Duval

RAYMOND DUVAL

A COGNITIVE ANALYSIS OF PROBLEMS
OF COMPREHENSION IN A LEARNING OF MATHEMATICS

ABSTRACT. To understand the difficulties that many students have with comprehension
of mathematics, we must determine the cognitive functioning underlying the diversity of
mathematical processes. What are the cognitive systems that are required to give access
to mathematical objects? Are thesesystems common to all processes of knowledge or,
on the contrary, some of them are specific to mathematical activity? Starting from the
paramount importance of semiotic representation for any mathematical activity, we put
forward a classification of the various registers of semiotic representations that are mobilized
in mathematical processes. Thus, we can reveal two types of transformation ofsemiotic
representations: treatment and conversion. These two types correspond to quite different
cognitive processes. They are two separate sources of incomprehension in the learning
of mathematics. If treatment is the more important from a mathematical point of view,
conversion is basically the deciding factor for learning. Supporting empirical data, at any
level of curriculum and for anyarea of mathematics, can be widely and methodologically
gathered: some empirical evidence is presented in this paper.1
KEY WORDS: cognitive paradox, figural organization, knowledge object, language,
mathematics learning, recognition, multifunctional and monofunctional registers, noncongruence, representation, representation conversion, semiotic representation, semiotic
system, thinkingprocesses, treatment

How can we understand the difficulties, frequently insurmountable, that
many students have with comprehension of mathematics? What is the nature of these difficulties? Where are they located? These questions have
taken on a particular magnitude and importance with the recent pressure
for more initial mathematical training to be given to all students in order to
prepare them toface a technological and computer-oriented environment of
perpetually increasing complexity. They are both an educational challenge
in classrooms and a theoretical challenge to research on the development
and learning of mathematical knowledge. The processes of mathematical
knowledge acquisition are so complex that quite different approaches seem
required. The most predominant, and sometimesopposite, are the epistemological and the educational. But they have in common the use of the
notion of representation to characterize the kind of phenomena that occur
in any knowledge process or that constitute it.
This basic notion of representation is very old and accurate. A representation is something that stands for something else. But at the same time this
notion can be elusive or tooformal. What is the nature of this “something
Educational Studies in Mathematics (2006) 61: 103–131
DOI: 10.1007/s10649-006-0400-z

C

Springer 2006

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RAYMOND DUVAL

standing for . . .”? You can get quite a wide range of answers, depending on
whether you consider the representations with regard to the concrete individual and his or her experiences, to the mind structures, or onthe contrary,
to the knowledge objects with their specific epistemological requirements
(Hitt, 2002). Thus, representations can be individuals’ beliefs, conceptions
or misconceptions to which one gets access through the individuals’ verbal or schematic productions. This answer, first developed in two major
studies of Piaget (1923, 1926), is now one of the major methodological
and theoreticalframeworks for investigating and explaining mathematical knowledge acquisition. But representations can also be signs and their
complex associations, which are produced according to rules and which
allow the description of a system, a process, a set of phenomena. There
the semiotic representations, including any language, appear as common
tools for producing new knowledge and not only for...