In school, children are first taught to count. Adding up is then presented as a manipulation that is implied in counting. Philosophically, this is not just a huge step, it is a questionable act. At first the groups of 3 and 4 units each had their own counting lines, each counting back to one. At the other side of the equation there suddenly is only one counting line.
Teachers like to present multiplication as just another case of counting.
The question what six times three is, is tackled by creating six groups of three units each. Such an equation is presented as a stretched-out exercise in counting, resulting in eighteen units to be counted in total. In this approach, the symbols 6x3 stand for a queue of units that are counted and add up to eighteen. Again, the separate counting lines within groups are magically transformed into a single counting line. Maths combines processes such as counting, adding and multiplying into a single system, as if all such processes are manipulations of numbers that are all lined up and named in a uniform way.