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Council chiefs say they need to seek "clarity" from the Scottish Government after the surprise manifesto pledge.
A decision to launch a statutory consultation on the plans for the Gaelic-medium education (GME) high school to be co-located with a new Liberton High was due to be taken last month, but the item was rescheduled for next week.
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Now it has been delayed again with a special meeting due to be called a week later. The council insisted the move would not hold up the consultation.
Many parents of pupils at Edinburgh's Gaelic primary school, Taob na Pairce in Leith, were delighted at the manifesto pledge of support for "the creation of a standalone GME secondary school in central Edinburgh" and additional GME primary schools.
The manifesto added: "Edinburgh City Council has taken forward important engagement on GME provision, but we will ensure that this is now incorporated within a new national strategic approach."
Supporters of a stand-alone school say research shows children learn a language better if they are in a "linguistic bubble", speaking it all the time they are at school, which is more difficult if they share a site with an English-language school.
Kenny Beaton, who has two children in Gaelic-medium education, lives five minutes’ walk from Liberton High but wants to see the new Gaelic school in a more central location.
He said: "Although it would be very convenient for me personally, I’m not convinced a shared campus school at Liberton is a good idea. Liberton is on the periphery of the city, and I’m concerned some families living near Taobh na Pàirce may not continue with Gaelic-medium education if the secondary moves here.
“In contrast, the SNP manifesto offers a dedicated stand-alone Gaelic school in a central location, similar to the Glasgow Gaelic School, which would clearly be much better for the growth of the language. The council should pursue that idea with the Scottish Government, and should drop their plans to move Gaelic to Liberton.”
He said he would also welcome a new GME primary in the Liberton area and another in the west of the city. “This would boost numbers of Gaelic pupils from families in the local areas and make a central high school the more sensible option, with feeder primary schools from different parts of the city.”
The existing secondary-age GME unit has to move from its base at James Gillespie's High because the school is at capacity with pupil numbers set to grow.
A survey last year found just 15 per cent of Gaelic parents supported the Liberton joint campus proposal.
A council spokesman said: “A report on our plans for a statutory consultation has been delayed by a week as we seek clarity from the newly-elected Scottish Government around their plans for future GME provision.
“This will still allow sufficient time for a statutory consultation process to be undertaken which if successful would allow a GME secondary school to be delivered by August 2025