This chapter presents first an overview of the topics pertinent to this study. The first section describes the role that multiple representations have played in the teaching and learning of mathematics. The second section includes the importance that the study of functions has in mathematics curricula. The last section discusses how technology has been used in order to enhance and promote a better understanding of mathematics. The statement of the problem, the purpose of the study and the research questions complete this chapter.
One of the most important issues that arises in mathematics education scenarios is the fact that ways need to be found to promote understanding in mathematics (Hiebert and Carpenter, 1992). In order to fulfill this goal, teachers, administrators, curriculum designers and researchers have suggested and implemented different ideas, based on mathematical learning theories. As cited in Porzio (1994) and based on research done by Hiebert and Carpenter, Kaput (1989a) and Skemp (1987), “an emerging theoretical view on mathematical learning that has been growing in significance is that multiple representations of concepts can be utilized to help students develop deeper, more flexible understanding” (p. 3).
The role and use of multiple representations have been constituted as an emerging research and extensive discussion area during the last years in the mathematics education community. Most recently, the National Council of Teachers of Mathematics (NCTM, 2000), facing a new millennium, has included the uses of representations as one of the new standards in mathematics teaching and learning. The representation standard states:
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 to use representations to model and interpret physical, social, and mathematical phenomena. (p. 67)
This educational guide, illustrated by this standard in its three aspects, confirms the important and transcendental role and the urgent need of using representations in teaching mathematics at all levels, grades K through 16.
It has been extensively discussed that mathematics, by its own nature, is one of the academic subjects where multiple representations are currently used (De Jong, et al., 1998). Mathematics as a “collection of languages” (Kaput, 1989a, p. 167) and characterized, the majority of the time by the presence of symbols and abstractions, is one of the fields where representations could be used widely due to their capabilities to enhance “understanding and for communicating information” (Greeno and Hall, 1997, p. 362). Due to this extensive use of symbols, abstractions, rules, definitions, it is also known that students in mathematics are confronting real troubles trying to understand, internalize, apply, and communicate important concepts in their mathematics school levels. Because of this, it is right and necessary to think about the ways that mathematical ideas are being currently represented, due to the understanding of these concepts and the use of the ideas depend on how these representations are being used (NCTM, 2000).
Dufour-Janvier, Bednarz, and Belanger (1987) have classified the term representation in two major categories: internal representations and external representations. Each of them possesses a considerable amount of sub-themes exposed to more and deeper research linked with other fields. According to them, the first category deals with “more particularly mental images corresponding to internal formulations we construct of reality”. The second area deals with “all external symbolic organizations” (p. 109), illustrated frequently in the forms of symbols, schema, and diagrams. Özgün-Koca (1998) states “multiple representations are defined as external mathematical embodiments of ideas and concepts to provide the same information in more than one form” (p. 1). On the other hand, NCTM (2000) affirms that the “term representation refers both to process and to product –in other words, to the act of capturing a mathematical concept or relationship in some form and to the form itself” (p. 67). This research project, in order to fulfill its objectives, proposes to limit the term representation to its external category.
The capabilities of using these representations in mathematics teaching and learning have also been discussed and illustrated by the literature. Özgün-Koca (1998) suggested that the use of multiple representations in mathematics could provoke an appropriate and healthy environment for students to abstract and understand major mathematical concepts. Moreover, Dufour-Janvier and colleagues (1987) expressed their motives for using external representations in mathematics. They argued that first, representations are an inherent part of mathematics; second, representations are multiple concretizations of a concept; third, representations are used locally to mitigate certain difficulties; and last, the representations are intended to make mathematics more attractive and interesting (p. 110-111). Porzio (1994) calls “obvious” all of the benefits that the use of multiple representations can give to mathematical teaching and learning (p. 47). In addition, as cited in the same study, Kaput (1992) says that the use of more than one representation or notation system help to illustrate a better picture of a mathematical concept or idea. “Complex ideas are seldom adequately represented using a single notation system. The ability to link different representations helps reveal the different facets of a complex idea explicitly and dynamically” (p. 542). In summary, mathematics at all levels needs the use of representations in order to communicate appropriately ideas, and more importantly, to transmit, meaning, sense and understanding.
Other studies have supported the use of representations in mathematics in order to enhance concept understanding. Hiebert and Carpenter (1992) state that the process of the learning of mathematics with understanding “extends beyond the boundaries of mathematics education” (p. 65). They define understanding as the way certain information can be represented and structured. Moreover, they affirm that “mathematics is understood if its mental representation is part of a network of representations” (p. 67). Kaput (1989a), as well as, Keller and Hirsch (1998) found that the use of multiple representations provide diverse concretizations of a concept, carefully emphasize and suppress aspects of complex concepts, and promote the cognitive linking of representations. Furthermore, Moschkovich, Schoenfeld and Arcavi (1993) explored in their research the fact that there are multiple ways to solve a given problem and that solving a problem calls for making connections across representations and for employing both the process and object perspectives (p. 94). In this way, NCTM (2000) states, “representations should be treated as essential elements in supporting students’ understanding of mathematical concepts and relationships; in communicating mathematical approaches, arguments, and understandings to one’s self and to others; in recognizing connections among related mathematical concepts; and in applying mathematics to realistic problem situations through modeling” (p. 67). In summary, it has been showed that the use of multiple representations is a useful tool to promote better understanding of key concepts in the mathematics curricula.
Functions have a key place in the mathematics curriculum, at all levels of schooling; particularly in secondary and college levels where they get their maximum expressions and representations. The concept of function has been usually introduced early in algebra courses, starting in the majority of the cases with the linear form. As a result, NCTM (2000) has placed the concept of function as one of the cornerstones of mathematics curricula: algebra. The algebra standard states that students from prekindergarten through twelfth grade should understand patterns, relations, and functions (p. 37). Thorpe (1989) proposed the use of functions “as the centerpiece of algebra instruction” (Gningue, 2000, p. 28). The literature in mathematics education possesses a vast amount of research concerning functions and their teaching and learning. Dubinsky and Harel (1992), and Cooney and Wilson (1993) have agreed to say that functions should be located at the center of the mathematics curricula. Lastly, Selden and Selden (1992) point out that functions play a central and unifying role in mathematics (Poppe, 1993, p. 2).
By their nature, functions are one of the best examples in which to use multiple representations in the teaching and learning process. Researchers have agreed that functions can be represented in the following forms: algebraic or formulas, tables, and graphs (Brenner, et al., 1997; Greeno & Hall, 1997; Iannone, 1975; Janvier, et al., 1993; Mevarech & Kramarsky, 1997; and others). “These forms of representation – such as diagrams, graphical displays, and symbolic expressions – have long been part of school mathematics” (NCTM, 2000, p. 67). In the same document, NCTM continues saying that one of the major goals of algebra is that students should “understand the relationships among tables, graphs, and symbols and to judge the advantages and disadvantages of each way of representing relationships for particular purposes” (p. 38). Furthermore, Leinhardt and colleagues (1990) and Moschkovich, et al. (1993) affirm that using multiple representations to teach functions, that is, numeric, graphic, and symbolic, will enhance a broad understanding of functions. In summary, the use of representations in mathematics consists of a rich and varied group of alternatives that students can use, whenever they want, in order to promote a better achievement of a particular topic.
Technology in all of its manifestations plays an important and primary role in introducing and supporting multiple representations in mathematics. It has served to engage students in a harmonious process of teaching and learning mathematics. Through the use of technology, multiple representations can be introduced more powerfully as well as, in an interactive and attractive way (Confrey, et al., 1991). Fey (1989) proposed the use of calculators and computers to introduce algebraic concepts like functions. Porzio (1994) assures that “instructional practices that involve the use of multiple representations are not employed simply because technology now makes multiple representations more readily accessible, but because of the potential benefits associated with their use” (p. 4). Fey (1989), Goldenberg (1987), and Kaput (1992) have agreed that due to the advancements and advantages of technology, the chance to provide students better access to the use of representations have considerably increased. In summary, the appropriate use of technology, represented in this case by graphing calculators, computers, software packages, like spreadsheets, without doubts, brings an invaluable direction to the acquisition and understanding of mathematical concepts, such functions, at the same time, emphasizing the varied representations that functions have (Schwarz, Dreyfus, and Bruckheimer, 1990; Browning, 1991; and Hart, 1991).
Following calls for reform according to Keller & Hirsch (1998), current precalculus and calculus reform projects are attempting to incorporate numeric, graphic, and symbolic representations into the curriculum. The Calculus Consortium at Harvard (2001), a group of recognized scholars established in the late 1980’s, started a revolution in the teaching and learning of mathematics, particularly in calculus courses at the college level. One of the guiding principles of this consortium is based on the ‘Rule of Four’ where mathematics topics are introduced geometrically, numerically, analytically, and verbally (Hart, 1991; Hughes-Hallet, 1991; Megginson, 1995 & Porzio, 1994).
During the past decade, with the purpose to “consider the needs of all undergraduates attending all types of United States two- and four-year colleges and universities”, the National Science Foundation (NSF) issued the report Shaping the Future on new expectations for undergraduate education in science, mathematics, engineering, and technology (George, et al. 1996, p. ii). The goal of this report was that:
All students have access to supportive, excellent undergraduate education in science, mathematics, engineering, and technology, and all students learns these subjects by direct experience with the methods and processes of inquiry. (p. ii)
As part of this report, the NSF emphasized the importance of the effective use of technology to enhance learning (p. iv) recommending to institutions of higher education its incorporation into the curriculum of science, mathematics, engineering, and technology.
The proposition that mathematics teaching and learning, at all levels of education, is divorced from major curricular trends is still alive. In many mathematics education scenarios, both processes are going in opposite directions, disregarding the calls and movements for reform. It is also true that antique methods and strategies that are strictly traditional instruction. In many instances they are based on the idea that teachers are the authority and transmitters of knowledge. And those students are but passive recipients predominates in our classrooms. Therefore, the mathematics curriculum continues to be strictly limited, in the majority of the cases, to the prescribed textbook, when available. The problem solving process is limited to the use of paper and pencil, without the initiatives to experiment with innovative changes such the use of technology like calculators and computers. Moreover, the textbooks currently used in some mathematics classrooms are not offering to students the use of multiple representations of transcendental concepts, like functions (Rodríguez-Ahumada, et al., 1997; Angel, 2000). In these traditional settings, teachers and students are experiencing functions without an appropriate emphasis on multiple representations, and moreover, the linking process that should exist between them is missing (Kaput, 1989a).
Greeno and Hall (1997) state that “under the pressure to cover the prescribed curriculum, teachers often feel that there is not enough time to teach students what representations are for and why the forms are useful and effective” (p. 362). Hart (1991) affirms that students who use multiple representations along with technology can acquire richer concept images than those who do not have the same experience (p. 45). In addition, Hart has shown that students exposed to the use of technology and representations “had better conceptual understanding than those students not having this exposure” (p. 46).
In summary, the literature on representations in mathematical teaching and learning has shown that the appropriate use of multiple representations, supported by technology, seems to be helpful in promoting understanding and the acquisition of a broader achievement of important mathematics concepts, like functions.
Functions are very important in the mathematics curriculum. The use of multiple representations of functions, strongly supported by technology, has not reached all corners of mathematics education. In many courses the uses of calculators and computers have been nonexistent. In other cases, where some kind of technology is implicitly allowed, it has been classified as optional.
The main purpose of this study is to develop computer-based algebra lessons using spreadsheets about linear functions and their related topics where multiple representations can be emphasized in order to determine if these learning activities can help college students achieve a broad understanding of linear functions.
In order to fulfill this purpose, an experiment was carried out in which a portion of subject matter dealing with linear functions was developed using multiple representations as basis for instruction. A control group was also used, wherein the same subject matter was taught. Figure 1 below shows how spreadsheets supporting multiple representations were handled in this study.
Figure 1. Multiple representations of a linear function using spreadsheets.
This study investigates the following research questions:
1. How did the students in the two groups, experimental and control, compare in the prior achievement and attitudes, and their experiences with technology?
2. What relationships appear to exist between attitudes and achievement in the learning of linear functions activities?
3. At which level and in what ways, can the use of multiple representations be supported by spreadsheets learning activities to better promote understanding of linear functions in students at college level algebra?
4. How well does the medium of a powerful spreadsheet like Excel, lend itself to promoting instruction through multiple representations?
The next chapter consists of a review of the research literature pertinent to this study. It will include a review about the uses of multiple representations in mathematics, learning theories dealing with multiple representations, technology and multiple representations, and functions and their representations.