Hi Carol! How are you doing? As Lauro mentioned earlier, Bertha Mulvihill did indeed mention seeing a mother and her children on the deck as the ship sank. The family was from steerage, but was the Rice family, not the Goodwins.
The newspaper article which I believe is the one in question, was published in The Providence Journal, April 20, 1912. This article misquotes Bertha Mulvihill as saying (off the top of my head, sometimes a dangerous proposition!) that she saw Mrs. Rice and her husband and children and that Mrs. Rice stayed behind because her husband was held back by an officer. Bertha Mulvihill did indeed say that she saw the Rice family on the deck before she left the ship and mentioned this in press accounts, to family and friends, but the part about the husband is almost certainly an invention of the press (as is the part of the article which among other things, inaccurately quotes her as saying that she saw Captain Smith brandishing a revolver, and that she heard 'Nearer My God To Thee' as the ship sank, statements that she denied making to her family).
Mrs. Rice’s husband had been killed in an accident well before the Titanic’s maiden voyage and was never aboard the ship. Bertha would have been well aware of this and was able to identify them, as she was acquaintances with Margaret Rice, who was also originally from Athlone, Ireland. Margaret, in a letter written to friends in Washington prior to the maiden voyage, mentions that she was going to be traveling to America with Eugene Daly and Bertha Mulvihill. The Rice family (Margaret and her five sons Albert, Arthur, Eric, Eugene, and George) was traveling to Spokane, Washington. All of them perished.
I hope that this answers your questions. More information about Bertha Mulvihill and the Rice family is given in an article that I wrote about Bertha which was just published in the Titanic Commutator. I would be more than happy to get you a copy of the 1912 Mulvihill press account in question somehow if you can't find it elsewhere. Hope this letter finds you well.