A developmental perspective on children's counting

27Is it necessary that a child recognize a group of four objects, let us say dots on a piece of paper, before he may be said to "know" four? Investigations have shown that the child may "know" four in some patterns of grouping and not in others … should the standard be recognition of four in any and all possible groupings? … Does a child really "know" four until he is able to assemble a group of four objects, to select four from a large number? Must he be able to distinguish four from three, from five, and all other numbers? Can he be said to have a "true" concept of four if he is not aware of all of its properties, e.g. that it is half of eight or a third of twelve, that it is twice two and the sum of three and one, and that it is the difference between ten and six and between five and nine? Must one not know that it is the square of two and the square root of sixteen, that it is the logarithm of 10,000? … It is clear that there is no limit which may be set to the extension or perfection of concept It is never complete, and the bounds of its development are limitless. (Douglass, 1925, pp. 444–445)