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Melissa Sue Anderson (Mary) doesn't appear in this episode until much later, in the baseball game scenes. She is oddly absent from all Ingalls family scenes before that.
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When Mr. Mumford throws his first practice pitch, it hits Mr. Edwards square in the face and knocks him down. In reality, a baseball going at that speed would have probably killed him--sadly, it has happened to other people before. Mr. Edwards, however, didn't even get so much as a measly bruise!
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Dr. Baker wears a uniform he says is from his college team, the "Mulligans". This is unlikely, as base ball was a gentleman's club sport when he would have been in college and the very first college challenge game (between Williams and Amherst) was in 1859 -- only about 15 years before this episode takes place.
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Nitpick: On the scoreboard, Laura writes a 2 on Walnut Grove's side. The shape of the 2 changes the next time they show the board.
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Goof: All the base ball (two words back then) players on the opposing team are wearing gloves when fielding. In the year depicted in this episode, only the catcher and occasionally the first baseman wore gloves. It wasn't until the late 1890s that all fielders wore them.