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.

Interdisciplinary Mathematics Education

“…Yet listening to discussions in the art class revealed language of space and lines, and the management of both, as the children overcame obstacles involved in scaling. Such challenges and related reasoning have been explored in mathematics education (for example, Streefland, 1984;Cox, 2013) without explicit attention to development of notions of identity and power, possible, in such contexts. In the art class, we describe the mathematics involved as embedded in actions and conversation that also supported the development of identity and power.…”

Learning About Spatial and Temporal Scale: Current Research, Psychological Processes, and Classroom Implications

“…Geometric scaling problems for complicated figures where one shape is embedded in another are more difficult for learners than typical scale factor problems, because the learner must preserve not only the proportional sizes of the objects but also the scalar relationships of the spaces between the figures (Cox, 2013). Thus, teaching proportional reasoning solely through the use of simple scale factor problems may be insufficient for the type of scaling problems encountered by geoscience learners.…”

“…Scaling problems in mathematics are often taught numerically using scale factors. In order to successfully complete scaling problems, students must realize they need to multiply or divide by the scale factor rather than add or subtract, but this concept is not easily mastered by all when completing geometric (Cox, 2013) or temporal (Cheek, 2012(Cheek, , 2013b scaling tasks. Working with scale factors in the metric system is more straightforward than it is in the U.S. customary system (Delgado, 2013a).…”

“…A stream table introduces multidimensional scale relationships such as the channel width-todepth ratio, if only in approximate terms. Scalar relationships between objects in the model may be challenging (Cox, 2013), and learners must be able to determine whether particular spatial relationships that exist in the referent are preserved or transformed in the stream table. Sediments in the model have less scalar variability than those in the referent, and stream tables use only unconsolidated sediments for pragmatic reasons.…”

Students' Concepts‐ and Theorems‐in‐Action on a Novel Task About Similarity

“…Moreover, although students in middle school are generally able to recognize similarity relationships, they may not have developed a systematic way to check for similarity (Cox et al., ). Research has examined how to support students' transitions from informal, visual understandings of similarity to more formal numerical approaches, using students' knowledge of magnification and scale to support the development of proportional reasoning (Cox, ; Lehrer et al., ). Our goal was to examine the interplay between students' use of geometric ideas and proportion by examining the concepts and theorems that students used implicitly in their work on a task about similarity.…”