Fable
The calculation trace and the workbench are not the same thing
At the foot of the mountain, the school had two kinds of paper. The first was called calculation-trace paper, on which students wrote every step of their reasoning while solving problems, so the teacher could see how they went from the question to the answer. The second was called workbench paper, on which students could draw circles, list objects and try things out, and did not necessarily have to hand it to the teacher in the end.
The new instructor mixed the two kinds of paper together. He required all workbench paper to be posted on the wall, saying this was the most transparent. Students soon dared not write freely, and even their trial and error looked neat, while their real thinking hid in their minds.
The instructor then went to the other extreme: only collecting answers, without looking at any process. Scores were graded quickly, but when it came to difficult problems, no one knew whether the student had guessed, copied, or taken a reusable path.
The old gentleman asked everyone to do a bridge problem. Someone tried three bridge arches on the workbench paper, and the first two collapsed; when handing in the paper, he wrote only the final key derivation on the scratch paper. That messy sheet helped him think, but it was not suitable as a public explanation.
Another student submitted only a polished set of calculations, every line looking as if he had known the answer all along. When the old master questioned him about why he had ruled out the old bridge material, he could not answer, because the real exploration had never left any trace that could be examined.
So the academy changed its rules: reasoning that needs to be shown to others is written as clear calculation traces; temporary work that helps one explore is left on the workbench. Important exams may inspect the workbench, but not every scribble is treated as a final reason.
The students gradually understood that process has two kinds of value. One lets outsiders judge where the answer came from; the other lets the problem solver temporarily set aside clues and mistakes. Mixing them together both crushes exploration and obscures the reasoning.
At the year-end exam, the school no longer posted all the messy papers on the table, nor did it accept only one cold answer. The teacher required handing in calculation traces sufficient for others to review, while allowing students to keep their temporary explorations. People who learn need a surface where they can make mistakes, and people who collaborate also need an explainable path; both have their place. Some scribbles serve only the current exploration and should not be treated by outsiders as formal justification; some key derivations must be clearly left behind, otherwise others cannot trust the answer. What the school taught was not to hide the process, but to distinguish which process should be made public. Over time, students dared to try, and could also explain the path that finally worked to others.