The Wider Picture
The general situation in early Victorian England was that school was for the children of the rich. There were few schools other than those provided by the churches and those that provided a very elementary service (little more than a child minding service). Even very young children could work and therefore be a source of income to their families. Therefore there was some reluctance for children to be sent to school. Possibly they would be allowed to attend a Sunday school. The British and Foreign School Society for the Education of the Labouring and Manufacturing Classes of Society of Every Religious Persuasion supported the building of the British Schools linked to the non-conformist churches. The National Society for Promoting the Education of the Poor in the Principles of the Established Church supported the building of National Schools, and these were linked to the Church of England. The 1870 Elementary Education Act dictated that in areas where there were insufficient school places then a School Board should be created to oversee the provision of a local non-denominated school. Parents still had to pay for their children to attend, but, where they were unable to the School Board would pay. These boards were financed through the local rates. School attendance for five to ten year olds became compulsory in 1880 although 'truancy' rates were high. The upper age limit was gradually extended. In 1891 elementary education was made free.
A detailed history of Malton School can be read in "A History of Malton School 1547 - 2006" by Stephen Fearnley. The author also maintains the Maltonian Web - a collection of material collated about the school and the pupils (known as 'Old Maltonians'). You'll find archive material and an insight into various bits of general Malton history.
Early Malton Schools
Trade directories and newspaper advertisements indicate that there were a number of schools or academies operating in the 1820s. Those wanting their children to attend would have needed to pay 'reasonable fees'. A Mrs Soulby ran a school in Yorkersgate for 'a limited number of Young Ladies to be instructed in English, History, ancient and modern, Chronology, and Work.' Fees were stated to be twenty six guineas per annum  W. Constable ran the Newbegin House School and advertised in 1880: The attention of Agriculturists, Tradesmen, and all engaged in Commercial pursuits, is directed to the above school, in which the course of instruction is particularly adapted to the educational requirements for the sons of the above classes at the present time. The chief features of the school are: Excellent Accommodation, Thorough Education, Diet Substantial, Terms Reasonable … 
-  Yorkshire Gazette, 11 May 1822
-  Driffield Times, 10 January 1880
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MRS. SOULBY takes this opportunity of calling the attention of the Public to her Establishment, for a limited number of Young Ladies, to be instructed in English, History, ancient and modern, Chronology, and Work. Terms, Twenty-six Guineas per annum - Entrance One Guinea - Music, Drawings, and French, &c, &c on the most reasonable Terms. The House is spacious and handsome, beautifully situated in a large Garden, to which the Pupils will have free access.
The School will be open on the 22nd JULY next.
Yorkshire Gazette Saturday, 11 May 1822
Malton Sunday Schools
The Wesleyans and Primitive Methodists had Sunday Schools operating in the 1820s. The Malton Wesleyan Sunday School celebrated its sixteenth anniversary in October 1840  The Primitive Methodist Sunday School celebrated its twenty-eighth anniversary in July 1854  There was also a Malton Church Sunday School, presumably under the wing of St. Michael's 
-  Yorkshire Gazette, 24 October 1840
-  Yorkshire Gazette, 5 August 1854
-  York Herald, 6 August 1842
PROSPECT HOUSE, MALTON
THE MISSES HALL (Successors to the Misses Spence) beg to announce that they receive YOUNG LADIES to board and Educate in the sound principles of a useful and accomplished Education. Their Establishment affords great and numerous advantages on Moderate Terms, and its healthy situation, good order, and general arrangements are well known and universally appreciated. References may be had from Parents of Pupils now under their charge. A PUPIL TEACHER will be required after the Vacation. YG Saturday 29 June 1867
Malton Adult Schools
The Malton Adult Schools were founded in 1875, and in 1882 a larger building was erected on the site of an old Friends’ Meeting-house. In 1906, an additional wing, containing billiard-room, classrooms etc., was erected  For more information about the Malton Adult School see this article transcribed from the Yorkshire Gazette, 30 December 1905.
-  Yorkshire Post and Leeds Intelligencer, 11 October 1906
Information in Four Categories
Lots of information about who lived here and where! Families, Malton Butchers Cricket Team, Malton People Database, Memories of Malton, Charles Dickens Connection, Town Bellman, Town Crier, Wives Wanted, Malton in the 1840s, Census including some transcriptions, 1858 List of Voters for St. Michael's, Earl Fitzwilliam Rent Accounts, Newspaper announcements of births, deaths and mariages
What was the town like in Victorian and Edwardian times? Where is/was that street? House numbering, Malton in the 1840s, Planning & Building Control, Toilets in Chancery Lane, Streets Butcher Corner, Castlegate, Greengate, Market Place, Middlecave, Newbiggin, Old Maltongate, St. Michael Street, Saville Street, The Mount, Wentworth Street, Wheelgate, Yorkersgate, York Road, Peasey Hill, Planning & Building Control, Town Guides Old Pictures and Maps, Walks around the town take in a bit of history! Land Hearth Tax, 1873 Return of Owners of Land, Property, North Riding Register of Deeds, Finance Act 1910
What businesses and industries were here? Trade Directories, Photographers, Undertakers, Apprentices, Banks, Breweries, Local Traders and Advertisements, Bankruptcy, Malton and Norton Cooperative Society, Longsters, Fitch & Co, Thomas Taylor, Public Benefit Boot Co., Shopping Week 1922, The Manure Company, Biscuit Mills, Milling, Iron Foundries, Ralph Yates, Industrial Safety
What happened here both locally and in response to wider events? Baker's Chronology, New Malton Spa, Emigration, The Telephone Comes to Malton, Cemetery, Horse Procession, Coronation of Queen Victoria, 1937 Coronation Celebrations, St. Michael's School, Racecourse, Emigration, Malton Golf Course, Sebastopol Cannon, Middlecave Windmill, Traffic, Theatre, Queen Victoria's Jubilee, Coronation of Edward Vll, Longster's Spa Garden, Curling, Talbot Hotel Newspapers Historical Background, Malton Messenger, Malton & Norton Gazette, including digital copies of the first few editions from 1855 Worship Three Ecclesiastical Parishes, 1857 locations, St. Michael's, St. Leonard's, Primitive Methodists, Unitarians, Independents, Catholics, Baptists, Wesleyans, Society of Friends, Congregationalists, 1851 Religious Census Police & Fire Police, law & order, prostitution, fire brigade, fires, Fire Brigade Friendly Society Pubs & Beerhouses Brewster Sessions, landlords, Temperance movement, closure dates Secret Orders Freemasonry, Friendly Societies, Oddfellows, Independent Order of Rechabites, Shepherds and Charities The Railway Victorian Heyday, Abolition of the Turnpikes, Accidents, Station Location, Railway Crossing & Bridge, Excursions, Railway Buildings, York & Scarborough line, Malton & Driffield line, Malton and Whitby line Health The sanitary condition of Malton 1854, nuisance removal, local board of health, the Dispensary, the Cottage Hospital, Apothecaries, Doctors and Surgeons, Dentists, Galvanism, 1932 typhoid outbreak, Health & Housing in 1909, Cholera, Smallpox, Smallpox Vaccination, Typhoid, 1918 Influenza epidemic Workhouse Settlement & Removals, managing the workhouse, state of the workhouse 1818, workhouse provisions, scandal at the workhouse, life in the workhouse, one of yours in the workhouse, masters & matrons, advertisements for staff First World War War is declared, the Wider War Effort, Employment issues for local tradesmen, Zeppelin Raids, Local Recruitment Advertising and Meetings, Military Tribunals, Medals and Bravery, Casualties, War Memorial, Armistice is Signed