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Question: What critical claims are related to Joseph Smith becoming "partial to the Methodist sect" near the time of the First Vision?
It is claimed that Joseph didn't become "partial to the Methodist sect" until at least 1823, after Alvin's death, or as late as 1838, rather than in 1820 as he claimed in his 1838 First Vision account
The Wikipedia article "First Vision" (as of May 18, 2009) contains the assertion:
While [Joseph] almost certainly never formally joined the Methodist church, he did associate himself with the Methodists eight years after he said he had been instructed by God not to join any established denomination.
In John A. Matzko, "The Encounter of Young Joseph Smith with Presbyterianism," Dialogue 40/3 (2007): 71., the author claims:
Although Joseph later wrote that his “Father’s family was proselyted to the Presbyterian faith,”—rather than emphasizing his mother’s membership—the death of Alvin and the arrival of Stockton seem to have driven both Smith and his father (who glided easily between religious skepticism and folk mysticism) farther from the Presbyterian church and its Calvinistic doctrine. It was probably during this period that Joseph “became partial to the Methodist sect,” whose opposition to Reformed doctrine was notorious.
It is entirely reasonable to conclude that Joseph was telling the truth when he said that he became "partial to the Methodist sect" in 1820. Critics who attempt to place this event later in Joseph's life do so in order to discredit the story of the First Vision.