Re: Swearing on the Quran
In that case, I think the best solution is to take the mother into confidence and politely try to know why does she not want her children to swear. Is there any hadith that convinced her in thinking so. May be she thinks that her children are very innocent and their character should not be suspected by making them swear by the Quran. May be she thinks that this act would have a bad effect on the children who might question why were they forced to do such an act which they are not fimiliar with. May by she thinks that this act might bring an evil thought in her children's mind that in future they might do anything wrong and use the Quran to justify themselves innocent. (This might be possible in the case where they know their mother is guilty / culprit)
It all depends on how her psychology works. The best solution can be given after we come to know what is going through her mind. I might be wrong.
Allah knows BEST.