In case it isn't obvious by the treatment they received, this is a low-income Black family. The mother lives in a housing project served by poor schools.
Aside from the ugly race and class discrimination here, this ruling is also bad news for families of any constellation other than legally married straight couples with biological children who all live in the home together. This wasn't a case of the family using the address of a random friend (in which case the district would have the right to tell them it didn't satisfy the residency requirements of the district, but which shouldn't warrant criminal charges). This family registered the children using the address of their own father. Unless the parents have opted to go to court for custody or child support reasons, the court has no business defining this family against their will. It isn't any of their business to get into where the children spend more time or who is supporting them more. Both homes are the children's homes. Like most families, they probably have good reasons they've made whatever arrangements they've made, like around finances or work schedules or health reasons, but again, none of that is anyone's business. If these children spent most of their time with a grandparent or a family friend, but without any legal arrangement, would the court say they couldn't use their mother's address either? This is why we need more legal protections for varying family structures, not laws or interpretation of laws that define families however it's convenient for the authorities.
So now this mother can't finish her degree and become a teacher, and both parents are facing stress and legal issues that will mess up their whole lives. All of this happened basically because they weren't married and living in the same home, or weren't the sort of people who draw up fancy legal documents stating that the children primarily live with their father.