I love the term 'approximating.' It makes so much sense, recognizing the value of getting near to a goal rather than seeing things as simply right or wrong. As parents, we are wise to help our kids get where they're heading step by step with support. To do this, we need to establish expectations, that toys are put away before bed, for example. We start with our own example, tidying up after ourselves and after them when they're very small..
Kids will naturally start to help, and we can support this by noticing that they are approximating the thing we do. Simply acknowledging their work without a big fuss or reward encourages them. There will be times that they'll resist the habit of clean up. We don't need to demand perfection, just help them back to the path of clean up by supporting their approximation.
This works at every age. We can approach helping a child do homework, teaching them to drive, encouraging them to be respectful toward other people, in this same way. It even works with ourselves when prepare a few extra seasonal vegetables each week rather than expecting ourselves to prepare perfectly healthy meals each night.
Kids (and all people) integrate responsibilities bit by bit with support. They seem to master a skill, then may regress for a few days or weeks. With support, they master it again. Mess ups are less of a problem when we recognize that we can support an incremental movement toward mastery rather than believing in an 'all or nothing' kind of mastery.
I'm a person on a messy journey--healing, learning, discovering and rediscovering. Sharing, writing, talking, and teaching help me to understand myself and the world more clearly. I hope it can help make your journey a easier, too.