This report opens by asking how operations in general, and addition and subtraction in particular, are characterized for elementary school students, and examines the "standard" instruction of these topics through secondary schooling. Some common errors and/or "sloppiness" in the typical textbook presentations are noted, and suggestions are made that these problems could lead to pupil difficulties in understanding mathematics. The ambiguity of interpretation of number sentences of the forms "a+b=c" and "a-b=c" leads into a comparison of the familiar use of binary operations with a unary-operator interpretation of such sentences. The second half of this document focuses on points of view that promote varying approaches to the development of mathematical skills and abilities within young children. The importance of meaning and understanding, particularly among young pupils, is promoted. An interpretation of symbolic notation within the domain of natural or whole numbers is emphasized which stresses "change-of-state" and is thought to be a neglected and potentially useful approach to number operations for young children. Investigations are called for to test instructional methods that focus on states and operators rather than operations per se, to see if the proposed change is indeed fruitful. (MP)