Concrete models of addition and subtraction have students working with physical items in their calculations. In these models students observe the process of putting the items together, and taking them apart. When teaching with these models emphasis should be placed on grouping and ungrouping in the base 10 system. 

It should be noted that adding and subtracting through use of a number line is considered a concrete method. 

In the representational, students are drawing base 10 models side by side with a standard algorithm. 

Another version of a representational algorithm is an expanded form model. In expanded form students are seeing how regrouping works, and lends itself to the understanding of the standard algorithm. 

The abstract model, is the standard algorithm of addition and subtraction. In this students "carry" and "borrow". To understand this students have to have a firm grasp on regrouping in the base 10 system. 

Emphasis should be placed on why we "carry", and why we borrow.