AcquisitionofMeaningforArithmeticStructuresWith thePlanner
2012
Different perspectives generate different conceptions of what it is to understand arithmetic. For this purpose, Ohlsson (1987) distinguished three aspects of arithmetic: theory, activity and language. The theory involves mathematical principles such as closure, associativity, or laws of distribution. Arithmetic in the activity sense enables the student to apply procedures to compute values of arithmetic functions. To understand arithmetic in the language sense is to understand its symbols and the expressions in which they appear. As in natural language, the study of the semantics of arithmetic symbols leads to the distinction between sense and reference. For instance, the operation of adding has several senses: combine expresses that two quantities are joined together, change means that a quantity is modified by adding to it a new quantity, and increase conveys that a certain quantity is bigger than another by a certain amount (see Carpenter, 1985). However, understanding the referential aspect of addition is being able to identify a referent for "a + b" when "a" and "b" are assigned to referents. Similarly, understanding the referential aspect of multiplication is being able to identify a referent for "a X b" (e.g., an area, or a space of possible couples [a, b]).