract concepts of two and four). On the other hand, business practices would not constitute a zcience in this sense, because introducing a new behavior (possible due to some research on business practices) might change the whole environment (including the business practices themselves). Naturally, I do not claim things which are not zcience in said sense are in any way inferior. This is just yet another characteristic.
- Math is build on assumptions. Once the axioms are stated, everything is done, all the inferences, effects, relations, properties, nothing changes. The set of axioms builds a whole new world and fixes it. It might be fascinating, but it is static, does not change (when moving to another problem we might change the world instead).
- Math is abstract. Some objects that are studied by mathematicians could not exists in reality (e.g. be material).
- Math is objective. There is no "my hypothesis is better than yours". Any proof is valid or not. You don't prove theorems "by authority".
- Math has a significant (subjective) art component. Any domain has an art component, but from all the sciences, mathematicians strive for (subjective form of) beauty the most. Unlike in some other domains, papers might be accepted solely by the elegance of the result or cleverness of the proof, despite how insignificant it may be. The distinction is that in other areas researchers investigate worlds which are already there, while mathematicians create their worlds (e.g. by assuming different axioms), similarly to writers, musicians, photographers, etc.
- Math has strong cognitive aspect (in the information-processing sense).
I would even say: stronger than any other domain excluding a few like physics, theoretical computer science, philosophy, etc., about which I'm not sure. I'm not suggesting that geography, cooking or farming do not require thinking, only that math requires thinking in different amounts. I'm afraid that quantifying this sentiment is difficult. First there is an issue of school/domain distinction (as presented by two comments reproduced below), second I see no way of measuring the cognitive work (not effort) objectively. Therefore, I cannot backup this point in any reasonable way and I present it as my whim. To indicate why I'm biased in this direction, I have met multiple mathematicians who could handle history (or other domains), but very few historians (and practitioners of other domains) who could handle mathematics (I know that's not only anecdotal, but also improperly measured, and yet there might be other factors; it still managed to skew my opinion).
- I said: "Do you claim that the understanding/memorizing ratio for history lessons (what an unfortunate example, I like history very much!) is as high as in math? I disagree. In particular you can derive the formulas for quadratic equation, but you can't infer the historical events with accuracy that is expected at tests and exams. It is possible to rebuild math from scratch, but it's impossible to recover history once it's lost (assuming irreversible dynamic of our world, no time machines, etc.)." which refers to the math as a domain.
- Benjamin Dickman said: "Since this is MESE and not HESE, let me limit myself to this last remark: I think that with regard to classroom requirements in HS or college, it is more likely that a student has to write history papers than mathematics papers. In this respect, I expect the nature of assignments (and standardized testing for HS) would result in students - in many cases - memorizing more for mathematics classes, and being given greater opportunities to integrate/synthesize material in meaningful ways in history classes. (As quid suggests: Further discussion is probably best directed to meta.)" which refers to subjects as they are done at schools/universities.
2. How does it affect the answers and questions?
- Math is a zcience. This basic effect this could have is that soft arguments, e.g. teacher's opinion or "this has always worked for me", have much less weight. For example, some psychologist might say that one therapy applies better than the other basing only on his instinct and prior experience. In math however, it is not the opinion that convinces someone, but the proof or counterexample. Therefore, answers which refer to using soft arguments might not apply. Also, it might be possible to suggest solutions which use hard arguments that would be impossible in other settings.
- Math is build on assumptions. This might create a curious class dynamic, because it has a kind of "what if" flair. There are no disputes like "The Earth is flat! No, the Earth is spherical!", instead there could be "Let's assume that the Earth is flat. Then..." One does not argue if the axiom of choice is true, only what consequences it has. It does not matter whether some theory aligns with reality, it might be interesting by itself (or not, but not because of reality). This could make some approaches infeasible (e.g. let's go to the math-zoo to observe Banach-Tarski paradox), while others viable (e.g. "imagine you are a two-dimensional entity that lives at a Möbius band"). Questions and answers need to take it into account.
- Math is abstract. This poses additional challenges like "how to present a Klein bottle" which does not appear in practical domains or "how to explain Yoneda lemma" which become trivial for any concrete instance. This suggests teaching-questions which aren't of interest in other fields.
- Math is objective. This changes the meaning of "authority" and forces the teachers into different mode of work, for example "because I said so" is not a valid argument. Moreover, students may present their own solutions which cannot be disregarded if correct. However, it also makes some parts easier like demonstrating why student's argumentation is invalid (a task much harder to do if the question is of the "state your opinion and back it up" form).
- Math has a significant (subjective) art component. This might help in some aspects, like motivating students, but also pose problems with teaching students to recognize the mathematical kind of beauty (there's hardly anything else like this). Hence, there might be questions and answers which wouldn't be of interest to the teachers of other domains.
- Math has strong cognitive aspect. This is one of the reasons why math causes difficulties to a huge percent of students, this is what makes it possible to pass a math exam without studying hard, this might be a source of joy and frustration to both students and teachers. MESE already has a number of questions that steam directly from this issue, e.g. Should I design my exams to have time-pressure or not?, How do you handle a wide ability range when delivering a 50 min tutorial with lots of material to get through? or Presenting a solution with a stroke of genius. I doubt this need more commentary.
3. Does it apply to the original question?
The examples below are just to explain why I think that some of the bullets are relevant to the original question (in particular, I made it up).
- Math is a zcience. It could cause satisfaction if some student would perform a successful independent research.
- Math is build on assumptions. It could cause satisfaction if some student came up with some really nice model/set of axioms, possibly with some unexpected consequences.
- Math is abstract. There might be some satisfaction when a student makes some far-reaching parallels between mathematical theories, but this is a huge stretch.
- Math is objective. It could cause some gratification if the student would start to thing for himself/herself, e.g. perhaps not taking what teacher said for granted, but verifying it themselves.
- Math has a significant (subjective) art component. This could be a great source of satisfaction, e.g. if some students arrives at especially nice solution.
- Math has strong cognitive aspect. This could be a source of satisfaction for example in case when we see the intellectual growth of the pupil which causes an additional development/progress in areas other than mathematics.
There are some general-education questions which might get significantly different answers if the context is mathematics. Some of them could be asked, for example, at academia.SE, however the answer/advice could be unsuitable. I think such questions are on-topic at MESE.
I think that the original post that caused this discussion describes one of such questions.
What do you think?