A New Look at the Representations for Mathematical Concepts: Expanding on Lesh's Model of Representations of Mathematical Concepts.

Author:Johnson, Elizabeth Lee

The Introduction

The mathematics that students need to know today is different from the knowledge needed by their parents. In our global society and a world of ever changing technology, our current students will become adults facing new demands for mathematical proficiency (Casner-Lotto & Barrington, 2006). Children in the United States are not developing acceptable levels of proficiency in mathematics (The Business Roundtable, 2005; Gonzales, 2007; National Mathematics Advisory Panel, 2008; The Nation's Report Card, 2011). Concern about achievement in mathematics of US students has grown, and research has shown that improvement in the teaching and learning of mathematics is needed (Ball, Hill, & Bass, 2005; Fleishman, et al, 2010; Gonzales, 2007). Results from the Trends in International Math and Science Study (TIMSS) Report (Gonzales, 2007), served to reinforce that concern regarding how mathematics is taught in the US. The scores from the 2009 Program for International Student Assessment (PISA) show 15year-old students in the US scoring below average in mathematics. Out of 34 countries, the US ranked 25th in math (Fleischman et al., 2009). Only 27% of US students scored at or above proficiency level 4, which is the level at which students can complete higher order tasks such as solving problems that involve spatial reasoning in unfamiliar context and carrying out sequential processes (The Nation's Report Card, 2011). These results have brought about a need for reform in how students are taught and how teachers are prepared to teach them. There needs to be an emphasis on representational fluency in our classrooms in order for our students to gain mathematical understanding on a much deeper level.

The Common Core State Standards of Mathematics (2010) stipulate that students need opportunities to solve problems and have multiple occasions to communicate mathematical ideas with others, and that teachers should focus on student understanding rather than on "right" answers. Researchers have recently examined instruction more closely by investigating the choice and use of various academic tasks (Johnson, 2015). Mathematics has many types and levels of representation which build upon one another as mathematical ideas become more abstract (Lesh, 2003). Physical representations serve as tools to mathematical thought and communication. They help illuminate ideas in ways that support reasoning and build comprehension. Mathematics requires representations. It is because of the abstract nature of mathematics that people use representations to access to mathematical ideas.

In my previous research observing the use of representations for mathematics instruction, much of my attention focused on Lesh's model of mathematical translations of representations (Lesh, 1987; Lesh, 2003). It investigates elementary teachers' use of pictures, words, symbols, concrete materials, and real world contexts to help their students make sense of mathematical ideas. During the course of this investigation, however, evidence surfaced that an additional model, technology, should be considered as an additional form of representation. Three sources of data were collected: video-recorded lessons, interviews, and a focus group. Original analyses indicated that although concrete representations were accessible to all three teachers, they were least used among the available representations. Verbal expression was most prominent, followed closely by abstract written symbols. Technology, however, which was not one of the mathematical representations reported, appeared regularly throughout lesson observations, interviews and the focus group. An implication that surfaced in this study is that technology is becoming more visible in classrooms as one form of mathematical representation. This article suggests an expansion of Lesh's model to include technology (or moveable pictures through the use of technology) as an additional means of representation.

The Big Picture

To gain better insight into this topic, I used constructivism as a theoretical foundation and Lesh's model of translations of representations (2003) to guide this study. Ideally, students move within and among five forms of mathematical representation in order to construct meaning of mathematical concepts (Cramer, 2003). Representational fluency is the ability to use several different representations and translate among these models with relative ease. This ability is foundational in students' mathematical proficiency (Fennell and Rowan, 2001; Goldin and Shteingold, 2001).

Mathematical thinking can be "represented" in many ways. It can be represented through drawings and pictures, written or oral words, through manipulatives and, all of these, alongside the abstract (numbers) (Lesh et al., 2003; Goldin, 2003; Kamii, Kirkland, & Lewis, 2001). In order for the reader to have clarity, I have defined each of the forms of representation as I use them within the study and the context of this article.

Manipulatives, also referred to as concrete representations, are objects designed to allow students to learn a particular mathematical concept by manipulating them (Reys et al, 2007; Van de Walle, 2005). The use of manipulatives allows students to learn difficult concepts in developmentally appropriate, hands-on, experiential ways (Reys et al, 2007). Some examples of manipulatives are: base ten blocks (which can be used for computational strategies with whole numbers and decimals); Geoboards (which can be used to explore two dimensional geometric shapes, area and perimeter, angles, etc.); pattern blocks (which are used in creating tessellations and exploring patterns in our world around us); fraction pieces (which can be used for computational strategies with fractions); and attribute blocks (which are effective in categorizing and organizing by characteristics).

Pictorial representations, also referred to as pictures, refers to anything hand-sketched or computer generated that represents concrete objects. It could be a photograph, a hand-drawn picture, tallies, graph, or chart. These may include any two-dimensional representations (Ainsworth, 1999; Tabachneck-Schijf & Simon, 1998).

Real-life representation refers to events and objects happening in the real world that allow students to make mathematical connections. Examples may include using money in a grocery store, measuring ingredients when cooking a recipe, or measuring wooden beams when building a garage, etc. (Lesh, 1987).

Symbolic representation, also referred to within this study as symbols or abstract...

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