The gains and the pitfalls of reification - The case of algebra

Algebraic symbols do not speak for themselves. What one actually sees in them depends on the requirements of the problem to which they are applied. Not less important, it depends on what one is able to perceive and prepared to notice. It is this last statement which becomes the leading theme of this article. The main focus is on the versatility and adaptability of student's algebraic knowledge. The analysis is carried out within the framework of the theory of reification according to which there is an inherent process-object duality in the majority of mathematical concepts. It is the basic tenet of our theory that the operational (process-oriented) conception emerges first and that the mathematical objects (structural conceptions) develop afterward through reification of the processes. There is much evidence showing that reification is difficult to achieve. The nature and the growth of algebraic thinking is first analyzed from an epistemological perspective supported by historical observations. Eventually, its development is presented as a sequence of ever more advanced transitions from operational to structural outlook. This model is subsequently applied to the individual learning. The focus is on two crucial transitions: from the purely operational algebra to the structural algebra 'of a fixed value' (of an unknown) and then from here to the functional algebra (of a variable). The special difficulties experienced by the learner at both these junctions are illustrated with much empirical data coming from a broad range of sources.