By the Education Act of 1872 school education was made compulsory for all children between ages of 7 and 13 (later (I think by 1882) extended to 5 to 13). The management of the schools was to be under the control of elected school boards in each town and parish (hence title Board Schools) and the costs to be bourne out of taxes.
In Peel the choice was probably between
Christian’s Endowed School—Mr. Wm. Fargher, head master ; Miss Jackson, girls ; Mrs. Cowley, infants.
Wesleyan Schools—Mr. Arthur John Ridge, master.
and most likely Christian's (or more commonly Cloth-workers) - the school still exists though on a different site thus I suspect it will still have any records