This research was designed to create computer-based algebra lessons using spreadsheets about linear functions with emphasis on multiple representations and to investigate possible effects of instructional uses of multiple representations on students’ outcomes (attitudes and achievement).  This chapter reviews literature relevant to this study and presents a theoretical framework for the research.

            The chapter is divided into four main sections.  The first section presents research concerning the use of multiple representations in mathematics.  It will briefly discuss research studies dealing with the following topics: (a) need to use representations in mathematics education, as well as, some of their strengths and weaknesses; (b) definitions and classifications of representations; (c) students’ preferences for using representations; (d) connections among representations; and (e) interpretation of representations.  The second section describes related learning theories that support the use of multiple representations. The third section discusses the role of technology and the use of representations in mathematics.  The fourth section of this chapter contains research studies supporting the teaching of functions using representations.

Multiple Representations in Mathematics Teaching and Learning

Needs to Use Multiple Representations in Mathematics

            The uses of multiple representations have been strongly connected with the complex process of learning in mathematics, and more particularly, with the seeking of the students’ better understanding of important mathematical concepts. Research done by Hiebert and Carpenter, (1992); Kaput, (1989a); and Skemp, (1987) illustrates that multiple representations of concepts can be utilized as a help for students in order to develop deeper, and more flexible understandings (Porzio, 1994).  As cited in Gningue (2000, p. 43), Kaput (1989a) “thinks that students learn through several modes of representations”.  Dufour-Janvier, Bednarz, and Belanger (1987) have described important elements about the uses of representations in mathematics. Dufour-Janvier and colleagues argue that representations are inherent in mathematics; they are multiple concretizations of a concept; they could be used to mitigate certain difficulties; and they are intended to make mathematics more attractive and interesting (pp.110-111).  Keller and Hirsch (1998) describe some potential benefits [italics added] related to the use of representations.  Among these benefits are: (a) provide multiple concretizations of a concept, (b) selective emphasis and de-emphasis different aspects of complex concepts, and (c) facilitate cognitive linking of representations (p. 1).  Kaput (1992) points out that the use of multiple representations or notations could be helpful at the time of present a clear and better picture of a concept or idea.

Complex ideas are seldom adequately represented using a single notation system… Each notation system reveals more clearly than its companion some aspect of the idea while hiding some other aspects. The ability to link different representations helps reveal the different facets of a complex idea explicitly and dynamically. (p. 542)


De Jong, et al. (1998) argue that in today’s educational processes, students have been “confronted with information from different sources (computer programs, books, the teacher, reality, the classroom, peers, etc.) and in many different representations that they have to evaluate, make a selection from, and integrate them into their personal knowledge construction process” (p. 9).  About this particular, Poppe (1993) says that the wide uses of mapping diagrams, graphs, and tables, provide a visual representation of the relationships between quantities. 

            The uses of representations in mathematics have not been a really new trend in educational practices. Porzio (1994) indicates that mathematics educators have made efforts for the last years, in order to use more than one representation to introduce mathematical concepts to students.  Janvier, Girardon, and Morand (1993) point out that educators and researchers have emphasized, through the years,  the roles of different forms of representation illustrated as: graphs, tables, diagrams, charts and figures.  Current reform efforts in various curriculum projects dealing with calculus instruction at college level, demonstrate that multiple representations have played a particular role in these processes.  As cited in Hart (1991, p. 2), “this emphasis on multiple representations fits the picture of calculus reform in which Tucker (1987, p. 16) sees ‘a vista of a more conceptual, intuitive, numerical, pictorial calculus’ as the calculus of tomorrow”.

            De Jong, et al. (1998) stated that there are three goals that multiple representations serve.  First, multiple representations are recommended to use due to the information that students learn has varied characteristics.  Second, multiple representations are good resources to induce in the students a particular quality in their knowledge. De Jong and colleagues say, “both approaches lead to a concurrent presentation of multiple representations” (p. 39).  And third, it is an assumption that the use of representations in sequence is beneficial for learning.  This last goal illustrates the transitional presentation of representation.  Furthermore, these researchers have identified four factors that  mediate the effects of using representations.

The type of test used is partly responsible for the effects… The type of domain in the learning environment may also be of influence… The type of learner using the environment also influences the effectiveness, and finally, the type of support present in the environment also plays a role. Most environments simply assume that the co-presence of more than one representation will prompt the learner to integrate the information. (p. 39)


            In addition, De Jong, et al. (1998) identified three reasons (explicitly or implicitly) about the uses of representations.  The first reason deals with what to do with the tuning of the domain information and the representation.  The second concerns the idea that the use of multiple representations will promote a flexible knowledge.  And lastly, the specific order that representations are introduced into learning will facilitate it.

The first reason for using multiple representations is that specific information can be conveyed in a specific representation, and that for a complete set of learning material, containing a variety of information, a combination of several representations is therefore necessary.  The main issue here is that of adequacy, which concerns the expressional possibilities of a representation.  A second aspect that can be involved here is efficiency, which concerns the expressional power of a representation. Within one level of adequacy, e.g., graphical representations, some may still be more efficient than others.  The second reason for using more than one representation is that expertise is quite often seen as the possession and coordinated use of multiple representations of the same domain.  In this theory expertise is viewed as being able to understand the domain knowledge from multiple perspectives… The third reason for using more than one representation is based on the assumption that a specified sequence of learning material is beneficial for the learning process. (pp. 32-33)


            Greeno and Hall (1997) call tools [italics added] the forms of representations in mathematics. They argue that students can learn to use them “as resources in thinking and communicating” (p. 362).  Porzio (1994) citing the research of Dufour-Janvier and colleagues (1987) says that it is desired that “students can perceive representations as mathematical tools for solving problems and helping students in the ‘construction’ of a concept by viewing common properties and differences between representations of the concept.  The research work group headed by Dufour-Janvier has explored three important categories concerning representations and have raised a group of questions dealing with each one of these categories: (a) how these tools have been used in mathematics instruction; (b) how are the expected outcomes achieved in the current teaching of mathematics; and (c) how should be the representations to be useful in mathematics. 

Dufour-Janvier, et al. (1987) have realized that mathematics teaching, together with all the elements including in its curricula have submitted students, of all ages and school levels, to a wide variety of representations.  At this point, these researchers propose the following questions:

What are the motives for using external representations in mathematics teaching? What are the expected outcomes that justify such a wide variety of representations? Are these outcomes achieved in current teaching of mathematics? To what extent is it possible that such representations are inaccessible to students and even detrimental? Can the teaching of mathematics be organized in such a way that learning is articulated with the representations children develop themselves? (p. 109)


            These recognized scholars have, also looked at the outcomes of the uses of representations in the learning of mathematics.  Dufour-Janvier and colleagues (1987) present some expectation concerning the uses of representations.  They expect first, that in particular mathematics problem situations, students could be able to reject one representation in order to choose another one, knowing the reasons because they are doing this selection.  Second, it is expected that students could pass from one representation to another, knowing the possibilities, limits and effectiveness of each one. Third, students should be able to select the appropriate representation taking into consideration the task.  Finally, through the use of multiple representations, students will be able to “grasp the common properties of these diverse materials and will succeed in constructing the concept” (p. 111).

            Another group of questions that Janvier and colleagues (1987) focused in their research were the following:

1.      Does the students “select” a representation? Among several representations presented to them, do they know which one to retain, which is the most appropriate to accomplish the task?


2.      Do the students see the same task in each of the representations given?

3.      Are the students convinced that regardless of the particular representation they use as an aid to solve a problem they will necessarily arrive at the same result?


4.      How do students develop the attitude of having recourse to representations in case they encounter difficulties? (p. 114)


The literature, supported by the extensive research done by Janvier, et al. (1987) has raised these questions, summarized above, and many others regarding the usefulness of representations in mathematics.  The base of their concern and many of their inquiries is in the fact that current teaching practices using representations are not fulfilling their objectives and moreover, their contribution to the learning process is almost null.  “Certain representations lead more to difficulties rather than functioning as aids to learning” (p. 116).

Greeno and Hall (1997) explored the argument about how representational forms should be made and used in innovative classroom settings.  First, they affirm that representations are constructed for specific purposes in order to attempt to solve problems and communicate with others about it.  Second, students frequently develop representations with the purpose of observing patterns and performing mathematical procedures, keeping in mind the fact that different forms provide different supports.  Lastly, students frequently use multiple representations in order to solve a problem.  Some of the representations used by students are constructed by themselves and they could differ considerably from the representations taught in the curriculum.

Some Weaknesses of Using Representations in Mathematics

Lines of research studies describe some weaknesses or disadvantages of the uses of representations in mathematics teaching and learning.  Poppe (1993) exploring the effects of differing technological approaches to calculus on students’ use and understanding of multiple representations when solving problems, found that although students realized that tables, graphs, and mapping diagrams were helpful, they did not use them in order to solve unfamiliar mathematical problems unless suggested to do so.  Dufour-Janvier and colleagues (1987) investigating the accessibility of representations concluded that the use of representations is sometimes abstract to students, and this could provoke a lack of meaning to them.  Also, they affirm that the inappropriate context use of representations, as well as the prematurely of their use, resulting in negative consequences to students.  “The use of such nonaccessible representations encourages a play on symbols, puts the emphasis on the syntactical manipulations of symbols without reference to the meaning.  The signified is absent!  Mathematics is reduced to a formal language” (p. 11).

Van Someren, et al. (1998) conducting research in multiple representations in teaching, affirm that the use of combined representations in mathematics “creates new problems for the learner” (p. 4).  They go beyond by saying that multiple representations are not a good thing per se [italics added].  These researchers claim that when information is presented to students in varied forms, it is particularly important to also teach the relations or connections between representations since, if students are left alone to construct them themselves, it will be difficult.  Finally, Van Someren and colleagues call for a need for a closer analysis between their semantic relations and performance characteristics, in order to appropriately use multiple representations in problem solving.

Definitions of Representations

Until this point, the research has showed the need to use representations in mathematics teaching and learning.  It is important to look at how the literature has defined representations.  There are few researchers who have attempted to define representations in mathematics.  The only clear definition comes from the work done by Özgün-Koca (1998) who stated that “multiple representations are defined as external mathematical embodiments of ideas and concepts to provide the same information in more than form” (p. 1).  Another definitions could not be found.

Classification of Representations

Nevertheless, the literature does show some research studies concerning the classification of representations.  Porzio (1994, p. 3), citing the work done by Dufour-Janvier and colleagues (1987), classifies representations as external and internal.

Internal representations concern most particularly mental images corresponding to internal formulations we construct of reality (we are here in the domain of the signified).  External representations refer to all external symbolic organizations (symbol, schema, diagrams, etc.) that have as their objective to represent externally a certain mathematical ‘reality’. (p. 109)


Lesh, Post, and Behr (1987) have said that external representations are the way by which mathematical ideas could be communicated and they are presented as physical objects, pictures, spoken language, or written symbols.

            The research group headed by Janvier (1993), a recognized scholar in this field, expanded the idea of classification of representations.

External representations act as stimuli on the senses and include charts, tables, graphs, diagrams, models, computer graphics, and formal symbol systems.  They are often regarded as embodiments of ideas or concepts.  The nature of internal representations is more illusive, because they cannot be directly observed. (p. 81)


They affirm that important concepts in a representation theory are “to mean” or “to signify” (p. 81).  In this way, Janvier and colleagues state that external representation, which they call signifier, and internal representation, called signified, should be linked.

            Cuoco (2001) affirms that:

External representations are the representations we can easily communicate to other people; they are the marks on the paper, the drawings, the geometry sketches, and the equations.  Internal representations are the images we create in our minds for mathematical objects and processes – these are much harder to describe. (p. x)


Goldin and Shteingold (2001) expand the discussion on the types of representation arguing that:

External systems of representation range from the conventional symbol systems of mathematics (such as base-ten numeration, formal algebraic notation, the real number line, or Cartesian coordinate representation) to structured learning environments (for example, those involving concrete manipulative materials or computer-based micro worlds).  Internal systems, in contrast, include students’ personal symbolization constructs and assignments of meaning to mathematical notations, as well as their natural language, their visual imagery and spatial representation, their problem-solving strategies and heuristics, and (very important) their affect in relation to mathematics. (p. 2)


            Janvier and colleagues (1993) emphasizing the classification of representations introduced the term “iconic”. They say that external representations could be iconic since “they can more or less suggest in their arrangement or configuration the internal representation to which they relate” (p. 82).  These researchers consider the term “symbolic” as equivalent to the word “noniconic”.  They explain that the symbolism of an external representation depends primarily on the arbitrary arrangement or the selection of elements, which constitute it.  When any other feature has not helped the interpretation process, it refers to as noniconic representations.  Janvier, et al. affirms that the majority of mathematics representations could be classified as noniconic.

            The psychologist Jerome Bruner, using some guidelines investigated by Piaget, has been considered as one of the first researchers who implicitly classified representations.  Bruner (1964) proposed three modes of representation: (a) enactive, (b) iconic, and (c) symbolic.  Using the modes of representation introduced by Bruner, Mason (1987a) he has presented the idea that teaching schemes are a spiral movement.  As they pass through the spiral, students will go from using manipulable external representations to gain a meaning of internal representations to symbolic representations.  Mason proposes that one aim should be to help students to construct internal representations strongly related to external representations where they feel confident.

As discussed by Janvier, et al. (1993) another line of research regarding classification of representations comes from the studies done by Bertin (1967) who used three categories. The first one maps, which includes the representation that keeps a fair degree of similarity with the special properties of the objects they represent.  The second, which shows the nature of the relations between variables, is called diagrams.  Familiar mathematical concepts such as data charts, graphs, belong to this category.  Lastly, networks refer to when representations of this class show the relationships between events, factors, or individuals (pp. 82-83).

            Janvier and colleagues (1993) have realized that the existence of many representations in mathematics is a cause of confusion on students.  Trying to relate internal and external representations in mathematics, they propose two important terms in their discussion: homonymy and synonymy [italics added].  The first phenomenon in mathematics is found when one representation has two different meanings.  That is, from an external representation there are two different internal representations.  The second term refers to when one mental object is denoted in many representations: from two different external representations there is one internal representation.  According to their findings, homonymy, as well as synonymy cannot be avoided in mathematics.  “They belong to it per se” (p. 88).

Students’ Preferences for Representations

            It is frequently observed that students in the classroom show certain preferences for one particular external representation.  The literature contains important research studies concerning preferences exhibited by students in order to select a representation.  Hart (1991), who developed extensive research concerning representations, explored their management.  She studied students’ preferred representations and how they vary the choice of representation depending on the problem.  Hart found that there are factors that influence students’ choice of representation.  Her findings are summarized in the following points:

1.      Students confident in their symbolic manipulation skills tend to use alternate representations only when unsuccessful at finding an answer symbolically.


2.      Students make a choice of representations depending on the complexity of the symbolic information provided.


3.      Some students do not use a certain representation because they do not recognize that it’s a viable choice.


4.      Students lack confidence in using certain representations.

5.      Students who do not have access to a graphing calculator do not typically choose to use the graphical representation.


Hart’s findings indicate that the representation used by students to solve problems is strongly influenced by their previous experiences.

            Research by Yerushalmy (1997) revealed, “normally, symbolic (formula or equation) representation is the more convenient representation for modeling situations with two independent variables.  However, the priorities for students who have not yet learned to manipulate symbols but have experienced modeling through various other representations could be different” (p. 432).

            Keller and Hirsch (1998) identified two types of research on students’ preferences for representation.  The first line of research deals particularly with the attempt to determine students’ preferences by the representation used to perform tasks.  LaLomia, Coovert and Salas (1988) conducted research regarding which of two types of representation – tables and graphs – students used most often to solve tasks.  Their findings show that students preferred tables when they had to locate particular numbers.  On the other hand, students only slightly used graphs with interpolation and forecasting tasks.  The second line of research dealing with preferences in representations concerns learning theories or cognitive styles.  About this second line of research on representations’ preferences, Turner and Wheatley (1980) explored the preferences of students in an elementary calculus course emphasizing two representations: graphical and linguistic.  They found that students exhibited strong preferences for each form.  Furthermore, there was a significant correlation between graphical representations and the students’ spatial performance.

            Keller and Hirsch (1998) identified several factors that influence the preference of representations.  These factors included: (a) the nature of students’ experiences with each representation, (b) the students’ perceptions of the acceptability of using a representation, and (c) the level of the task.  Another theories concerning representations’ preferences comes from the research done by Donnelly (1995), Dufour-Janvier, et al. (1987), Eisenberg and Dreyfus (1991), Poppe (1993), Porzio (1994), and Vinner (1989).  Özgün-Koca (1998, p. 5) summarized the previous findings of research in reasons for students’ preferences for representations.  These reasons were classified in two sections: internal and external effects.  In the first sections are: personal preferences, previous experience, previous knowledge, beliefs about mathematics, and rote learning.  Under external effects there are: presentation of problem, problem itself, sequential mathematics curriculum, dominance of algebraic representation in teaching, and technology and graphing utilities.

Connections Among Representations

            An issue widely discussed in the consulted literature has been the connections between representations.  Other authors have referred to it as translations and linking processes among representations.

            Dufour-Janvier and colleagues (1987) realized that often students confront problems to see the same task when different representations of the same problem are given.  Students think that there are equal numbers of problems as there are representations.  These researchers presented the following situation:

A child resolves a problem using a representation.  We then show him to the same problem resolved by someone else using a different representation.  When we deliberately show him the answer of this other child (the answer happens to be incorrect); a number of children are not at all disturbed and find this quite natural because in their view the first problem was done one way and the second done in another way. (p. 114)


With this example it has been shown “that students do not see all of the representations accompanying a single task as different ways for tackling the same situation” (Porzio, 1994, p. 45).  The literature shows that students are able to work with different types of representations.  The troubles start when they try to relate similar information provided by different representations.  Lesh, Post, and Behr (1987) have stated that the connections between representations are particularly important in order to solve problems.

            Porzio (1994) conducted research exploring the students’ abilities to see or make connections between graphical, numerical, and symbolic representations in the context of problem situations, using three different approaches: (a) traditional approach, (b) graphic calculator approach, and (c) Calculus & Mathematica software.  He found that in the traditional course where symbolic representations were emphasized, students belonging to this group exhibited the most difficulty of all the students in recognizing connections between different representations and different forms of the same non-symbolic representation.  In the group where graphics calculators were used and graphical and symbolic representations were emphasized, students seemed to consider the main emphasis to be on graphical representations.  Finally, students who used the Calculus & Mathematica software, where multiple representations which were illustrated in the majority of the times as symbolic and graphic, were better than the other students at recognizing connections between different representations and varied forms of the same representation.  Also, students often used graphical/symbolic and symbolic/numerical representations.  The research results from Porzio can be summarized as follow:

Students are able better able to see, or make, a connection between different representations when one or more of the representations is emphasized in the instructional approach that they experienced and [underlined by the author] when then instructional approaches includes having students solve problems specifically designed to explore or establish the connection(s) between the representations. (p. 443)


            Kaput (1989a), one of the recognized researchers in the field has introduced the concept of linked representations [italics added].  He describes the cognitive potential of dynamic links between representations.  Kaput said: “multiple, linked representations of mathematical ideas likewise provide a form of redundancy, a redundancy that can be exploited directly in a multiple, linked representation learning environment” (p. 179).  According to him, one of the advantages of using linked representations is that they enable students to repress some aspects of complex ideas and give more attention to others, supporting the varied ways of the learning and reasoning process.

            Janvier and colleagues (1993) have introduced the term translation in the discussion of representation in mathematics.  They argue that the process of translation from one representation to another is possible as the result of the synonymy phenomenon presented earlier.  These researchers think that in order to teach the translations skills efficiently it is necessary that students view the translations from both directions.  Janvier, et al. suggests, for example, that opposite translations, that is “graph à formula” and “formula à graph” should be tackled in pairs (p. 98).

            Hiebert and Carpenter (1992) have conducted extensive research dealing with teaching and learning mathematics with understanding.  They have devoted some sections in their research to connections between representations.  They argued that connections between external representations of mathematical concepts could be constructed by the student “between different representations forms of the same mathematical idea or between related ideas within the same representation form” (p. 66).  Hiebert and Carpenter said that the connections between different representations are possible if they are based on the relationships of similarity (“these are alike in the following ways”) and in the relationships of difference (“these are different in the following ways”) (p. 66).  The particular connections between representations can be constructed, according to these researchers, looking carefully at how they are the same and how they are different.  Finally, Hiebert and Carpenter affirm that the process of connections between representations plays a particular role in learning mathematics with understanding.

Representations and Understanding

            Understanding and meaning are two key terms in mathematics teaching and learning.  They have been reinforced in the current reform movements.  On this topic, Goldin and Shteingold (2001) affirm “conceptual understanding consists in the power and flexibility of the internal representations, including the richness of the relationships among different kinds of representation” (p. 8).  Janvier et al. (1993) mentions that in any discussion about theories of representation, two terms are transcendental: “to mean” and “to signify” [italics added] (p. 81).

            Porzio (1994) points out that the theoretical framework that support the use of multiple representations in mathematics comes from the research done by Hiebert and Carpenter (1992).  These researchers affirm that understanding can be described in terms of internal knowledge structures.  They define understanding in mathematics as follows:

A mathematical idea or procedure or fact is understood if it is part of an internal network.  More specifically, the mathematics is understood if its mental representation is part of a network of representations.  The degree of understanding is determined by the number and strength of the connections.  A mathematical idea, or procedure, or fact is understood thoroughly if it is linked to existing networks with stronger and numerous connections.  (Hiebert and Carpenter, 1992, p. 67)


Based on this definition of understanding, Porzio says that one of the principal goals of mathematics teaching and learning is to provide tools and opportunities to students in order that they can develop large and well-connected internal networks of representations.

            Goldin and Shteingold (2001) remark that:

A mathematical representation cannot be understood in isolation.  A specific formula or equation, a concrete arrangement of base-ten blocks, or a particular graph in Cartesian coordinates makes sense only as part of a wider system [italics added by the author] within which meanings and conventions have been established.  The representational systems important to mathematics and its learning have structure, so that different representations within a system are richly related to one another. (p. 2)


            Kaput (1989b) describes as epistemological sources of mathematical meaning the connections that could be possible between representations.  He identifies the following factors as the epistemological sources of mathematical meaning:

1.      transformations within and operations on a particular representation system;

2.      translations across mathematical representation systems;

3.      translations between non-mathematically described situations and mathematical representation systems; and


4.      consolidation and reification of actions, procedures, or webs of related concepts into phenomenological objects that can then serve as the bases of new actions, procedures, and concepts at a higher level of organization. (p. 106)


Porzio (1994) points out that the first three sources of mathematical meaning identified by Kaput correspond to the many kinds of connections that can be made between distinct forms of the same type of representation and between different kinds of representations.

Interpretation of Representations

            The final topic concerning multiple representations deals with the interpretation of these representations in mathematics.  This topic is one of the most widely discussed.  As cited in the work by Janvier and colleagues (1993, p. 81), Von Glasersfeld (1987, p. 216) affirms: “A representation does not represent by itself – it needs interpreting; to be interpreted, it needs an interpreter”.  Greeno and Hall (1997) mention that in order to interpret representations, students should be involved in a learning environment where complex practices of communication and reasoning are emphasized.

            The literature agrees in finding that graphs, tables, pictures, and diagrams, among others, do not constitute a representation by themselves.  Greeno and Hall (1997) citing the research done decades ago by Charles Sanders Peirce (1955) said, “for a notation to function as representation, someone has to interpret it and thereby give it meaning” (p. 366).  Peirce identified three factors involved in representation: (a) something that is represented, the referent; (b) the referring expression that represents the referent; and (c) the interpretation that links the referring expression to the referent.  Following Peirce’s principle, Greeno and Hall say that notations such as tables, equations, and graphs are considered as potential representations.  They become representations per se when someone gives them meaning by interpreting them.

            Greeno and Hall consider equations, Cartesian graphs, and tables as standard forms of representations and they have frequently shared conventions of interpretation.  These researchers indicate that the process of learning these conventions are important for students in order to encounter, construct and communicate their ideas.

Standard instructional practices in mathematics provide students with opportunities to learn the conventions of interpretation of standard representational forms at an operational level.  Teachers explain how to construct and interpret tables, graphs, and equations, and students are asked to construct representations of given information in these forms and to interpret representations that they are given.  In these activities students can learn to follow the standards conventions of interpretation for the forms, and with this learning the forms function as representations for the students. (p. 366)


According to these researchers, a practice like this one is now promoting the recognition of interpretation as an essential part of representations in mathematics.  These activities serve to give students the opportunity to learn how to follow standard conventions of interpretation, and moreover, how to understand how representations work.

Learning Theories Supporting Multiple Representations in Mathematics

            As stated earlier, the use of multiple representations in mathematics is strongly linked to the learning of important mathematics concepts.  This section will describe some research of theorists and their contributions to this field.

            One of the most recognized researchers in this field is Zoltan P. Dienes.  His extensive work in theories of learning has impacted mathematics teaching and many of his ideas are still been applied today in educational settings (English and Halford, 1995).  As cited in Gningue (2000, p. 59), “Dienes (1971) believes that abstraction results from the passage of concrete manipulations of objects to representational mapping of such manipulations and then to formalizing such representations into rule structures”.  Based on this belief, Dienes elaborated his four general principles for teaching concepts.

            The two first Dienes’ principles are the Dynamic Principle and the Constructivity Principle.  He thinks that the best way to teach a new concept is through the formulation of a particular situation where students are lead to constructive, rather than analytical thinking and understanding (Gningue, 2000).  The third principle is the Mathematical Variability Principle.  It states “concepts involving variables should be learned by experiences involving variables should be learned by experiences involving the largest possible number of variables” (Dienes, 1971, p. 31).  Lastly, the Perceptual Variability Principle or Multiple Embodiment Principle “demands a richness of concrete experiences with the same conceptual structure, so that children may glean the essentially abstract mathematical idea, which must be learned.  To allow as much scope as possible for individual variations in concept-formation, as well as to induce children to gather the mathematical essence of abstraction, the same conceptual structure should be presented in the form of as many perceptual equivalents as possible” (pp. 30-31). 

This principle suggests that the learning of a mathematical concept reaches its maximum expression when students are exposed to a concept using a variety of physical materials or embodiments [italics added].  Resnick and Ford (1981) said: “multiple embodiments are viewed as facilitating the sorting and classifying process that constitutes the abstraction of a concept.  Seeing a principle operating similarly even when different materials are used seems to help children discover what is and is not relevant to the concept” (p. 121).  These researchers point out that the students’ familiarity with the various mathematical materials is an assumption of presenting concepts using multiple embodiments.  Resnick and Ford argue that if this familiarity process does not occur first, the use of embodiments will be “counterproductive” (p. 121) since students should learn the materials and a new mathematical principle at the same time.  According to Dienes, as cited in Resnick and Ford’s research, multiple embodiments should look different from each other in order that children can observe the structure from many different perspectives and construct a vast amount of mental images about each concept.  The use of these embodiments should allow manipulation of all variables related with the concept under study.

Dienes (1973) clarified his four principles by pointing out six stages of teaching and learning mathematical concepts.  Similar to the intellectual developmental stages introduced by Piaget, Dienes affirmed that the learning of mathematical concepts occur through sequential stages.  These stages are: (a) free play, (b) games, (c) searching for communalities, (d) representation, (e) symbolization, and (f) formalization.  As mentioned in Gningue (2000), the first three stages are described as components of the first Dienes’ principle.  The second phase of the learning cycle promoted by Dienes constitutes the transition process from manipulative materials to abstract representations.  These representations are illustrated initially as pictorial models and graphs, and finally as mathematical symbols.  The beginning of this second phase is the fourth or representation stage.

The child needs to develop, or to receive from teacher, a single representation of the concept that embodies all the common elements found in each example.  This could be a diagrammatic representation of the concept, a verbal representation, or an inclusive example.  Students need a representation in order to sort out the common elements present in all examples of the concept.  A representation of the concept will be usually more abstract than the examples will bring students closer to understanding the abstract mathematical structure underlying the concept.  (Gningue, 2000, p. 64)


The fifth stage described by Dienes is where the students describe the representation of the concept verbally and using mathematical symbols.  Dienes suggests that the teacher should supervise the use and construction of symbols.  Students can use their own symbols, but they should be aligned with those included in the textbook.

Janvier and colleagues (1993) affirm that students do not always appreciate and accept that two or more external representations belong to the same concept.  Rather, students have exhibited the preference to work mainly “on a one-to-one correspondence basis” (p. 91).  Janvier et al. mention that opponents of Dienes’ principles state that adding more embodiments to concept instruction is not a guarantee that students will get a better and more meaningful internal representation of the concept.

Constructivism has had an enormous impact on current education learning theories, and mathematics instruction is no exception.  De Jong and colleagues (1998) said “modern education learners are encouraged to construct their own knowledge, instead of copying it from an authority, be it a book or a teacher” (p. 9).  Hart (1991) mentions “constructivist theory suggests that knowledge is actively constructed out of one’s experiences” (p. 4).  Noddings (1990) explains that constructivism has basically two main characteristics: (a) a cognitive position, and (b) a methodological perspective.  This review will focus on the first characteristic of constructivism.  She affirms: “as a cognitive position, constructivism holds that all knowledge is constructed and that the instruments of construction include cognitive structures that are either innate or are themselves products of developmental construction” (p. 7).

Noddings (1990) in her extensive work in the field, has summarized in the following points the current constructivists views:

1.      All knowledge is constructed.  Mathematical knowledge is constructed, at least in part, through a process of reflective abstraction.


2.      There exist cognitive structures that are activated in the processes of construction.


3.      Cognitive structures are under continual development.  Purposive activity induces transformation of existing structures.


4.      Acknowledgement of constructivism as a cognitive position leads to the adoption of methodological constructivism.


a.       Methodological constructivism in research develops methods of study consonant with the assumption of cognitive constructivism.


b.      Pedagogical constructivism suggests methods of teaching consonant with cognitive constructivism. (p. 10)


Technology and Multiple Representations

            Technology has the potential to completely change current trends in teaching and learning of mathematics.  Researchers as De Jong and colleagues (1998) have agreed with the need for technology in mathematics scenarios.  They said:  “Technology plays a major role in implementing these new trends in education” (p. 9).  As cited in Gningue (2000), Fey (1989) proposed the use of a vast amount of technological resources, such as calculators, computers, and computer software to teach concepts in algebra.  According to him, “the most obvious implication of computer tool software is the opportunity to rebalance the relationship among skill, understanding, and problem-solving objectives in algebra” (p. 204-205).  Findings from research studies conducted by Orton (1983a, 1983b) and Tall (1985) indicate that the use of technology is advantageous in order to promote conceptual understanding.

            One of the main advantages to the uses of technology in mathematics education is, without a doubt, the capability to present information in multiple representations.  Mathematical concepts can be introduced through the use of tables, graphs, equations, and other representations.  Keller and Hirsch (1998) affirm that the incorporation of multiple representations supported by technology is an important topic in mathematics curricula.  Important lines of research conducted by recognized scholars such as Fey (1989), Goldenberg (1987), Kaput (1992), and Porzio (1994) indicate that the access to multiple representations of mathematical concepts has increased with the advancements of technology.

Use of technology in the classrooms appears to affect student learning in a positive way.  Those students using technology to access multiple representations may have “richer” concept images than those who do not have the same experience…

Technology can provide a means for presenting concepts via multiple representations and for students to work within multiple representations.  A review of the literature indicated there may be some positive effects from the use of technology, capable of graphing and/or symbolic manipulation, in the classroom. (Hart, 1991, pp. 45-46)


            Several mathematics reform projects have been developed nationwide in order to promote teaching mathematical concepts using multiple representations supported by technology.  One of them is the Harvard Calculus Project, also called “Rule of Three”, which emphasizes the use of three representations: graphical, numerical, and symbolic (Hart, 1991; Porzio, 1994).  Hughes-Hallet (1991) indicated:

[The philosophy of the project] is based on the belief that in order to understand an idea, students need to see it from several points of view, and to build web connections between the different viewpoints.  I believe that in calculus most of the ideas should be presented in three ways: graphically and numerically, as well as in the traditional algebraic way.  Technology is invaluable here. (p. 33)


Porzio (1994) points out that there are differences between students participating in curriculum projects, specifically in calculus (Tucker, 1990), where they are using computers, calculators and representations where graphics and symbols are also emphasized, students using the Calculus & Mathematica software where technology is used intensively, and students from traditional approaches.  Nevertheless, he states that there is little evidence of the effectiveness of these technological approaches.

            Fey (1989) affirms that the use of numerical, graphic and symbol manipulation is a powerful technique for mathematics teaching and learning.  He identified several ways in which computer-based representations of mathematical ideas are unique tools for problem solving.  These are:

1.      Computer representations of mathematical ideas and procedures can be made dynamic in ways that no text or chalkboard diagram can.


2.      The computer makes it possible to offer individual students a work environment with representations that are flexible, but at the same time constrained to give corrective feedback to each individual user whenever appropriate.


3.      While some multiple embodiment computer programs might be viewed as poor simulations of more appropriate tactile activity, it has been suggested that this electronic representation plays a role in helping move students from concrete thinking about an idea or procedure to an ultimately more powerful abstract symbolic form.


4.      The versatility of computer graphics has made it possible to give entirely new kinds of representations for mathematics.


5.      The machine accuracy of computer generated numerical, graphic, and symbolic representations makes those computer representations available as powerful new tools for solving problems. (p. 255)




Functions and their Representations


            The concept of function is one of the key topics in mathematics.  It dominates the mathematical panorama and is present in a vast part of the instructional activities developed at secondary and college levels.  Thorpe (1989) proposes the study and the use of functions as “the centerpiece of algebra instruction because functions are at the very heart of calculus” (p. 11).  Selden and Selden (1992) coincide with Thorpe, when they say that functions play a central and unifying role in mathematics.  As cited in Hart (1991, p. 10), Vinner and Dreyfus (1989) introduced the Dirichlet-Bourbaki definition of what a functions is.  It says: “a function is a correspondence between two nonempty sets that assigns to every element in the first set (the domain) exactly one element in the second set (the codomain)” (p. 357).  This definition has been kept and taught in the majority of the mathematics curricula (Lloyd and Wilson, 1998).

            Further, the concept of function has the capability of being taught using different representations.  The literature illustrates functions in several ways, such as mapping diagrams, tables, graphs, and equations.  All of these representations are primarily intended to promote a better understanding of the concept.  Research done by Sfard (1987) indicates that in order to get a good concept of functions, students should develop an operational before a structural concept.  After this, students will benefit from the introduction of functions using the different representations, such as mappings, tables, and graphs (Poppe, 1993, p. 26).

            According to Poppe (1993), tables, graphs, and mapping diagrams are representations of functions that can be used to create mental structures.

The computational processes of creating tables, graphs, and mapping diagrams would afford the students an opportunity to develop an operational conception of function.  The exploration of the function idea in a concrete context using tables, graphs, and mapping diagrams provides the students with a richer foundation for development of the variable concept. (p. 25)


            Thomas (1975) examined the aspect of understanding of functions in students from seventh and eighth grades, identifying five stages in the development of the concept of function:

1.      Finding images in mapping. Simple interpretations of arrow notation.

2.      Identification of instances of mapping with finite domains.

3.      Operational ability in finding images, pre-images, range, and domains where the mappings are given by some display of the set of ordered pairs.


4.      Identification of noninstances of mappings with finite domains.

5.      Composition of mappings and the translation from one representation of mapping to another. (Poppe, 1993, p. 21)


Markovits, Eylon, and Bruckheimer (1986) found that most students understood that a function would have more than one representation.  They stated that almost fifty percent of their study population was able to identify two functions, one in algebraic form and the other in graphical form, as being the same.  In addition, several studies have been done comparing the difference between the uses of two or more representations of functions.  Iannone (1975) compared tabular approach and mapping diagrams of functions.  Results show that the best way to represent the function concept is through the use of mapping diagrams.  Poppe (1993) conducted research in this specific area and found that students were aware of the uses of tables, graphs, and mapping diagrams, and tables were helpful in finding generalized patterns.  On the other hand, students found tables, graphs, and mapping diagrams helpful. In conclusion, the use of tables, graphs, and mapping diagrams aided instruction.  Students had the opportunity to see the same information in different ways.

Results from Markovits et al. (1986) also show that difficulties arose when students managed more than one representation of functions at the same time.  They pointed out, for example, that students changed domain and codomain of some functions.  Goldenberg (1988) affirms that confusion may occur trying to relate information provided by two different representations.  He suggests an appropriate transfer process between the representations.  Hart (1991) introduced the term compartmentalization [italics added] when students do not relate several representations for the same function.  A lack of connections between two representations –graphical and algebraic– was found in research conducted by Dreyfus and Eisenberg (1988).  Ferrini-Mundy and Graham (1991) found similar results when students managed algebraic and graphical contexts as separate worlds.  Recognizing troubles shown by students trying to relate representations of the same function, Poppe (1993) affirmed that: “students needed more opportunity working with the different representations” (p. 98).



            The previous sections have described research studies and current trends on the uses of multiple representations in mathematics teaching and learning.  Preferences, connections, among others, were also discussed.  Theories of learning that support the use of representations in mathematics were introduced and discussed.  Further, research studies dealing with how the available technologies have been used to promote understanding through representations in mathematics were included in this chapter.  Finally, the importance of functions in the curricula and a view of their representations were discussed.

The next chapter will present the methodology of this research project, including participants, settings, and instruments used to obtain data.  The procedures followed in the instructional activities will also be described, as well as the statistical tests used to answer the questions of this investigation.


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