> } ʰbjbj55 8__z~UUyyylDW=:wwww0.0.0.VVVVVVVY3\jVy4-^0.44VUUww[iW@@@4UwywV@4V@@STw@h<SVW0WS\x@d\,T\yTD0.J0@1420.0.0.VV@0.0.0.W4444\0.0.0.0.0.0.0.0.0. : an overarching theory for research in visualization in mathematics education
Norma Presmeg
Illinois State University
The visual aspects of teaching and learning mathematics have gained increased attention in the last three decades. As lenses for interpreting the results of empirical investigations in this field, several theories have proved useful. This paper synthesizes theoretical developments regarding visual aspects of mathematics education in terms of taxonomies of types of imagery and inscriptions. In particular, insights from Peircean semiotics are used to discuss external and internal signs, descriptive and depictive signs, polysemic and monosemic signs, and related aspects from linguistic theory that include metaphor and metonymy. Examples from recent research investigations involving the learning of trigonometry and geometry are used to illustrate the developing theory.
the need for an overarching theoretical framework
The need for a unified view of the structure and components of a theory of the role of visual representation in mathematics education has been evident for some time. Presmeg (2006a) identified the small number of publications that have addressed such a needand possible directions that theory development might takeas follows.
Already in 1992, in his plenary address at PME-16 [the 16th conference of the International Group for the Psychology of Mathematics Education], Goldin outlined a unified model for the psychology of mathematics learning, which incorporated cognitive and affective attributes of visualization as essential components in systems of representation in mathematical problem solving processes. More specifically, also in a plenary address, Gutierrez (1996) posited a framework for visualization in the learning of 3-dimensional geometry. More recently, from a review of the extant literature, Marcou and Gagatsis (2003) developed a first approach to a taxonomy of mathematical inscriptions based on distinctions between external and internal, descriptive and depictive, polysemic and monosemic, autonomous and auxiliary representations as used in mathematical problem solving. This work is not yet fully available in English but gives promise of valuable theory development. I see some aspects of their taxonomy relating to the triadic semiotics of Peirce (1998): descriptive and depictive systems are reminiscent of symbolic and iconic signs respectively. Peirces indexical signs, with their emphasis on context and metaphor, might add an element to the taxonomy of these authors, who did not make a connection with semiotics. The further development of theory concerning the use of visualization in mathematics was also suggested by Kadunz and Strsser (2004), and this development could include connections with semiotics regarding gestures and other signs (Radford et al., 2003; 2005). The need for ongoing theory development is clear. (p. 226)
In identifying 13 significant questions for research on visualization in mathematics education, Presmeg (2006a, p. 227) included the following: What is the structure and what are the components of an overarching theory of visualization in mathematics education?
In this paper I shall take up the suggestion to expand the initial taxonomy suggested by Marcou and Gagatsis (2003), in terms of Peirces triadic semiotics, illustrating categories with data drawn from an ongoing investigation of high school students trigonometric connections as well as other mathematical topics. It is important to note that I consider visualization to be a significant aspect of all branches of mathematics (and not merely of obviously visual branches such as geometry). Symbolism may in and of itself entail spatial characteristics, thereby implicating visualization.
introduction to the development of a theoretical framework
The first dichotomy in Marcou and Gagatsis (2003) taxonomy, that of external and internal representations (based on distinctions made by Goldin and others), is not pursued in this development of theory. The reason for omitting this distinction is that I prefer to follow Piaget and Inhelders (1971) claim that visual imagery (internal representation) underlies the creation of a drawing or spatial arrangement (external representation). Thus it does not seem fruitful to separate these modes of representation. The illustrations provided in this paper from trigonometric data concern almost exclusively the external mode, called inscriptions here (cf. Roth, 2004), because the term representations acquired some ambiguous and controversial associations in the changing paradigms of the last few decades. Thus the following is an attempt to develop and illustrate a taxonomy of inscriptions, although many of the categories might well be applied to the corresponding forms of visual imagery as well. To summarize the latter, types of visual imagery that were identified by Presmeg (1985, 2006a) were as follows:
concrete imagery (having characteristics of a picture);
pattern imagery (pure relationships stripped of concrete details);
memory images of formulas;
kinaesthetic imagery (involving physical movement);
dynamic imagery (in which the image itself is moved or transformed). (2006a, p. 208)
These five types of imagery were identified in the transcripts of mathematical task-based interviews with 54 high school learners over a complete year when they were in grade 12 (Presmeg, 1985). These categories are roughly comparable to Drflers (1991) kinds of image schemata, which he characterized as figurative, operative, relational, and symbolic (Presmeg, 2006a). However, in this paper I shall concentrate on interpretations of inscriptions, although these interpretations might also be associated with various kinds of imagery or image schemata.
Two of Peirces triads
I shall begin with a brief introduction to the way that I am using the terminology of semiotics, and to two of Peirces ten semiotic triads.
Semiotics is the study of activity with signs (Colapietro, 1993; Whitson, 1997). But what is a sign? Although in this paper I shall follow the triadic model of Peirce (1998) with basic components that he designated as object, representamen (standing for this object in some way) and interpretant (the result of interpreting this relationship), even in Peirces own writings at various periods there is ambiguity in the sense in which he used the word sign. Thus it is necessary to specify how I am using the word. I shall take a sign to be the interpreted relationship between some representamen or signifiercalled the sign vehicleand an object that it represents or stands for in some way. In mathematics, the objects we talk about cannot be apprehended directly through the senses: for instance, point, line, and plane in Euclidean geometry refer to abstract entities that we can never see, strictly speaking, as in Sfards (2000) virtual reality. We apprehend these objects, see them, and communicate with others about them, in a mediated way through their sign vehicles, which may be drawn or written by hand or through dynamic geometry software, labeled in conventional ways, moved and manipulated for multiple purposes. We work with these sign vehicles as though we were working with their objects: in Ottes (2006) terms, we become accustomed to seeing an A as a B. It is this interpreted relationship between a sign vehicle and its object that constitutes the sign.
Another useful trichotomy, according to Peirce (1998), is that signs may be iconic, indexical, or symbolic. These types are not inherent in the signs themselves, but depend on the interpretations of their constituent relationships between sign vehicles and objects. To illustrate by using some of Peirces examples, in an iconic sign, the sign vehicle and the object share a physical resemblance, e.g., a photograph of a person representing the actual person. Signs are indexical if there is some physical connection between sign vehicle and object, e.g., smoke invoking the interpretation that there is fire, or a sign-post pointing to a road. The nature of symbolic signs is that there is an element of convention in relating a particular sign vehicle to its object (e.g., algebraic symbolism). 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, indexical, or symbolic respectively, according to their interpretationsthus effectively generating three different signs, as is illustrated later in this paper for the case of the quadratic formula.
A TAXONOMY OF INSCRIPTIONS
In addition to their external-internal distinction regarding representationswhich will not be pursued furtherthe initial approach of Marcou and Gagatsis (2003) included the following pairs of characteristics:
(a) descriptivedepictive;
(b) polysemicmonosemic;
(c) autonomousauxiliary.
Each of these dichotomies is examined in terms of Peirces triad of iconic, indexical, and symbolic signs.
Descriptive and depictive sign vehicles
Descriptive and depictive sign vehicles would appear to correspond to words and pictures respectively, or to the symbols of mathematics roughly contrasted with the visual forms that capture its structure (bearing in mind the point made previously, that symbols have a visual element too, e.g., as illustrated in memory images of formulas). Words and symbols have an element of convention, both in their linguistic form and in the choices of mathematical symbolism, thus the relations of these descriptive sign vehicles to their mathematical objects may be interpreted as symbolic. In contrast, depictive visual forms are iconic because they bear a physical resemblance to the structure of the mathematical objects they are representing. However, Peirces triad indicates that there is a third formnot taken into account by Marcou and Gagatsis in their classificationthat of indexical signs. The interpretation of the relation of the sign vehicle with its mathematical object is neither conventional nor iconically structural in this case, but depends on the context (see the following example). This characterization yields a broad theoretical classification of symbolic (descriptive), iconic (depictive), and indexical sign vehicles. However, in practice the distinctions are more subtle because they depend on the interpretations of the learnerand therefore the distinctions may be more useful to a researcher or teacher for the purpose of identifying the subtlety of a learners mathematical conceptions if differences in interpretation are taken into account.
As an example, let us examine the quadratic formula in terms of this triad. The roots of the equation ax2+bx+c = 0 are given by the well known formula
______
x = -b "b2- 4ac .
2a
Because symbols are used, the interpreted relationship of this inscription with its mathematical object may be characterized as symbolic, involving convention. However, depending on the way the inscription is interpreted, the sign could also be characterized as iconic or indexical. The formula involves spatial shape. In my original research study of visualization in high school mathematics (Presmeg, 1985), many of the students interviewed reported spontaneously that they remembered this formula by an image of its shape, an iconic property. However, the formula is also commonly interpreted as a pointer (cf. a direction sign on a road): it is a directive to perform the action of substituting values for the variables a, b, and c in order to solve the equation. In this sense the formula is indexical. Thus whether the inscription of the formula is classified as iconic, indexical or symbolic depends on the interpretation of the signa theme that is elaborated in the following sections.
Metaphor and metonymy
The literary forms known as metaphor and metonymy are useful in providing a finer grain of classification of iconic, indexical, and symbolic signs in mathematics education. Each has a specific structure. Metaphor implicitly compares two domains of experience, giving meaning to one of these domains (the target domain) by reference to structural or practical similarities in the other (the source domain). The target and source domains are sometimes called the tenor and the vehicle of the metaphor respectively (Presmeg, 1998). There are always some elements that are common to the two domains (called the ground) and some elements that are different (the tension). The constituent relationships are depicted in figure 1. In mathematics, metaphoric relationships are often iconic, e.g., in trigonometry, triangles in the unit circle are a bow tie is a useful metaphor to help learners apprehend the constituent relationships and their shape (in a teaching unit following Brown, 2005).
SHAPE \* MERGEFORMAT
A teacher is a gardener
Target (tenor) Source (vehicle)
Figure 1: Structure of metaphor.
The structure of metonymy is different from that of metaphor. There are two forms, namely, metonymy proper and synechdoche, both of which are used frequently in mathematics (Presmeg, 1998). Because context is important in interpretation of metonymy, these forms are indexical. Metonymy proper uses one sign vehicle to stand for another, taking the context into account (figure 2).
SHAPE \* MERGEFORMAT
Figure 2: Structure of metonymy, using Lakoffs (1987) example.
Synechdoche uses part of a sign to stand for the whole, or the whole to stand for a part. For example, a drawing of a triangle is used to stand for a general triangle, i.e., an element of the class is used to stand for the whole class of triangles (figure 3).
SHAPE \* MERGEFORMAT
Figure 3: Structure of synechdoche: triangle example.
These literary forms have the potential to provide a finer grain in unpacking the structure of iconic, indexical, and symbolic signs in mathematics education research.
Polysemic and monosemic sign vehicles
Polysemy and monosemy comprised the next category in the preliminary classification of representations by Marcou and Gagatsis (2003). However, there is a third, possibility, that of homonymy. I would like first to clarify the distinction between polysemy and homonymy. Lakoff (1987) provided useful examples to illustrate the difference between these terms, as follows.
The bank of a river and the bank in which money is kept are examples of homonyms, because although the same word is used there is no connection between the objects represented by these sign vehicles. The same sign vehiclethe word bankrefers to two different and unrelated objects.
I work for the newspaper and The early newspaper is already being delivered exemplify polysemy, because there is a relationship between the news organization and the newspaper that it produces. The two objects underlying the same sign vehiclethe word newspaperin these two sentences, are related by the fact that the newspaper organization produces the paper, i.e., the objects are related and their sign vehicle produces different but related interpretants.
In a semiotic analysis of mathematical ideas it is likely that polysemy will be involved, as in the three related interpretants for the quadratic formula in the previous section. However, there are also cases in which the same sign vehicle is used for unrelated mathematical objects. Vinner and Dreyfus (1989) described the compartmentalization that results when the same sign vehicle is interpreted by an individual as standing for two different and more or less unrelated mathematical objects, which they termed the concept definition and the concept image. Thus the theoretical formulation should be open to the possibility of homonymy.
Again, whether or not sign vehicles are monosemic, polysemic, or an example of homonymy, will depend crucially on the interpretant constructed by an individual. Because of differing interpretants, mathematical sign vehicles are seldom monosemic. However, if teachers do not recognize that they are polysemic, and take them to be monosemic, confusion may ensue for the learner. Working in three countries (Cyprus, Italy, and Greece), Gagatsis and Elia (2003) showed clearly that the number line is used by children in polysemic ways in conducting arithmetic operations. Other examples, from more advanced mathematical content, are the symbols and +", which may be apprehended in at least two different ways: they may be taken as mere conventional symbols for a sum and an integral respectively; or they may be understood as indices pointing to the need to perform the operations of finding a sum or integrating. Within these interpretations, there may be further polysemic nuances of meaning. The interpretation could be merely procedural and instrumentalcarrying out an algorithmor it could be conceptual and relational (Skemp, 1976) if the underlying structure is apprehended. Moschkovich, Schoenfeld, & Arcavi (1993) described two fundamentally different perspectives for the concept of a mathematical function, which they identified as a process perspective and an object perspective. The nature of the former was characterized as algorithmic, involving a process of algebraic substitution, whereas the latter could be seen as a more holistic apprehension of a function in its entirety, including its iconic relationships. The fundamentally different natures of these two perspectives indicate the polysemy of these interpretations, both of which are occasioned by the notion of function, i.e., the same sign vehicle.
Autonomous and auxiliary sign vehicles
On the one hand, although sign vehicles such as and +" are polysemic, they may be regarded as defining particular mathematical concepts in their own right, i.e., they are autonomous rather than auxiliary. On the other hand, when diagrams are drawn in attempting to make sense of a mathematical problem in which no diagram is given, such diagrams are auxiliary if the problem could have been solved without them. All of the possible diagrams in response to the 24 questions in Presmegs (1985) Mathematical Processing Instrument (MPI) are of this nature. The instrument measures high school students and teachers preference for using visualization, rather than ability to do so. (Someone with high visual or spatial ability may nevertheless choose to do these problems purely algebraically.) Use of a diagram, or merely a picture in the mind, according to self-report on an accompanying questionnaire, results in scoring for a visual solution, and the total visuality score enables individuals to be placed on a continuum for preference for use of visualization (Presmeg, 1985, 1986, 1997). The results of use of this instrument confirm Krutetskiis (1976) claim that visualization is distinct from the use of logical reasoning in mathematics, which defines mathematical ability. According to his writings, visualization is often useful, but it is not essential to high achievement in mathematics. In fact, in some cases visualization may hinder mathematical thinking (Krutetskii). In Presmegs (1992, 1997) research, all of the difficulties experienced by the 54 visualizers in her study could in one way or another be related to problems of generalization: visual sign vehicles are specific, but they need to be interpreted as standing for general mathematical objects. In a two-dimensional plane, Krutetskiis model could be characterized as placing verbal/logical skills on the continuum of the x-axis and use of visual/spatial aspects on the continuum of the y-axis. According to Presmegs instrument, individual scores may fall into categories in all four quadrants of this plane. In this way, the auxiliary types of sign vehicles may influence the style of mathematical thinking, and the ways they are used may determine its efficacy.
examples from a trigonometry investigation
Some of the results of a collaborative investigation of the classroom learning of trigonometry (with Susan Brown, following on from her dissertation research in 2005) were reported elsewhere (Brown, 2006; Presmeg, 2006b). The original research of Brown (2005) identified different ways that students interpreted the conversions involved in moving from trigonometric definitions using right triangles, to those in the coordinate plane, to the use of a unit circle, and finally to the graphs of the sine and cosine functions. Brown did not use a semiotic framework in this study. However, in spring of 2006, Presmeg investigated some aspects of students learning in a trigonometric course taught by Brown, and analysed data from a series of interviews with four students in the class, using a semiotic lens. Space permits only a brief summary, but the data provide evidence of students interpreting sign vehicles iconically, indexically and symbolically. The presence of polysemy was strongly evident, as were metaphors (both those introduced by the teacher, and idiosyncratic metaphors) and metonymies.
For instance, the definitions of the trigonometric relationships in a right triangle were understood both iconically (depictive use) in terms of the diagram, and symbolically (descriptive use) because students had memorized them using the mnemonic SOH CAH TOA. Polysemy was evident particularly in moving from the triangle definitions to those on the coordinate plane. For some learners, the gap between the sine of as a sign vehicle for the triangle definition, and as a sign vehicle for the relationships in a unit circle or in a coordinate plane, provided huge problems in synchronizing their polysemic interpretations. In this case, the bow-tie metaphor introduced by the teacher turned out to be a powerful aid.
Within the structure of the coordinate plane, the same sign vehicle, presented iconically as a point P (x, y) on the unit circle in the second quadrant, was interpreted polysemically by different students and in different tasks in all of the following ways:
with a focus on the numerical values of the coordinates x and y;
in terms of horizontal and vertical distances;
dynamically in terms of movements from the origin;
holistically using the structure of the plane according to positive and negative numerical values.
In the preliminary analysis of the whole data corpus, the following aspects were identified as significant. (The numbers in parenthesis indicate how many of the four students interviewed provided evidence for the phenomenon.)
Compartmentalization and prototypes (4).
Aspects of generalization and logic: the need for a concrete case (3);
a mathematical cast of mind (1).
Compression, unification, and encapsulation (1).
Metaphors: those taught by the teacher (3);
those constructed spontaneously and idiosyncratically (3).
Sliding interpretations of notation (2).
Gestures as sign vehicles (2).
Spontaneous visual sign vehicles: icon helpful (3);
icon unhelpful (1).
The power of falsification (3).
Use of a tool as a sign vehicle (1).
There is evidence from the data to support the conclusion that all of the above aspects are significant in individual students learning of trigonometry. The structure of the theoretical framework outlined in this paper has the potential to introduce a finer grain in a second round of data analysis, which is ongoing. In analysing the data further, there is the possibility that the theoretical lens itself may need modification. Each theory is developed with a particular purpose in mind; the purpose of the theory developed in this paper is to capture in summary form notions that have been useful in this researchers analysis of data in investigations of the strengths and pitfalls of visualization in mathematics education. A different purpose would of necessity require modification of the theoretical lens, or use of a different lens.
In summary, the theoretical formulation described in this paper identified the following components as a lens for research of aspects of visualization in mathematics education.
a) Sign vehicles that are interpreted iconically, indexically, or symbolically (embracing the descriptive-depictive dichotomy of Marcou and Gagatsis, 2003).
b) Metaphor, and metonymy, which may in turn be classified as metonymy proper or synechdoche.
c) Homonymy, polysemy, and monosemy.
d) Autonomous and auxiliary sign vehicles.
It is likely that a basic theoretical framework such as the one described in this paper could provide starting points for interpreting phenomena of mathematical visualization in many areas of teaching and learning mathematics, and at many levels. My early research on visualization at the high school level (Presmeg, 1985) involved the teaching and learning of topics from algebra, Euclidean geometry, analytical geometry, trigonometry, vectors, and calculus. It remains to be seen whether the ideas put forward in this paper are useful in the issues and concerns of the Topic Study Group on visualization. I am certain that the theory will continue to evolve, and the need for modification in different research studies will be ongoing. These ideas are all open to further empirical investigation.
References
Brown, S. A. (2005). The trigonometric connection: Students understanding of sine and cosine. Unpublished Ph.D. dissertation, Illinois State University.
Brown, S. A. (2006). The trigonometric connection: Students understanding of sine and cosine. In J. Navotna, H. Moraova, M. Kratna, & N. Stehlikova (Eds.), Proceedings of the 30th Annual Meeting of the International Group for the Psychology of Mathematics Education, Vol. 1, p. 228.
Colapietro, V. M. (1993). Glossary of semiotics. New York: Paragon House.
Drfler, W. (1991). Meaning: Image schemata and protocols. In F. Furinghetti (Ed.), Proceedings of the 15th Annual Meeting of the International Group for the Psychology of Mathematics Education, 1, 17-32.
Gagatsis, A. & Elia, I. (2003). The use of number line in conducting arithmetic operations: A comparative study between Cypriot, Italian and Greek primary pupils. In J. Navotna (Ed.), Proceedings of the International Symposium in Elementary Mathematics Teaching (pp. 70-73). Prague, The Czech Republic: Charles University.
Goldin, G. A. (1992). On the developing of a unified model for the psychology of mathematics learning and problem solving. In W. Geeslin & K. Graham (Eds.), Proceedings of the 16th Annual Meeting of the International Group for the Psychology of Mathematics Education, 3, 235-261.
Gutirrez, A. (1996). Visualization in 3-dimensional geometry: In search of a framework. In L. Puig & A. Gutierrez (Eds.), Proceedings of the 20th Annual Meeting of the International Group for the Psychology of Mathematics Education, 1, 3-19.
Kadunz, G., & Strsser, R. (2004). Image metaphor diagram: Visualization in learning mathematics. In M. J. Hines & A. B. Fuglestad (Eds.), Proceedings of the 28th Annual Meeting of the International Group for the Psychology of Mathematics Education, 4, 241-248.
Krutetskii, V. A. (1976). The psychology of mathematical abilities in schoolchildren. Chicago, Illinois: University of Chicago Press.
Lakoff, G. (1987). Women, fire, and dangerous things: What categories reveal about the mind. Chicago: University of Chicago Press.
Marcou, A., & Gagatsis, A. (2003). A theoretical taxonomy of external systems of representation in the learning and understanding of mathematics. In A, Gagatsis & I. Elia, (Eds.), Representations and geometrical models in the learning of mathematics (Vol. 1, pp. 171-178). Nicosia: Intercollege Press (in Greek).
Moschkovich, J., Schoenfeld, A. H., & Arcavi, A. (1993). Aspects of understanding: On multiple perspectives and representations of linear relations and connections among them. In T. A. Romberg, E. Fennema, & T. P. Carpenter (Eds.), Integrating research on the graphical representation of functions (pp. 69-100). Mahwah, NJ: Lawrence Erlbaum Associates.
Otte, M. (2006). Mathematical epistemology from a Peircean semiotic point of view. Educational Studies in Mathematics, 61(1-2), 11-38.
Peirce, C. S. (1998). The essential Peirce. Volume 2, edited by the Peirce Edition Project. Bloomington, Indiana: Indiana University Press.
Piaget, J., & Inhelder, B. (1971). Mental imagery and the child. London: Routledge & Kegan Paul.
Popper, K. (1974). The philosophy of Karl Popper. The Library of Living Philosophers, edited by Paul Arthur Schilpp. La Salle, Illinois: Open Court.
Presmeg, N. C. (1985). Visually mediated processes in high school mathematics: A classroom investigation. Unpublished Ph.D. dissertation, University of Cambridge.
Presmeg, N. C. (1986). Visualisation in high school mathematics. For the Learning of Mathematics, 6(3), 42-46.
Presmeg, N. C. (1992). Prototypes, metaphors, metonymies and imaginative rationality in high school mathematics. Educational Studies in Mathematics, 23(6), 595-610.
Presmeg, N. C. (1997). Generalization using imagery in mathematics. In L. D. English (Ed.), Mathematical reasoning: Analogies, metaphors and images (pp. 299-312). Mahwah, New Jersey: Lawrence Erlbaum Associates.
Presmeg, N. C. (1998). Metaphoric and metonymic signification in mathematics. Journal of Mathematical Behavior, 17(1), 25-32.
Presmeg, N. C. (2003). Ancient areas: A retrospective analysis of early history of geometry in light of Peirces commens. Paper presented in Discussion Group 7, Semiotic and socio-cultural evolution of mathematical concepts, 27th Annual Meeting of the International Group for the Psychology of Mathematics Education, Honolulu, Hawaii, July 13-18, 2003. Subsequently published in the Journal of the Svensk Frening fr MatematikDidaktisk Forskning (MaDiF) (Vol. 8, pp. 24-34), December, 2003.
Presmeg, N. C. (2006a). Research on visualization in learning and teaching mathematics. In A. Gutirrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education: Past, present and future (pp. 205-235). Rotterdam, The Netherlands: Sense Publishers.
Presmeg, N. C. (2006b). A semiotic view of the role of imagery and inscriptions in mathematics teaching and learning. Plenary Paper. In J. Navotna, H. Moraova, M. Kratna, & N. Stehlikova (Eds.), Proceedings of the 30th Annual Meeting of the International Group for the Psychology of Mathematics Education, Vol. 1, pp. 19-34.
Presmeg, N. C. & Nenduradu, R. (2005). An investigation of a preservice teachers use of representations in solving algebraic problems involving exponential relationships. In H. Chick & J. Vincent (Eds.), Proceedings of the 29th Annual Meeting of the International Group for the Psychology of Mathematics Education, 4, 105-112.
Radford, L., Bardini, C., Sabena, C., Diallo, P., & Simbagoye, A. (2005). On embodiment, artifacts, and signs: A semiotic-cultural perspective on mathematical thinking. In H. L. Chick & J. L. Vincent (Eds.), Proceedings of the 29th Annual Meeting of the International Group for the Psychology of Mathematics Education, 4, 113-120.
Radford, L., Demers, S., Guzman, J., & Cerulli, M. (2003). Calculators, graphs, gestures and the production of meaning. In N. Pateman, B. J. Dougherty, & J. Zillox (Eds.), Proceedings of the 27th Annual Meeting of the International Group for the Psychology of Mathematics Education, 4, 56-62.
Roth, W.-M. (2004). Towards an anthropology of graphing: Semiotic and activity-theoretic perspectives. LMZ[tv ;
B
=BOP}~_ddlm=EJRHW&o{]euh*9h:gh/_6h^e3h/_6h& h/_6h/_hmAhm9he$hm9h8
hevZh8
hfh h7
nh^hahAwhY3hY=hK=M[uv~dmSsC%!`!gd`xgdw#
^`gdJ0gd/_!`!gdm9gd8
gdm9gdJ;@Schqs<FZ_! # % 2 ^ k !
!!!!!""8$A$$$[%a%c%l%q%y%Q&W&&&''))))))j++̵̹̹̮ЪhY#h6hhhfh/_haxh*9h/_6h`h/_hhh/_hwhvhvhmAhvhthvhwhwhwhhevZh>%))****j++122222^7t7P;m;n;o;p;q;;gdgdh$gd7
n!`!gdh$gdY#gdJxgdt!`!gd`+,",],,,//0&060<0N0W0122422222222222222222>5ɺyj^ZhJhfmH nH sH tH h
#UhJmH nH sH tH hfh
#UmH nH sH tH "h7
nhJ>*H*mH nH sH tH h7
nhJ>*mH nH sH tH hJhJ>*mH nH sH tH hJhJmH nH sH tH h
#UmH nH sH tH hY"h hh$H*hhJh5h>.hwhShJh!>5?56747J7P7T7U7\7]7^778@8H8I88889 9s9999::F:::#;%;B;O;P;Q;h;i;j;k;l;m;n;q;;;;;;;ƻh
h>6h>hGRjhdU"jhTUmHnHsH tH uhjhUhh.O.hOuh
h6hh6h>.h hk6 h.O.6hkh]h.O.hhh2;;;R=o=p=q=r=s=t==>>>>>>>???QAnBADF!`!gdh$gdY#gd`ygd
/gdJgdgd;;;;;<%<<<<<<<<D=O=Q=R=S=j=k=l=m=n=t====>>>>>>>>>>?????ůͣů|tplhY#hOuhfh`y6h6h`yj,hdUh> h
/h
/ h
/6hfh6h.2jhdU"jhTUmHnHsH tH uhjhUhJh> h> h> 6h`yh}dh> hfhfhGR6hh>h>*??6@@@@@QAUAYAmAqAAAB'B-B.B/B2B3B?BMBlBBBBBSCdCeCxCzCCCDDDD%D1D?D@DADDDEGETEEpFrFFFbG;HsHHKž𦺢𪞪hDLBhXhuJnhlEh)Xh_h h6hfhRhRhRhRhR6hRh6hY#h>*hY#h6hY#hhhY#hOuh:K&KKK0LLLLM>MMSU̷̾~paphh6H*]mH sH hh6]mH sH hmH sH hhmH sH hhrhrH*hrhY6hYhYh
NhYhYhah@:h}dh}d6]mH sH h}dh}dmH sH h@:mH sH h}$h}$mH sH h}$mH sH h}$h}$6]mH sH $EFHlowx݇އ߇&'ݬP\^ڭP+oŽԯŽ̽㨽Ũ㢞hh$hh$hI-@hI-@hh$6hh$ hS6h]hSUhc<hYhr6H*h9hYhrhY6hYhYhYhr6hrhr6H* hr6hYhhevZmH sH hh]mH sH 3Dordrecht: Kluwer Academic Publishers.
Sfard, A. (2000). Symbolizing mathematical reality into being or how mathematical discourse and mathematical objects create each other. In P. Cobb, E. Yackel, & K. McClain (Eds.), Symbolizing and communicating in mathematics classrooms: Perspectives on discourse, tools, and instructional design (pp. 37-98). Mahwah, New Jersey: Lawrence Erlbaum Associates.
Skemp, R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77, 26-29.
Vinner, S. & Dreyfus, T. (1989). Images and definitions for the concept of function. Journal for Research in Mathematics Education, 20(4), 266-356.
Whitson, J. A. (1997). Cognition as a semiosic process: From situated mediation to critical reflective transcendance. In D. Kirshner & J. A. Whitson (Eds), Situated cognition: Social, semiotic, and psychological perspectives (pp. 97-149). Mahwah, New Jersey: Lawrence Erlbaum Associates.
ground
tension
communication
Government
of Russia
Drawing of a triangle
Government
of the USA
stands for
WASHINGTON is talking with MOSCOW
stands for
stands for
General
triangle
Drawing of a triangle
Figure of a
triangle
stands for
opͯίϯٯگ]^ablmpq|}ǰɰʰѴѴh6hh3hh.2h6hJhhhJh6 h6hTh
h6hc<hYhYhYhY-ϯگ^_`abmnopq|}gdǰȰɰʰ"gdYgd6P&P {. A!n"n#$%DpDd 2J
3@@#"?Dd 2J
3@@#"?Dd 2J
3@@#"?^'666666666vvvvvvvvv666666>6666666666666666666666666666666666666666666666666hH6666666666666666666666666666666666666666666666666666666666666666662 0@P`p2( 0@P`p 0@P`p 0@P`p 0@P`p 0@P`p 0@P`p8XV~_HmH nH sH tH V`V
JNormal$dx7$8$a$CJ_HaJmH sH tHVV0 Heading 1$$dp@&a$5;CJ KH\aJ NN0 Heading 2$$d@x@&a$ 5;\@@0 Heading 3$d@@&5\DA`D0Default Paragraph FontRiR0Table Normal4
l4a(k (
0No List^/^Heading 1 Char*5CJ KH OJQJ\^JaJ mH sH tH`/`Heading 2 Char,56CJOJQJ\]^JaJmH sH tHZ/ZHeading 3 Char&5CJOJQJ\^JaJmH sH tH0"00Endnoted@6B260 Body Text
]L/AL0Body Text CharCJ^JaJmH sH tH:UQ:0 Hyperlink>*B*^JphFYbF0Document Map-D M
Z/qZ0Document Map Char CJOJQJ^JaJmH sH tHNON0PME Author/Institution$a$:O:0PME Abstract6]^o^0
PME Normal$d@x7$8$a$CJ_HaJmH sH tH\O\0
PME Heading 1$$x@&a$5;CJ KH\aJ POP0
PME Heading 2$$x@&a$ 5;\HOH0
PME Heading 3$$@&a$5\DOD0 PME Quote!d^!CJaJ>O>0PME FigTitle$xa$pp0PME Numbered transcript
I`d^I``CJaJVV0PME Transcript!I`d^I``CJaJVO"V0PME References"!d^!`CJaJVO2V0
PME Bullet'#
&F
hyy^y`JVAJY=0FollowedHyperlink>*B*^JphRRR
&60Balloon Text%dCJOJQJ^JaJ^/a^%60Balloon Text Char#6CJOJQJ^JaJmH sH tHPK![Content_Types].xmlj0Eжr(Iw},-j4 wP-t#bΙ{UTU^hd}㨫)*1P' ^W0)T9<l#$yi};~@(Hu*Dנz/0ǰ$X3aZ,D0j~3߶b~i>3\`?/[G\!-Rk.sԻ..a濭?PK!֧6_rels/.relsj0}Q%v/C/}(h"O
= C?hv=Ʌ%[xp{۵_Pѣ<1H0ORBdJE4b$q_6LR7`0̞O,En7Lib/SeеPK!kytheme/theme/themeManager.xmlM
@}w7c(EbˮCAǠҟ7՛K
Y,
e.|,H,lxɴIsQ}#Ր ֵ+!,^$j=GW)E+&
8PK!Ptheme/theme/theme1.xmlYOo6w toc'vuر-MniP@I}úama[إ4:lЯGRX^6؊>$!)O^rC$y@/yH*)UDb`}"qۋJחX^)I`nEp)liV[]1M<OP6r=zgbIguSebORD۫qu gZo~ٺlAplxpT0+[}`jzAV2Fi@qv֬5\|ʜ̭NleXdsjcs7f
W+Ն7`gȘJj|h(KD-
dXiJ؇(x$(:;˹!I_TS1?E??ZBΪmU/?~xY'y5g&/ɋ>GMGeD3Vq%'#q$8K)fw9:ĵ
x}rxwr:\TZaG*y8IjbRc|XŻǿI
u3KGnD1NIBs
RuK>V.EL+M2#'fi~Vvl{u8zH
*:(W☕
~JTe\O*tHGHY}KNP*ݾ˦TѼ9/#A7qZ$*c?qUnwN%Oi4=3ڗP
1Pm\\9Mؓ2aD];Yt\[x]}Wr|]g-
eW
)6-rCSj
id DЇAΜIqbJ#x꺃6k#ASh&ʌt(Q%p%m&]caSl=X\P1Mh9MVdDAaVB[݈fJíP|8քAV^f
Hn-"d>znǊ ة>b&2vKyϼD:,AGm\nziÙ.uχYC6OMf3or$5NHT[XF64T,ќM0E)`#5XY`פ;%1U٥m;R>QDDcpU'&LE/pm%]8firS4d7y\`JnίIR3U~7+#mqBiDi*L69mY&iHE=(K&N!V.KeLDĕ{D vEꦚdeNƟe(MN9ߜR6&3(a/DUz<{ˊYȳV)9Z[4^n5!J?Q3eBoCMm<.vpIYfZY_p[=al-Y}Nc͙ŋ4vfavl'SA8|*u{-ߟ0%M07%<ҍPK!
ѐ'theme/theme/_rels/themeManager.xml.relsM
0wooӺ&݈Э5
6?$Q
,.aic21h:qm@RN;d`o7gK(M&$R(.1r'JЊT8V"AȻHu}|$b{P8g/]QAsم(#L[PK-![Content_Types].xmlPK-!֧6+_rels/.relsPK-!kytheme/theme/themeManager.xmlPK-!Ptheme/theme/theme1.xmlPK-!
ѐ' theme/theme/_rels/themeManager.xml.relsPK]
6Mdp
6Mdp
+>5;?KZot~oʰDFHIKLMOQRTUY%;FfyϯʰEGJNPSZ[P2h2k2R4j4m4555___8 !@ (
n
*
3 "<?t
c$X99?#"n
*H2
B
#"~z
H2
#"~"z
\
##"
3
c"$?S
3c"$?G"
'<
h
##"7E,
h
#
#"
3 c"$?A
a
v n
*
3 "<?t
c$X99?#"n
*H2
B
#"~z
H2
#"~"z
\
##"
3c"$? j
3c"$?G"
m)<
h
##"E'
h
##"
3c"$?A
a
\
##"
h
##"A'
\
##"
##
3 c"$?s#am)
B n
*0
3 "<?t
c$X99?#"n
*0H2
B
#" mH2
#"m\
B
##" m 0
3c"$?R'
3c"$?!R%&
B
S ?i2k45%t %tt%tt&"##̈$Ř%̡&#'(˘)j*,-#+q,#-.</#021
2D3*45(67l-#84+9:4;t<=|6>6?6@4k"Atk"Bk"Ct^D^E^F4_Gt_ c cccrdrdjjjj(jjjjjk7nDnMnMnUnn=o=oGoQoooo8uGuz)|1|=|I~Q~]~
!"#$%cc"c"czdzdjj$j/j/jjjjjkBnKnTn_n_nnEoOoOo[ooooAuRuz/|;|E|O~[~e~
!"#$%9""*urn:schemas-microsoft-com:office:smarttagsState9%%*urn:schemas-microsoft-com:office:smarttagsplace=$$*urn:schemas-microsoft-com:office:smarttags PlaceType=&&*urn:schemas-microsoft-com:office:smarttags PlaceName&%$$"%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%SZ$V]CINV
",3&-ajr|io(e"k"p"x"""""&&&&'',,u0|0C3N33344445566666666I7O7T7\7777799;;]<c<====>>>>??s?{?????AA&B+B_BjBlBvBzBBjDrDEENEFFHH8I?IIIJJJJ.K9KL'LLLNN^PePQQQQQQRRTS\STToUyUUU:^D^F^Q^^^^^_
__$_*_2_``ccccccccAdKdddddXe`egekeffffg#ggghhhhhhii$i-iii7j=jjjjjRkZk`kdkkkkkllllllTmXmmmun}nnnnn4o;o]odoppp$popvpqqqqfrmrssstt t
ttt&t(t-tTt[tttttfumuuuuv
vvv!vvvvvvvxxx x(x1xMxVxfymyyyyyzzzz{{Z|_|||}}}}}}}}z~|[e$!!""""""**l*q*"V&VcVeVVVVV
X2XXXXXYYS8
ah
DRY3lEvOY"Y#/$e$h$y).O.
/J0{0.2w23^e3p4Y;7m9):@: K:!L:Y=I-@XFBDLBC0J9FL
NyOGRTvT
#UevZ]^R^/_Pa O_
y6bxw*9hy;RY)XmAf&c<GK I]aR>.,axd9:gFrSc
+aGS9}d0Jmuh mf4]XK5`6Yz~|~@|*+!@!A!E!FSTzh@h2hh@hJh@hPh@h`h@hX@UnknownG* Times New Roman5Symbol3.* Arial7K@Cambria5.*aTahoma?= * Courier New;WingdingsA BCambria Math"A*F*F\Uk@k@!nxx20:~:~2$P92!xxTemplate PME28
A B FuglestadNorma,Oh+'0
@LX
dpxTemplate PME28A B FuglestadNormalNorma2Microsoft Office Word@@`$@^h@^hk՜.+,0hp|
HiA@:~Template PME28Title
!"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\^_`abcdfghijklmnopqrstuvwxyz{|}~Root Entry F,
hData
]1Tablee\WordDocument8SummaryInformation(DocumentSummaryInformation8CompObjy
F'Microsoft Office Word 97-2003 Document
MSWordDocWord.Document.89q