Number concepts must support arithmetic inference. Using this principle, it can be argued that the integer concept of exactly ONE is a necessary part of the psychological foundations of number, as is the notion of the exact equality - that is, perfect substitutability. The inability to support reasoning involving exact equality is a shortcoming in current theories about the development of numerical reasoning. A simple innate basis for the natural number concepts can be proposed that embodies the arithmetic principle, supports (...) exact equality and also enables computational compatibility with real- or rational-valued mental magnitudes. (shrink)

This article defends a continuity position. Infants can abstract numerosity and young preschool children do respond appropriately to tasks that tap their ability to use a count and cardinal value and/or arithmetic principles. Active use of a nonverbal domain of arithmetic serves to enable the child to find relevant data to build knowledge about the language and use rules of numerosity and quantity.

The meaning and function of counting are subservient to the arithmetic principles of ordering, addition, and subtraction for positive cardinal values. Beginning language learners can take advantage of their nonverbal knowledge of counting and arithmetic principles to acquire sufficient knowledge of their initial verbal instantiations and move onto a relevant learning path to assimilate input for more advanced, abstract understandings.