Several consequences can be drawn from the fundamental piagetian idea
that knowledge results from adaptation. One of them has been drawn by
Piaget himself, that we should study development to understand what kn
owledge consists of. Further consequences are the idea that activity i
s the main instrument of adaptation, and also the idea that teaching s
hould offer students as many opportunities as possible for them to dev
elop operational schemes. The cognitive status of the knowledge contai
ned in schemes changes when it is worded and symbolised; Vygotsky stre
ssed that point more than Piaget did. But neither Piaget nor Vygotsky
paid enough attention to the specific difficulties and processes raise
d by the learning and development of specific concepts. This is what d
idactics tries to do. This paper analyses with some detail what scheme
s are made of and what they address; also the variety of domains in wh
ich we develop schemes. The example of additive structures is briefly
analysed, as a conceptual field involving quite different classes of s
ituations and several interactive concepts: they enable students to de
velop a complex network of theorems-in-action over a period of ten yea
rs or more. Several other conceptual fields are mentionned, such as mo
rals, history, physical education, physics and mathematics. An importa
nt consequence of studying the development of specific conceptual fiel
ds rather than logical structures is that cognitive development concer
ns adults as well as children and adolescents. It takes as much time f
or adults to become professionnals (and experts) as it takes children
to master elementary arithmetics and algebra. The conclusion stresses
three key-ideas for education: transposition, mediation and conceptual
ization.