ࡱ>  @ bjbj00 RbRb0000000D8 <<D*N*P*P*P*P*P*P*$,R.t*0###t*00*.*.*.*#v00N*.*#N*.*.*00.*x T ".$.*N**0*.*/(/.*DD0000/0.* F.*t*t*DD *DD A DEWEYAN PERSPECTIVE ON AESTHETIC IN MATHEMATICS EDUCATION Andr Mack Department of Science and Mathematics Education, Oregon State University, Weniger 249, Corvallis, OR 97331, USA When artistic objects are separated from both conditions of origin and operation in experience, a wall is built around them that renders almost opaque their general significance, with which esthetic theory deals. John Dewey, Art as Experience, 1934, p. 3. Some mathematics educational researchers (Papert, 1980; Davis & Hersh, 1981; cite) make a provocative assertion first posited by the nineteenth century French mathematician, Henri Poincar, claiming that the essence of the mathematical experience is aesthetic. While some have adequately characterized mathematical aesthetics (Hanna, 1989; Nelson, 2000; 1993; Otte, 1990; Winchester, 1990a; 1990b; Wheeler, 1990; Davis and Hersh, 1981), work in this area has proven to be of little use to the mathematics education community beyond a cursory acknowledgement of the existence of specific mathematical aesthetics (NCTM, 2000; 1989). Furthermore, the pervasiveness of aesthetic perception in the practice of mathematics (Davis and Hersh, 1981) makes its limited role in mathematics education somewhat of an unfortunate anomaly. Clearly, there exists some inherent characteristic of mathematics from which mathematicians and a small minority of mathematics students derive pleasure. Yet this particular nature of mathematics is rarely ever transferred in the course of mathematics instruction. Too few students are given the opportunity to experience mathematics from this perspective (Wang, 2001; Sinclair, 2001; Sinclair & Watson, 2001). This assumption forms the heart of oft-ignored ideological questions about the aesthetics of mathematics: What is the nature of mathematical beauty, and in what ways can it be instantiated in the context of pedagogy? This thesis stakes out the philosophical stance on contemporary educational practice first articulated by Dewey (1934), who argues that the aesthetic connection to any particular field of study be retained and sought explicitly within pedagogy. Dewey wanted to restore continuity between the refined and intensified forms of experience that are works of art and the everyday events, doings, and sufferings that are universally recognized to constitute experience. He emphasized the necessity of such an undertaking with a powerful metaphor. He writes: Mountain peaks do not float unsupported; they do not even just rest upon the earth. They are [emphasized] the earth in one of its manifest operations. It is the business of those who are concerned with the theory of the earth, geographers and geologists, to make this fact evident in its various implications. The theorist who would deal philosophically with fine art has a like task to accomplish (pp.3-4). In as much as the practice of mathematics has inherent qualities of aesthetics and artistic beauty (Papert, 1980; White, 1993; Davis and Hersh, 1981), Deweys illustration compels mathematics educators and researchers to recognize and make explicit the artistry of their discipline. Eisner (2002; 1985) also presents practical benefits of educational artistry that seem to resonate with more naturalistic, emotional components of learning. He writes: Artistry is important because teachers who function artistically in the classroom not only provide children with important sources of artistic experience, they also provide a climate that welcomes exploration and risk-taking and cultivates the disposition to play. To be able to play with ideas is to feel free to throw them into new combinations, to experiment, and even to fail. It is to be able to deliteralize perception so that fantasy, metaphor, and constructive foolishness may emerge. For it is through play that children eventually discover the limits of their ideas, test their own competencies, and formulate rules that eventually convert play into games. This vacillation between playing and gaming, between algorithms and heuristics, between structure-seeking and rule-abiding behavior, is critical for the construction of new patterns of knowing. (Eisner, 2002, p.162) Eisner connects the objectives of the educational practices with the methodologies of both artistic creation and art education. Here student play and experimentation are paramount. Current research suggests, however, that the practical tasks of these and other education reforms are both tenuous and daunting for classroom teachers, because of the pedagogical uncertainty they produce (Floden, 1997; Carpenter, 1998; Frykholm, 2000; 1998). That aside, the notion of educational artistry has hardly taken root in a significant way in the context of mathematics instruction. I claim that artistic perception could serve as a circumspect and unifying theme for mathematics education, producing efforts for successful reform on various levels, especially those with more affective aims. The shift in educational perspective towards aesthetics casts mathematics instruction from a purely mathematical vantage point. It establishes potentially beneficial connections between the practice of mathematics pedagogy and that of mathematics in general. More specifically, this rest on the idea that "if mathematical and scientific structures are seen to fully participate in the social plane, then not only are they structured by the social plane but they also structure social activity, including learning and teaching (Stoup et al. 2002, p.195)." In this case, it is mathematical aesthetic that is seen as a sublime structuring agent for classroom discourse. This article creates a conceptual framework for a research agenda designed to answer a critical question about the nature of the mathematics in mathematics education: What aesthetic qualities exist in school mathematics? To initiate this task, I draw from educational and philosophical literature, namely educational artistry (Dewey, 1934; Eisner, 2002) and mathematical aesthetic (Wang; Sinclair; Davis and Hersh, 1981). These are organized around a central theme, the projection of domain aesthetic vis--vis pedagogy. The philosophical foundation formed here is not merely new nomenclature to add to an increasingly divergent conversation about mathematics education reform. Rather it is a tightly connected framework of ideas from which to extract a theoretical model, and from that, an educational research designone that amplifies mathematical aesthetics. In the first section, I begin by defining mathematical aesthetic relative to artistic practices, and in the second, I show how it is linked to mathematics education and constructivist views of learning. Defining mathematical aesthetic In his treatise on artistic experience and aesthetic theory, Dewey (1934) creates a set of general assumptions about the common substances of art, which I find intriguing relative to my own experience as a long-time student of pure mathematics. He states as follows: Apart from some special interests, every product of art is matter and matter only, so that the contrast is not between matter and form but between matter relatively unformed and matter adequately formed (p. 191)." Here Dewey suggests that the practical emphases in aesthetic pursuit are form and organization. In an analogous way, mathematics also entails form and organization (Davis & Hersh, 1981; et al). Although the media of such organization and structuring may be somewhat more abstract than was intended by Dewey's original assertion, mathematics is said to share commonalities with artwork in general and has its own aesthetic qualities (White, 1993). This is not only interesting philosophical repartee, but a unique view of mathematics that can have powerful implications for pedagogical practices. Not unlike that of other art forms, mathematical aesthetic manifests itself through the adequate formation and coordination of a medium. According to Dewey (1934), every art uses some manipulation of the media, space, time, or space-time to project its aesthetic. In mathematical work, however, space-time is not limited to constraints of known physical reality, perhaps making it all the more difficult for one to see mathematics as an artistic experience. Rather mathematics practice uses a larger, sometimes infinite-dimensional, but well-defined medium to constitute its forms. So while qualitative distinctions in the physical artistic space-time can be easily matched to similar physical forms, as with drawings or music, some mathematical structures can only be evaluated by attaching them to structurally equivalent constructs that exist only in the imagined mathematical space. If none exists, then one may be created arbitrarily. Such breadth and versatility of the mathematical space is further indication that it is a purely abstract space. While it is important to acknowledge that the medium for mathematics is pure abstract thought (Davis & Hersh, 1981), I call specific attention to the fact that "[media] are the middle, intervening, things through which something now remote is brought to pass (Dewey, 1934, p.197). Dewey extends this idea, writing, Every art so uses its substantial medium as to give complexity of parts to the unity of its creations (p.202). So not only do aesthetic qualities emerge from the media of art forms, it is the specific arrangement of the medium into one coherent construct by which an aesthetic achievement comes to pass. Similarly in mathematics, artistry is revealed as much in the ways that mathematicians use reason and logical arguments to explain the connections between the various ideas, as in the ideas themselves (Hanna, 1989). Characterizing aesthetic pursuit as an attempt to create one "beautiful" object by coordinating a set of diverse parts is at the heart Dewey's argument. Variations within each dimension of media and the transformations thereof can create an aesthetically appealing unified form (Dewey, 1934). This is strikingly similar to Herstein's (1964) explanation of the basic building block of abstract algebra, the group. He suggests that much of this important branch of modern mathematics is concerned with finding and explaining how different objects, which have obvious qualitative differences, can be lumped into one equivalence class with a few, very general commonalities among them. They are all different in some respects. On another level, however, a group theoretical level, they are simply variations of the same form. Once the layer of qualitative adornment is lifted, the objects are indistinguishable. Herstein explains: The systems chosen for study are chosen because particular cases of these structures have appeared time and time again, because someone finally noted that these special cases were indeed special instances of a general phenomenon, because one notices analogies between two highly disparate mathematical objects and so is led to a search for the root of theses analogies. To cite an example, case after case after case of the special object, which we know today as groups, was studied toward the end of the eighteenth, and at the beginning of the nineteenth century that the notion of an abstract group was introduced. The only algebraic structures, so far encountered, that have stood the test of time and have survived to become of importance, have been those based on a broad and tall pillar of special cases. Amongst mathematicians neither the beauty nor the significance of the first example, which we have chosen to discuss--groups--is disputed. (p.27) In other words, this one topic in mathematics, Group Theory, has its aesthetic in finding ways to construct a single form from a seemingly diverse set of objects by finding their general commonalities in what Herstein termed the root of analogies. Yet Davis and Hersh (1981) argue much more generally that mathematical aesthetic itself is derived from the interplay between variation, randomness and diversity on one the hand and singularity, precision and unity on the other. While many other mathematics connoisseurs (Nelson, 2000; 1993; Tymoczko, 1993) give ideas about specific instances of mathematical beauty, Davis and Hersh (1981) broaden the notion of aesthetic pursuit in mathematics to a more minimally-defined task of finding precise order or patterns from an unstructured medium. They write, A sense of strong personal aesthetic delight derives from the phenomenon that can be termed order out of chaos. To some extent the whole object of mathematics is to create order where previously chaos seemed to reign, to extract structure and invariance from the midst of disarray and turmoil (p. 172). It follows that mathematical work, like other aesthetic pursuits, produces aesthetically-pleasing artifacts. Although they may be much more abstract and nebulous that the usual, these artifacts are themselves media for artistic expression in mathematics. Dewey (1934) states that artwork refers to both the process and the end result of aesthetic pursuit. Likewise, mathematics practice entails a similar coalescence of means and ends. Mathematical objects such as groups, geometric constructions, theorems, and proofs fit well within general conceptions of artwork, because they carry aesthetic qualities similar to that of other forms of expression in the arts (Eisner, 2002; 1985; Dewey, 1934). For example, the aesthetic quality of mathematical proofs is captured in the organization and coordination of facts into a consistent and detailed explanation of a structure and/or space (De Villiers, 1999; Hersh, 1993; Hanna, 1989; Van Dormolen, 1977). It is specifically the elegance, simplicity and clarity of the explanations of "good" proofs, which contains so much of the aesthetic appeal for both mathematics and mathematics education (Knuth, 1999, 2000; Mack, 2000). In this sense, it is a didactic aesthetic. If the beauty of mathematical ideas is revealed, at least in part, through the quality of its explanation, then there exists some basis for critique in the art of mathematics" (Tymoczko, 1993). Not only is the theorem itself left behind as an artifact to be critiqued, but the proof of the theorem as well. Furthermore, it is in this explanatory role of proof that I find glimpses of a possibility for thinking about an aesthetic in mathematics pedagogy. Some proofs, after all, are clearly better explanations of a phenomenon than others. It is without question that part of the work of mathematics teaching is to choose between alternative explanations, seeking the most powerful and elegant for a specific instructional taskappealing to a pedagogical aesthetic of mathematical proof (Knuth, 1999, 2000; Mack, 2002). Personal intuition, creativity, and cognitive payoffs account for variations between proofs, making each one qualitatively distinct (Wheeler, 1990a; b). This raises a pressing question for researchers. What knowledge is entailed in teachers detecting and critiquing such differences? The dualistic nature of pedagogical aesthetic of mathematics suggests that it takes more than simply understanding mathematical constructs. Indeed, some (Sinclair, 2001; Ball and Bass, 2000; Knuth, 2000a; b) have described the work of mathematics teaching as requiring an appreciation of those constructs and their origin in the aesthetic of elegant explanations. From this perspective not on is the practice of mathematics entirely humanistic and artistic, but so is mathematics pedagogy. Like Winchester (1990a) explains, mathematical thought in general is part of the dialectical thought of humankind and possesses the vagueness, the complexity, and the progress and regress of our thought in general (p. v). The implication of his argument for the field of mathematics education is that humanistic mathematics demands student initiative, student independence, indeed creativity of both teacher and student in the mathematics classroom (Hersh, 1993, p. 15). As was previously demonstrated, mathematics practice may be defined, at least partly, in terms of artistic connoisseurship and critique (Papert, 1981). The central issue is whether not this includes the mathematics that emerges (or not) in the course of schooling. Some have already observed that the mathematical ability needed to teach mathematics from the ideological perspective of reform requires both the technical skills needed to understand mathematics from a mathematicians perspective and the sensibilities needed to make the necessary choices and delicate adjustments to explanations that will allow the most number of students to access its meaning (Ball & Bass, 2000; 1999; Ball, 2000; Mack, 2002). Connoisseurship, as well as the creation of an artifact of aesthetic involves the ability to manipulate form (media) in order to express and create within the observer the desired emotional response (Dewey, 1934; Eisner, 1985). Yet, according to Eisner, the artist and the art connoisseur must share a particular refinement of sensibilities in order to perceive the aesthetic. Unfortunately, very few have the necessary skills and sensibilities to perceive mathematical aesthetics. As such, these have often been both the creators and the sole connoisseurs of mathematical aesthetic (Winchester, 1990b). As with other artwork, artifacts of mathematics activity are shaped and designed by constraints dictated by aesthetic sensibilities kept in mind by the mathematician. Some (Davis & Hersh, 1981; Davis & Reuben, 1998) argue that aesthetic sensibilities are part and parcel of the mathematics epistemology. To illustrate the point, Davis and Reuben (1998) explain that mathematics is replete with both existence theorems and uniqueness theorems. Yet existence simply implies that there must be a certain solution to a problem. Uniqueness, on the other hand, implies a much more well-defined construct, complete with detailed explanation of its make-up. Uniqueness theorems are therefore, more aesthetically pleasing to mathematicians because of their explanatory power (Davis & Hersh, 1998). Even when existence is the only clear image that can be characterized by a theorem, the field of mathematics has sought to suppress the ambiguities created by mere existence theorems, creating families or superentities that are themselves unique and viewable in closer detail (p.118). Appreciation of the aesthetic difference between the two, existence and uniqueness theorems, requires certain mathematical sensibilities. But where do such sensibilities originate? How are they formed? My contention is that very nature of mathematical aesthetic raises questions that are most relevant to mathematics education reform efforts. Educational researchers continue to have success in their quest to ground mathematics education research more deeply within the domain of mathematics itself (Shulman, 1987; Ma, 1999; Ball & Bass, 2000); however, I argue that a vital component remains absent from traditional research perspectives, namely the aesthetic. I conjecture that school mathematics includes some variant of aesthetic (quality notwithstanding), just as inherent in mathematics pedagogy as in mathematics proper. Yet provocative ideas and titles such as Humanistic Mathematics (White, 1993) and Creativity, Thought and Mathematical Proof (Winchester, 1990) bring the notion of aesthetic to the foreground without clearly defining mathematical beauty for the purposes of educational research or practice. It is my claim that notions of proof and reasoning are partly determined by ones sense of beauty in mathematics, and for teachers this include an aesthetic perception of mathematics teaching as well, since it too is mathematical in nature (Ball & Bass, 2000; Lampert, 1998; Carpenter, 1990). It was Papert's (1980) assertions about mathematics practices being rooted in aesthetic pursuit that first led me to question the nature of aesthetic in mathematics pedagogy: To what extent can the mathematical practices and artifacts entailed in teaching and learning be seen as aesthetic constructs? How do teaching and learning mathematics compare with artistic experiences as expressed by Dewey (1934) and Eisner (2002)? Answers to these and other similar questions are important for establishing a methodology for analyzing data from the present case study. Such analysis will be used as evidentiary support for and instantiation of the mathematical aesthetic embedded in the practice of instruction. Mathematical aesthetic, epistemic seeing and constructivism Some researchers (Wang, 2001; Foshay, 1991) have noticed a dearth of research literature on aesthetics in mathematics education. A possible result of this is that there exists a contradiction in the way that mathematics education describes mathematics as a rigorous" activity while a more robust point of view of it prevails amongst mathematicians as imaginative thinking. Despite its reputation as strict and rigorous, mathematicians often speak of mathematical experiences as being filled of transcendent, powerful, astonishing, majestic ideas (Sinclair & Watson, 2001). Translating this experience to students through pedagogy, however, has proven to be a difficult challenge for the system of mathematics education (Wang, 2001). Conceptual and curricular sequencing in mathematics education is patterned after the same purely logical sequence as that of formal mathematics structure (Steiner, 1987), despite the fact that historically, the process of mathematical inquiry sometimes followed a different path (Dennis, 2000). That path of advancement of ideas by mathematicians reflects the stop-and-go vacillation that characterizes the account in Lakatos (1973) Proofs and Refutations: the Logic of Mathematical Discovery. One implication of this is that the learning of mathematics itself may not proceed along purely logical lines, despite proposition to the contrary made by the curricular structure of traditional schooling. If that is so, then the philosophical grounding of the curriculum and instruction of contemporary mathematics education is weakened by this erroneous assumption and leaves gaps between theory and practice in mathematics education reform. On the other hand, learning sometimes follows a more anthropological/historical path (Schaverien & Cosgrove, 2000; 1999), where a students cognitive development often retraces the line of progressive thinking of past mathematicians or even whole civilizations (Ball & Bass, 2000). The two alternate trajectories for mathematics education, logical versus historical, reflect the difference between utilitarian and aesthetic bases for natural mathematical inquiry. Brady and Kumar (2000) espouse an educational philosophy that entails the motive of the inventor [of scientific and mathematics ideas] as a way to access what is the essence of mathematical discovery and inquiry. They write: In using ideas to create new knowledge, the mathematician does not structure them in the same formal way that they would be structured for communication. In the process of the mathematicians learning something new, process and content are inextricably linked. What Clairaut called the motive of the inventor cannot simply be written down and learned from reading. Clairaut implied that mathematics cannot be adequately learned unless one is searching for and discovering mathematics in the process of working on problems. Clairauts distinctions between creating and communicating mathematics are, like Bourbakis, slippery. He asserted that the originators of mathematics are motivated to present their discoveries in a formal manner, but leaves the mathematics learner in the realm of continuous involvement with problems. (p. 60) This applies not only to mathematics learning, but has implications for teaching as well. As teaching often requires leaning the mathematics of others, it is an instantiation of real mathematical inquiry. It follows, then, that designing instruction to reveal the true motive of the inventor is a way to both support and model the practices of mathematicians. According to mathematicians who write about it, the motive of mathematicians is aesthetic (Tymoczko, 1993; Papert, 1980). In a related way, this argument has been extended to science learning and the process of scientific inquiry (Brady & Kumar, 2000). Emphasis on aesthetic influence in the process of scientific inquiry as it is done in the scientific community, though seldom expressed by educators, gives student a clearer understanding of the science. Brady and Kumar argue that the more teachers are aware of the interconnection of science and aesthetic the more likely they are to improve the pedagogical process as a whole by moving it closer to a truer representation of science. They cite the fact that Noble Prize winners and other prestigious figures in science often attribute scientific breakthroughs to aesthetic inspiration. The interaction between the two, science and aesthetic, is a part of their experience in science. Sinclair and Watson (2001) also espouse the use of mathematical aesthetic in education to inspire in students a sense of wonder about mathematics. This rephrases the original question of the nature of mathematical aesthetic in pedagogy to a more practical one: In what ways are the phenomenological experiences of mathematicians reproduced in mathematics pedagogy? Furthermore, they argue for the structuring [of] students experiences of mathematics so that results which might otherwise seem commonplace emerge as surprising special cases (p. 40). Commenting on Fishers (1998) book Rainbows, Wonder, and the Aesthetic of Rare Experiences, they write: Attention to wonder provides a pathway to seeing something spiritual in learning about mathematical structure, in seeing mathematical results as surprises, in following up first instances of wonder with small steps of further wonder, in exploiting the visual and the intuitive, both fueling and being fuelled by curiosity. (p. 40) Sinclair and Watson assume that wonder is a common-enough experience for us to offer situations which we and others find wonderful and learn, from our students, about the universality (or not) of wonder (p. 40). This raises key questions for the research community and mathematics educators alike. How do students learn that in which mathematicians find aesthetically pleasing? What instructional designs give students authentic mathematical experiences, including aesthetic? With pedagogical goals framed in these terms, mathematics instruction becomes a much more complex venture. Levels of analysis of instructional designs as well as the focus of that analysis begin to take on a theme of educational connoisseurship as described by Eisner (2002; 1985). Wang (2001) has attempted to answer the aforementioned questions. He prescribes a pedagogy that would allow students enough time immersed in a mathematical experience to allow transcendent understanding of mathematic. He also stresses the importance of creating opportunities for students to experience mathematical aesthetic by placing education of the imagination as the highest curricular priority in mathematics education. However, neither the research community nor mathematics educators have come up with a practical framework to achieve have this same priority. This connection between mathematics instruction and aesthetics casts the teacher as a connoisseur of both mathematics and mathematics pedagogy. On the connoisseurship of art in general, Dewey (1934) asserts, For art is a selection of what is significant, with rejection by the very same impulse of what is irrelevant (p. 208). This assertion suggests an operational definition of connoisseurship, even for mathematics. It is perceptive skill specifically that accounts for the ability to find what is the salient aesthetic of mathematics as an art form. However, the research community has made few attempts to clarify what is meant by mathematical aesthetic, leaving open the question of mathematical connoisseurship. What particular sensibilities are entailed in perceiving mathematical aesthetic? The answer to this question is important in establishing an epistemology in mathematics education that includes connoisseurship of mathematics. I hypothesize that the expertise of mathematics educators is revealed partly by their perception of mathematics practice as an aesthetic pursuit, replete with emotionally moving experiences. Dewey further states: In ordinary perception we depend on contribution from a variety of sources for our understanding of the meaning of what we are undergoing. The artistic use of a medium signifies that irrelevant aides are excluded and in one sense quality is concentratedly and intensely used to do the work usually done loosely with the aid of many. (p.201) Deweys quote highlights the special, non-ordinary nature of artistic perception. Likewise, mathematics requires of educators a specialized perceptive ability to find mathematical aesthetic as well as to project it to their students. The notion of specialized perception is also reflected in Eisners (2002) educational connoisseurship, reiterating the need for artistically-minded teachers who function as efficient designers of the social space for the purpose of mathematical creations. Here the focus of perception is not so much on the mathematics per se, but on various students understandings of mathematics. Eisner describes this epistemic seeing, shifting between alternative vantage points and various levels of perspective. This facet of educational connoisseurship involves what Eisner termed epistemic seeing. Epistemic seeing, he explains, allows the perceiver to envision alternatively viable constructions of a subject. Variations in perception occur both in different qualities and levels of perspective. Epistemic seeing, then, is a way of perceiving the world through multiple lenses from various perspectives, as many as possible. This idea resonates with the radical constructivist ideology of Von Glasersfeld (1989). He argues that cognition is in fact ones own private perception of reality. According to radical constructivist views, cognition and therefore, perception of reality, even that of mathematical constructs, varies from person to person. Moreover, in any classroom, there exist as many different epistemologies as students. In addition to variations between students, there also exist variations in one students perception over time, adding yet another level of complexity to the problem of mathematics teaching. As epistemic seeing entails perception and heightened sensitivity, it is, thereby, an exercise of making meaning from real world experiences. Epistemic seeing, as Eisner defines it, uses constructivist ideologies as a point of departure. And although constructivism has been studied in many varied forms of philosophy and learning theories, it is at its core an epistemic process. Constructivism, particularly the radical variety, is based upon the knowers perception (Von Glasersfeld, 1989). Subsuming Piagets (1975) notion of cognitive change vis--vis accommodation, whereby one is compelled to reconstruct his personal version of reality in order to accommodate a novel experience, constructivist accounts of learning also entail the shifting of perception and perspective. Inasmuch as artistry and aesthetic pursuits entail the perception of the doer, they are also products of epistemologies. The notion of epistemic seeing begins with constructivism in that it presumes that perception and furthermore, learning are not just receiving, but rather making. Perception is, in the end, a cognitive event. What we see is not simply a function of what we take from the world, but what we make of it (Eisner, 2002, p. xii). Artistic ways of knowing seek to liberate the imagination in such a way as to broaden the space for more possibilities of meaningful connections. [Work in the arts] is a way of creating our lives by expanding our consciousness, shaping our dispositions, satisfying our quest for meaning, establishing contact with others, and sharing culture (Eisner, 2002, p. 3). As mathematics is, in a sense, a pursuit of al aesthetic and work in the arts (Papert, 1980) it also expands ones space for thinking. This description of intellectual growth is also reflected in the work of Duckworth (1996), who writes, knowledge is broadened, in the sense that [one] can conceive of (which means that he can act on) the world in more varied ways; it is deepened, in the sense that he can know more aspects of one given object or situation (p.41). In a similar way, Eisner describes how epistemic seeing also fosters intellectual growth and development. Although some readers may find it somewhat paradoxical to equate the extremely rational mathematics discipline with the very emotional idea of aesthetic, Scheffler (1976) articulated well the notion that human cognition is never fully bifurcated into its rational and emotional parts. He writes, The growth of cognition is thus, in fact, inseparable from the education of the emotions (p. 15). From this idea he developed the construct of cognitive emotions, which include rational passions, perceptive feelings, and theoretical imaginations. Successful mathematical proofs, I argue, have a similar quality of connectedness between cognition and emotion. I posit that Schefflers construct, cognitive emotions, is the most appropriate classification for the notion of aesthetic in mathematical reasoning. One might refer to this as a cognitive aesthetic, whereby In as much as mathematical inquiry is motivated by wonder and curiosity, it points specifically to the interconnection between rational thought and emotions, as described by Scheffler (1976). Surprise and awe are generated by ones desire to make sense (in a logical or common sense way) of mathematical constructs and ideas, which satisfies the sometimes emotionally unsettling cognitive perturbations (Piaget, 1975). This idea is another connection between aesthetics and constructivism. Mathematical aesthetic, I would argue, is an account of the convergence of both the cognitive and emotional parts of the mind. Conclusion In this essay, I have attempted to characterize mathematical knowledge in a way that combines content and pedagogy. Mathematical aesthetic is offered here as one possible combination. I argue that, in some sense, pedagogical decisions are already informed by the educators perception of mathematical aesthetic, in particular the didactic aesthetic of explanations, mentioned earlier. Unfortunately, there is no known framework for evaluating and critiquing the mathematics that emerges in the classroom, relative to it aesthetic value. This leaves open these pressing questions of our mathematics educational system: What is the nature of school mathematics? How does its aesthetic value compare to that of mathematics proper? These questions of the mathematical aesthetic in mathematics pedagogy are, at their core, questions of the quality of contemporary mathematics education, in general. This is a reflection of both the mathematical thinking and sensibilities of mathematics teacher, as well as curriculum designers. 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