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Running head: MATHEMATICAL REPRESENTATIONS

Mathematical Representations

Gwenanne M. Salkind

George Mason University

EDCI 857 Preparation and Professional Development of Mathematics Teachers

Dr. Margret Hjalmarson

Spring 2007

Introduction

*Representation received increased attention when it was added as a new process standard in the National Council of Teachers of Mathematics’ (NCTM) Principles and Standards for *

School Mathematics (2000). “The ways in which mathematical ideas are represented is fundamental to how people can understand and use those ideas” (National Council of Teachers of Mathematics, 2000, p. 67). Students use representations as tools to support their mathematical understandings. Teachers’ presentations and uses of representations in the context of teaching influence students’ knowledge of and uses of representations. This, in turn, is related to students’ mathematical achievement. Therefore, it is important to examine the nature of mathematical representations and their uses in mathematics instruction.

The purpose of this literature review is to answer the following questions:

1.

What are representations?

2.

How do representations affect students’ mathematical thinking and learning?

3.

What should teachers know about representations?

4.

What representations should teachers use in teaching?

5.

How should teachers use representations in teaching?

Representations and Student Learning

*General Types of Representations *

“A representation is a configuration that can represent something else in some manner”

(Goldin, 2002, p. 208). People develop representations in order to interpret and remember their experiences in an effort to understand the world. Bruner (1966) found three distinct ways in which people represent the world: (a) through action, (b) through visual images, and (c) through words and language. He called these kinds of representations enactive, iconic, and symbolic,

respectively. Most researchers agree that these three types of representations are important in human understanding. Other researchers have reduced the three types to two categories (Clark &

Paivio, 1991; Marzano, 2004; Marzano, Pickering, & Pollock, 2001) or included additional categories (Lesh, Landau, & Hamilton, 1983). Dual coding theory maintains that there are two systems of representation ( verbal and visual ) that allow the brain to process and store information in memory (Clark & Paivio, 1991). The interconnectivity of the verbal and visual coding systems allows information retrieval to occur easily. These two systems have also been called linguistic and nonlinguistic (Marzano, 2004; Marzano, Pickering, & Pollock, 2001). Lesh,

Landau, and Hamilton (1983) found five kinds of representations which are described in more detail in the following section.

*Types of Mathematical Representations *

Some researchers have specifically looked at representations used to understand mathematics (Abrahamson, 2006; Goldin, 2003; Goldin & Shteingold, 2001; Kilpatrick,

Swafford, & Findell, 2001; Lesh, Landau, & Hamilton, 1983; Lesh, Post, & Behr, 1987).

“Mathematics requires representations. In fact, because of the abstract nature of mathematics, people have access to mathematical ideas only through the representation of those ideas”

(Kilpatrick, Swafford, & Findell, 2001, p. 94). Representations of number include objects, actions, pictures, symbols, and words. These could be likened to Bruner’s three types of representations, with objects and actions being enactive, pictures being iconic, and symbols and words being symbolic.

Lesh, Landau, and Hamilton (1983) found five kinds of representations that are useful for mathematical understanding: (a) real life experiences, (b) manipulative models, (c) pictures or diagrams, (d) spoken words, and (e) written symbols. These categories could be considered to be

an expansion of Bruner’s three categories. Real life experiences and manipulative models are enactive representations, pictures and diagrams are iconic representations, and spoken word and written symbols are symbolic representations.

Goldin and Shteingold (2001) wrote of two systems of representation. External systems of representation include conventional representations that are usually symbolic in nature.

Internal systems of representation are created within a person’s mind and used to assign mathematical meaning. Our numeration system, mathematical equations, algebraic expressions, graphs, geometric figures, and number lines are examples of external representations. These representations have been developed over time and are widely used. External representations also include written and spoken language. Examples of internal representations include personal notation systems, natural language, visual imagery, and problem solving strategies.

*Students’ Mathematical Learning *

“The power of a representation can . . . be described as its capacity, in the hands of a learner, to connect matters that, on the surface, seem quite separate. This is especially crucial in mathematics” (Bruner, 1966, p. 48). Researchers have found that representations can be powerful aids to student learning (Bruner, 1966; Clements, 1999; Cuoco & Curcio, 2001; Fennema &

Franke, 1992; Flevares & Perry, 2001; Goldin & Shteingold, 2001; Greeno & Hall, 1997;

Kilpatrick, Swafford, & Findell, 2001). Representations are useful tools that support mathematical reasoning, enable mathematical communication, and convey mathematical thought

(Kilpatrick, Swafford, & Findell, 2001). Students use representations to support understanding when they are solving mathematical problems or learning new mathematical concepts. In addition, the use of nonspoken representations (objects, pictures, symbols, and gestures) have

been found to be helpful in clearing up students’ mathematical confusions (Flevares & Perry,

2001).

*Goals for Mathematics Instruction *

Traditionally, mathematics instruction has involved students learning to understand external systems of representation and use those systems to solve mathematical problems

(Goldin, 2002). In order to understand external systems, students must process them internally , therefore, developing efficient internal systems of representation that connect to established external systems of representation is an important goal of mathematics instruction.

Greeno and Hall (1997) criticize mathematics instruction that teaches forms of representations as ends in themselves rather than as tools used to aid in problem solving, conceptual understanding, and communication. In school, students learn to write equations, construct graphs, and interpret tables in order to pass high-stakes tests. In contrast, real-world representations are used as tools in the work-place to solve problems, communicate with coworkers, and justify solutions. In order to prepare students for the work force, educational goals should include having students learn to employ representations as tools in communication, weigh advantages and disadvantages of different representations, and create and use multiple forms of representation flexibly in problem-solving.

NCTM’s representation standard says “instructional programs from prekindergarten through grade 12 should enable all students to create and use representations to organize, record, and communicate mathematical ideas; select, apply, and translate among mathematical representations to solve problems; [and] use representations to model and interpret physical, social, and mathematical phenomena” (National Council of Teachers of Mathematics, 2000, p.

67).

*Sequence of Mathematics Instruction *

Which types of representations are useful for school mathematics, and when should they be introduced to students? Bruner (1966) believed that different stages of human development emphasize different representational systems. Young children learn through manipulation and action (enactive representation), older children learn through perceptual organization and imagery (iconic representation), and adolescents learn through the use of language and symbolic thought (symbolic representation). This idea has become a staple of school mathematics instruction with teachers knowing that students must begin with concrete experiences (enactive), move to pictorial representations (iconic), and finally progress to abstract understanding

(symbolic). In recent years, Clements (1999) has suggested that all three types of representation should be used in parallel to facilitate student learning. When students make connections among concrete, pictorial, and symbolic representations, their learning is enhanced and increased. This idea is similar to others already discussed in connecting internal and external representations

(Goldin & Shteingold, 2001) and dual coding theory (Clark & Paivio, 1991).

*Students’ Difficulties with Representations *

Abrahamson (2006) proposed that mathematical representations are conceptual composites (i.e., they include two or more connected ideas). The composite ideas may be hidden within the representations, and, therefore, not easily understood by students. As a result, students can use representations procedurally without fully understanding them. Abrahamson suggests that classroom discussions can help students understand the ideas embedded within mathematical representations.

Other studies have found that student differences in reading and spatial ability may limit the usefulness of visual representations in promoting students’ mathematical understandings.

When mathematical problems include information embedded in text and diagrams, students tend to disregard the information presented in the diagrams (Diezmann & English, 2001). Similarly, diagrams have been found to be useful for students with high spatial ability, but not students with low or average spatial ability (Pyke, 2003).

Clark and Paivio (1991) examined differences in people’s ability to create and use mental images. Some people tend to use visual imagery and do so spontaneously, while others use it rarely and with difficulty. Students can be encouraged to use visual imagery, however, and students who are instructed to use mental images remember more than students who are not given those instructions.

Teachers and Representations

*Teachers’ Knowledge of Representations *

Effective teachers need to know how mathematical ideas can be represented in order to facilitate students’ understandings of those ideas. Shulman (1986) identified pedagogical content knowledge as a specialized form of content knowledge that teachers need for teaching. Along with knowledge of the topics of instruction within one’s subject area, an understanding of what makes those ideas easy or hard to grasp, and students’ frequent misconceptions within those topics; pedagogical content knowledge includes “the most useful forms of representation of those ideas, the most powerful analogies, illustrations, examples, explanations, and demonstrations—in a word, the ways of representing and formulating the subject that make it comprehensible to others” (Shulman, 1986, p. 9). Shulman saw representations as an important part of pedagogical content knowledge.

Studies of teachers’ knowledge of mathematics for teaching (Ball, Hill, & Bass, 2005;

Hill & Ball, 2004; Hill, Schilling, & Ball, 2004) have identified a kind of mathematical

knowledge that is important in the teaching of mathematics. “Mathematical Knowledge for

Teaching” is a deep understanding of mathematics that allows teachers to explain why common algorithms work, evaluate students’ problem-solving strategies, anticipate students’ misconceptions, and analyze students’ errors. Teachers who scored high on tests designed to measure mathematical knowledge for teaching had students who made gains in measures of student achievement (Ball, Hill, & Bass, 2005). An important part of mathematical knowledge for teaching is the ability to generate and use representations. Teachers need to be able to translate complicated mathematical ideas into representations that students can understand

(Fennema & Franke, 1992; Orton, 1988). In order to do so, they must have facility with a repertoire of representations that are useful for teaching mathematics which would include story problems, pictures, situations, and concrete materials (Ball, 1990). Teachers also need to understand the strengths and weaknesses of different representations and how they are related to one another (NCTM, 2000).

While representations are an important part of teachers’ knowledge, researchers have discovered than many U.S. teachers do not possess such knowledge. Ma (1999) found that U.S. elementary school teachers lacked knowledge of representations in the teaching of division of fractions. Ball (1990) found similar results in a study of preservice teachers. Of 25 elementary school teachers, none could generate an appropriate representation for 1 ¾ divided by ½, ten teachers supplied an inappropriate representation, and fifteen teachers were unable to generate any kind of representation.

Ma (1999) pointed out that teachers need to have a deep knowledge of mathematical content in order to generate mathematically correct representations of that content. In interviews

with 94 Chinese and U.S. elementary teachers, Ma found that teachers with inadequate subject matter knowledge could not correctly represent mathematical ideas.

Most of the [Chinese] teachers generated at least one correct and appropriate representation. Their ability to generate representations that used a rich variety of subjects and different models of division by fractions seemed to be based on their solid knowledge of the topic. On the other hand, the U.S. teachers, who were unable to represent the operation, did not correctly explain its meaning. This suggests that in order to have a pedagogically powerful representation for a topic, a teacher should first have a comprehensive understanding of it. (Ma, 1999, p. 83)

*Teachers’ Uses of Representations *

Studies of teachers’ uses of representations bring to light important ideas about what representations should be used in mathematics classrooms and how they can be used effectively.

Studies of U.S. and Chinese teachers found differences in the kinds of representations used to teach elementary mathematics (Cai, 2004; Cai & Lester, 2005). Chinese teachers typically used symbolic representations, while U.S. teachers used verbal explanations and pictorial representations. Not surprisingly, the representations that teachers used influenced the representations that students chose to use for problem-solving. Chinese students used more symbolic representations than their U.S. counterparts and had better scores on problem-solving tests. It is widely known that Chinese students outperform U.S. students in mathematics achievement (Hiebert et al., 2003). It is not hard to imagine that teachers’ choices of representations are a factor in students’ levels of achievement. These studies suggest that the use of symbolic representations may be more useful for students’ mathematical understanding than verbal explanations or pictorial representations.

Lamon (2001) made a distinction between models of presentation that teachers use to explain mathematical concepts and models of representation that students use to show their mathematical thinking. She suggested that teachers can evaluate whether or not students understand mathematical ideas by examining the representational models students choose to use.

If a student’s representation is different from the presentational model the teacher used, then it can be assumed that the student understands the concept. Students that use the exact same representations may be parroting the teacher without real understanding. Similarly, Goldin

(2002) found that effective teachers pay attention to students’ interactions with external representations in order to understand their internal systems of representation. In this way, teachers examine students’ conceptions and misconceptions of the mathematical ideas being taught.

Leinhardt (1989) discovered that experienced teachers use representations that students already know to teach new content, while novice teachers introduce new representations alongside new content. She also found that expert teachers tend to use the same representations to teach multiple content topics. In addition, novice teachers often struggle to explain topics using representations because they are not familiar with the representations. Implications of this study suggest that it is important for students to understand the representations that teachers use, familiar representations can be useful for teaching new content, and one representation may be valuable for teaching multiple content topics. Furthermore, as previously discussed in this paper, teachers must be well versed in the representations they use to illustrate mathematical ideas.

Flevares and Perry (2001) found that first grade teachers often use representations in combination. For example, a teacher might point (gesture) to beans (object) on a ten-frame

(picture) while giving a verbal explanation. This suggests that presenting combinations of

representations that include verbal explanations along with objects and pictures may be better than presenting visual representations in isolation. This idea supports the work of Abrahamson

(2006) who suggested that teachers facilitate student discussions around mathematical representations and Clements (1999) who suggested that the three types of representations

(concrete, pictorial, and symbolic) should be taught in parallel.

Conclusion

This paper examined the uses of representations in school mathematics. There is a considerable amount of research on general types and uses of representations. Most researchers agree that representations are useful, if not necessary, in the teaching of mathematics. Goals for student learning include developing internal systems of representation, understanding traditional external systems of representation, and creating and using representations as tools for communication and problem solving. One theme that emerged is that student learning involves forming connections among the different types of representations: concrete, pictorial, and symbolic; verbal and visual; and internal and external.

The representations that teachers choose to use and the way in which they use them are important factors in student learning. Teachers’ knowledge of mathematics content impacts their ability to use representations effectively. Along with a deep knowledge of the mathematics content that they teach, teachers must have a repertoire of representations that are most useful for teaching that content.

Few studies have investigated the nature of teachers’ knowledge of representations.

Further research should be done to examine what representations teachers use in teaching elementary mathematics and how they use them. In addition, studies should be conducted to determine the most useful representations for teaching school mathematics. What representations

should be part of a teachers’ repertoire? Which representations are useful for teaching a variety of content topics? It is also not clear how teachers become proficient at using a variety of representations. Research on professional development activities that facilitate teachers’ knowledge of representations would contribute to the research literature on mathematical representations.

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