Calculation isn't just figuring out the result of an exercise: it is figuring out the decimal representation of the result. Therefore, the ability to calculate is in fact tantamount to a profound understanding of the decimal system. This is one of the reasons why calculation is so important, and why it should not be replaced by a calculator.
Arithmetical operations can be calculated in many ways. The methods currently taught in school are the result of generations of thought, and much wisdom has been invested in them. Most are based on writing the exercises vertically, so that the ones digits are one above the other, the tends digits are one above the other and so forth.
Calculations are based on the knowledge of the addition and multiplication tables -- the sums and products of numbers smaller than 10. These must be memorized. The addition table should be well established in the first grade, and the multiplication table in the second or third grade. In addition, the children should be familiar with the rules that govern the operations, such as the distributive law and the rules of change.
The operation of division is the most difficult to calculate. On the other hand, the algorithm of division, called "long division," includes fundamental principles and therefore it should not be passed over.
Arithmetic for Parents: A Book for Grownups about Children's Mathematics
by Ron Aharoni