If I'm not mistaken, Prussia had the first national public education system. Minister of Education Wilhelm Humboldt began designing and implementing the system in 1809, in the wake of Prussia's humiliating defeat to Napoleon. It was the first to feature universal elementary and secondary education, with centrally controlled texts and examinations.

Humboldt was part of a generation of reformers, working under King Frederick William III and affiliated with Chancellor Hardenberg, who sought to make the government more responsive to the will of the people. Like the liberalizing militia laws sought by Minister of War Boyen (ultimately repressed by the army high staff), Humboldt's reform aimed at building a new generation of citizens that would be able to participate in their own government.

Unlike the army reforms, Humboldt's education system endured as a legacy of the reform generation, though Humboldt himself was ultimately forced out of office by an aristocratic reaction led by Prince Wittgenstein. Future leaders like Frederick William IV and Bismarck would maintain an unrepresentative and illiberal parliamentary system. Nevertheless, by the time of the Austrian war in 1866 and the Franco-Prussian war in 1871, Prussia had the highest public literacy rate in Europe. A Prussian soldier was expected to keep a record of his shooting - something the average recruit in other European nations would never have been capable of. Christopher Clark, writing in "Iron Kingdom," thinks that might, in part, explain the excellent performance of the average Prussian soldier in battle during the unification wars.

My interpretation is that other nations, seeking to emulate Prussia, began creating their own public education systems from the late nineteenth century - but someone jump in to correct me!

My source for the above is Christopher Clark's Iron Kingdom, the Rise and Fall of Prussia. Information about the army reforms comes from Gordon Craig's Politics of the Prussian Army, 1640-1945.