Riot police fired tear gas to stop protesters from forcing their way past a cordon outside parliament, and later when demonstrators tried to push away a barrier near Prime Minister Alexis Tsipras' office.
No arrests were reported. Police said an internal investigation would be held into the injury of two demonstrators.
Public Order Minister Olga Gerovassili said some riot police had behaved in an "unacceptable and politically dangerous" manner, accusing them of indiscriminately attacking the protesters.
About 1,500 people took part in the demonstration.
The government has promised to hire 15,000 new state school teachers in the next three years but teachers' unions say that's too little, too late.
When the march stopped outside parliament, some demonstrators damaged a wreath that had been laid at the Unknown Soldier's monument by visiting German Chancellor Angela Merkel.
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