What was the process for the naturlization of children in the 1930's?

Hi,

I found my mother's certificate of citizenship (U.S.) in her papers, and it has raised more questions. Hopefully those with a better understanding of the process can clarify what happened. Below are the facts I know —-

1. At 5 years old she arrived in the U.S. in Oct. 1938. She was born in Norway.
2. The certificate says she became a U.S. citizen in Mar. 1934.
3. She lived in Michigan in 1938.
4. The certificate then said she was issued a replacement certificate in July 1952. (I am guessing this was due to a name change after marrying in 1941.) She later divorced.
5. The certificate I found in her papers was dated July 1958. (I am guessing, again, that this is a replacement, too, as she remarried & changed her name in 1957.)

Her father arrived in the U.S. in Mar 1927, and became a citizen in Mar. 1934. Her mother became a citizen in Oct. 1935. So, it appears to me that my mother became a citizen at the same time as her father. Does that sound right based on my information?

I have not been able to find ANY records showing her 'original' naturalization, or her initial replacement certificate. Were there no documents recorded to show that she became a citizen because she was a minor? Does the certificate I found in her papers supercede all other documents, and then regard her as not being a U.S. citizen until 1958? If current day documents ask me to state when she became a citizen, which date do I actually use —- 1934, or 1958?

I have had little luck finding much out about the process of children getting citizenship and would appreciate any insights and clarification from those who might know. Thank you for your assistance with this personally perplexing turn of events.