Further to Shellyesq's response, James in the 1920 census has a niece (23) and nephew (18) as members of the family along with his adopted son aged 8. It is not beyond the bounds of possibility that the niece, Laura M Shaw could be the mother.
Though Shellyesq is Laura one of the daughters from 1910? In 1920 Letitia is shown as a widow so may have been the widow of a brother of James Luther Shaw? The other thought I had was that Paul was the son of Letitia
and her husband Her husband died in 1903 and this would not be possible, and was adopted by James Luther Shaw to be his heir/carry the family name. I 'wonder' if in fact Paul L was the son of James L & Letitia? or Letita and perhaps XXXXX Siegfried.
https://www.familysearch.org/ark:/61903/1:1:M6YH-H81I will keep looking.
ETA in the 1930 census Paul is still showing as age 10 but this cannot be correct if he was born in 1911.
ETA2 in 1940 census Laura is shown as a widow.
ETA3 Looking at the 1900 census it seems that Letitia may have been married to John W? Shaw born 1870 and they had been married for 7 years in 1900 and had 2 living children (Sarah E 6yrs & Laura 4 yrs ) at that stage. They lived in a large extended family group with John & James widowed mother and her brother and other siblings.
ETA4 John W Shaw died in 1903 and was the husband of Laetitia who died in 1942. They have a joint gravestone
https://www.findagrave.com/memorial/139874241/john-w-shawETA5 When looking at the Findagrave record of John Shaw and Letitia they are showing only one daughter Sarah Emma Shaw.
https://www.findagrave.com/memorial/139874241/john-w-shawNo mention of Laura Shaw?