Our first aim is to make the study of informal notions of proof plausible. Put differently, since the raison d'étre of anything like existing proof theory seems to rest on such notions, the aim is nothing else but to make a case for proof theory; ...

As to the misconceptions: In the first place, the existence of "undecidable propositions" or "unsolvable problems" has only remote connections with the failure of excluded middle. More precisely, from the fact that a certain problem is unsolvable, one cannot infer that the affirmative and negative answers to that problem are both incorrect. Both Gödel's and Church's theorems were originally proved for systems with the excluded middle, i.e. for systems in which 'p or not p' is provable for every proposition 'p'; (...) though in the light of Gödel's theorem there are propositions 'p' such that neither 'p' nor 'not p' is provable in those systems. Careful distinction should be made between the following locutions. (shrink)

This paper investigates teachers' use of prior knowledge in whole class teaching contexts and draws on data from an ESRC-funded study. The paper explores how teachers conceptualise prior knowledge, principally as that which has been taught in school. It demonstrates strong teacher awareness of how the teaching under consideration fits with learning previously undertaken by the class, but less awareness of how the learning might build on prior learning outside school. The paper considers how teachers make connections between new learning (...) and prior learning, and how those connections can variously support or confound children's acquisition of new knowledge and understanding. (shrink)

This paper investigates teachers' use of prior knowledge in whole class teaching contexts and draws on data from an ESRC-funded study. The paper explores how teachers conceptualise prior knowledge, principally as that which has been taught in school. It demonstrates strong teacher awareness of how the teaching under consideration fits with learning previously undertaken by the class, but less awareness of how the learning might build on prior learning outside school. The paper considers how teachers make connections between new learning (...) and prior learning, and how those connections can variously support or confound children's acquisition of new knowledge and understanding. (shrink)