Questions:

Do you distinguish between a valid and an invalid norm?  If so, are invalid norms developed by different mechanisms than valid ones?   Is there a normative character shared by valid and invalid norms, but developed in a different manner? Or are there only valid norms in your perspective, and anything else is not really a norm--just something having the "appearance" of a norm?

Answer1:

Thanks for these instructive questions - the main answer is "Yes". Q3 concerns norms. The royal road to their empirical invetsigation is through normative reasoning - which can be sound or unsound, valid, or invalid. The main problem facing any developing individual is exactly that of demarcating one from the other in an endless cycle through variable causal contingencies, contexts, cultures.

In a 1922 paper, Piaget specifically referred to the problem of pseudo-necessity, and this notion generalises over all mental states, concepts - including pseudo-norms. A pseudo-norm is recognised by an individual as a norm, when this is not the case [cf. false-positives]. Their converse is necessity-blindness, norm-blindness, ie: the disregard and non-recognition of a valid norm [cf. false-negatives]

Pseudo-Norm example
One aspect of conservation reasoning has not been noticed in previous research in that, in this study, five per cent of the responses were  °more blues than whites ±. The standard reason for non-conservation is that there are more in the line which has been lengthened, in this case the whites. Yet these children reckoned that it was the other way round. Reasoning for this response was distinctive, apparently based on a common line of argument.  In three cases, the reasoning  seemed to be familiar in research on conservation in that the children ¯s argument was based on the increased length of the white line:
q more blues because you moved (whites) one on there and on there q more in the blues because the whites have been stretched q when they were like that, they were just the same, but you spreaded them (whites) out But this can ¯t be familiar reasoning. These children had used a premise about the lengthening of the white line to justify their conclusion that the blue line now had more. Was this miscounting on their part? Apparently not so:
q more blues because that one ¯s got six, and this has five, no four q because there is 1, 2, 3, 4, 5 and (pointing to whites) there ¯s four here q more blues because there ¯s only four (whites) left What is interesting in these cases is not miscounting but misconception. From their perspective, the white line after transformation had four counters which these children correctly counted. The children had deliberately not counted the  °other ± two whites since, in their judgment, they were no longer part of the white line. All is clear in two further justifications:
q more blues because you ¯ve taken two away q less [whites] because these two aren ¯t there Quite simply, these children believed you ¯ve taken two away. The white line was no longer the same since these two aren ¯t there. This is despite the fact that the two end whites were open to direct observation in front of them. An analogy might help here. Just as a player in a football game may be sent off as a result of which the team is reduced by one member, so the  °team ± of whites had lost two members in the lengthening. Therefore: there were more blues than whites.

This is a spectacular argument central to which is the defining criterion of line. A defining criterion lays down the properties which must be met by all of its instances, and so what could not be a member if the properties are not met. That is, a defining criterion concerns  °what must be ±.   If a puppy is defined as an individual who is (a) young and (b) canine, then all infant dogs are (and must be) puppies, and no old cats are (nor can be) puppies. Similarly, if a line is defined as something (a) equal,  (b) in length to any other line in that array, then the blue and white lines should be equal in length. These children had invoked in their reasoning the principle that two lines should equal, i.e. their properties should be the same. This is comparable to the principle invoked by children in moral reasoning, namely the  ®principle of...equal action; that is, everyone should get the same treatment under any circumstances ¯ (Damon, 1977, p. 75). This principle is invalid in the moral domain (Turiel, 1983, pp.158-59). It is equally invalid in the mathematical domain. The decisive contribution made by the children who used this argument is their disqualification of the protruding white counters which infringe their criterion of equality in line length. However, these children did not use modal language to express their belief, nor did their reasoning have an explicit modal force. Patently, this defining criterion is false applied to lines in mathematics: it is not a mathematical requirement that two lines should be equal. There can be unequal lines may be in the same array. Further, these children assigned more importance to spatial identity than to numerical equality, which amounts to privilegisation of the former over the latter.

For Discussion, see
Smith, L. (2002). Reasoning by mathematical induction in children ¯s arithmetic. Advances in Learning and Instruction series. Oxford: Elsevier Science Pergamon Press.
See pp67-69
Smith, L. (2003). Teaching reasoning in a constructivist epistemology. Educational and Child Psychology, 20, 31-50 Smith, L. (2006/in press). Norms and normative facts in humna development. In L. Smith & J. Voneche (eds). Norms in Human development; Cambridge: Cambridge University Press.
Smith, L. (2006/in press). Developmental psychology of normative facts. In M. J. Roberts (Ed.). Integrating the mind. Hove, UK: Psychology Press.