In the story "Rules of the Game," what does Waverly's mother want for her sons and daughter?
She simply wants what most parents desire for their children--success and happiness--and as an immigrant from China, she believes that America is the only place where that is possible for her children.
Yes, her perspective might be flawed, such as her putting intense pressure on Waverly to learn chess seemingly so that she can brag about her daughter, but in reality, she is just like any other proud mother who wants her daughter to feel successful. America, for Waverly's mother, represents so much in the way of opportunity that her mother views wasting any of that opportunity to hone one's skills or to develop new skills as unacceptable.
In addition to her very typical maternal goals for her children, Waverly's mother wants them to assimilate into American culture, something that she does not feel that she has done or will ever be able to do. Her comment to Waverly about the chess rulebook demonstrate this desire. She tells Waverly and her brothers:
" 'Every time people come out from foreign country, must know rules. You not know, judge say, Too bad, go back.' "
Her statement reveals the only difficulty she experienced as an immigrant with no source of help or advice and also shows why she is so concerned about her children finding their own place in America.