From patterns to theories: conditions for conceptual change $ Carolyn A. Maher a, *, Amy M. Martino b a Robert B. Davis Institute for Learning, Graduate School of Education, Rutgers University, 10 Seminary Place, New Brunswick, NJ 08901, USA b Conover Road School, Colts Neck, NJ, USA Abstract This research builds on earlier work of the development of mathematical proof by young children. In this paper, we see the 9-year-old Stephanie extending earlier understanding of argument by cases to argument by mathematical induction, as she investigates, with other classmates, a particular theory. D 2000 Elsevier Science Inc. All rights reserved. 1. Introduction Two, four, eight, sixteen ... that's weird! Look! 2 Â 2 is 4, 4 Â 2 is 8 and 8 Â 2 is 16. It goes like a pattern! You have the 2 Â 2 equals the 4, the 4 Â 2 equals the 8 and the 8 Â 2 equals the 16. (Stephanie, Grade 4, March 6, 1992) $ This research is supported in part by a grant from the National Science Foundation #MDR-9053597 to Rutgers, the State University of New Jersey. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the National Science Foundation. This is a revision of the paper that first appeared as: Maher, C. A., & Martino, A. M. (1997). Conditions for conceptual change: from pattern recognition to theory posing. In H. Mansfield & N. H. Pateman (Eds.), Young children and mathematics: concepts and their representation. Sydney, Australia: Australian Association of Mathematics Teachers. It is reproduced here with the permission of The Australian Association of Mathematics Teachers. * Corresponding author. Tel.: +1-732-932-7971; fax: +1-732-932-1318. E-mail address: cmaher3@home.com (C.A. Maher). Journal of Mathematical Behavior 19 (2000) 247 ± 271 0732-3123/00/$ ± see front matter D 2000 Elsevier Science Inc. All rights reserved. PII:S0732-3123(00)00047-X