Reasons to study English

I'm married and I have two children: a 3-year-old boy and 2-year-old girl.
We live in Japan,
but our children go to an international school to study English.
They will be able to speak English well, but I can only speak Japanese.
So between housework I started to study English. (This is fine, but I think "I've started studying english between housework" is more natural)
I do understand English a little now, but I would like to be able to speak better.

My son started English when he was 1-years-old.
He is getting better at English.
He is already better than me. He’s only three!
I think that my daughter can improve at English in a short period too. 
Children are quick learners.
But I think I'm improving little by little. (This is correct, but it's more natural to say "But I think I'm improving little by little too")
I would like to improve with my children.

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Good work! Good luck with your studies!