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reproduction on microﬁlms or in any other physical way. BC Canada Luis Radford École des Sciences de l’Education Gert Kadunz Université Laurentienne Department of Mathematics Sudbury. adapt. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland . recitation.Norma Presmeg Wolff-Michael Roth Department of Mathematics Lansdowne Professor of Applied Cognitive Illinois State University Science Normal. that such names are exempt from the relevant protective laws and regulations and therefore free for general use. All rights are reserved by the Publisher. with respect to the material contained herein or for any errors or omissions that may have been made. Open Access This book is distributed under the terms of the Creative Commons Attribution- NonCommercial 4. and any changes made are indicated. service marks. reuse of illustrations. unless indicated otherwise in the credit line. IL University of Victoria USA Victoria.org/licenses/by-nc/4. reprinting. whether the whole or part of the material is concerned. or by similar or dissimilar methodology now known or hereafter developed. a link is provided to the Creative Commons license. The images or other third party material in this book are included in the work’s Creative Commons license. and reproduction in any medium or format. as long as you give appropriate credit to the original author(s) and the source. Neither the publisher nor the authors or the editors give a warranty. speciﬁcally the rights of translation. users will need to obtain permission from the license holder to duplicate. registered names. or reproduce the material. adaptation. even in the absence of a speciﬁc statement. ON Alpen-Adria Universitaet Klagenfurt Canada Klagenfurt Austria ISSN 2366-5947 ISSN 2366-5955 (electronic) ICME-13 Topical Surveys ISBN 978-3-319-31369-6 ISBN 978-3-319-31370-2 (eBook) DOI 10. the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication.1007/978-3-319-31370-2 Library of Congress Control Number: 2016935590 © The Editor(s) (if applicable) and The Author(s) 2016. express or implied.0 International License (http://creativecommons. The publisher. etc. distribution. This book is published open access. The use of general descriptive names. and transmission or information storage and retrieval. This work is subject to copyright. which per- mits any noncommercial use. in this publi- cation does not imply. if such material is not included in the work’s Creative Commons license and the respective action is not permitted by statutory regulation. computer software.0/). broadcasting. electronic adaptation. duplication. trademarks.
• Influential theories of semiotics. • Various types of signs in mathematics education. • Applications of semiotics in mathematics education. v .Main Topics • Nature of semiotics and its signiﬁcance for mathematics education. • Other dimensions of semiotics in mathematics education.
. .1 The Relationship Among Sign Systems and Translation . . .2 Further Applications of Semiotics in Mathematics Education . . . . . . . . . . . . . . . 1 1. .1 A Summary of Inﬂuential Semiotic Theories and Applications . . . . . . . . . . . . . . .2 Linguistic Theories and Their Relevance in Mathematics Education .. . . . . . . . . . . . . .. . . . . . . .1 Saussure . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3. . 3 2 Semiotics in Theory and Practice in Mathematics Education . . . . . . . . . . . . . . 5 2. . . . . . . . . . . . . . . . . 31 References. . . . .4 Other Dimensions of Semiotics in Mathematics Education.. . . 26 2. . . . . . .. . . . . . . . . . . . . . . . . 22 2. . . . . 28 3 A Summary of Results . . . . . . . . . . . 7 2. . ..4. . . . . . . . . . . . . . . . . . . . . .1. . 5 2. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 5 2. . . .. . . 22 2. .2 Purpose of the Topical Survey on Semiotics in Mathematics Education . .1 The Role of Visualization in Semiosis .3 Semiotics as the Focus of Innovative Learning and Teaching Materials. . . . . . . . . . . .4. . . . . . . .. . . . . . . . . 2 1. . .4. . . . . . . . . . . . . . . .1 Embodiment. . . . . . . . . . .. . 33 vii . 15 2. . . Gestures. . . . . . . . . 28 2. . . . . . . . . . . . . . . . 24 2. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . .. . . . . .1. . . . . .. .3. . . . . . . . . . . . . . . . . . . . . . . . . . . . and the Body in Mathematics Education . . . 26 2. . . . . . . . .. . . . . .Contents 1 Introduction: What Is Semiotics and Why Is It Important for Mathematics Education? . . . . . .1. . . . . . . . . . . . . . . . . 10 2. . . . . . . . . . . . . .2 Semiotics and Intersubjectivity.3 Vygotsky . . . . .3 The Signiﬁcance of Various Types of Signs in Mathematics Education . . . .2 Peirce . .
and more recently. de Freitas and Sinclair 2013. Sáenz-Ludlow and Kadunz 2016).g. Roth 2010). a theory of how signs signify. the American philosopher who originated pragmatism. it is one segmentation of the material continuum in relation to another segmentation (Eco 1986). Saussure 1959. a Swiss linguist generally recog- nized as the founder of contemporary linguistics and the major inspiration for structuralism. Sáenz-Ludlow and Presmeg 2006. semiotics is a general theory of signs or. semiotics has gained the attention of researchers interested in furthering the understanding of processes involved in the learning and teaching of mathematics (see. A sign is “something that stands for something else” (p. The study of signs has long and rich history. 2011. 2003. 179). In addition to these two research traditions. 2014a. a theory of sign-iﬁcation. Semiotics has long been a topic of relevance in connection with language (e. Vygotsky 1997).Chapter 1 Introduction: What Is Semiotics and Why Is It Important for Mathematics Education? Over the last three decades. Components of some of these theories are elaborated in what follows. social semiotic (Halliday 1978). semiotics is a contemporary ﬁeld originally flowing from two independent research traditions: those of C. Semiotics. is “the study or doctrine of signs” (Colapietro 1993.. ICME-13 Topical Surveys. Presmeg. Vergnaud 1985. Peirce to designate any sign action or sign process: in general. Radford 2013a. this use is ubiquitous in every branch of mathematics. It could not be otherwise: the © The Author(s) 2016 1 N. then. theories of embodiment that include gestures and the body as a mode of signiﬁcation (Bautista and Roth 2012.S.g. by Peirce). 2008.g.. as Eco (1988) suggests. DOI 10. p. Semiotics in Mathematics Education.1007/978-3-319-31370-2_1 . several others implicate semiotics either directly or implicitly: these include semiotic mediation (the “early” Vygotsky 1978). But what is semiotics. 2013). 179). e. p. Font et al. However.. de Saussure. the activity of a sign” (Colapietro 1993. various theories of representation (Goldin and Janvier 1998. Radford 2009. relationships amongst sign systems (Duval 1995). Anderson et al. 178). that is. Radford et al. and F. The signiﬁcance of semiosis for mathematics education lies in the use of signs. as a self-conscious and distinct branch of inquiry. Sometimes designated “semeiotic” (e. Peirce. and why is it signiﬁcant for mathematics education? Semiosis is “a term originally used by Charles S.
An elementary example is a drawing of a triangle—which is always a particular case—but which may be used to stand for triangles in general (Radford 2006a). Peirce sometimes used the word “sign” to designate his whole triad. This principle of “seeing an A as a B” (Otte 2006. which 1 A note on terminology: The term “sign vehicle” is used here to designate the signiﬁer. or microscopes. It was Thomas Mitchel’s dictum that the linguistic turn is followed now by a “pictorial turn” or an “iconic turn” (Boehm 1994). “sign vehicle” is used for the representamen/signiﬁer. the mathematical concepts are the result of the continuing reﬁnement of physical objects Greek craftsmen were able to produce (Husserl 1939). 2006a. general in nature. Thus semiotics. and to represent them—to others and to oneself—and to work with them. which allow us to see nearly inﬁnite distant objects. The con- centration on visualisation in cultural sciences is based on their interest in the ﬁeld of visual arts and it is still increasing (Bachmann-Medick 2009). In this regard images became a major factor within epistemology. craftsmen were producing rolling things called in Greek kulindros (roller). As long as these structures were not visible we could only speculate about them. To avoid confusion. a limit object that does not bear any of the imperfections that a material object will have. Such new developments. 1.1 which are not the mathematical objects themselves but stand for them in some way. Radford 2002a). when the object is the signiﬁed. Wartofsky 1968) is by no means straightforward and directly affects the learning processes of mathematics at all levels (Presmeg 1992. Children’s real problems are in moving from the material things they use in their mathematic classes to the mathematical things (Roth 2011). For example. We can say that their ontological status has changed. medical imaging allows us to see what formerly was invisible. now we can debate about them and about their existence.1 The Role of Visualization in Semiosis The sign vehicles that are used in mathematics and its teaching and learning are often of a visual nature (Presmeg 1985. 2014). As a text on the origin of (Euclidean) geometry suggests. it is necessary to employ sign vehicles. The signiﬁcance of semiosis for mathematics education can also be seen in the growing interest of the use of images within cultural science. object [signiﬁed]-representamen [signiﬁer]-interpretant. has the potential to serve as a powerful theoretical lens in investigating diverse topics in mathematics education research. . but sometimes Peirce used the word “sign” in designating the representamen only. Other examples could be modern telescopes. in several traditional frameworks.2 1 Introduction: What Is Semiotics and Why Is It Important … objects of mathematics are ideal. With the help of these machines such tiny structures become visible and with this kind of visibility they became a part of the scientiﬁc debate. which led to the mathematical notion of the cylinder. which bring the inﬁnitely small to our eyes. For example. But more inter- esting for our view on visualisation are developments within science which have introduced very sophisticated methods for constructing new images.
126) and “Rhetoric of the image” (p. The introduction to “Logik des Bildlichen” (Hessler and Mersch 2009). focusses on the meaning of visual thinking. In this book. 1. Even in the theory of organizations.. e. Among these questions we read: epistemology and images. (3) The signiﬁcance of various types of signs in mathematics education. the order of demonstrating or how to make thinking visible. And again theoretical approaches from semiotics are used to interpret empirical data: In “Powell’s point: Denial and deception at the UN. (2) Further applications of semiotics in mathematics education. The anthology The visual culture reader (Mirzoeff 2002) presents in its theory chapter “Plug-in theory. perspectives and issues that have been the focus of research in mathematics education are presented. and to open thought to the potential for . Mitchell 1987. caused substantial endeavour within cultural science into investigating the use of images from many different perspectives (see. charts and diagrams. which we can translate as “The Logic of the Pictorial”.” the work of several researchers well known for their texts on semiotics. semiotics is used as means for structuring: In his book on Visual culture in organizations Styhre (2010) presents semiotics as one of his main theoretical formulations. including Jaques Lacan and Roland Barthes.” Finn makes extensive use of semiotic theories. There are four broad over- lapping subheadings: (1) A summary of influential semiotic theories and applications. (4) Other dimensions of semiotics in mathematics education. Arnheim 1969. Let’s take a further look at a few examples of relevant literature from cultural science concentrating on the “visual. they for- mulate several relevant questions on visualisation which could/should be answered by a science of images.1. Within each of these sections. They show in detail the role of diagrams as means to construct knowledge and interpret data and equations.1 The Role of Visualization in Semiosis 3 can only be hinted at here. 135).” In their book The culture of diagram (Bender and Marrinan 2010) the authors investigate the interplay between words. images in action (Finn 2012) devotes the fourth chapter to questions which concentrate on maps. Visual communi- cation and culture. and formulas with the result that diagrams appear to be valuable tools to understand this interplay. Another relevant anthology. pictures. The structure of the next section is as follows. to give an introduction to what has already been accomplished in this ﬁeld. with their respective texts “What is a picture?” (p. the purpose of this Topical Survey is to explore the signiﬁcance—for research and practice—of semiotics for understanding issues in the teaching and learning of mathematics at all levels.g.2 Purpose of the Topical Survey on Semiotics in Mathematics Education Resonating with the importance of semiotics in the foregoing areas. Hessler and Mersch 2009).
Open Access This chapter is distributed under the terms of the Creative Commons Attribution- NonCommercial 4.4 1 Introduction: What Is Semiotics and Why Is It Important … further developments. users will need to obtain permission from the license holder to duplicate. or reproduce the material. if such material is not included in the work’s Creative Commons license and the respective action is not permitted by statutory regulation. for greater depth and detail.0/). . as long as you give appropriate credit to the original author(s) and the source. adapt. which cannot be fully comprehensive. distribution and reproduction in any medium or format.org/licenses/by-nc/4. duplication. and interested readers are encouraged to read original papers cited. adaptation. unless indicated otherwise in the credit line. The images or other third party material in this chapter are included in the work’s Creative Commons license. a link is provided to the Creative Commons license and any changes made are indicated.0 International License (http://creativecommons. This Survey is thus an introduction. which permits any noncommercial use.
Semiotics in Mathematics Education. a linguistic sign is the result of coupling two elements.1. Because these differ in a signiﬁcant aspect—a three-fold relation in the case of the former. In both instances. a two-fold relation in the case of the latter—Peirce’s version goes under semiotics. In the ﬁrst. These signiﬁers are arbitrary. or even an intonation. in a close. It is noteworthy that both components in this dyad are psychological1: the acoustic image is a psychological pattern of a sound. a concept and an acoustic image. a phrase. the Latin word arbor [tree] (on the bottom) and the French «arbre» [tree] (on top) form a sign. whereas de Saussure’s version often is referred to as semiology.1007/978-3-319-31370-2_2 . In the second diagram. ICME-13 Topical Surveys. the adjective psychological is the better choice because it allows for bodily knowing that is not mental in kind (e. insep- arable relationship (metaphorically..1 A Summary of Influential Semiotic Theories and Applications Both Peirce and de Saussure developed theories dealing with signs and signiﬁca- tion. in 1 De Saussure uses the French psychique [psychical] rather than mental. where the former is the signiﬁer and the latter the signiﬁed. Roth 2016b). 2. Ferdinand de Saussure’s (1959) semiology was developed in the context of his structural theory of general linguistics. arbor is retained as the signiﬁer but the drawing of a tree takes the place of the signiﬁed. which could be a word. Presmeg. To anticipate ambiguities de Saussure proposed to understand the sign as the relation of a signiﬁed and a signiﬁer.Chapter 2 Semiotics in Theory and Practice in Mathematics Education 2. In this theory. like the two sides of a single piece of paper. as he suggests).g. He uses two now classical diagrams to exemplify the sign. just as Vygotsky will use psixičeskij [psychical] rather than duxovnyj [mental]. © The Author(s) 2016 5 N.1 Saussure The basic ideas of this semiotic theory are as follows. DOI 10.
This theory has applications in mathematics education. Lacan (1966) had inverted this relationship. and also became important in Presmeg’s research in the 1990s using chains of signiﬁcation to connect cultural practices of students. This version of semiology was used by Walkerdine (1988). with the canonical mathematical ideas from the syllabuses used by teachers of classroom mathematics (Presmeg 1997).6 2 Semiotics in Theory and Practice in Mathematics Education the sense that there is no logical necessity underlying them—which accounts for humanity’s many languages—but they are not the product of whim because they are socially determined. The dyadic model of Saussure proved inadequate to account for the results of Presmeg’s research. the signiﬁed. but particularly in how ideas change—in the processes involved as stu- dents engage over time with mathematical objects (diachrony). p. which emphasizes that “the signiﬁer does not mark a thing” but “marks a point of pure difference or movement in a discursive chain” (Brown 2011. Whitson pointed out that for Saussure. and of Vygotsky (in his earlier notion of semiotic mediation). although there was interplay between the signiﬁed and signiﬁer (denoted by arrows in both directions in his diagrams). The synchronic view is a snapshot in time. As Fried (2007. A useful botanical metaphor is that syn- chrony refers to a cross-section of a plant stem. as the top element of the dyad. hence both of these distinct viewpoints are useful in semiotic analyses. The Lacanian version also is central to a recent conceptualization of subjectivity in mathematics education. sign vehicles play a signiﬁcant role in standing for mathematical objects. in a chapter titled “Cognition as a semiosic process: From situated mediation to critical reflective transcendence” (Whitson 1997). placing the signiﬁer on top of the signiﬁed. while diachrony takes a longitudinal section. . appeared to dominate the signiﬁer. de Saussure’s notions of synchronicity and diachronicity are particularly useful in clarifying ways of looking at both the history of mathematics. but there are aspects of Saussure’s theory that are highly signiﬁcant. and both are necessary for a full under- standing of a phenomenon (Fried 2007). In both the synchronic and diachronic views. in a series of steps. and was later replaced by a Peircean nested model that invoked the interpretant (Presmeg 1998. 2008) points out. 112). Saussure’s ideas were brought to the attention of the mathematics education community in the 1990s in a keynote presentation by Whitson (1994). The theoretical ideas of de Saussure have not been used as extensively in mathematics education research as those of Peirce. 2006b). where one sign–referent relation replaces another sign–referent relation leading to inﬁnite (unlimited) semiosis (Nöth 1990). In mathematics education we are interested not only in understanding what is taught and learned in a given situation (syn- chrony). creating a chain of signiﬁers that never really attain the signiﬁed. while a dia- chronic analysis is a longitudinal one. This movement from signiﬁer to signiﬁer creates an effect similar to the interpretant in Peircean semiotics. and the processes involved in teaching and learning mathematics. These views are complementary. and by Kirshner and Whitson in the context of a book on situated cognition.
1. p. and so determines an effect upon a person. Peirce developed several typologies of signs. I recognize three Universes. it may be useful to look at some of Peirce’s examples. that the latter is thereby mediately determined by the former. thus effectively creating different signs for the same object. indexical.1 A Summary of Inﬂuential Semiotic Theories and Applications 7 2. a photograph of a person representing the actual person. (p. or a sign-post pointing to a road. and every higher number can be formed by mere complications of threes. indexical. In an iconic sign. why stop at three?” he wrote (Peirce 1992. 251) Accordingly. or symbolic respectively. and symbolic signs. I deﬁne a Sign as anything which is so determined by something else. which are distinguished by three Modalities of Being. but also in the types of each of these components. the distinctions may be useful to a researcher or teacher for the purpose of identifying the subtlety of a learner’s mathematical conceptions if differences in interpretation are taken into account.. The relationship leads to three kinds of signs: iconic.g. 1908. without introducing something of a different nature from the unit and the pair. which effect I call its Interpretant. Maybe the best known typology is the one based on the kind of relationship between a sign vehicle and its object. e.g. The nature of symbolic signs is that there is an element of convention in relating a particular sign vehicle to its object (e.. In a letter to Lady Welby on December 23. but depend on the interpretations of their con- stituent relationships between sign vehicles and objects. smoke invoking the interpretation that there is ﬁre. “But it will be asked.g. In practice the distinctions are subtle because they depend on the interpretations of the learner—and therefore. e. (Peirce 1998. . p. indexical. viewed in this way. By his own admission. he used triads not only in his semiotic model including object. and his reply to the question is as follows: [W]hile it is impossible to form a genuine three by any modiﬁcation of the pair. tri- chotomic is the art of making three-fold divisions. four. the sign vehicle and the object share a physical resemblance. 251). To illustrate the differences among iconic. and in- terpretant. These types are not inherent in the signs themselves. because I despair of making my own broader conception understood. These distinctions in mathematical signs are complicated by the fact that three different people may categorize the ‘same’ relationship between a sign vehicle and its object in such a way that it is iconic. 478) It follows that different individuals may construct different interpretants from the same sign vehicle.2 Peirce The basic ideas of this theory are as follows. According to Peirce (1992). called its Object. algebraic symbolism). according to their interpretations. ﬁve. and symbolic. which stands for the object in some way. he wrote as follows.. My insertion of “upon a person” is a sop to Cerberus. Signs are indexical if there is some physical connection between sign vehicle and object.2. he showed a proclivity for the number three in his philosophical thinking. representamen [sign vehicle].
secondness. and which I cannot think away. positioned. We become aware of things because we are able to recognize their own quale. secondness. Peirce argued that there are three and only three categories: ‘He claims that he has look[ed] long and hard to disprove his doctrine of three categories but that he has never found anything to contradict it. However. Rather. the former deals with the nature of being and the latter with the phenomenon of conscious experience. (Sáenz-Ludlow and Kadunz 2016. then—as in teaching—there are three kinds of interpretant. A hard fact is of the same sort. and thirdness as either ontological or as phenomenological categories Sáens-Ludlow and Kadunz (2016) mention the following: Peirce’s semiotics is founded on his three connected categories. The existence of these three categories has been called Peirce’s theorem. It has to do with the qualia of the thing. p. Commenting on the sub- tleties of the interrelationships amongst ﬁrstness. p. p. as it were. and thirdness. as follows: . In an act of communication. it results from a linkage between facts. Qualia—such as bitter. It is a moment of actuality or occurrence. it is something which is there. 4) Peirce’s model includes the need for expression or communication: “Expression is a kind of representation or signiﬁcation. A quale is the distinctive mark of something. we enter secondness: We ﬁnd secondness in occurrence. This new level (thirdness) requires the use of symbols. but am forced to acknowledge as an object or second beside myself. making it possible to note that something is there.418)—account hence for the possibility of expe- rience. Here. Now. Firstness has to do with that which makes possible the recognizance of something as it appears in the phenomenological realm. 44). and he extends to everyone the invitation to do the same’ (de Waal 2013. noble (CP. (Peirce CP 1. because an occurrence is something whose existence consists in our knocking up against it.25). requires us to enter into a level that goes beyond quality (ﬁrstness) and factuality (secondness).224). Thus. “Each quale is in itself what it is for itself. “The mode of being a redness. the subject or number one. without reference to any other” (Peirce CP 6.… He considers these categories to be both ontological and phenomenological. what allows us to perceive a red rose is the quality of redness.8 2 Semiotics in Theory and Practice in Mathematics Education Peirce also introduced three conceptual categories that he termed ﬁrstness.358) Because we have reached awareness. the very eruption of the object into our ﬁeld of perception marks the indexical moment of consciousness. 1. which can be differentiated from each other. and which forms material for the exercise of my will. hard. tedious. the object now becomes an object of knowledge. 281). But knowledge is not an array of isolated facts or events. quale is not perception yet. before anything in the universe was yet red. A sign is a third mediating between the mind addressed and the object represented” (Peirce 1992. and this link. Were we to be left without qualia. that is to say. It is its mere possibility: it is ﬁrstness—the ﬁrst category of being in Peirce’s account. Peirce argues. and which cannot be reduced to one another. we would not be able to perceive anything. regardless of something else (it is its suchness). heartrending. in the boundaries of consciousness (Radford 2008a). was never- theless a positive qualitative possibility” (CP 1.
The results may be understood in terms of the deﬁnition of the sign as relation between two segmentations of the material continuum (Eco 1986). which is a determination of the mind of the utterer”. and symbolic sign vehicles.1 A Summary of Inﬂuential Semiotic Theories and Applications 9 • the “Intensional Interpretant. The commens proved to be an illuminating lens in examining the history of geometry (Presmeg 2003).g. The Peircean approach also was central to a study of how professionals. the study pointed out that the signs did not just exist. many of the 54 students interviewed reported spontaneously that they remembered this formula by an image of its shape. the interpreted relationship of this inscription with its mathematical object may be characterized as symbolic. However.. and • the “Communicational Interpretant. read graphs (Roth and Bowen 2001).2 ¼ : 2a Because symbols are used. these needed to emerge from the interpretive activity before they could be related to a biological phenomenon. Applications in mathematics education are as follows. which is a determination of the mind of the inter- preter”. b. depending on the way the inscription is interpreted. In this sense the formula is indexical. p. the value of a function or its slope at a certain value of the abscissa) were taken as a sign that referred to some biological phenomenon. indexical. Importantly. and c in order to solve the equation. Instead. the formula is also commonly interpreted as a pointer (cf. the sign could also be characterized as iconic or indexical. As an example. let us examine the quadratic formula in terms of the triad of iconic. an iconic property. The roots of the equation ax2 þ bx þ c ¼ 0 are given by the well-known formula pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ b  b2  4ac x1. involving convention. The complexity and subtlety of Peirce’s notions result in opportunities for their use in a wide variety of research studies in mathematics education. his emphasis) It is the latter fused mind that Peirce designated the commens. In that study. or symbolic depends on the interpretant of the sign.” (Peirce 1998. such as changes in population size. a direction sign on a road): it is a directive to perform the action of substituting values for the variables a. However. which is a determination of that mind into which the minds of utterer and interpreter have to be fused in order that any communication should take place. or say the Cominterpretant. As a study of the transformations within a scientiﬁc research group shows. The formula involves spatial shape. sci- entists and technicians. Thus whether the sign vehicle of the formula is classiﬁed as iconic. In Presmeg’s (1985) original research study of visualization in high school mathe- matics. not the material matters to signiﬁcation but the form of this . certain aspects of graphs (e. • the “Effectual Interpretant. indexical. The phenomenological classiﬁcation is of importance.2. 478.
thinking. and the rise of Stalin to the top of Soviet political leadership. In a paper from . The central subject of the book suggests a psychology oriented to essential human ques- tions. each one marking different emphases that cannot be attributed to a premeditated clear intention: Differing emphases that characterize moments in Vygotsky’s work did not come about purely as a result of clear intentions. It is in the second moment that we ﬁnd Vygotsky elaborating his concept of sign. (González Rey 2011b. irreducible to behavior or to an objectivistic view of human beings … in Psychology of Art. etc. during which the world saw the succession of the Russian Revolution.10 2 Semiotics in Theory and Practice in Mathematics Education material (Latour 1993). The main work of Vygotsky’s ﬁrst moment is his 1925 book The psychology of art (Vygotsky 1971). p.) from dif- ferent angles. p. Vygotsky’s concept of sign was influenced by his work on special education (Vygotsky 1993). emotions. Roth et al. instead.3 Vygotsky Basic ideas Vygotsky’s writings spanned a short period of time (from 1915 to 1934). Vygotsky’s focus here is on the active character of the mind. 2002).1. interpretation is not observed. users see right through the sign as if it were transparent thereby giving access to the phenomenon itself (Roth 2003a. During this period. 258) The ﬁrst moment covers approximately from 1915 to 1928. special edu- cation.” Yasnitsky (2011) identiﬁes three main interrelated domains of research that occupied the “Vygotsky circle” (the circle of Vygotsky and his collaborators): (a) clinical and special education studies. as an individual whose psychical processes have a cultural-historical genesis. Roth and Bowen 2003. in reading. will. Those moments were also influenced by the effects of the turbulent epoch during which his writings were brought to life. Contemporary Vygotskian scholars suggest a rough division of Vygotsky’s work in terms of domains and moments. (González Rey 2011b. In the case of familiar signs that appear in familiar cir- cumstances. and action. cultural child development. and culture). and (c) studies around affect. in his article “The Vygotsky that we (do not) know. the First World War. emotions and phantasy. (b) philological studies (covering problems of language. 259) The second moment goes roughly from 1927 to 1931. cognitive functions. the basis was created for a psychology capable of studying the human person in all her complexity. Vygotsky tackled different problems (creative thinking. 2. González Rey (2011a) suggests an approach to the understanding of Vygotsky’s work in terms of three moments. Taking a critical stance towards the current chronology of Vygotsky’s works.
g. The transformation of the human mind that signs effectuate is related to their social-cultural-historical role. Thus. “The inclusion in any process of a sign. in principle. This is the idea behind Vygotsky’s famous genetic law of cultural development. Speech is at ﬁrst a means of contact between the child and the surrounding people. and later. as an inter-psychological function.2. in a paper read at the Institute of Scientiﬁc Pedagogy at Moscow State University on April 28. a special system of cultural signs and symbols” (p. the manner in which signs alter the human mind is not related to signs qua signs. (p. because it is too commonplace and we are therefore blind to it. but the deaf or blind child achieves the goals of a normal child by different means and by a different path” (p. That is. It is from here that Vygotsky developed the idea of the sign both as a psychological tool and as a cultural mediator. Vygotsky (1993) argued that “A child learns to use certain signs functionally as a means to fulﬁlling some psychological operation or other. elementary and primitive forms of behavior become mediated cultural acts and processes” (p. As a result. are not merely aids to carry out a task or to solve a problem. and a blind person will never master writing. be equated with a normal child. And how does education do it? Vygotsky’s answer is: by “creating artiﬁcial. hence. 296). but when the child begins to speak to himself. signs are not characterized by their repre- sentational nature. “remodels the whole structure of psychological operations just as the inclusion of a tool remodels the whole structure of a labor operation” (Vygotsky 1929. Signs are rather characterized by their functional role: as external or material means of regulation and self-control. they alter the way children come to know about the world and about themselves. We do not notice this fact. 95) . The most striking example is speech. cultural techniques. Signs serve to fulﬁll psychological operations (Radford and Sabena 2015).. Signs. then as an intra-psychological function. 421). a blind or deaf child may. in Vygotsky’s account.1 A Summary of Inﬂuential Semiotic Theories and Applications 11 1929 he stated that “From a pedagogical point of view. on the individual level” (Vygotsky 1978. it depends on how signs signify and are used collectively in society. which he presented as follows: “Every [psychic] function in the child’s cultural development appears twice: ﬁrst. The special child may achieve her goal in interaction with other individuals. a deaf-mute child will never learn speech. as a certain way of behaving. Braille dots) compensate for differences in the child’s sensorial organization. 60).” he noted. This two-fold idea of signs allowed him to account for the nature of what he termed the higher psychological functions (which include memory and perception) and to tackle the question of child development from a cultural viewpoint. Vygotsky thought of these compensating means as signs. p. Commenting on this idea. Vygotsky (1997) offered the example of language: When we studied the processes of the higher functions in children we came to the following staggering conclusion: each higher form of behavior enters the scene twice in its devel- opment—ﬁrst as a collective form of behavior. that is. “Left to himself [sic] and to his own natural development. In other words. By becoming included in the children’s activities. 1928. this can be regarded as the trans- ference of a collective form of behavior into the practice of personal behavior. However. on the social level. auxiliary material cultural means (e. 168). 57). p. Thus. In this case education comes to the rescue” (p. 168).
his hands. Consequently.” that is. . Only later. (p. Instead. this gesture is nothing more than an unsuccessful attempt to grasp something. remain poised in the air. which still represents a very objectivistic approach to the comprehension of the psyche. a turn in which the subjective dimension that was at the heart of Vygotsky’s ﬁrst moment shades away to yield room to the study of “internalization of prior external processes and operations” (p. emotions. 64) In the third moment (roughly located during the period from 1932 to 1934). He continues: Vygotsky explained the transition from intermental to intra-mental. Roth 2016b). he pointed out that the genetic origin of all higher psychological functions was a soci(et)al relation (Vygotsky 1989). When the mother comes to the child’s aid and realizes his movement indicates something. and problems of personal experience to an instrumental investigation of higher psychological functions. it is when s/he assumes all parts of the relation that the higher function can be ascribed to the individual (e. This instru- mental investigation revolved around the notion of signs as a tool and the con- comitant idea of semiotic mediation. Pointing becomes a gesture for others. p. Vygotsky returned with new vigor to some ideas of the ﬁrst moment. His ﬁngers make grasping movements. the soci(et)al relation itself is the higher function. personality. Initially.12 2 Semiotics in Theory and Practice in Mathematics Education To account for the process that leads from a collective form of behavior to an intra-psychological function Vygotsky introduced the concept of internalization. At this initial stage pointing is represented by the child’s movement. a movement aimed at a certain object which designates forthcoming activity. Moving away from the mechanist or instrumental turn of the second period. 56) To sum up.. He did not write that there was something in the relation that then was transferred mys- teriously into the person. through internalization. This comprehension of that process does not lend a gener- ative character to the mind as a system. the situation changes fundamentally. which seems to be pointing to an object—that and nothing more. 63). does he begin to understand this movement as pointing. He wrote: “We call the internal reconstruction of an external operation internal- ization” (Vygotsky 1978. and the interrelationship of social context and subjective experience. the developing individual already contributes to the realization of the higher function. a speciﬁcally psychical ﬁeld. The child’s unsuccessful attempt engenders a reaction not from the object he seeks but from another person. such as the unity between cognition and emotion. During those years. recognizing it only as an internal expression of a formerly inter-mental process. To illustrate the idea of internalization Vygotsky (1978) provided the example of pointing gestures: A good example of this process may be found in the development of pointing. That is. in the second moment of Vygotsky’ work there is a shift from imagination. when the child can link his unsuccessful grasping movement to the objective situation as a whole. The child attempts to grasp an object placed beyond his reach. González Rey (2009) qualiﬁes this moment as an instrumentalist “objectivist turn. 56). (p. phantasy. questions of the generative power of the mind that we ﬁnd in his study of Hamlet came to the fore again (Vygotsky 1971). stretched toward that object. Several Soviet psychologists also criticized the concept of internalization in different periods.g. the primary meaning of that unsuccessful grasping movement is established by others.
Based on the idea of inner contradictions. the Cartesian division between body and mind or psychic-physical parallelism (Mikhailov 2004). we read: “the problem of meaning was already present in [our] older investigations. 27. for all of these things are subjectively ‘everyone’s’” (Mikhailov 2001. but. not the signs in themselves. as often assumed. The very notion of a mediator is the result of. Even if a person writes into a diary. but is a reality for two. signs generally and language speciﬁcally—generally theorized as the medi- ators between subject and material world or between two subjects—“are given to the child not as an ensemble of mediators between the child and nature. Leontyev 1981). The problem of meaning is tackled again in his later work. subjectivity is the result of participation in relations with others. To overcome the dangers of the split between body and mind. Mikhailov 2006). s/he still is relating to herself as to another. or gives rise to. One indication of this move is noticeable at the very end of the posthumously published Thinking and speech (Vygotsky 1987). Vygotsky was turning to a Spinozist idea. where he notes that the word is impossible for an individual. now our task is to demonstrate the difference that exists between them” (pp. relations that take place in a semiotic ﬁeld. To do so. in the second period.2. We should not think for instance that. where material bodies and culture (mind) are but two (contradictory) manifestations of one substance. Some Russian scholars in the cultural-historical tradition now suggest that towards the very end of his life. 85). Vygotsky was moving away from the idea of sign mediation. We may change the signs but the meaning will be preserved” (p. Applications to mathematics education Vygotsky’s work has inspired mathematics education researchers interested in the question of teaching and learning. signs are strictly thought of as mediators per se. p. underline added).1 A Summary of Inﬂuential Semiotic Theories and Applications 13 Although the aforementioned moments are relevant in the understanding of Vygotsky’s ideas and in particular the understanding of Vygotsky’s semiotics. Already in his work on special education Vygotsky (1993) noted that “Meaning is what is important. this time in the context of a communicative ﬁeld that is common to the participants in a relation (Roth 2016a). original emphasis. Thus. El’konin 1994. it is a modality of the semiotic speech ﬁeld. Marxist psychologists have shown a possible evolutionary and cultural-historical trajectory that led from the ﬁrst cell to the human psyche of today.g. including its languages and tools (Holzkamp 1983. we should not think that the problems that Vygotsky tackled were marked differently from one moment to the other. In some notes from an internal seminar in 1933—hence a short time before Vygotsky’ death—a seminar in which Vygotsky (1997) summarized his group’s accomplishments and new research avenues.. Whereas before our task was to demonstrate what ‘the knot’ and logical memory have in common. as subjectively his own. These moments may be understood in terms of focus. developing instead the idea of a semiotic or intersubjective speech ﬁeld (e. Arzarello and his collaborators have been interested in the evolution of signs. in fact. they have developed the theoretical . 130–131). This insight implies that intersubjectivity is not problematic. they were associated with meaning too. Instead.
2009). Arzarello et al. 754). Bartolini Bussi and Mariotti (2008) have focused on the concept of semiotic mediation. Within this perspective. something that is mediated. and drawing systems. Arzarello et al. 752) Within this context. Possibly the teacher too participates to this production and so the semiotic bundle may include also the signs produced by the teacher. in particular in the case of artifacts and signs.14 2 Semiotics in Theory and Practice in Mathematics Education construct of semiotic bundle (Arzarello 2006. and considering them as semiotic resources in teaching and learning pro- cesses. 2009. in particular related to the artifact used. Duval 2006. and not just because of its being mainly linguistic—the feature that. in Bartolini Bussi and Mariotti’s (2008) account.’s work is located within a broader context of multimodality that they explain as coming from neuroscience studies that have highlighted the role of the brain’s sensory-motor system in conceptual knowledge. She wrote as follows: Human communication is special. and also from communication and multiple modes to communicate and to express meanings. on the other hand. This notion encompasses signs and semiotic systems such as the contemporary mathematics sign systems of algebra. seems to be extremely rare. (Arzarello et al. Typically. but also gestures. Arzarello and collaborators track the students’ learning through the evolution of signs in semiotic bundles. On the one hand. p.g. 100) Paying attention to a wide variety of means of expression. and. Anna Sfard (2008) has also drawn on Vygotsky in her research on thinking. from the standard algebraic or other mathematical symbols to the embodied ones. a semiotic bundle is made of the signs that are produced by a student or by a group of students while solving a problem and/or discussing a mathematical question. writing. and the circumstances for mediation (Hasan 2002). which she conceives of as the individualized form of interpersonal communication. speaking. “any artifact will be referred to as tool of semiotic mediation as long as it is (or it is conceived to be) intentionally used by the teacher to mediate a mathematical content through a designed didactical intervention” (p. It rests on the truly recognizable relationship between particular artifacts and particular signs (or system of signs) directly originated by them. like gestures and gazes. they may be related to the content that is to be mediated … Hence. Mediation involves four terms—someone who mediates (the mediator). Cartesian graphs. the concept of semiotic bundle goes beyond the range of semiotic resources that are traditionally discussed in mathematics education literature (e. Semiotic mediation. if not lacking altogether. In their seminal paper they distinguished between mediation and semiotic mediation. appears as a particular case of mediation: Within the social use of artifacts in the accomplishment of a task (that involves both the mediator and the mediatees) shared signs are generated. the link between artifacts and signs overcomes the pure analogy in their functioning in mediating human action. in animals. a semiotic bundle is deﬁned as a system of signs […] that is produced by one or more interacting subjects and that evolves in time. some- one or something subjected to the mediation (the mediatee). The ability to coordinate our activities by means of interpersonal communication is the basis for our being social . these signs are related to the accomplishment of the task. (p. Ernest 2006). It is the role communication plays in human life that seems unique..
p. she deﬁnes thinking as follows: “Thinking is an individ- ualized version of (interpersonal) communicating” (p.2 Further Applications of Semiotics in Mathematics Education The summary of influential semiotic theories conducted in the previous section provided an idea of the impact of these theories in mathematics education. Such a conception of signs and artefacts is consubstantial with the conception of the dialectical materialist idea of activity. 90). and one in which signs are conceptualized as mediating tools (Radford 2014b). and the narratives that are being constructed and labelled as “true” or “cor- rect.” (Sfard 2010. We commu- nicate in order to coordinate our actions and ascertain the kind of mutuality that provides us with what we need and cannot attain single-handedly. b). in which signs are essentially representation devices. p. p. They include “artifacts produced specially for the sake of communication (p. which are “perceptually accessible objects with the help of which the actor performs her prompting action and the re-actor is being prompted” (Sfard 2008. depending on how signs are conceptualized: a representational one. and material culture in general that is featured in the theory of objectiﬁcation (Radford 2006b. Activity (Tätigkeit in German and deyatel’nost’ . This is the approach to signs. nor do they mediate it (Radford 2012). 26). The latter line of thinking reduces activity to a functional conception: activity amounts to the deeds and doings of the individuals. two different approaches can be distinguished within semiotics. visual mediators and the ways they are operated upon. 2013b. As previously mentioned. 2. 81) From this viewpoint. 2008b.2. routine ways of doing things. 2015a. learning mathematics means changing forms of communication. yet they do not represent knowledge. This conception of activity is very different from usual conceptions that reduce activity to a series of actions that an individual performs in the attainment of his or her goal. artifacts. 90).1 A Summary of Inﬂuential Semiotic Theories and Applications 15 creatures. An important role is ascribed to communication mediators. There is still a third approach— a dialectical materialist one—in which signs and artefacts are a fundamental part of mathematical activity. More precisely. 81). (p. that is. Within this context. In this section we discuss in more detail the impact that semiotics has had in speciﬁc problems of mathematics teaching and learning. 217) In the next section we turn to semiotics in mathematics education. Sfard conceptualizes learning as changes in discourse. “as one’s com- munication with oneself” (Sfard 2001. even our ability to stay alive is a function of our communicational capacity. The change may occur in any of the characteristics with the help of which one can tell one discourse from another: words and their use. Activity in the theory of objectiﬁcation does not merely mean to do something. And because communication is the glue that holds human collectives together.
As mentioned before. as being simply busy with something (Roth and Radford 2011). Activity as Tätigkeit does not have the utilitarian and selﬁsh stance that it has come to have in capitalist societies. artefacts. The semiotic dimension of the theory of objectiﬁcation is apparent at different levels: (1) The ﬁrst one is the level of the material culture (signs. It seeks to study the manners by which the students become progressively aware of historically and culturally constituted forms of thinking and acting. and (2) deﬁnite forms of material and spiritual production. rational development. the fact that signs and artifacts are bearers of human intelligence does not mean that such an intelligence is transparent for the student who resorts to them. the students are conceived of as encountering and becoming gradually aware of culturally and historically constituted forms of mathematics thinking. Activity as Tätigkeit is the endless process through which individuals inscribe themselves in society. It allows one to see teaching and learning activity not as two separate activities. 55). that is. indeed. Activity as Tätigkeit should not be confounded with activity as Aktivität/ aktivnost’. signs and artefacts are not considered representational devices or aiding tools. in a deﬁnite way. and aesthetic enjoyment” (Donham 1999. they are bearers of human intelligence and speciﬁc historical forms of human production that affect. Activity as Tätigkeit is a social form of joint action through which individuals produce their means of subsistence and “comprises notions of self-expression. and how. hear with our ears. in this theory. etc. The theory of objectiﬁca- tion is an attempt to understand learning not as the result of the individual student’s deeds (as in individualist accounts of learning) but as a cultural-historical situated processes of knowing and becoming. (2) The second one is a suprastructural level of cultural meanings that shape and organize joint labor. The concept of joint labor allows one to revisit classroom teaching and learning activity. Now. or perceive with our eyes. More precisely. the manner in which we come to know about the world. But neither are they considered as the mere stuff that we touch with our hands. Leont’ev (1968) notes: . it is a form of life. The concept of joint labor is central to the theory of objectiﬁcation: It is.). one carried out by the teacher (the teacher’s activity) and another one carried out by the student (the student’s activity).16 2 Semiotics in Theory and Practice in Mathematics Education in Russian) refers to a dynamic system geared to the satisfaction of collective needs that rests on: (1) speciﬁc forms of human collaboration. To avoid confusions with other meanings. teachers and students position themselves in mathematical practices. They are considered as bearers of sedimented human labor. through joint labor that. but as a single and same activity: the same teachers-and-students joint labor. The joint-labor bounded encounters with the historical forms of mathe- matics thinking are termed processes of objectiﬁcation. as subjectivities in the making. Activity as Tätigkeit is termed joint labor in the theory of objectiﬁcation. p. That is.
3). These answers suggest that the students were focusing on numerosity. The sufﬁx–tiﬁcation comes from the verb facere meaning “to do” or “to make” (p.2 shows two paradigmatic answers provided by two students: Carlos and James. 2. of research studies in this paradigm. and so on. such as Terms 12 or 25. the following is one example.) objects. The students worked by themselves more than 30 min. rhythm. 2. 311). they become a central part of the processes through which students encounter culturally and historically constituted forms of thinking and acting. speech. so that in its etymology. Figure 2. gestures. objectiﬁcation becomes related to those actions aimed at bringing or throwing something in front of somebody or at making something an object of awareness or consciousness (Radford 2003). Radford (2010a) discusses an example of pattern generalization in which Grade 2 seven-to-eight-year-old students were invited to draw Terms 5 and 6 of the sequence shown in Fig. she engaged them in an exploration of the patterns in which a spatial structure came to the fore: to see the terms as made up of two rows (see Fig. The history should restart from the beginning. but there would be no one who would reveal their use to the young generations. (p. The machines would be idle.1 The ﬁrst terms of a sequence that Grade 2 students investigated in an algebra lesson . the corporeal position of the students and the teacher. Such a strategy may prove difﬁcult to answer questions about remote terms. 550). All the semiotic resources that students mobilize in order to become aware of such historical forms of thinking and action are termed semiotic means of objectiﬁcation (Radford 2002a. An example of research In order to show the pragmatic implications of the theoretical ideas presented in the forgoing. 2. signs and artefacts have to become an integral part of joint labor. 29) To fulﬁl their function and to release the historical intelligence embedded in them. alphanumeric formulas and sentences. In doing so. in more detail. Semiotic means of objectiﬁcation may include material mathe- matical signs (e. whose origin derives from the Latin verb obiectare.2. p. to throw before” (Charleton 1996. the books would not be read. written language. meaning “to throw something in the way. When the teacher came to see the students.2 Further Applications of Semiotics in Mathematics Education 17 If a catastrophe would happen to our planet so that only small children would survive. but the history of humanity would inevitably be inter- rupted. The treasures of material culture would continue to exist. perceptual activity.g. The term objectiﬁcation has its ancestor in the word object. 2003). etc. which was in fact the case in this classroom. artistic productions would lose their aesthetic function. graphs.1. the human race would not disappear. Term 1 Term 2 Term 3 Term 4 Fig.
3. it . body position. each one going from the bottom row of Term 1 to the bottom row of Term 4.2 Left Carlos. and 8. Figure 2. In Term 1 (she points with her two index ﬁngers to the bottom row of Term 1. the mathematical ﬁgures. 2. left. that is.18 2 Semiotics in Theory and Practice in Mathematics Education Fig. the non-perceptually accessible Terms 5. sensuous. 7. shows the beginning of the ﬁrst sliding gesture. Right James’s drawing of Term 5 Fig. They are at work in a crucial part of the students’ joint labor process: the process of objectiﬁcation. and also.3.” The teacher and the students continue rhyth- mically exploring the bottom row of Terms 2. Then.” At the same time. counting aloud.3 Left The teacher pointing to the bottom rows. “We will just look at the squares that are on the bottom. points sequentially to the squares in the top row of Term 3. that is to say. the progressive. an object of history and culture. one of the students answers “one. and rhythm. The theory of objectiﬁcation posits the subject and the object as heterogeneous entities. see Fig. 6. 2. The theory of objectiﬁcation is a dialectical materialist theory based on the idea of Otherness or alterity. the teacher makes three consecutive sliding gestures. they turn to the top row. How many [squares] are there?” Pointing. left). Right Students and the teacher counting together The teacher says. Not the ones that are on the top. and 4. This short excerpt illustrates some semiotic means of objectiﬁcation: gestures. Middle Carlos’s drawing of Term 5. In encountering the cultural object. The teacher continues: “Only the ones on the bottom. through gestures and words. perceptual activity. 2. 3. Learning is to encounter something that is not me. words. to visually emphasize the object of attention and intention. and material encountering and making sense of a historically and culturally form of thinking mathematically.
as the example suggests. However. I come to cognize it. (Radford et al. an algebraic structure in the terms of the sequence. or conscious of. sensuously. perception. and rhythm as an encom- passing sign that links gestures. The semiotic node is in this case a collective phenomenon out of which the algebraic structure appears in sensible. the semiotic means of objectiﬁcation do not operate isolated from each other. 2003) are two theoretical constructs that attend to different things. the corporeal position of the students and the teacher. they operate through a complex coordination of various sensorial modalities and semiotic registers that the students and teachers mobilize in a process of objectiﬁcation. 2003). A semiotic bundle is made of the signs produced by teachers and students in problem solving. This encounter of the object is what objectiﬁcation is about (and from whence the theory takes its name). The segment of joint labor where such a complex coordination of sensorial modalities and semiotic registers occur is called a semiotic node (Radford et al.2. The students start discerning a new way of perceiving out of which an algebraic numerical-spatial structure becomes apparent and can be now applied to other terms of the sequence that are not in the students’ perceptual ﬁeld. Etymologically speaking. and symbols. in the poetic encounter of the object and myself. grasp and become aware. In this example. within the teacher-students joint labor occurs a process of objectiﬁcation in the course of which the students start noticing a culturally and historically constituted theoretical way of seeing and gesturing— seeing and gesturing algebraically. the words that the teacher and the students pro- nounce simultaneously. even if I do not agree with it. the teacher’s sequence of gestures. semiotic nodes focus on joint labor (Activity as . in press) A semiotic node is not made of signs. In the example under consideration. while semiotic bundles focus on signs. imagination is central in the entailed process of objectiﬁcation. speech. when the students are counting along with the teacher. A semiotic bundle attends to the evolution of signs: Looking at the evolution of the students’ signs. In the previous example. not only cognitively but emotionally. the semiotic node is a segment of joint labor where signs and sensuous modalities cooperate in order for the students to notice. The segment of joint labor that constitutes the semiotic node includes signs on the activity sheet. Semiotic bundles (Arzarello 2006) and semiotic nodes (Radford et al. it means that I feel it as something alien and. It is a segment of the teacher’s and students’ joint labor where a complex coordination of sensorial modalities and semiotic registers occurs in a process of knowledge objectiﬁcation. Thus.2 Further Applications of Semiotics in Mathematics Education 19 objects me. the teacher can gain clues with respect to the students’ understanding: the multimodal aspects of the activity can therefore help her decide whether or not to intervene in order to support the students. collective consciousness. Imagining the non-perceptually accessible terms is a fully sensuous process out of which an algebraic sense of the functional relations between the number of the terms and the number of squares in their bottom and top rows starts to emerge. the coordinated perception of the teacher and the students. On the contrary.
becoming aware. grasping. Nor is it the connections of its separate elements. for as Voloshinov (1973. Thinking and speech. becoming conscious. has how many [squares] on the bottom? 2.) the theory of objectiﬁcation just collapses (as the theory of didactic situations (Brousseau 1997) would do if you removed from there the concept of situation or if you remove the idea of autonomous child from constructivism). Teacher: Now it’s Term 8! (The teacher comes back to Term 1. “consciousness… is ﬁlled with signs. 31) stated in 1925: “consciousness must be seen as a particular case of the social experience. Of course. consciousness is open to empirical investigation. 137). from the famous “Consciousness as a problem in the psychology of behavior” 1925 paper (Vygotsky 1979) up to his last works (e. In other words. only in the process of social interaction”. Let us come back to the previous example. p. This is why semiotic nodes focus on attention. Man’s activity is the sub- stance of his consciousness.. She points again with a two-ﬁnger indexical gesture to the bottom row of Term 1) Term1. . p. (p. The encounter is described in terms of noticing. As Vygotsky (1979. Consciousness becomes consciousness only once it has been ﬁlled with… (semiotic) content.” The structure of consciousness “is the relation [of the individual] with the external world” (Vygotsky 1997. awareness. in dialectical materialism. etc. con- sciousness is a concrete theoretical construct. The theory of objectiﬁcation would simply collapse because it deﬁnes learning as processes of objectiﬁcation. It is a segment of joint labor in which a progressive encounter with mathematics occurs. Students: 1. It is not a flat surface. Vygotsky dealt with the concept of consciousness throughout his academic life.11) put it. Leont’ev (2009) insisted on the idea that con- sciousness cannot be understood without understanding the individual’s activity (Tätigkeit): Man’s (sic) consciousness… is not additive. and these are a problem of consciousness. and meaning making. consciousness is not the metaphysical construct of idealism—something buried in the depths of the human soul. making sense. to a passage where the teacher and the students are exploring the bottom row of the terms—a passage that occurred a bit later than the previous one. It is the internal movement of its “formative elements” geared to the general movement of the activity which effects the real life of the individual in society. If we remove the concept of consciousness (and stop talking about noticing. which informs the theory of objec- tiﬁcation. Vygotsky 1987). p.g. From the dialectical materialist perspective adopted in the theory of objectiﬁcation. 26). The teacher points rhythmically to the terms one after the other and says: 1. the fabric of consciousness is semiotic. nor even a capacity that can be ﬁlled with images and processes. consciousness is a central concept. Within dialectical materialism philosophy. consequently.20 2 Semiotics in Theory and Practice in Mathematics Education Tätigkeit). intention. From this perspective. There are many theories of learning that do not need to refer to consciousness.
Teacher: (Pointing as above) Term 7? 15. indeed. Students: 2! 5. Joint labor.4 The teacher pointing the bottom row of Term 5 . Although it is unpredictable and cannot be Fig. 2.4). Teacher: (Pointing as above) Term 6? 13. Students: 4! 10. Teacher: (Pointing as above) Term 8? 17. Students: 8! 18. Teacher: (Pointing as above) Term 5 (see Fig. Teacher: (Pointing with a two-ﬁnger indexical gesture to the bottom row of Term 2) Term 2? 4. Sandra: There would be 8 on the bottom! This segment of the students-and-teacher joint labor is the semiotic node. We can turn now to the second semiotic dimension in the theory of objectiﬁ- cation. 2. Teacher: (Pointing with a two-ﬁnger indexical gesture to the hypothetical place where the bottom row of Term 4 would be) Term 4? 9. 2. Students: 5! 12.2)? 11. Students: 3! 8. The excerpt allows us to see the complex coordination of sensorial modalities and semiotic registers that collaborate in the students’ awareness or consciousness of a functional relationship between the number of the terms and the number of squares on the top row of the terms (Fig. to the bottom row of Term 3) Term 3? 7. We mentioned that this dimension has to do with a supra-structural level of cultural meanings that shape and organize joint labor.2 Further Applications of Semiotics in Mathematics Education 21 3. Students: 7! 16. 2. is not something that unfolds spontaneously. Students: 6! 14. Teacher: (Pointing with a two-ﬁnger indexical gesture 6.
g. This is the case of the so-called “process-object” theories.3.. What are their roles? The teaching and learning of mathematics differ as we move from one historical period to another. Unpredictable as it may be. although an increasing effort is been made to expose and discuss them (see. 2008b) these meanings are considered to be part of a symbolic suprastructure that is called Semiotic Systems of Cultural Signiﬁcations. in the Renaissance schools of Abacus and today. 2010. it is shaped culturally. but can help the student to produce more. that is. The mathematics to be taught and learned conveys views about the purpose and the nature of mathematic (e. One devoted to the Western Late Middle Ages and Renaissance and the other to the Buwayhid period of medieval Islam. in Plato’s time. The teacher appears as a ﬁnancial advisor (Radford 2014c). theories that conceive of thinking as . joint labor is nonetheless shaped by forms of human collaboration and modes of knowledge production that ﬁnd their meaning in culture and society and that are fostered by the school. Some of the theories of embodiment in mathematics education have been influenced by Piaget’s genetic epistemology and the Kantian idea of the schema. how it should be taught and learned. Gestures. ﬁnd their legitimacy in cultural meanings that go in general undisputed. e. etc. Pais 2013. Alrø et al. mathematical methods of investigation and production. Our understanding of ourselves as humans conveys views about the child.1 Embodiment. 2.3 The Signiﬁcance of Various Types of Signs in Mathematics Education 2. Radford and Empey (2007) resort to these systems in order to investigate the historical creation of new cultural forms of mathematical understanding and novel forms of subjectivity. For instance. and ultimately with our understanding of ourselves as humans. the teacher did not know how the students would engage by themselves in the gen- eralizing tasks.g. the conception of the teacher and the idea of the student. Skovsmose 2008). the relationship between mathematics and the empirical world). Popkewitz 2004. teaching and learning of mathematics were very different in the Mesopotamian House of Scribes. the nature of mathematical truth. and the Body in Mathematics Education Embodiment has gained a great deal of attention in the past few years.22 2 Semiotics in Theory and Practice in Mathematics Education anticipated in all its details. She cannot produce for the student. nor did she know how the students would react to her invitation to explore the sequence in terms of rows. the legitimacy of the methods of investigation. They present two case studies. but also the conception of mathematics. and also about the teacher. and speciﬁcally the student. In the previous example. Those meanings of mathematical truths. What is different is not only the content. These meanings have to do with conceptions about the mathematics to be taught and learned. In Radford (2006b.. as well as the meanings of students and teachers.
that relate the mind to the material world (e. “what is taken to be sign initially has had to be accessible in it and has to be captured prior to being made sign” (Heidegger 1977. she related the graphs to a condition coefﬁcient. which is calcu- lated by dividing ﬁsh weight by the cube of the length and multiplying the result by 100. she then showed on the graphs which of the ﬁshes were short and fat versus those that were long and skinny (Roth 2016b). For instance. in press). The problem of these embodiment theories is that something else is assumed as prior to the movement. 81).3 The Signiﬁcance of Various Types of Signs in Mathematics Education 23 moving from the learner’s actions to operation knowledge structures. In one study of graphing in a ﬁsh hatchery. 32). Moreover. Before the sign—understood as the relation between two segmentation of matter— can be read (transparently) or interpreted.g. Two exam- ples are APOS theory (Dubinsky 2002. In this case.2. Dubinsky and McDonald 2001) and the “three worlds of mathematics” (Tall 2013). something that then is enacted by the body. The meaning of the term embodiment in the “three worlds of math- ematics” approach is explained as something that is “consistent with the colloquial notion of ‘giving a body’ to an abstract idea” (Tall 2004. Embodiment theories coming from the ﬁeld of cognitive linguistics generally assume a mediator. Using examples from university physics lectures. a ﬁsh culturist with high school certiﬁcation was looking at two distributions. an approach to semiotics has been proposed in which bodily movement takes precedence over the schemas or concepts ordinarily taken as that which comes to be enacted in gestures and other signifying body movements and positions (Roth 2012). for example. such as (bodily) schemas.. and grounded in the works of a largely forgotten French philosopher P. The study proposes a model where body movements self-affect so as to lead to a bifurcation in which the sign is born as the relation between two movements. p. one representing the weight of 100 ﬁsh. As a result. 16). it actually has to come into being (Roth 2008). p. 14). which builds on perception and action to develop mental images that “become perfect mental entities” (Tall 2013. The ﬁrst world of mathematics refers to conceptual embodiment.g. embodiment remains a general category. Johnson 1987. The study showed that over the years of literally handling and inspecting . and this assumption conjures again the specter of Cartesianism (Sheets-Johnstone 2009).000. the source-path-goal schema mediates between the mathematician’s concept of conti- nuity and the bodily gestural expression. Maine de Biran (e. the fate of embodied actions is to be superseded by flexible actions with symbols (Radford et al.. the other one the lengths of the same ﬁsh. one study ana- lyzes the hand/arm gestures that a mathematics professor produces in the course of a lecture on the mathematical concept of continuity (Núñez 2009). Thus. p. 1841) and the uptake of his work in material phenomenology (Henry 2000). “the number line develops in the embodied world from a physical line drawn with pencil and ruler to a ‘perfect’ platonic construction that has length but no thickness (Tall 2008. p. the study shows how the signs (parts of a graph) are the endpoint and the distillate of movements that generate a ﬁeld and perceptual structures. The schemas themselves are the result of bodily engagements with the world that are developed into new concepts by means of metaphorization (Lakoff and Núñez 2000). Lakoff 1987). From a phenomenological perspective. Following up on this critique.
He illustrates this in articulating how to take the verb “to signify. then is a placeholder for all the situations in which s/he has encountered and made use of it (e. then. those required to prove Gödel’s theorem) are mobilized to do work and to formulate the work of doing (Livingston 1986).g. signs involving iconic relations) develop into arbitrary relations. the ﬁsh culturist had developed a feel for both graphs and the ﬁshes. for example. Wittgenstein uses many cases to exemplify that the use and function of signs matters rather than some metaphysical concept or idea.. ergotic gestures) or found something out by means of sense (i.e. This program is taken up in ethnomethodological research on mathematics.e. making constative statements. An application of this in mathematics education shows that signs denoting “concepts” can be taken concretely. is grounded in and indexes all those concrete situations in which it has found some appropriate use. . Throughout his book. when the master builder shows the helper another instance of the same sign. and in the transition to symbolic function.2 Linguistic Theories and Their Relevance in Mathematics Education An important contribution to a theory of signs can be found in the Philosophical Investigations (Wittgenstein 1953/1997). in asking questions.g. motivated signs (i.. 2. Vilela 2010). A subsequent study of the emergence and evolution of sign systems (Roth 2015) suggested that signs initially are immanent to the work activity. the focus is on use rather than meaning..” used by a child. or contesting observation categoricals of others). Here. In the process.e. In fact. which studies the actual living work of doing mathe- matics and how signs (e. This non-metaphysical approach to signs is taken up.g. 3 [§2]). Roth and Lawless 2002).24 2 Semiotics in Theory and Practice in Mathematics Education ﬁshes. had symbolic function.. The sign “cylinder. these may be replaced by other signs.. Once there are symbolic func- tions. Knijnik 2012.” He suggests marking a tool used in building something with a sign. in studies of ethno- mathematics (e. epistemic gestures) sometime later. This is why Wittgenstein deﬁnes a language-game as a whole that weaves together a concrete human activity and the language that is part of the work of accomplishing the work. they transcend the activity. These studies show a progression from hand/arm movements that either did work (i. Wittgenstein notes that the “philo- sophical concept of meaning has its place in a primitive idea of the way language functions” (p. it is apparent that the usage of signs is tied to concrete situations. In this pragmatic approach to the sign..3. and entering their lengths and weights into a spreadsheet that immediately plotted graphs. A sign. referring us to the many concrete ways in which some sign ﬁnds appropriate use (Roth in press). Longitudinal studies among high school students provided insights about the emergence of signs from work generally and hands-on activities speciﬁcally (Roth 2003b. the latter will get the tool and bring it to the master.
Mikhailov 2001.. is not a unitary thing (Roth 2006. signs (language) change at the very moment of their use (Bakhtin 1981. The sign. Braathe and Solomon 2015. “Individuals do not speak or write simply to externalise their . as the commodity in a dialectical materialist approach to political economy. 1981). where the “monologism of the content begins to destroy the form of the Socratic dialogue” (Bakhtin 1984. 220) Indeed.g. Researching from this perspective Morgan (2006) notes: An important contribution of social semiotics is its recognition of the range of functions performed by use of language and other semiotic resources. This allows us to address a wide range of issues of interest to mathematics education and helps us to avoid dealing with cognition in isolation from other aspects of human activity. Moreover. 2009. Vološinov 1930). dialogue with others is the origin of individual speaking and thinking (Vygotsky 1987).. in this line of inquiry.3 The Signiﬁcance of Various Types of Signs in Mathematics Education 25 Another important language theory was developed in the circle surrounding the literary critic and philosopher M. signs (language) live. attitudes and beliefs. Every instance of mathematical communication is thus conceived to involve not only signiﬁcation of mathematical con- cepts and relationships but also interpersonal meanings.. The Bakhtinian approach is found particularly suited in studies of mathematical learning that focus on mathematical learning (e.g.. The most important aspect of the dialogical approach is that it inherently is a dynamical conception of language and sign systems generally and of ideas speciﬁcally.M. in use. But dialogue always presupposes familiarity with the situation and the purpose of speaking. That is. On the other hand. In both theories. (p. 110).. as exempliﬁed in the works of the late Plato. there is an intention to go beyond the traditional view that reduces individuals to the cognitive realm and that reduces the student to a cognitive subject. often referred to as dialogism. Thus.g. Roth 2013).2.g. and monologue may occur even in the exchange between two persons. Bakhtin (e. This theory. Solomon 2012). in this theory dialogue does not require two or more persons. Another line of inquiry comes from Halliday’s social semiotics (Morgan 2006. and therefore such speech is never ﬁnal. signs and the ideas developed with them evolve (Bakhtin 1984). p. but because they live. 2015) and in studies of the narrative construction of self and the subject of mathematical activity (e. the talk involving two individuals may simply be a way of expounding a pre-existing. Vološinov 1930). Kazak et al. This dialogue may occur within one person. the sign is conceived as a phenomenon harboring an inner contradiction that manifests itself in the different ways that individuals may use a sign (word). whenever we use signs.g. 2014). ﬁnalized truth. Thus. has fundamental commonalities with the approach to language taken by the last works of Vygotsky (e. even though some authors appear to be unaware of the fundamentally dia- logic approach in Vygotsky (e. dialogical speech requires the relation between two voices that build on and transform one another. as exempliﬁed in Dostoevsky’s novella Notes from the underground. Barwell 2015. dialogue requires the sign (word) to be a reality for all par- ticipants (Vygotsky 1987. whether in dialogue with others or in dialogue with ourselves. there is a pri- macy of the dialogue as the place where linguistic competence emerges. 2012). Barwell 2015). Instead. For Bakhtin. Radford 2000.
the thinking and meaning making of individuals is not simply set within a social context but actually arises through social involvement in exchanging meanings. 221).26 2 Semiotics in Theory and Practice in Mathematics Education personal understandings but to achieve effects in their social world” (Morgan 2006. The ﬁrst one is the relationship among sign systems (e. such a uniformity is relative and needs to be nuanced in order to account for the variety of responses. (p. behaviors.4 Other Dimensions of Semiotics in Mathematics Education In this section we discuss briefly three interesting questions that have been the object of scrutiny in semiotics and mathematics education research. “Studying language and its use must thus take into account both the immediate situation in which meanings are being exchanged (the context of situation) and the broader culture within which the participants are embedded (the context of culture)” (Morgan 2006.1 The Relationship Among Sign Systems and Translation Duval’s (2000.4. like the Semiotic Systems of Cultural Signiﬁcations in the theory of objectiﬁcation alluded to before. however. This formulation of context of culture suggests a uniformity of culture both between and within the partici- pants” (Morgan 2006. meaning-making. 2. As a result. diagrams. The context of culture theoretical construct is oriented. as Morgan (2006) argues. pictorial and alphanumeric systems) and the translation between sign systems in mathematics thinking and learning. 221). history and orga- nizing concepts that the participants hold in common. p. The third one is about semiotics as the focus of innovative learning and teaching materials. p. 2006) studies have been very important in showing the complexities behind the relationship between sign systems and the difﬁculties that the students encounter when faced with moving between semiotic registers. towards the understanding of classroom practices as loci of production of subjectivities within the parameters of culture and society: “The context of culture includes broader goals. and language use that are found in individuals of a same culture: [T]he notion of participation in multiple discourses will be used as an alternative way of conceptualising this level of context. Yet. Importantly. natural language. 221) 2.g. an investigation into the meaning that students ascribe to their ﬁrst alge- braic formulas expressed through the standard algebraic symbolism suggested that their emerging meanings are deeply rooted in signiﬁcations that come from natural .. The second one concerns semiotics and intersubjectivity. 221). values. p. In this line of thought.
e. to elaborate a symbolic expression for Kelly and Josée. In the case of the translation of statements in natural language into the standard algebraic symbolism. Radford suggests the term symbolic narrative. arguing that what is ‘translated’ still tells us a story but in mathematical symbols. Radford suggests that one of the difﬁculties in dealing with problems involving comparative phrases like “Kelly has 2 more candies than Manuel” is being able to derive non-comparative. This he terms the collapse of narratives. the students were invited to designate Manuel’s number of candies by x. Vol. then. All together they have 37 candies. adding The collection of similar terms means a rupture with their original meaning. (Radford 2002b).g. Manuel has 4 candies. in Problem 1. Latour (1993) shows how soil samples are translated into a sign system that is translated into other another sign system until. He discusses a mathematical activity that was based on the following short story: “Kelly has 2 more candies than Manuel. He shows that some of the difﬁculties that the students have in operating with the symbols are precisely related to the requirement of producing a collapse in the original stated story. The relation between two sign systems is not inherent or natural but is established through work.” During the mathematical activity. (Radford 2002b. If. the story problem has to be re-told.2. In each case. In a study of biological research from data collection to the published results. All the efforts that were made at the level of the designation of objects to build the symbolic narrative have to be put into brackets. 4. The whole symbolic narrative now has to collapse. which have symbolic notations as their material . There is no corresponding segment in the story-problem that could be correlated with the result of the collection [addition] of similar terms. the adjective has to be referred to in some way. and. the assertive phrase would take the form «Kelly (Subject) has (Verb) 6 (Adjective) candies (Noun)». leading to what has been usually termed (although in a rather simplistic way) the ‘translation’ of the problem into an equation.. Josée has 5 more candies than Manuel. In using a letter like ‘x’ (or another device) a new semiotic space is opened. Radford 2010b). the adjective is not known (one does not know how many candies A has). p.4 Other Dimensions of Semiotics in Mathematics Education 27 language and perception. assertive phrases of the type: “A (or B) has C”. verbal statements about the biological system are made. the sign system consists of a material base with some structure. say. In Problem 2. In the case of algebra. Incidentally. Radford (2002b) noticed that the ﬁrst algebraic statements are not only imbued with the meanings of colloquial language. to write and solve an equation corresponding to the short story. As a result. 87) The longitudinal investigation of several cohorts of students in pattern gener- alization point to a similar result: One of the crucial developmental steps in the students’ algebraic thinking consists in moving from an indexical mode of desig- nation to a symbolic one (see. a similar articulation was offered to understand how students relate a winch to pull up weights and mathe- matical (symbolic) structures. In this space. the students were invited to designate Kelly’s number of candies by x while in Problem 3 the students were invited to designate Josée’s number of candies by x. at the end. but also colloquial language lends a speciﬁc mode of designation of objects that conflicts with the mode of designation of objects of algebraic sym- bolism.
4. instructional materials integrating interactive dia- grams. One recent study exhibits the different types of work students need to accom- plish to relate natural phenomena and different computer-based. ‘Spinozan’ works [where] the idea of semiotic mediation is supplanted by the concept of the intersubjective speech ﬁeld” (Mikhailov 2006.4. animations and video as instructional tools with new technologies. Brown 2011).. Semiotics helps to understand the challenges driven by these materials. 35). To assist students in learning the relation between mathematical graphs and the physical phenomena they are investigating. weight vectors) corresponding to the weights. • Roles of diagrams. • Design of activities and tasks that are based on interactive visual examples.2 Semiotics and Intersubjectivity A different take on intersubjectivity apparently arose from Vygotsky’s “last. p. Subjectivity is a signiﬁ- cant topic in its own right (e. interactive visual examples and visual demonstration animations have constituted a privileged terrain of research in mathematics education. The authors suggest that these complexes of sign systems may be difﬁcult to unpack because relationships emerge only when students structure each layer in a particular way so that the desired relationships then can be constructed. some textbooks layer different sign systems with the apparent intention of offering students a way to link particular aspects of one system to a corresponding aspect in the other (Roth et al. Because children always already ﬁnd themselves in an intersubjective speech ﬁeld.. dynamic sign systems—images and graphs—that are used to stand in for the former (Jornet and . • Patterns of reading. there exists a “dynamic identity of intersubjectivity and intrasubjectivity” (p. 36). Using an example from a Korean science textbook.28 2 Semiotics in Theory and Practice in Mathematics Education (Greeno 1988). using and solving with interactive linked multiple representations. a graph exhibiting Boyle’s law relating the volume V and pressure of an ideal gas (V * 1/p) is overlaid by (a) the images of a glass beaker with different weights and (b) differ- ently sized arrows (i. As a result. 2.3 Semiotics as the Focus of Innovative Learning and Teaching Materials Digital mathematics textbooks. Some important threads are as follows: • Innovative visualization tools for teaching and learning. 2005).e. the world and language are given to them as their own.g. 2. which can only be mentioned here.
The structures of the different systems may then be related and compared.2. often leading to a revision in the structuring process. distribution and reproduction in any medium or format. In each case. adapt.org/licenses/by-nc/4. users will need to obtain permission from the license holder to duplicate. a link is provided to the Creative Commons license and any changes made are indicated. which enables new forms or relations between the dif- ferent systems Open Access This chapter is distributed under the terms of the Creative Commons Attribution- NonCommercial 4.4 Other Dimensions of Semiotics in Mathematics Education 29 Roth 2015). . which permits any noncommercial use. adaptation. unless indicated otherwise in the credit line. duplication. or reproduce the material. if such material is not included in the work’s Creative Commons license and the respective action is not permitted by statutory regulation. in natural phenomenon and sign systems. struc- turing work is required to get from the material base to a structure.0/). that is.0 International License (http://creativecommons. as long as you give appropriate credit to the original author(s) and the source. The images or other third party material in this chapter are included in the work’s Creative Commons license.
The images or other third party material in this chapter are included in the work’s Creative Commons license. • Relevant theoretical notions such as objectiﬁcation and communicative ﬁelds. users will need to obtain permission from the license holder to duplicate. © The Author(s) 2016 31 N. Presmeg. unless indicated otherwise in the credit line. as long as you give appropriate credit to the original author(s) and the source. ICME-13 Topical Surveys. we have of necessity concentrated on the many theoretical constructs that are relevant to semiotics in mathematics edu- cation. intersubjectivity. semiotic bundles. duplication. • The roles of visualization and language in semiosis. the creation of innovative learning and teaching materials. and other seminal thinkers. • Other dimensions: sign systems and translations among them. Interested readers may follow the rich empirical results contained in many of the references cited. Open Access This chapter is distributed under the terms of the Creative Commons Attribution- NonCommercial 4.org/licenses/by-nc/4. Vygotsky. Applications are mentioned. but we have included more details of only one of the many empirical research studies that have been conducted. • Embodiment and gestures in semiosis. Semiotics in Mathematics Education.Chapter 3 A Summary of Results Within the constraints of this Topical Survey. • Semiotic chains. DOI 10. Peirce.0/). or reproduce the material. • Basic ideas and applications of theories of de Saussure.0 International License (http://creativecommons. which permits any noncommercial use. distribution and reproduction in any medium or format. adapt. adaptation. a link is provided to the Creative Commons license and any changes made are indicated.1007/978-3-319-31370-2_3 . and semiotic nodes. if such material is not included in the work’s Creative Commons license and the respective action is not permitted by statutory regulation. Each of the topics in the following list of items surveyed has the potential to generate questions for further empirical research.
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