LIORA LINCHEVSKIand JULIAN WILLIAMS USING INTUITION FROM EVERYDAY LIFE IN ‘FILLING’ THE GAP IN CHILDREN’S EXTENSION OF THEIR NUMBER CONCEPT TO INCLUDE THE NEGATIVE NUMBERS ABSTRACT. We report here an instructional method designed to address the cognitive gaps in children’s mathematical development where operational conceptions give rise to structural conceptions (such as when the subtraction process leads to the negative number concept). The method involves the linking of process and object conceptions through semi- otic activity with models which first record processes in situations outside mathematics and subsequently mediate activity with the signs of mathematics. We describe two experiments in teaching integers, an interesting case in which previous literature has focused on the dichotomy between the algebraic approach and the modelling approach to instruction. We conceptualise modelling as the transformation of outside-school knowledge into school mathematics, and discuss the opportunities and difficulties involved. 1. I NTRODUCTION This paper reports an instructional method aimed at helping to overcome the cognitive gaps in children’s extensions of their number schema. Sfard (1991) identified major intuitive gaps in children’s cognitive development with hurdles she observed in the historical development of mathematics, where mathematical processes had to be transformed into mathematical objects. This transformation historically involved long periods in which processes were constructed, encapsulated and finally reified; the final stage was a relatively sudden ontological shift which occurs when the familiar processes are finally understood to be mathematical objects in their own right. This transformation involves an unavoidable paradox; it requires that mathematicians manipulate the processes instrumentally as objects before they are able to mentally grasp them as such. (See also Gray and Tall, 1994; Sfard and Linchevski, 1994). Our approach addresses this paradox; we claim that with the benefit of hindsight, one may find learning path- ways which do not recapitulate the historical or logical development of mathematics. In our instructional method, we build on Realistic Mathematics Educa- tion (RME) developed by Freudenthal’s followers at the Freudenthal Insti- tute (Treffers, 1987; Gravemeijer, 1994). We appeal to children’s everyday common sense and intuition. Meaning is given to the manipulation of ob- jects through the siting of the objects within familiar contexts. The children Educational Studies in Mathematics 39: 131–147, 1999. © 1999 Kluwer Academic Publishers. Printed in the Netherlands.