As teachers of elementary mathematics I suppose we are all familiar with the Circular 711 of the Board of Education. That circular is now out of print, and its place has been taken by Circular 851, which, I think, is not so well known. The purpose of the latter circular is to state the matter afresh, to explain it more fully, and to justify the attitude adopted. The main practical suggestion of these two circulars has been so largely adopted in present-day teaching in this country that I propose at the outset to base my remarks largely upon it.