I've found another couple of bits from the Scotsman which bring a little more details to William:
Association of Directors of Education
At a meeting of Director of Education and Executive Officer under the various Educational Authorities of Scotland, held in Moray House, Edinburgh last night, it was agreed to form an Association of Directors of Education. Mr William Malcolm, M.A., Lanarkshire, was appointed Chairman and Mr James Kemp, Executive Officer Midlothian, Secretary.
18 Sept 1920
There is also a further reference on 8 June 1923, to William being the Director of Education for the Education Authority of Lanarkshire.
Monica