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The development of patterning strategies during the year prior to formal schooling was studied in 53 children from 2 similar preschools. One preschool implemented a 6-month intervention focusing on repeating and spatial patterns. An interview-based Early Mathematical Patterning Assessment (EMPA) was developed and administered pre- and postintervention, and again following the 1st year of formal schooling. The intervention group outperformed the comparison group across a wide range of patterning tasks at the post- and follow-up assessments. Children from the intervention group demonstrated greater understanding of unit of repeat and spatial structuring, and most were also able to extend and explain growing patterns 1 year later. In contrast, most of the comparison group treated repeating patterns as alternating items and rarely recognized simple geometrical patterns. The findings indicate a fundamental link between patterning and multiplicative reasoning through the development of composite units.

In this study, an intuitive model was defined as an internal mental structure corresponding to a class of calculation strategies. A sample of female students was observed 4 times during Grades 2 and 3 as they solved the same set of 24 word problems. From the correct responses, 12 distinct calculation strategies were identified and grouped into categories from which the children's intuitive models of multiplication and division were inferred. It was found that the students used 3 main intuitive models: direct counting, repeated addition, and multiplicative operation. A fourth model, repeated subtraction, only occurred in division problems. All the intuitive models were used with all semantic structures, their frequency varying as a complex interaction of age, size of numbers, language, and semantic structure. The results are interpreted as showing that children acquire an expanding repertoire of intuitive models and that the model they employ to solve any particular problem reflects the mathematical structure they impose on it. Several recent studies have shown that students can solve a variety of multiplicative problems long before formal instruction on the operations of multiplication and division. For example, Kouba (1989) found that 30% of Grade 1 and 70% of Grade 2 students could solve simple equivalent group problems and Mulligan (1992) found a steady increase in success rate on similar problems from over 50% at the beginning of Grade 2 to nearly 95% at the end of Grade 3. More recently Carpenter, Ansell, Franke, Fennema, and Weisbeck (1993) found that even kindergarten students could learn to solve multiplicative problems. Students use a range of solution strategies to solve multiplication and division word problems, and from this it has been inferred that they acquire various intuitive models of multiplication and division (Fischbein, Deri, Nello, & Merino, 1985; Kouba, 1989; Greer, 1992). The interest in intuitive models lies in the proposition that they are formed early on in elementary contexts and can strongly influence students' understanding of more complex multiplicative situations in secondary school and adulthood, often negatively (Fischbein et al. 1985; Graeber, Tirosh, & Glover 1989; Simon, 1993). However, it is not clear what intuitive models young children form, how they are related to the semantic structure of the problems to be solved, and how the models develop over time. The present paper attempts to throw light on these questions using data from a longitudinal study of students in Grades 2 and 3.

Children's solution strategies to a variety of multiplication and division word problems were analvsed at foor interview stages in a 2-year longitudinal study. The study followed 70 children Erom Year 2 into Year 3, from the time where they had received no formal instruction in multiplication and division to the stage where they were being taught basic multiplication facts. Ten problem structuren, five for multiplication and five for division, were classified on the basis of differences in semantic structure. The relationship between problem condition (i..e. small or large number combinations and use ofphysical objects or pictures), on performance and strategy use was also examined.The results indicated that 75% of the children were able to solve the problems using a wide variety of strategies even though they had not received formal instruction in muitiplication or division for most of the 2 year period. Performance level generally increased for Bach interview stage, but few differences werefound between multiplication and division problems except for Cartesian and Factor problems.Solution strategies were classified for both multiplication and division problems at three levels: (i) direct modelling with counting; (ii) no direct modelling, with counting, additive or subtractive strategies; (iii) use of known or derived facts (addition, multiplication).A wide range of counting strategies were classified as counting-all, skip counting and double counting. Analysis of intuitive models revealed preference for a repeated addition model for multiplication, and a 'building-up 'model for division.In recent years there has been a steady growth in mathematics education research investigating how children develop mathematica) concepts and processes (Bel)

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