School governor Alison Streatfeild-James said it was an "utter tragedy".
"It's a sombre mood at the school today. All the children were informed in assembly and prayers were said. It's a terrible piece of news to break," she said.
"Children are intelligent, they understand what has happened. All we can do is offer them support in an appropriate way."
Headmaster John Milne said: "Alexander was a well liked and respected boarder who will be sorely missed by his fellow pupils and staff."
Mr Coakley, a founding director of Aberdeen-based oil industry company Project Design and Management Services Limited, was working on a survey in Brazil.
Mrs Coakley, from Whitby, said the news had devastated their sons, Dominic, 31, and Patrick, 29, and daughter Mise, 25.
"They are distraught. They absolutely idolised and worshipped their father," she said.
Mrs Coakley added: "He's not going to come back. Yesterday I was really optimistic, today maybe more realistic. I would love him to come back."
Aberdeen-based undersea engineering firm Subsea 7 said one of its employees, Mr Gardner, from Gourock, was on the plane.
Joyce Gardner, 51, paid tribute to her "loving, caring" husband.
She said: "Graham was very supportive and always there for me when I needed him.
"Although we don't have children, he was very much a family man and loved playing with all the children in our extended family. He was just a big kid at heart.
Missing man Arthur Coakley’s wife, Patricia, and his business partner Ken Pearce
"I'm sure I can speak for everyone that knew him, whether family, friend or colleague, in saying that he will be missed, dearly."
The company's chief executive Mel Fitzgerald said: "Graham is a well respected and highly regarded employee of Subsea 7 who has made a valuable contribution to the company over many years. Our deepest thoughts are with his family".
Dancer Eithne Walls, 26, originally from Ballygowan, County Down, was among the Irish people on board.
She spent a year with a production of Riverdance on Broadway before studying medicine at Trinity College, Dublin, and working as a doctor in the Irish capital.
Ms Walls was travelling home from holiday with fellow doctors Aisling Butler, from Roscrea, County Tipperary, and Jane Deasy from Dublin.
The Irish Republic's foreign affairs minister, Micheal Martin, said his officials had spoken to the families of the Irish passengers thought to be on the plane.
Flight AF 447 left Rio at 1900 local time (2200 GMT) on Sunday
Airbus A330-200 carrying 216 passengers and 12 crew
Contact lost 0130 GMT
Missed scheduled landing at 1110 local time (0910 GMT) on Monday in Paris
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