JMB JOURNAL OF MATHEMATICAL BEHAVIOR, 16 (2), 125-132 ISSN 0364-0213. Copyright 0 1997 Ablex Publishing Corp. All rights of reproduction in any form reserved. How Far Can You Go With Block Towers? CAROLYN A. MAHER zyxwvutsrqponmlkjihgfedcbaZYXWVUT Rutgers University ROBERT SPEISER Brigham Young University This report focuses on the development of combinatorial reasoning of a lCyear-old child, Stephanie, who is investigating binomial coefficients and combinations in relationship to the bino- mial expansion and the mapping of the binomial expansion to Pascal’s triangle. This research reports on Stephanie’s examination of patterns and symbolic representations of the coefficients in the binomial expansion using ideas from earlier explorations with towers in grades 3-5 to examine recursive processes and to explain the addition rule in Pascal’s triangle. This early work enabled her to build particular organization and classification schemes that she draws upon to explain her more abstract ideas. This teaching experiment is a component of a longitudinal study of the development of mathematical ideas in children. Attention has been given to studying how children build mathematical ideas, create models, invent notation, and justify, reorganize, and extend their ideas. We have been observing Stephanie doing mathematics for nine years. Stephanie’s early work in combinatorics began in grade 2, building models to justify her solutions and validating or rejecting her own ideas and the ideas of others on the basis of whether or not they made sense to her. In the earlier studies Stephanie simultaneously referred to and monitored the strategies of other group members and integrated the ideas of her partner into her own representations. This enabled her to keep track of her data and to cycle through constructions, thereby producing more powerful representations (Davis, Maher, & Martino, 1992; Maher & Ma&no, 1991, 1992a, 1992b). In grade three, Stephanie was introduced to investigations with block towers1 which enabled her to build visual patterns-such as the local organization within specific cases based on ideas like “together,” “separated, ” “how much separated”-to show us her ideas. She recorded her tower arrangements first by drawing pictures of towers and placing a sin- gle letter on each cube to represent its color, and then by inventing a notation of letters to represent the colored cubes. Stephanie’s working theories about the towers provided strik- ing and effective ways of working with mathematical ideas. They triggered, for her, the spontaneous use of heuristics (guess and check, looking for patterns, think of a simpler problem, etc.); the development of arguments to support a component of a solution; and the extension of arguments to build more complete solutions. Direct all correspondence ?o: Dr. Carolyn A. Maher, Rutgers University, Graduate School of Education, 10 Sem- inary Place, New Brunswick, NJ 08903. 125