What are the theories and methods guiding research on the teaching of mathematics?

Mathematics presents some difficulties of comprehension which are not found in other subjects. The choice of theories and methods to collect and analyze students’ work depends on the perspective taken to study them. This paper examines and compares the mathematical, cognitive and pedagogical perspectives to analyze the process of comprehension and acquisition of mathematical knowledge. With this aim, four issues are approached: a) are the comprehension criteria the same under the three perspectives?; b) does mathematical success with a problem mean comprehension of a cognitive viewpoint?; c) how can types of error be distinguished in order to analyze the causes of lack of comprehension?; d) do progressions built on “pre-requisite” analyses favor or ignore the specifi c cognitive development which leads to the comprehension of mathematics? The paper shows that the particular importance of the cognitive perspective stems from the fact that the mathematical activity has always two sides.