The radical union, which goes by the initials CNTE, is opposed to the mandatory testing of teachers as part of the government's sweeping education reform and is also protesting the arrest of union leaders on money laundering and other charges.
Yesterday's clashes involved federal and state police. An Associated Press journalist saw riot police firing on protesters in municipality of Nochixtlan in southern Oaxaca state in an attempt to dislodge them.
Isabel Garcia, a member of the CNTE's political commission, said three people supporting the protests were killed. She provided no further details.
In a statement, federal police said protests have been the most aggressive in Nochixtlan, north of Oaxaca City, where protesters even took a police officer hostage.
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Clashes lasted hours in the municipality of San Pablo Huitzo, also north of the state capital. And clashes were continuing yesterday evening outside Oaxaca City, where protesters burned federal police installations.
On Saturday night, police removed protesters who were blocking a major highway on the isthmus of Tehuantepec.
Over the past week, unionized teachers have blockaded streets, a shopping mall and even train tracks in the western state of Michoacan.
Federal prosecutors accuse union leaders of setting up an illegal financial network to fund protests and line their own pockets. They allege the scheme operated in 2013-2015, when the union effectively controlled the payroll of Oaxaca's teachers.
Following the arrest of some if its top leaders, the union called for a revolt against Mexico's government.
Ten years ago, the teachers started a six-month takeover of Oaxaca that didn't end until police stormed the barricades.