Similarity in Middle School Mathematics: At the Crossroads of Geometry and Number

Abstract

The mathematical idea of similarity is typically taught to students across the middle school years between ages 11 and 14. In this study, students' understanding of presimilarity is examined based on a set of clinical interviews of 21 students aged 12–13 years. Students were asked to scale a series of geometric figures and were found to use a variety of strategies including some that incorporated both geometric and numeric reasoning. Tasks were developed that manipulated the characteristics of figures that students were required to attend to in order to explore the boundaries of numeric reasoning and to maximize the degree to which visual reasoning could be brought to bear on the task. Contrary to the literature, student use of visual reasoning did not indicate less developed conceptions of similarity. In fact, visually-based strategies supported students as they reflected on and sought to improve wholly numeric strategies. Analysis of the interview data indicated that providing students with tasks that required them to scale more complex geometric figures improved their capability to attend to the quantifiable features of shape and to the numeric relationships between them.