[quote=@Crimmy]
Non-Gemstone Espers get their powers through the Power Curriculum Program rather than discovering that they're naturally superpowered.
[/quote]

Being an esper means owning your own version of reality, which often happens during a child's developmental years. Many espers DO gain their powers in school, but usually the basis is already in place, and it's not unheard of for a child to learn to use it themselves. Notably, Accelerator could use his ability since before his school years because of Reflection's instinctive nature.