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.
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.