The development of algebra: Confronting historical and psychological perspectives

Why did the development of algebra lag behind geometry for so many centuries? Why do today’s pupils have difficulties with even the simplest word problems? What prevented generations of mathematicians from accepting the idea of the irrational and the negative numbers? What are the roots of the difficulties experienced by students confronted with the concept of complex number for the first time? It is neither by chance, nor by mere carelessness, that my list of questions is a mixture of psychological and historical puzzles. As different as they seem at first glance, these two sets of problems may in fact have much in common. Indeed, there are good reasons to expect that, when scrutinized, the phylogeny and ontogeny of mathematics will reveal more than marginal similarities. At least, this is what follows from the constructivist view according to which learning consists in the reconstruction of knowledge. Piaget-one of the first and most outspoken protagonists of constructivism, and thus of the thesis that “the historical-critical and psychogenetic studies [should] converge” (Garcia & Piaget, 1989, p. 108)-grounds his position in the claim that

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