I I l l I D JOURNAL OF MATHEMATICAL BEHAVIOR, 1"/' (1), 25-32 ISSN 0364-0213. HV! Copyright © 1998 Ablex Publishing Corp. All rights of reproduction in any form reserved.

METAPHORIC AND METONYMIC SIGNIFICATION IN MATHEMATICS

NORMA C. PRESMEG

The Florida State Universi~

This paper explores the roles of metaphor and metonymy in making sense of the ambiguities inher- ent in representation of mathematical constructs. Using the metaphors of "chains of signification" and "descent into meaning" for metonymies and metaphors respectively, these literary figures are discussed with regard to their use in mathematics. Synonymy, homonymy and polysemy are viewed as explanatory constructs in an analysis of ways in which metaphor and metonymy aid learners and mathematicians alike in making sense of mathematical ideas and resolving ambigu- ities.

What are metaphors and metonymies, and why are these literary constructs important in making sense of mathematical ideas? These questions pertain to the roles of symbolism of various kinds in mathematics and to the complex processes involved in using and moving among these systems of signification.

In describing his use of the term representational systems, Goldin (1992, p. 241) wrote of the "essential presence of ambiguities in representational systems" (his emphasis). It is these ambiguities which are at the heart of metaphoric and metonymic signification in mathematics as well as in literature. Before exploring some aspects of these ambiguities, a few words about terminology and epistemology are necessary.

1. T E R M I N O L O G Y AND E P I S T E M O L O G Y

There is ample evidence in the literature on representation in mathematics over the past few years (Cobb, Yackel, & Wood, 1992) to substantiate von Glasersfeld's (1987, p. 216) claim that "A representation does not represent by itself---it needs interpreting and, to be inter- preted, it needs an interpreter." A recent study by Sproule (1994) gives clear evidence that the interpretation of three-dimensionality from two-dimensional diagrams has less to do with cues in the diagrams, which were long thought to be the key issue (Serpell, 1976), than it has to do with the experiential context and the culture of the interpreter. Accordingly, I shall take it for granted when I use words like "representation," "signification" and "sym- bolism" that I am referring to interpretive action by a cognizing being, and not to any essential and intrinsic properties of diagrams, figures, words or other symbols. This use of

Direct all correspondence to: Norma C. Presmeg, Curriculum & Instruction, Florida State University, Tallahas- see, FL 32306 <npresmeg@garnet.acns.fsu.edu>.

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26 PRESMEG

the verb "to represent" has the sense of connotation rather than denotation in the English language. Of the four German verbs which are relevant in this regard (von Glasersfeld, 1987), it is vorstellen and bedeuten rather than darstellen or vertreten, which suggest this sense of active interpretation. However, the meanings of all these words are context depen- dent. I agree with Vergnaud (1994) that the complexity of the processes involved makes it desirable to go beyond his tripartite model of signifier, signified, and referent, but since reduction is inevitable in any model, these terms may still be useful in distinguishing between various aspects of signification while recognizing the central role of human cog- nitive activity in all of these.

This article starts by describing some aspects of the construction of mathematical objects by means of metaphor and metonymy. I suggest that metaphors, metonymies, and the imagery and symbolism which accompany them are essential components in the repre- sentation of mathematical constructs for an individual, because they help the individual to make sense of the construct amid the ambiguities inherent in its representation.

1.1 Metaphor as a Type of Analogy

Metaphors may be implicit in all areas of human experience. When Jakubowski's (1990) teachers analyzed their beliefs about teaching mathematics, some of the implicit metaphors were as follows:

• a teacher is a policeman; • a teacher is a mother hen; • a teacher is an entertainer; • a teacher is a gardener.

The metaphors created a ground which influenced the teachers' decisions and actions in their classrooms. The two domains which are linked by this ground are called in much of the traditional literature (Leino & Drakenberg, 1993), the tenor (the teacher in this case) and the vehicle (e.g., mother hen). The vehicle contributes to the structuring of beliefs about the tenor.

'Teaching metaphors' such as these are often not stated as directly as these are, or remain completely unstated and unconscious, but they may nevertheless implicitly influ- ence decisions and actions. This was also the case in the three mathematical metaphors described later in this section, until the researcher questioned and probed and the students being interviewed became aware of images which had metaphors associated with them.

In both metaphor and metonymy, a person uses one construct to stand for another. The difference between the two (which is elaborated in what follows) is roughly that metaphor links one domain of experience with another seemingly disparate domain--and creates meaning from the connection--while metonymy uses one element or salient attribute of a class to stand for another element or the whole class.

A useful distinction between metaphor and metonymy was made by Pimm (1991) based on ideas from Dick Tahta. Pimm (1991, p. 44) quoted Tahta as follows: "Linguists have called the movement 'along the chain of signifiers' metonymic whereas 'the descent to the signified' is metaphoric." Tahta pointed out that relations along the chains of signifiers (metonymy) as well as descent into meaning (metaphor) are both important aspects of

METAPHOR AND METONYMY 27

analogy

Source: Leino and Drakenberg, 1993, p. 30. FIGURE 1. Metaphor, Analogy and Simile.

(implicit) metaphor

(explicit) simile

learning mathematics, and they may be stressed at different times and for different pur- poses. Using this distinction, Pimm provided a rationale for the view that construction of meaning for mathematical symbols is complementary to development of fluency and auto- maticity of symbol manipulation, and that both processes are necessary. For the purposes of the present paper, Tahta's statement provides a useful starting point in elaborating what metaphor and metonymy are, and why they are necessary in using mathematical symbol- ism. The views of metonymy in terms of a (perhaps horizontal) chain of signifiers, and metaphor in terms of a (vertical) descent into meaning, are metaphors in themselves, which provide just one way of representing these constructs.

The structure of metaphor as a type of analogy will be discussed first.

2. METAPHOR

The noun metaphor is defined as "Application of name or descriptive term to an object to which it is not literally applicable" (Concise Oxford Dictionary). The word is derived from the Greek, metaphora, meaning 'transfer' or 'carry over.' Leino and Drakenberg (1993) in the course of a survey of theories of metaphor which are relevant to education, point out that metaphor can be considered to be an implicit form of analogy, while simile is an explicit form. The relationship is depicted in Figure 1.

Both these forms of analogy involve a comparison of two domains of experience, but while simile would specify, 'domain A is like domain B,' metaphor would state that 'domain A is domain B.' In both cases it is understood that the analogy refers only to some elements of the two domains: these similar elements constitute the ground of the compari- son, while the dissimilar elements constitute the tension. The ground and the tension are both essential elements of a metaphor. It is this recognition of simultaneous similarity and dissimilarity which gives metaphor its special power to structure new experience in terms of old, and which allows for the ambiguity which was referred to in the opening paragraphs as an unavoidable element of representation.

An example of a metaphor used in school mathematics is the following, taken from data collected by the author in an imagery project with high school students in Florida in Fall, 1991. Mark had been asked to find the sum of the first 30 terms of the sequence

5,8,11 . . . .

After some dialog with the interviewer, he solved the problem correctly by adding the first and 30th terms, then the second and 29th terms, and so on, until he realized that there would be 15 of these sums. He obtained the answer by multiplication, and explained when questioned, "First I saw, like, a dome, where it's going like this, and it's continuing" (draw-

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ing concentric arcs) "until we got down to the two very middle terms. And I just deducted [sic] that if we have 30 terms, and we're taking two each time, then it has to be 15. So I multiplied by 15." Incidentally, this example also illustrates the efficacy of alternating use of visual imagery and logic (Pallascio, Allaire, & Mongeau, 1993; Presmeg, 1986). A dif- ferent student in the same project saw Mark's "dome" as a "rainbow." In these cases, the dome or rainbow image was the vehicle of a metaphor in which the tenor was the process of finding the sum of a number of terms the 'Gaussian' way.

Another example, which I have described elsewhere (Presmeg, 1992) involved Alli- son's "water level" metaphor, accompanied by an image of a ship sailing, which reminded her that if she was trying to find the key acute angle in trigonometry, she must use 180 or 360 degrees, not 90 or 270. For each of these three students, the metaphoric image was an idiosyncratic way of representing a mathematical principle. There was "descent into mean- ing," in Tahta's words. In these examples, imagery is central to each metaphor. However, the image itself is not the metaphor. I agree with Leino and Drakenberg (1993, p. 31) when they quote Paul Ricoeur's view that "image occurs centrally in metaphor, but the image itself is not metaphor, it only provides 'figurability to the message'."

One additional striking example of a metaphor which furthered the development of mathematics as a discipline was presented by Sfard (1994b). When Leibniz was grappling with the existence or non-existence of the square roots of negative numbers, he made sense of this ambiguity by means of a metaphor: he saw these entities as "amphibian between non-being and being." Again, an image of a creature such as a frog, snake or other reptile, may be central to the metaphor--but it is not the metaphor. It merely provides "figurabilty" to the metaphor, in which the vehicle is an amphibian, moving freely between land and water in an implicit analogy with the square root of negative unity (the tenor) moving between non-being and being. In this example the tenor is a mathematical concept--an imaginary number--which may be just as difficult for students to construct meaningfully, as it was initially for Leibniz. The metaphoric connection of the mathematical construct with the image of a frog, which is at once a land creature and a water creature, might help students to construct meaning for complex numbers which have parts that are real and parts that are imaginary.

In their model of a dynamic theory of the growth of mathematical understanding, Pirie and Kieren (1994) included the categories of image making, image having and property noticing, on the way to a level of understanding called formalizing. In their theory, it is the category image having which includes metaphoric representation. They wrote, "At the image having level we see the learners as working with metaphors. For them, mathematics is the image that they have and their working with that image" (p. 40, their emphasis). Their examples of eight-year olds learning about fractions by folding paper illustrate that when attention shifts from the "objects of metaphor" to properties of those objects (property noticing level), then simile is the kind of analogy used, not metaphor: "is" becomes "is like." Their conclusion is that "understanding at the image having level is fundamentally metaphoric, while understanding at the prop- erty noticing level entails the use of similes" (p. 40). This conclusion suggests a kind of literal identification view of metaphor which facilitates the understanding of the thinking of these eight year olds, but which does not do justice to the power of meta- phor to encapsulate and resolve ambiguity, as in Leibniz' example. One does not need to say "is l ike"--"is" will suffice--when it is fully understood that the analogy is only

METAPHOR AND METONYMY 29

a partial one, and that it is, after all, only an analogy. (Consider the metaphor, "an equa- tion is a balance.") A similar point could be made concerning Greer and Harel's conclu- sion (this volume) that analogy should be used to reinforce already existing conceptions rather than to build new conceptions. This important and valid conclusion needs to be understood in the context of the evidence on which it is based (use of manipulatives in place value numeration and comparable mathematical activities), and not generalized to cast doubt on the value of individual construction of connections which aid in sense making through simile and metaphor. The power of metaphor lies in its use in making sense of new conceptions in terms of already existing conceptions-- thus in "building new conceptions." Sfard (1994a) makes this point very convincingly in her description of metaphors used by practicing mathematicians. Geometric or spa- tial metaphors for new structures were common, as was personification: a saturated model (tenor) was seen in terms of afar man, a "padded guy" (vehicle, p. 48).

After a brief discussion of the nature of various types of metonymy, the issue of meton- ymy and metaphor in relation to ambiguity in representation will be revisited.

3. METONYMY

3.1. Two Types of Metonymy

The noun metonymy, from the Greek metonymia denoting change of name, is defined as "a figure by which one word is put for another on account of some actual relation between the things signified" (Webster). The dictionary example is "We read Virgil," that is, his poems or writings. We say, "Washington is talking with Moscow," meaning that individuals representing the governments centered in those two cities are communicating. Once again, ambiguity is an unavoidable element in the representation, an ambiguity which is resolved by experience with the context and conventions of the metonymy. The foregoing are examples of what Johnson (1987) called "metonymy proper", in which a salient or related attribute or entity is taken to stand for another entity.

A second type of metonymy is the figure of speech, synechdoche, in which a part is made to represent the whole or vice versa. One element of a class may be taken to stand for the whole class. Leino and Drakenberg (1993, p. 31) quote Eisner's well-known example in the field of education, "the enlightened eye." It seems to me that this example goes beyond synechdoche to metonymy proper, since "eye" is repre- senting "mind" rather than the body of which eye is a part. In mathematics, examples of synechdoche abound: an illustration of a triangle or circle is taken to represent respectively the classes of all triangles or circles; a variable is taken to stand for all or for some of the elements of a class of numbers, for instance, the reals, and so on. In these examples, too, the signification may go beyond synechdoche to metonymy proper, since the signifier (a drawing of a triangle, or a letter of the alphabet such as x) is not an element of the class which is represented: these elements of classes are mental constructs (signified) and an act of interpretation is involved in setting up the meton- ymy.

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/ / / ~ r internal representation 1 (sitmified) e.g., principle for finding sum of terms

in an arithemetic progression external representation (signifier) e.g., drawing of a rainbow

~ v internal representation 2 (signified) ehicle e.g., image of a rainbow broken

up into strips of various colors FIGURE 2. Metaphor as Homonymy.

external representation I (signifier) e.g., drawing of straight line graph

external representation 2 (signifier) e.g., equation of the form y = mx + c FIGURE 3. Metonymy as Synonymy.

internal representation (signified) e.g., concept of a straight line

3.2. Ambiguity

Adda (1982), and more recently, Janvier, Girardon and Morand (1993) have pointed out ways in which synonymy and homonymy may create difficulties for students in their learn- ing of mathematics on account of the ambiguities inherent in symbolism of various kinds. Janvier et al. define a homonym as "a single word having several different meanings" (p. 84), and they give many examples of mathematical representations--in graphs, alge- braic symbolism, and diagrams--which behave like homonyms. In fact, several of the examples given by these authors are instances ofpolysemy rather than homonymy. The dis- tinction is as follows. Homonymy occurs "where two words with two totally different meanings happen to be pronounced the same way" (Lakoff, 1987, p. 416), as in the bank of a river and the bank in which money is deposited. On the other hand, polysemy happens "where there is one lexical item with a family of related senses;" for instance in different but related senses of the words window, run, open, and newspaper ("Do you have yester- day's newspaper?" vs. "I work for the newspaper.") The example, "x 2 + y2 = r 2 may describe a Pythagorean relation or define the property of each point (x, y) belonging to a circle of radius r centered at the origin" (Janvier et al., 1993, p. 85) is an instance of poly- semy because the two senses (the two signifieds) are not unrelated: the Pythagorean rela- tionship underlies the coordinates of points on a circle too.

However, Janvier et al. (1993) have provided an illuminating structure for understand- ing the respective roles of metaphor and metonymy in terms of mathematical homonymy (or polysemy) and synonymy. According to these authors, an external representation may correspond to two (or more) internal representations in the case of homonymy. In synon- ymy, the opposite is the case: two (or more) external representations are associated with

METAPHOR AND METONYMY 31

one internal representation. Thus homonymy and polysemy provide a structure for under- standing metaphor, while synonymy provides a structure for metonymy. The relationships may be thought of as in Figures 2 and 3.

In Figure 3, a generic example was given. It is more likely that this metonymy as syn- onymy would occur in terms of a specific straight line with its diagram and equation (sig- nifiers) and the concomitant concept of a specific straight line (signified). This characterization of metonymy makes it easy to see why Tahta wrote of this in terms of movement along a chain of signifiers, and also why Pimm (1991) considered fluency and automaticity of expression in various forms of symbolism to be important in the learning of mathematics. The metonymy depends on the property that each or any of the signifiers may be a mathematical "object" (Sfard, 1991) which stands for the signified, or for any of the other signifiers, or for both signifiers and signified. This ambiguity is not a disadvan- tage here because it facilitates movement along the chain of signifiers. In this movement, an act of translation is involved (Kaput, 1987, 1991). Fluency is facilitated by being con- versant with various signification systems. A "shift of attention" is involved in the transla- tion (Mason, 1994), and there are concomitant shifts and slides of meaning as attention is centered on different signifiers as the objects of metonymy. I would say, then, that meton- ymy does not consist in movement along the chain of signifiers, but that metonymy facili- tates this process of translation.

Systems of signification are an essential part of mathematics. Understanding of the roles of metaphor and metonymy in this regard may contribute to an epistemology of mathemat- ics, since it appears that these "literary" figures are used by students at school in their learn- ing of mathematics, as well as by practising mathematicians. I agree with Mason (1994) that as in research, it is learning "from the inside" which enables students and mathemati- cians alike to construct the satisfying personal meanings and relationships which cause them to want to continue to do mathematics. Metaphors and metonymies are indispensable in this regard.

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