In 1989 the National Council of Teachers of Mathematics presented a ground breaking effort: the publication of its Curriculum and Evaluation Standards for School Mathematics. This document presented a vision of K-12 school mathematics for the end of the twentieth century. Included in this vision was not only content standards which described a lofty mathematics curriculum for all students but also four process standards which described the core of doing mathematics common to all grade levels: 1) problem solving, 2) reasoning, 3) communication, and 4) connections. Perhaps one of the most innovative aspects of the original Standards was that they were not just a description of what mathematical content should be taught. Instead the Standards also focused on the ways in which students should learn: "what a student learns depends to a great degree on how he or she has learned it" (NCTM, 1989, p. 5).

     As the new millennium began to approach, NCTM continued its work to "ensure quality, indicate goals, and promote positive change in mathematics education"(1998, p. 11). In October 1998, NCTM released the draft version of Principles and Standards for School Mathematics (Standards 2000). In this draft a fifth process standard, representation, was added to the original four. By adding this fifth process standard, Standards 2000 continues to convey that the ways students learn and represent mathematics are of value. A discussion of representation and an argument for its inclusion in mathematics standards is important for three reasons: 1) fluency in translation between different representations forms a basis for being able to form concepts and think mathematically; 2) the way mathematical ideas are represented by teachers has an impact on the mathematics learned; and 3) students need practice in constructing their own representations to become flexible and powerful problem solvers.

     Although it may be difficult to imagine that the representation standard would be controversial as the draft of Standards 2000 was released, there may be some arguments against its inclusion. One criticism is that many see representation as synonymous with mathematical modeling. Because of this view of representation, many may argue that it is a subset of another content standard. For example, if representation is thought of as mathematical modeling, it could easily fit under Standard 2: Patterns, Functions, and Algebra. Indeed Standards 2000 explicitly states in the Patterns, Functions, and Algebra standard description that "students [should] use mathematical models and analyze change in both real and abstract contexts" (NCTM, 1998, p. 56). Others may argue that representation is only part of the process of problem solving and therefore belongs in the problem solving standard.

     Similarly, an argument against the prominence of representation as a process standard may be that it is not important enough to be a process standard. Some may argue that the other process standards of problem solving, communication, reasoning, and connections cut across all content standards while representation seems to be more limited in scope. Others may see representation as a subset of cognitive development and believe that a discussion of cognitive science does not warrant such a prominent role within the presentation of mathematical standards.

     The difficulty of where to include representation, if at all, stems from the fact that the term representation is ambiguous. The meaning of the word, as Standards 2000 admits, has many facets (NCTM, 1998, p. 94). From the perspective of many mathematics educators and cognitive researchers, representation involves much more than using symbolic notation to translate a situation into mathematical syntax (Goldin & Kaput, 1996). It is much more than the physical product of abstraction. It is also a cognitive process that occurs internally. Representation is the act of interpreting a concept or problem by giving it meaning. Although we cannot directly observe this internal process of representation from our students, we can infer mental configurations in our students from what they say or do.

     Representation impacts how and what mathematics is learned and can be extended to all content areas of mathematics. If representation were only a physical or symbolic product that could be taught explicitly, it should be listed with the other learning goals in the other content standards. However, representation is as much process as content. It is this deeper and broader concept of representation and its implications on teaching and learning which makes its inclusion in Standards 2000 even more necessary and which I wish to explore further in this paper.

Representation as Means to an End

     With these anticipated objections in mind, a more detailed examination of what representation is and how it affects teaching and learning is in order. First, it should be understood that representations have always been taught. This is nothing new. For example, when asked to think of types of representations, ones such as symbols and equations, graphs, and charts immediately may come to mind. One can examine almost any mathematics textbook and find those typical technical representations. However, too often in the past these types of representations have been taught as ends in themselves (Greeno and Hall, 1997). They have not been taught in a meaningful and connected way.

     It may be that the key to students' success in solving problems (or lack thereof) lies in the way that we use and teach different representations in the classroom. Seeing a concept through lenses of different representations may be a requirement for moving from the abstract general notion of a problem to a specific solution. Empirical studies suggest that mathematics problem solving competency depends on one's ability to think in terms of different representational systems during the problem solving process (Pea, 1987). Pea further suggests that " problem solvers can exploit particular strengths of different representational forms depending on the requirements of the problem in order to reach a solution. As an example, consider the three representational forms above. Sometimes relationships that are not obvious in terms of verbal descriptions, equations, or tables of data can become clear when a well-designed graph is constructed. Therefore, one's ability to move from one representation to another can dramatically affect one's ability and efficiency of problem solving. Standards 2000 echoes this sentiment and suggests that different representations clearly highlight different features of a situation, and students will benefit from experiences depicting a mathematical situation in multiple ways. (NCTM, 1998).

     Other researchers have examined this fluency in translating among representational systems in terms of the development of a concept. For instance, Kaput (1990) and Thompson (1994) have worked extensively in the field of development of the concept of the algebraic function. Insights into mathematical relationships, such as algebraic functions, can be enhanced by portraying them graphically rather than as only symbolic equations. However, according to Thompson, the physical representations of a function are not the " that we as math educators want to achieve with our students; it is the sense of the connections between the representations. He explains that "the core concept of 'function' is not represented by any of what are commonly called the multiple representations of function, but instead by our making connections among representational activities" (p.39). Therefore, it is important that representations not be taught just as ends in themselves. It is the interplay and connections between these different representations that develop the concept.

Implications of Representation on Teaching

     Attention to the representations that we as educators use in the classroom is also vital. If one agrees with the premise above that the way information is represented and translated within the student has an impact on learning and problem solving, then one must also agree that the way content is presented to the student has a impact on what and how the student learns. In other words, the internal representation is influenced by interactions with the external representation in the way it develops (Goldin & Kaput, 1996). We as teachers must be careful to be cognizant of this as we choose representations for mathematical concepts in our classrooms. This is especially important as Standards 2000 continues to support the use of concrete models, manipulatives, technology, and other learning strategies.

     As an illustration, consider the familiar idea of the presentation of fraction concepts (NCTM, 1998). It is quite common to see textbooks or teachers introduce and develop fraction concepts using visual representations such as pictures of partially eaten pies and shaded rectangles or concrete models such as pattern blocks and fraction strips. On the surface this seems to be a nice way to develop the part-whole relationship meaning of a fraction. That is, if four out of six congruent sections of a pentagon are shaded, then four sixths of the pentagon is shaded. This representation also easily shows that four sixths of the pentagon is the same as two-thirds of the pentagon and develops an understanding of addition of fractions with the same denominator.

     However, by using only this representation to develop the concept of fractions, students are not developmentally conscious of other interpretations of a fraction. The visual and physical representations above do not capture the different nuances of fractions such as indicated division, ratio, or fraction as a number. These representations of fractions also do little to enhance the understanding of addition of fractions with unlike denominators or show the connection to decimals and percents. In fact, many visual representations "unwittingly contribute to a conceptual structure that reinforces (rather than discourages)" student misconceptions. (Hiebert and Lefevre, 1986, p. 17) Other types of representations may be better at showing these properties of fractions. Furthermore, it should be understood that it is not that the area model of fractions is inadequate or misrepresentative; it is that many representations of fractions must be included in order to teach the broader interpretation of fractions.

     Hiebert and Carpenter (1992) also warn against inappropriate models and representations in the classroom. They point out that concrete models may be ineffective if the distance between the concrete material and the mathematical relationships that we intend to represent is too far. They propose an example of the place value system using concrete materials: base-10 blocks and colored chips. To represent the quantity of 124 with base-10 blocks using the small block as the unit cube, students could take advantage of the special grouping built into the representational system and represent 124 as four small cubes (units), two long blocks (tens), and one flat block (hundred). The total number of objects is still visible, but it has been grouped in a special way. Using the colored chips, values of ten and hundred would have to be arbitrarily assigned to different colors to represent the place value. The number 124 could be represented as four blue chips, two green chips, and one yellow chip. The point they make is that the "contextual support"contextual distance" is greater when using the chips rather than using the base-10 blocks. The chips provide no physical clues about their values. The values have been assigned arbitrarily. Therefore, the base-10 blocks may be more appropriate for developing initial representation of our base-10 number system.

     It should be noted, however, in this example that using the colored chips as representational models provides for a closer contextual distance to the value of 124 than do the individual numerals 1, 2 and 4. The numerals are abstract in purest form while the colored chips do at least provide some contextual support. This is said to emphasize that once again the colored chips' representation, as in the visual and physical fraction model, is not misrepresentative or inadequate. It is a more abstract level of representation through which students may need to progress on their way to understanding and building a representation of the base-10 number system.

     An example of how classroom instruction by the teacher can affect representations that we wish students to develop concerns the representation of multiplication as an area model. By developing the concept of multiplication as a rectangular array composed of two factors in elementary school, junior high and high school algebra students will have a representation of the meaning of multiplying two binomial expressions in terms of a rectangular array. The study of the two (often unnecessarily) distinct topics of arithmetic and algebra can be linked. This example also shows how representations that teachers use with students can build the connections between the different mathematical strands as discussed in the previous Standards as well as Standards 2000. (NCTM, 1989, 1998).

     Historically in mathematics education, most attention has been paid to symbolic representation of concepts. Our main objective has been for students to internalize the base-10 system of numeration along with its associated procedures. Those basic procedures then were extended to symbolic representations of fractions, decimals and algebraic systems (Goldin and Kaput, 1996). However, as the age of technology progresses, there are many other types of representational systems that will be available. Graphing calculators and computer-based algebra systems allow a translation between the more traditional representations of symbols, tables, and graphs. Computer software such as spreadsheets and Logo introduce a new and powerful way of examining mathematics of change. Technological advancements have caused educators to reflect on how the advent of new representational tools may dwarf the limited ones of base-10 arithmetic and traditional algebra (Wenger, 1987). Educators must be ready to use these new tools in the classroom.

     Based on these examples, it is clear that decisions about instructional strategies, materials, and tools must be examined carefully if we wish students to represent knowledge in an appropriate way. It is for this reason that inclusion of the representation standard is important. Educators must be aware of how their interactions with students in terms of instruction, content, and materials affect the learning process. We must not be promoting use of concrete materials for the sake of using them. Our choices of representations must be made in regard to future mathematics and connections we wish students to achieve. Choices of problems, classroom activities, and external representations must be examined with regard to the kinds of internal representations we wish students to develop (Goldin and Kaput, 1996).

Using Representation in the Process of Learning

     In addition to using representations for the final learning outcome, people also use representations to aid in understanding as they are working on a problem. For example, one may draw pictures, make charts, write notes, or construct equations while working on a problem. These representations may help keep track of progress or help to organize ideas. However, these representations are often nonstandard or invented representations and are constructed for immediate use in understanding. In other words, they are representations of something as well as representations for something.

     Greeno and Hall suggest that this contrasts with a common practice in school where students learn to construct representations of information such as graphs, charts, or equations "without having a real purpose" (1997, p. 365). Research indicates that as successful students solve problems they construct representations for specific purposes and these representations often match the processes of solving the problem. Again this is contrary to what usually happens in the classroom: students are instructed to represent problems with standard forms that depend on classification of problem types rather than on the process of solutions.

     Research along these lines has shown that students often represent their thinking in forms that help them see patterns or perform calculations, even though these are not conventional forms. Thus there is an interactive process of solving problems in which students construct representations based on partial understanding and then use those representations to refine their understanding. This, in turn, leads to a more refined representation, and so on. Furthermore, Greeno and Hall say that students use multiple forms of representations that sometimes differ from the forms explicitly taught in the curriculum. Using hybrids of part conventional, part inventional representations, successful students "can use representational material constructively as they build their understanding."(p. 365)

     There is evidence to support that problem solvers use forms of representations that are novel in approach in addition to those that have been taught. A study was conducted that examined the practices of students, teachers, and advanced undergraduate engineering students in solving typical algebra word problems (Hall, 1989). These problems were the traditional type of " word problems that involved two trains leaving at the same time in opposite directions. A specific example of a problem presented for study was

                          Two trains leave the same station at the same time.
                            They travel in opposite directions. One train travels
                            60 km/h and the other 100 km/h. In how many hours
                            will they be 880 km apart?

     The traditional approach of teaching these types of word problems is through the use of algebraic equations or formula charts. However, the results of the study showed that all three groups, the students, the teachers, and the advanced engineering majors, used as many non-conventional systems of representations in solving the problem as they did ones that were more traditional and explicitly taught, such as algebraic equations.

     A more in-depth analysis of the study showed that all three types of problem solvers used the nonstandard representational forms to construct models of the problem's structure. Also, it should be noted that when correct inferences were made about the problem, it was more likely that a nonstandard representation had been used rather than a standard representation such as an algebraic equation. Hall states that the significance of this is that the features of nonstandard representations lend themselves to inference-making and calculations to solve a problem and that representations are constructed for specific purposes. Furthermore, when standard forms of representations were used and errors were made, nonstandard forms were used to correct the errors. In terms of teaching practices and curriculum, we miss many of these nonstandard possibilities when we restrict our problem solving instruction exclusively to the more traditional standard forms of representation.

      Many reform minded mathematics educators might flinch at the problem above. This type of canned algebra word problem, they might argue, is the type of problem for which the Curriculum and Evaluation Standards (NCTM, 1989) would argue against. Word problems "by type" should receive decreased attention (p. 127). However, a problem such as the one above is not intrinsically bad as long as students are able to solve the problem using a variety of representations. Many types of representation, whether they be equations, modeling, narratives, drawings, or tables of quantities, should be available as tools of problem solving. Standards 2000 supports this idea of linking multiple representations in order to provide ownership of understanding (NCTM, 1998). The most important reason for including representation as a process standard is that teachers must become more aware and flexible in understanding different student representations and in using these student representations in turn to develop deeper understanding.

     It is important that learners of mathematics be able to construct their own representations. The goal of constructing representations within mathematics instruction should be to provide capacity for the learner to extend this creative construction to new and unfamiliar situations. Emphasis on internal representational systems provides a means for characterizing the outcomes of learning in a much more valuable way (Goldin & Kaput, 1996). To nourish this capacity for using representations, Greeno and Hall (1997) say that we must address the need through a situative perspective. In other words, as students participate in learning, they learn the prevailing practices of learning and knowing that occur within their classroom. Students must see the multiple representations of others and be able to judge when different representations are better at communicating and representing the situation.

     To stress the inclusion of using these nonstandard representations in the classroom many educators have called for the use of student invented algorithms (Mokros, Russell, & Economopoulos, 1995). The argument is that for basic operations, different and equally efficient algorithms have been taught in other countries and at different times in our country. Constructing effective algorithms that make sense to the constructor is in itself an important element of mathematical thinking. On the other hand, using someone else's algorithms to solve a problem when the algorithm has no meaning to the user is not meaningful mathematics. Standards 2000 also addresses this point very cautiously asking that educators "consider the role that other, less conventional representations might play in the mathematics classroom" (NCTM, 1998, p. 95).

     It should not be assumed, however, that the traditional, conventional forms of representations are no longer important. Standards 2000 is clear on that point. Conventional forms of representation have meaning in our society, advantages for the mathematical community, and effective communication properties (NCTM, 1998, p. 95). However, it is the presentation of these conventional representations without regard to idiosyncratic representations that leads students to think of mathematics as a collection of unrelated and arbitrary skills and processes.

     Brizuela (1997) addresses the difference between inventions and conventions. She asserts that although one is created by the subjects (invention) and the other is "discovered or learned by the environment" (convention), there does not need to be a dichotomy between the two. Quite the contrary is the relationship (p. 2). There exists a complementary relationship between the two forms that is cooperative, collaborative, and interactive. She asserts that inventions are of critical importance in knowledge development, but at the same time conventions are important because they provide a support for the development of inventions. It is the process of inventing that makes the acceptance of convention all the more tolerable.

     The addition of representation as a fifth process standard is needed to address the goal of each student gaining mathematical power. Being able to translate between representations and use those translations as a means to an end impacts the mathematics that is learned. The external representations of tools, instructional practices, and materials within the classroom provided by the teacher has an impact on what and how mathematics is represented by the student. Finally, the process of inventing representations during learning and problem solving aids in the actual learning of more standard mathematical representations because it structures the mathematics on a skeleton of meaning. By focusing on these interactions among representations and cognition of mathematics, educators will be better prepared to offer students a meaningful mathematics program that prepares them to be mathematically literate and powerful. Representation is very important to mathematics education as it enters the 21st century and should play a prominent role in NCTM's vision of mathematics education.

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