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Respect for learner's meanings and prior knowledge. Knowledge and un-
derstanding of mathematics depends on the learner's prior knowledge of
language and sense-making. Instead of discounting prior learning and im-
posing a completely new discursive practice of formal school mathematics,
which disregards and discounts the value of children's out-of-school knowl-
edge (but still implicitly depends on it), a social constructivist pedagogy
should consciously build on that knowledge and meaning. Thus the oral di-
mension is vitally important, as is getting children to describe their ideas,
interpretations, methods, strategies and out-of-school contexts and meaning
worlds. Attention should be paid to developing and extending their vocabu-
lary and the associated meanings for terms like large, small, next to, be-
tween, above, angle, how many, different, alike, less, more, number, shape,
and so forth.
Building on child-methods through the negotiation of knowledge. Much
of the symbolism, conventions and knowledge within school mathematics
(and mathematics in general) is arbitrary and depends on the decision of the
appropriate community and its utility in the pursuit of certain goals. These
processes should be made explicit in a number of ways, including the fol-
lowing: (a) The setting up of didactical situations in which learners develop
their own algorithms for solving problems, then, through discussion, com-
pare and streamline them, then compare them with standard algorithms.
Walther (1984) provides an example in which the multiplication algorithm
is so developed, (b) Offering young learners the opportunity to develop their
own representations and symbolism, as in Hughes (1986), where pre-school
children developed their own numeral notation. Through processes of social
negotiation they can then be presented with standard notation. (c) The ex-
plicit recognition of the arbitrary definitions of mathematics and the under-
lying rationale for them is needed: for example how are 
defined
and the pattern they support; and why 0° is not. (d) In the solution of math-
ematical problems, there may be no unique "right" answer or method, just as
in an English language essay. Correctness usually only applies to matching
a convention or its consequences and is less relevant to higher-level or cre-
ative work in mathematics as in any other school subject.
The inseparability of mathematics and applications (and the centrality of
motivation and relevance). Mathematics teaching, especially at the higher
levels, needs to result not only in learner knowledge of symbolism, algo-
rithms and formal methods and systems. Following Sneed (1971), scientific
knowledge can be understood as comprising a repertoire of interpretations
and applications in a variety of domains of human practice as well as formal
theory. Steiner (1987) recounts Jahnke's extension of this notion to mathe-
matics, and Dowling (1991) likewise has theorized a range of contexts of
application and practice as central to mathematical knowledge. A social
constructivist pedagogy would not separate the intellectual tools of mathe-
PHILOSOPHY OF MATHEMATICS
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matics from their uses. Thus the curriculum would treat concepts, methods
and other tools in the light of (a) their historical and cultural origins and the
problems they serve; (b) current uses and applications, including the mas-
tery of a chosen central selection; and (c) contexts of use of direct meaning
to the lives and interests of the learners. Mellin-Olsen (1987) provides ex-
amples of such projects from Norway.
6.2 A Social Constructivist Theory of Learning Mathematics
A sketch of a social constructivist theory of mathematics learning and
school mathematical activity related to current philosophical work (Ernest,
in press) is offered here as a final didactical consequence. This has three
levels: the social context (including classroom, teacher, learners, etc.), the