"In solving a problem, be it one in the calculus, in algebra, or in the second year of arithmetic, we begin by substituting for the actual things certain abstractions represented by symbols; we think in terms of these abstractions, aided by symbols, and finally from our result we pass back to the concrete and say that we have solved the problem. It is all a matter of "one to one correspondence," it being easier for us to work with the abstract numbers and their corresponding figures than to work with the actual objects. Fundamentally the process is something like this: 1. By abstraction we pass to numbers. 2. Thence we pass to symbols, and we make an equation, either openly, as in algebra, or concealed, as in many forms of arithmetic. This equation we solve, the result being a symbol. 3. We find the number corresponding to this symbol, and say that the problem is solved. All this does not mean that primary number is to be merely a matter of symbols. It means that in mathematics we find it more convenient to work purely with symbols, translating back to the corresponding concrete form as may be desired. And so those teachers who fear lest the child shall drift into thinking in symbols instead of in number, are really fearing that the child shall drift into mathematics."