Academic journal article The Mathematics Enthusiast

# Pursuing Coherence among Proportionality, Linearity, and Similarity: Two Pathways from Pre-Service Teachers' Geometric Representations*

Academic journal article The Mathematics Enthusiast

# Pursuing Coherence among Proportionality, Linearity, and Similarity: Two Pathways from Pre-Service Teachers' Geometric Representations*

## Article excerpt

Introduction

Arithmetic and geometry, which have their own code, means, and symbol system, are complementary (Otte, 1990). Mathematical objects disclose their essence in different forms. A single form of representation does not represent the essence of a mathematical object comprehensively. Multiple representations of a mathematical concept help students build a rich connection around the concept and develop insights into the concept (Even & Lappan, 1994). Thus instruction ought to be designed to allow students to create and use various representations and relate them (NCTM, 2000). For instance, students should be encouraged to present their understanding of a proportional relationship not only numerically (e.g. a/b=c/d) but also geometrically (e.g. a straight line passing through the origin or similar triangles).

It is also important to help students recognize connections among mathematical ideas. Building a connection among different ideas and topics aids the development of mathematical maturity (Lester et al, 1994). Away from the traditional view of mathematics as a set of isolated facts and procedures that causes difficulties in learning mathematics (Carpenter & Lehrer, 1999; Hiebert, 2003), students should leam how various mathematical ideas interconnect and build on one another and thus develop a coherent understanding (Common Core State Standards Initiative (CCSSI), 2010). When it comes to proportion, it is imperative to help students understand proportionality, linearity, and similarity as a coherent whole. Isolating proportionality from other subjects and lacking the visualization of it prevent students from seeing proportionality as a concept that connects many topics they leam (Streefland, 1985).

If preservice teachers do not see proportionality and its related geometric ideas as a coherent whole, they would have little chance to guide their future students toward a comprehensive understanding of proportion. With that in mind, we asked preservice teachers in our problem solving class to use geometric representations to solve a problem that requires proportional reasoning. The students' representations varied in terms of the degree of sophistication of their thinking. We also asked them to sequence different representations of their peers' as well as their own from a developmental perspective. In this article, we describe how our task helped preservice teachers be aware of (1) the importance of seeing proportionality and its related ideas as a whole and (2) the significance of sequencing the works of others and their own from a developmental perspective.

Proportional reasoning includes comparing ratios or establishing an equivalent relationship between ratios (Toumiaire & Pulos, 1985). It has played an important role in the development of mathematics in history (Radford, 1996). In school mathematics, proportion is the capstone of the elementary school curriculum and the cornerstone of algebra and beyond (Lesh, Post, & Behr, 1988). It is also considered a unifying theme in a sense that it involves using numbers, graphs, and equations to think about quantities and their relationships (NCTM, 2000).

Among the connections between algebraic and geometric aspects of proportional reasoning, linearity that presents a common ratio to make a line passing through the origin is particularly essential (Karplus, Pulos, & Stage, 1983; NCTM, 2000). Those who think proportionally have a sense of covariation so that they can analyze the quantities that vary together and determine the relationship that remains unchanged (Lamon, 1999). A proportional relationship between two quantities can be formalized using an equation y=kx, which is represented geometrically with a straight line through the origin (see Figure 1).

Proportionality, Linearity, and Similarity

Proportionality and similarity have a deep connection in nature. In Book VI of Elements, Euclid defined similarity based on proportion (Heath, 1956). …

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