Conference Theme

Phrases like ‘mathematics is the language in which God has written the universe’ (Galileo Galilei) or ‘all things in nature occur mathematically’ (René Descartes) express the idea that if we want to understand the world, then we need to use mathematics. But can we use mathematics without understanding? John von Neumann once said ‘Young man, in mathematics you don't understand things. You just get used to them.’ One way to interpret this statement would be to say you could use mathematics (with success) without understanding it. Or, perhaps we can speak of a kind of understanding that is merely instrumental instead of relational (Skemp, 1976) or intuitive, or formal (Byers & Herscovics, 1977). Another different way to read von Neumann’s statement is to take it as a clarification that understanding is not a white or black issue. There may be degrees of understanding. And there may also be understanding that impedes better understanding. In the words of Richard Skemp, “to understand something means to assimilate it into an appropriate schema. This explains the subjective nature of understanding, and also makes clear that this is not usually an all-or-nothing state” (Skemp, 1971, p. 46). Pragmatically, the power of adaptability of a schema results from its connection to a greater number of concepts, but it may happen that what is an appropriate schema at one particular time may be obsolete and turn into an obstacle later on (Brousseau, 1997).

Let’s get back to René Descartes: ‘All things in nature occur mathematically’. A different idea implied by this saying would be that to understand mathematics we need to use the world. This idea is at the base of the concept of mathematization, or, more precisely, horizontal mathematization (Freudenthal, 1991). Also concurring with this idea is the belief many have that Mathematics is a cultural product based on human experiences, such as counting, measuring, locating, designing, explaining, and playing (Bishop, 1988). Nevertheless, mathematical understanding has to do with both the learning of invariants and the acquisition of cultural tools in which children can represent mathematical ideas, in a dynamic and interconnected process (Nunes & Bryant, 1997). This idea is in line with a recent formulation of understanding in epistemology, in which understanding of a given phenomenon has to be maximally well-connected and it may have degrees of approximation (Kelp, 2015).

Concerning the learning and teaching of mathematics in the complexity of our world, we can revalue the ideas of Galileo, Descartes and Von Neumann on the central role of mathematics in the context of the genetic approach of epistemology proposed by Piaget to the logico-mathematical dimension of the construction of scientific knowledge. Piaget proposed to replace the positivist hierarchization of science with an interdisciplinary cyclic epistemology. This approach to epistemological interrelationships in the context of learning, conceived in the digital environment of education, calls into question not only the connections of mathematics as a scientific discipline, but also the connections of mathematics as an academic subject. How is it possible to make the presence of mathematics visible in the understanding of other school subjects? How to collaborate with other teachers of mathematics and of other courses? This question of interdisciplinarity is in close interaction with the learning and teaching of the complexity and variety of the natural and social phenomena of our era.