Why Mechanical Manipulation?

Just how difficult it is to purge our thinking mechanism of early implanted misconceptions is brutally manifest in Mr. Richter's splendid and most timely article in the February issue of The Mathematics Teacher. He states that it has been found that those aspects of algebra which involve thinking are retained to a much greater degree than those involving mechanical manipulations. To Mr. Richter and to most of us, who studied our high school algebra before the days when an over indulgent public became possessed with the mistaken notion that somehow the schools could literally implant education into a neutral mind, there really are no mechanical manipulations in algebra. No mental growth occurs from a manipulation that is not in direct response to some reasoned decision. Lapses in adherence to this principle in the forms of rules, such as invert divisor, transpose, change signs, cancel, and the like, are the cancerous infestations which, during our recent prodigious experiment in the field of untrained and inadequately and inexpertly supervised teaching, have brought down upon us the present widespread and generally warranted criticism.