Character Analysis
Despite their notorious rep, these kids aren't juvenile delinquents drinking moonshine in the graveyard and tripping old ladies. No. Jane and Michael are pretty good kids. They're just misunderstood.
Their first nanny thinks they've "run away," at the beginning of the movie, and promptly quits her job. But Michael and Jane were just chasing their kite after it got away. These kids aren't bad eggs—they just really like their kite.
Their father thinks that they need an even stricter nanny, an industrial-strength dose of nanny. This sounds like a bad idea, and Jane and Michael think so too. They just want a sweet, kind lady who will play games with them, and not be horrible. So, they write their own advertisement for a nanny:
JANE: If you want this choice position
Have a cheery disposition
Play games, all sorts
You must be kind, you must be witty
Very sweet and fairly pretty
Take us on outings, give us treats
Sing songs, bring sweets
Never be cross or cruel, never give us castor oil or gruel
Love us as a son and daughter
And never smell of barley water
If you won't scold and dominate us
We will never give you cause to hate us
We won't hide your spectacles so you can't see
Put toads in your bed or pepper in your tea
That's pretty good amateur poetry for a duo of juvenile delinquents.
So, interestingly, Jane and Michael actually know what's better for them than their father does. It goes against the whole "Father Knows Best" ethos of their time period. In fact, it's a little like what David Bowie said in "Changes":
These children that you spit on / As they try to change their world / Are immune to your consultations / They're quite aware of what they're really going through. (Source)
The Tuppence Comeuppance
But George Banks isn't buying it. He tears up their note and chucks it in their fireplace—where it blows away and somehow reaches Mary Poppins, who does listen to the children and does value their ideas.
So it's clear: these are good kids. They don't really need to change all that much. Rather, the adults around them need to change and re-evaluate how they're relating to them. Mary Poppins is the only one who really gets them.
At the same time, Mary Poppins points out some of their flaws. Her magical measuring tape says that Michael is "extremely stubborn and suspicious" and Jane is "rather inclined to giggle, doesn't put things away."
So, they're not perfect—and they need to learn find the fun in tidying up the nursery and all that stuff. Still, there's no major crisis in the souls of Michael and Jane—they just need more of their parents' love.
They embrace Mary's lessons about lightening up and showing love. Her song about the old lady who sells seeds to feed pigeons for a "tuppence a bag," inspires Michael to cling to his tuppence when his father takes him and Jane to the bank where he works. Dawes Sr., head banker, tries to convince him to use his money to invest it in a savings account:
DAWES SR.: And just how much money do you have, young man?
MICHAEL: Tuppence. But I want it to feed the birds.
Michael wants to feed the birds—not invest his tuppence in a savings account, clinging miserably to a tiny amount of money. When Dawes tries to seize the tuppence from him, Michael fights back, causing a panic in the bank. This leads to George getting fired.
Fortunately, that works out really well for the kids. Michael and Jane's own tenacity, sticking to their own ideals and not letting adults boss them around, actually ends up being good for the adults around them. True, Michael accidentally helps gets George fired—but this is good for George, since it makes him reckon with his own relationship to his kids, and lighten up.
So, the kids successfully convert their Dad into a fun-loving guy, who wants to fly kites with them. They help change him—and they find the strength and courage to do it with help from Mary P. As the British would say, "Good show, Gov'ner."
Jane and Michael's Timeline