Fable
The intermediate calculation marks left by old Mr. Abacus
The granary settled accounts every month, and Old Abacus could always give the total. The young clerk admired him and copied only the final answer into the ledger. Small accounts were fine, but as the large accounts piled up, no one knew where an error had occurred even when there was one.
One autumn harvest, three warehouses, five grain prices and two discounts were mixed together. The old gentleman reported a number, and when the county magistrate asked how it was calculated, the clerks could only stay silent. The magistrate had only wanted a credible total, but found that the big ledger was like a long alley: the last lamp being lit did not mean every corner before it had been walked correctly.
They first asked the old gentleman to be more careful. He did slow down somewhat, but the answers were still occasionally wrong. Without an intermediate process, it is hard to check whether he is being careful or not.
Later, the old gentleman began writing down each step on rough paper: first merge the warehouses, then convert units, then deduct losses, and finally calculate tax. The clerks saw how he broke a big problem into small problems. The first version of the rough paper was too casual; some people wrote guesses into it, and some omitted intermediate mental calculations. The reviewer saw a pile of traces but could not tell which ones actually supported the answer.
At first everyone thought the longer the draft paper, the better. Soon, many useless side branches appeared on the paper, and the real errors were buried instead.
The old gentleman then decreed that intermediate calculation traces should serve only problem-solving: split when splitting is needed, check when checking is needed, stop when stopping is needed. It is not for showing off to outsiders, only for helping complex judgments stabilize. So the accounting office set the format: key conversions must be written, sources of evidence must be written, repetitive arithmetic need not fill the page; when private accounts are involved, keep only reviewable markers.
For some accounts, writing out the intermediate process is clearly more accurate; for some simple additions, calculating directly is faster. The clerks began deciding whether to unfold the steps based on task difficulty. The county magistrate also learned not to treat scratch paper as absolute truth. Intermediate processes can help with checking, but they can also be written to look like reasons. Key accounts still had to go back to the original vouchers, and could not be let through just because the steps looked complete.
Later they also learned to manage draft paper. The ledger shown to the county magistrate displayed only the necessary explanations, while internal draft paper was used for review and training newcomers, avoiding the leak of details that should not be made public.