The archetypes is presented in five pages: cover, two pages of description and two of OGL. It could use more context for the archetype such as suggestions for where they might fir into a game world and what sort of culture would inspire them. Also, and this is very common for such materials, is a lack of support material such as a new spell, magic item, place or even a legend to inspire adventure would make them some much more interesting and easier to fit into a campaign.
Wizard's Apprentice is odd . . . Are you not an apprentice before you gain your first level? Perhaps student of the arcane would be a better name? Teacher's Pet gives you some extra spells in your spellbook and a cantrip known, so it is very good (and may give the DM some clues to where the character wants to go). But Beginner's Spellcasting is entirely a negative (and the 10th level ability is partly mitigating it but nothing else) while it could lead to some fun roleplaying, mostly I see it leading to frustration in play. Perhaps having their spellcasting generate odd side effects would be more playable? Their capstone ability of Advanced Spellcasting is good but could use a little more punch or flexibility or both.
Again, strong concept but the execution does not quite work.
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