In real life, Laura Ingalls Wilder wasn't very fond of teaching. She did it to help her family financially.
Laura is shown signing her contract, but, in the 1800s, no married woman would have been able to legally sign any paper. Almanzo would have had to legally sign it.
Miss Plum's fake bun is a different color than her hair.
Most women in the late 1800's considered piercing ears a barbaric act therefore it's unlikely Laura would have had hers pierced.
In the late 1880's it was highly inappropriate for a woman and man who were not married to be alone together let alone working on a book. Especially after Laura knew he was hitting on her as a married woman.
In Matthew's office near the 20 min mark, Laura has perfectly manicured nails with nail polish. Nail polish did not come to America until 1920.