Just over 90 per cent of those who voted backed the action, with further walk-outs seen as likely unless a compromise can be reached.
Jim Halfpenny, of the local branch of the Educational Institute of Scotland (EIS), said: “It shows that teachers are really angry. They say management is treating them with total contempt. They have failed so far to take the views of teachers seriously and are intent on imposing something teachers completely disagree with.”
There was potential for another day of strike this week, he added.
Members picketed St Peter the Apostle High, Clydebank High, Dumbarton Academy, Our Lady and St Patrick’s High, and Vale of Leven Academy, affecting a total of about 5,000 pupils.
The rise of so-called faculties has been contentious across Scotland over the past few years as some councils have combined subjects such as history, English and geography into joint departments under one senior teacher.
Teachers have opposed the trend because disparate subjects come under one umbrella, with faculty heads lacking understanding of some subjects for which they are responsible.
Larry Flanagan, general secretary of the EIS, said a fundamental gap remained between the council’s position and the view of teachers.
“It is particularly regrettable that the ruling Labour party administration within the council has chosen to remain at arm’s length from the negotiations rather than engaging in positive discussions in an attempt to understand teachers’ concerns and work towards an agreement,” he said.
“Teachers do not want to strike and the EIS has been prepared to make concessions, but in the absence of any movement regarding the damaging proposals, our members are left with no option but to continue their industrial action campaign.”
A council spokeswoman said: “We acknowledge the disruption this action is causing and would once again like to apologise to our pupils and parents. We’re really disappointed to see the prospect of further industrial action at our secondary schools.”
West Dunbartonshire had offered a “significant concession” to appoint an extra 16 principal teachers and to complement that with nine measures to reduce workload, she added.
“We would like to reiterate that we remain committed to reaching a resolution for the benefit of everyone and will continue to speak with the EIS in an effort to achieve this.”