However, 57% agreed with linking salaries to appraisals carried out by headteachers, while 31% were opposed.
Headteachers were not immune to stress either, with 960 days lost for that reason in 2009.
Ms Hyslop said she has "stuck her neck out" for headteachers in the past.
But the Association of Headteachers and Deputes in Scotland, which represents primaries, has backed the curriculum.
Headteachers I had been speaking to told me they had all had incompetent teachers in their schools.
The headteachers' union ASCL said quality of learning was more important than hours spent in the classroom.
Gordon Smith, from the Association of Headteachers and Deputies, called for more cases to be brought forward.
East Lothian, Eilean Siar, Inverclyde, Orkney, Moray and the Scottish Borders also gave their headteachers less independence.
BBC: NEWS | UK | Scotland | Heads could be given more powers
However, the Headteachers' Association Scotland (HAS) said it wanted local authorities to continue to play a role.
BBC: NEWS | UK | Scotland | Heads could be given more powers
That headteachers didn't want the spotlight on their school and so were avoiding using the competency process altogether.
The secondary headteachers union suggests the curriculum is in theory a good idea but in many ways impractical.
Councils would run the system, but the notices could be issued by headteachers, police and community support officers.
The worst offenders are Clackmannanshire, Fife and Highland which devolve less than 60% of education spending to headteachers.
BBC: NEWS | UK | Scotland | Heads could be given more powers
The primary headteachers' association said it welcomed changes but it wants the executive to consider extending the one-year course.
BBC: NEWS | UK | Scotland | Teacher training to be 'flexible'
The same theme emerges from another Scottish Labour proposal, to devolve the management of 90% of school budgets to headteachers.
He called on ministers to think again and said a way forward would be to survey headteachers to see what they wanted.
The improvement plan includes a commitment for better "performance management" of headteachers.
BBC: Leighton Andrews outlines reforms for schools standards
But headteachers' unions are pushing for pupils' improvement to be measured against their previous performance - and not against a national target.
The Association of Headteachers and Deputies (AHDS) has said momentum would be lost if the Scottish government does decide on a delay.
Headteachers should have authority to decide which pupils to accept, which to turn away, and to exclude unruly pupils, it will say.
All secondary headteachers agreed in principle to develop a working group to explore the benefits and efficiencies of implementing the 33-period week.
BBC: Scottish Borders Council school timetable moves on hold
David Hart, general secretary of the National Association of Headteachers, thinks so.
International research indicates that schools do improve if headteachers have reasonable autonomy.
BBC: NEWS | UK | Scotland | Heads could be given more powers
The decision to review the fees has been welcomed by some headteachers, including Brendan Wall, from St Simon Stock Catholic School, in Maidstone.
But its extra provisions, which make pupil-teacher affairs illegal, have been described by the headteachers' union as a "sledgehammer to crack a nut".
Scottish headteachers would be likely to have more control over staffing and spending although they would not have as much independence as their English counterparts.
BBC: NEWS | UK | Scotland | Heads could be given more powers
The NUT's campaign against the linking of pay to performance comes as a headteachers' union has called for a slowing down of the pace of reform.
Their curricula are rigidly laid down by state governments, and schools are among the last bastions of union influence in Australia: headteachers have few powers of discipline.
Headteachers in the borough were furious at the manner of her departure, though critics locally say she was hampered in the post by having no social work background.
The Headteachers' Association of Scotland said it welcomed the plans.
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