Cardinal numbers indicate the number of elements in a set, i.e. the number of elements in a group with some common identifiable quality (not necessarily identity!) Even very young children will identify 2 apples, which of course are not identical in their appearance in a mathematical sense, as 2 apples.
Ordinal numbers characterize the consecution of elements in a set. Young children will count a number of apples in a row as 1, 2, 3 and identify a certain element as 3, meaning it is the third one.
To the skilled adult this differentiation blurs when calculating. When asked 3 + 5 he will immediately answer 8, as if having added the cardinal numbers 3 and 5.
A young child solves the same problem openly using ordinal numbers, starting with counting the first row 1st, 2nd, 3rd, and continuing in the second row 4th, 5th, 6th, 7th, 8th. Alas! The result is 8.
For children of about 3 years it is a difficult task to calculate with ordinal numbers even for very small numbers. Yet they love to do it as a game. It takes a lot of repetition and exercise to climb from 1 + 1 (which is grasped very quickly) to 4 + 1 , and it is very charming how they use their fingers, open or concealed, to "cheat" by counting.
Quite surprisingly one never gets much beyond 5 in the spontaneous perception of cardinal numbers. Any skill to do fast addition beyond that small number range is the result of memorizing the outcome of numerous combinations and of rules, and hence of hard work. Insofar persistent exercise in basic calculation and the learning of multiplication tables is no boring ill treatment of kids, but the basis of an indispensible cultural skill.
Most adults do not realize that their skill in the perception of cardinal numbers does not go beyond that of young children (and that of well trainable animals, by the way), and that it cannot really be trained. Their skill (if they have it!) is based on a history of memorization of results and rules.