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abstract algorithm all-or-none analysis Appendix Attend automaton models basic behavioral psychology Beilin Bloomfield Campbell and Stanley Cartesian product characterized child Chomsky Chomsky's column addition concept of stages correct response curriculum Descartes detailed developmental developmental psychology discussion educational research elementary arithmetic elementary mathematical error parameters estimates evidence example existence of stages experimentation experiments finite automata finite automaton fundamental theory given grammar grid ideas important incorrect intellectual internal program language large number learning models likelihood function linear linear regression linguistic mathematical proofs matter negative proof nonsense syllables number of correct one-column addition exercise output error Patrick Suppes perceptual processing performance models Piaget principles probabilistic automaton probability problem proofs psychological theory reading register machines regression model relevance remarks restricted second-order fantasy segments seriation serious simple statistical structural feature student subroutine SUMR symbol systematic theoretical theory of competence theory of learning unsupported variable verbal instruction vertical addition