This week I show my first steps, as a confirmed non-expert in accounting, in tracking a mutual fund in hledger. Read about it at https://hledgerfan.com/accounting-for-a-mutual-fund/.
I'm liking this new series! 
The value of the shares will fluctuate daily according to the market, but the number of shares will change only when you buy or sell shares.
This is not entirely true. They might change on corporate actions, and on dividend payouts with accumulation funds.
An argument could be made that recording dollar amounts is easier (you need one number from your statement or Internet banking screen), and given that you put money in and get money out, shares are an implementation detail you can ignore, unless you need them for tax work.
Thank you for your comment, which caused me to think of a specific time when the number of mutual fund shares could change without buying or selling. Specifically, this is if the mutual fund split its shares. According to what I could find, this doesn’t happen very often with mutual funds, but it does happen from time to time.
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