For children age 6 and up, and also for adults, the most important thing is to Keep It Simple.
Writing down passwords is actually a good thing for adults, as long as the passwords are written down in a secure place. A note in your wallet qualifies, as you know how to keep your wallet secure (right?). This is even more secure than a password safe on your smartphone: inputting a strong password is a pain (and easily observed), and witht it your sm artphone becomes a prime target for theft (if it isn't already).
For children of 6 years old and older (I'm assuming a US centric view here, triggered by the word 'elementary'), the situation is not that much different. The only problem is that children at this age usually do not have a wallet.
This is then the only problem to solve: creating a secure place to write down passwords.