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Large numbers of pupils and staff are also off school waiting for test results; there are not enough supply teachers to provide cover and heads are reaching the end of their tether.
The union says too many parents and others assume everything is back to normal.
But it has written to all councillors in the Capital, warning: "Many schools and many staff are close to breaking."
Official figures from the council show that Covid numbers in Edinburgh schools have risen over the last three weeks. In the week ending November 7, 205 pupils and 35 staff tested positive; the following week, 285 pupils and 60 staff had the virus; and in the week ending November 21, 363 pupils and 55 staff were confirmed with Covid.
The council said positive cases were at their highest in mid-September when there were a total of 937 cases in one week.
EIS Edinburgh secretary Alison Murphy said education faced a comparable situation to the health service. "We have analogous pressures and we have analogous legacy issues - we're trying to help the kids to catch up; no-one wants to go back to closures."
In the letter to councillors she set out some of the pressures on staff, including:
– Dozens of staff members becoming infected, leading to increased anxiety amongst colleagues who worry about the same happening to them.
– Demand for supply teachers far outstripping availability, meaning headteachers and other staff are having to cover classes.
– Teachers having to provide online learning for children who are off waiting for test results while still having to teach the rest of the pupils who are still in class.
– Educational psychologists and specialist support staff being drafted into special schools to serve as teaching assistants just to keep the schools open.
Ms Murphy added: “I have never spoken to as many headteachers who are considering early retirement, or resigning, or who are on the verge of signing off sick, as I have done this term.
"Many school leaders tell me they feel there is little insight from politicians, parents or the public about the day-to-day reality in schools, and that this is leading to totally unrealistic demands being placed on them, and to them feeling isolated and unsupported.
"We are very much in a recovery phase – indeed, many would say we are still in the midst of the pandemic, with some pressures being worse now than they were last year.
"We need to make efforts to dramatically reduce the burden on staff, and particularly, headteachers.”
She said Education Scotland had announced school inspections were to restart and the EIS was planning a national campaign opposing the move.
"Given the staffing pressures in Edinburgh schools, we would welcome a strong commitment from the council to back such a campaign, and to do all in its power to prevent inspection activity in Edinburgh schools. The idea that, at the same time as educational psychologists and others are rolling their sleeves up and going into schools to simply keep the doors open, someone else is going to stand in the back of a classroom with a clipboard is insanity.”
Ms Murphy told the Evening News that Edinburgh council had been “pretty supportive” and was working hard. “But there is a lot of pressure on them to make it business as usual.
"What ought to be happening is dropping everything non-essential. But because so many people on the outside seem to think that it's back to normal and everything is fine there is huge pressure on people to carry on doing everything.”
Education convener Ian Perry praised staff for their efforts in the face of the pandemic.
He said: “It’s clear from the high number of positive cases in our schools Covid is still very much with us. We’re continuing to support our schools to keep following public health guidance to help stop the spread of the virus, help keep everyone safe and our schools open.
“Since March last year all our education teams have worked tirelessly to deliver teaching and learning so our young people receive the education they are entitled to, especially having missed so much face-to-face learning during the pandemic.
"I can’t praise them enough for all their efforts in the face of the immense challenges they are facing. So it’s really important we all keep working together and doubly renew our collective efforts to slow the spread of the virus in our communities by following health advice so our schools can stay open.”
And vice-convener Alison Dickie added: “Only the wonderful staff across this city’s amazing schools truly understand the sheer scale of what is involved on a daily basis to keep our schools open in the current environment. From the additional workload of preparing learning and catch up support for pupil absences to leadership teams providing teaching cover in the classrooms, they are working in conditions that will go down in educational history to ensure our children and young people are not further impacted, and that learning and teaching continues.
“The health and well-being of our teachers and support staff is extremely important to us and we must do everything we can to ensure they receive the support that they need. Our health and wellbeing guidance for schools has been updated and regular training sessions are delivered to school leaders. Most of all though, we need to increasingly listen to our teaching staff to discuss and resolve any concerns that they may have so that they feel supported.”
Read more: The 12 areas of Edinburgh with the highest numbers of positive Covid-19 tests