As a corollary to that, I absolutely hated spelling bees at school because while I was excellent at spelling, I simply could not do it by visualising a series of letters in my head in order. I always had to write a word down, then correct the spelling (or not) depending on whether it 'looked' right to me. I had to see the full word before beginning the process of analysis and correction. I could not build the word (accurately) letter by letter in consecutive order.
I presume that my brain is just wired differently and processes information differently. Which is why nowadays I object to standardised methods of teaching and testing young people. Not everyone fits in the same thinking box IMO.
Communication is an important skill for scientists, it's not enough to discover some new science, you must also get the point across to your peers and the general public (and of course the examiner in an exam situation). There are accepted forms (although some textbooks are really lousy). I keep telling my students that and am known to take points off for poor writing or unclear reasoning. It's something I learned from my teachers.
This doesn't just arise from students guessing. For any "difficult" problem, most students will end up doing useless steps, or possibly going down rabbit holes before finding the correct path. I don't think I have ever seen a proffesor that appreciated seeing all of this work (unless you couldn't find any path, in which case they often seemed to appreciate seeing what you did try).
I totally disagree about the "fictitious" part though. I think without proper working most interesting problems are impossible to solve.
I had six years of Saxon math, and probably did upwards of 700 30-problem problem sets. After that kind of repetition and consistent grading, showing work was instinctive, I'd seen it lesson after lesson presented the same way, and had to do it thousands and thousands of times. Other math curricula I've seen require far, far less repetition and practice at doing the mechanics.
From a software engineer's perspective, that's all I really want/need to do for my colleagues. I don't need to "show my work" because they're making sure I didn't copy/paste, they're trying to move quickly and don't have time to work things out on their own about how I came up with the solution.
Maybe "show your work" should be taught; less because it's a gateway into the mind of the solution provider, and more because it's helpful to others to see how someone arrived at a conclusion.
lets say youre building a car, maybe you take the engine from a ford and mount it on the chassis of renault with nissan steering, just because you didnt design those parts from the ground up doesnt make the end result any less valid as a final product (obviously thats a very loose analogy but you get the picture)
software is a bit different though, rather than showing that you followed a rigorous formula its about letting people know how the formula you came up with works
Many of my students came from schools where they learned a method called "guess and try," where you plug answers into the problem and see if one works. This is a speedy way to dispatch multiple choice tests.
To my students, "show your work" meant showing some evidence that they had solved the problem themselves. They thought I was policing them, when I really would have liked to engage them at a bit higher level.
In my view, "show your work" means, loosely speaking, to create a fictitious chain of reasoning and present it in a style learned from the textbook and classroom presentations. I call it fictitious because they might have guessed the answer and then worked backwards from it to obtain the steps. I could handle saying that a bright student should be able to do this, but that it should be taught.