Basic Irony: Examining the Foundations of School Mathematics With Preservice Teachers

BRENT DAVIS

BASIC IRONY: EXAMINING THE FOUNDATIONS OF SCHOOLMATHEMATICS WITH PRESERVICE TEACHERS

ABSTRACT. This article reports on an inquiry into what tends to be taken for grantedwith regard to the teaching and learning of mathematics. The inquiry, undertaken in thecontext of a course on methods for mathematics teaching, was developed around an exam-ination of the mathematical notions that infuse conventional theories of cognition and thatpermeate the structures and practices of school mathematics. In particular, concepts drawnfrom or aligned with Euclidean geometry were examined. Specifically, alternatives drawnfrom fractal geometry were explored. The importance of interrogating the often-transparentfigurative underpinnings of our thinking about thinking is highlighted.

THE PROBLEM OF THE BASICS

As a mathematics educator and a teacher educator, among the things I havefound most troubling in my professional life are teachers’ and prospectiveteachers’ thoughts on the nature of mathematics and on the role of math-ematics in their lives – or, more precisely, their lack of thoughts on thesetopics.

The issue was brought home in a recent introductory level course onmethods for teaching mathematics. Early in the term, I requested a groupof 19 future secondary teachers to compose for presentation in class a briefresponse to the question,What is mathematics?The assignment includedthe qualification that I was not after a formal definition. Rather, the inten-tion was to spark their thinking about what it was they would later beteaching. Because all of the teacher candidates had completed a baccalau-reate degree in mathematics or a mathematics-related field, and most ofthem had taken a course in the history and philosophy of mathematics, Iexpected that their responses would prompt some interesting discussionand debate.

The following week’s class began with the reading of the descriptions,during which I collected key points and common themes on the chalkboardfor later discussion. It quickly became apparent, however, that there wasnot going to be much to mull over, as the contributions were easily cat-egorized according to three repeated ideas: mathematics as “the study of

Journal of Mathematics Teacher Education2: 25–48, 1999.© 1999Kluwer Academic Publishers. Printed in the Netherlands.

26 BRENT DAVIS

relationships”, a notion that had been suggested by another instructor theprevious week; mathematics as “the study of pattern”, which had comeup in an assigned reading; and formal definitions drawn verbatim fromdictionaries of mathematics. Not one person had undertaken to composeher or his own description. My impression was that only a few suspectedtheir contributions might be lacking in any way. Whether due to misin-terpretation of my intentions or to some other factor, their descriptionsfailed to demonstrate an awareness of the complexity of thought and thediversity of opinion that are present in discussions of what constitutesmathematics.

Personally convinced that there is little hope for substantive changein the culture of school mathematics without deepened appreciations ofenacted beliefs on the nature of the discipline – and, of course, of therationales for teaching the subject matter that fall out of those beliefs –I set about to prompt more critical understandings of both the nature ofmathematical knowledge and the relevance for teachers of engaging withthat issue. This article represents, in some ways, a report of my ongoingefforts toward these ends.

The article is developed around a teaching strategy I call abasic irony,borrowed from Rorty’s (1989) notion of irony. In brief, Rorty describedan irony as the “opposite of common sense” (p. 74) which involves thedeliberate interrogation and reformulation of mindsets and worldviews. Asan intellectual stance, to be ironic is to be willing to question the familiar,to trouble the taken-for-granted, to seek out the transparent prejudices thatinform perception.

Irony, in this sense, is a deliberate effort to turn something onto itself.In his analysis, Rorty (1989) focused on the development of new figura-tive language as a means of revealing and interrupting the metaphors andimages that are used to make sense of various concepts and phenomena.Such efforts, he argued, render commonsensical notions more opaque andmore available for critical examination. By way of example, a popularmetaphor in discussions of education and psychology isbrain as computer.Although originally intended as an analogy that might offer insight intoparticular cognitive processes, the figurative dimension of this notion isoften forgotten. In fact, brain as computer is now commonly presented asa literal truth. In recent years, however, there has been considerable effortdevoted to revealing the limited and limiting nature of this metaphor, adiscussion which, ironically, relies on the invention and deployment of newmetaphors.

For Rorty (1989), such ironies simultaneously depend on and revealthe play of language. In the process, they highlight the uncertainty and

BASIC IRONY: EXAMINING FOUNDATIONS 27

volatility of our language-dependent interpretations of the world. Ironies,however, need not be strictly focused on vocabulary. Gödel’s (1931)Incompleteness Theorems, for example, are instances of ironies in thecontext of mathematics. In essence, Gödel used a privileged form of argu-mentation to reveal its own limitations. In a similar, but less sophisticatedmanner, this paper is intended to be ironic. In it, I use a linear argument tochallenge the appropriateness of thinking about mathematics teaching inlinear terms.

In my teaching methods course, the focus of the exercise in irony wasthe manner in which mathematics has permeated modern and Westernsensibilities. The context for the exercise was an introductory study offractal geometry. In brief, I sought to adapt the strategy of turning thingsonto themselves through an examination of the history and some of thesurprising results of this recent branch of mathematical inquiry. My hopewas to engage class members in a process of uncovering and recognizinglong-established mathematized notions that have come to infuse uncriticalunderstandings of the nature of the subject matter and that have seepedbeyond the conventional bounds of the discipline to serve as common sensenotions for describing and explaining the world.

My use of the termbasicto qualify irony is also intended to be ironic.As will be developed, one of the preoccupations of mathematics educa-tion has been with the basics, albeit the notion is subject to a diversityof interpretations. I employ it here both to point to the ongoing worriesabout basics and as a reminder that even our basics seem to have a basis.I maintain that it is the more fundamental category, the one that tends tobe accepted uncritically, that should be the subject of our basic concerns.I personally find it ironic that a field that is so preoccupied with basicsis, for the most part, so out of touch with the foundations of its ownactivity.

I use a teaching episode to frame this discussion, but my intention indoing so is not to prescribe a course structure for others. As I hope will beclear, the classroom happenings were dependent on a host of contingen-cies, including who was present, what some had seen on television, andwhat other instructors were suggesting. Moreover, references to my ownteaching should not be read as attempts to present evidence in support ofan argument. They are, rather, for the purpose of developing the notion of abasic irony as a pedagogical device – one which, in its capacity to uncoverunconscious knowings and doings, may help to transform how we thinkabout mathematics, education, learning, and teaching.

Although not a principal purpose of this article, I feel it important toattempt some articulation of the attitude toward teaching and formal edu-

28 BRENT DAVIS

cation that infuses this writing. In brief, I do not regard schooling as aninnocent activity. It is always and already implicated in the transformationof culture whether or not we are consciously aware of it. Moreover, aswith all realms of human activity, it is complicit in the manner in whichhumanity understands and acts out its relationship to what is perceivedas the non-human part of the universe. Mathematics has played a keyrole in the dichotomization of humanity and nature, just as more recentdevelopments in mathematics have worked to problematize the distinctionwe draw between our species and the rest of the biosphere. As such, Iwrite from the perspective that educators must be more deliberate partic-ipants in making culture, and this entails conscious efforts to erase theperceived and enacted distance between ourselves and the complex livingworld.

USING FRACTALS TO EXPLORE THE ISSUE OF BASICS

There were two reasons for choosing the topic of fractal geometry as thecontext for this inquiry. First, fractal geometry is a relatively new domainof mathematical inquiry. As such, it provides a location for examining thedynamic character and unpredictable directions of mathematics research.Given the unprecedented number and the currency of the histories of thisbranch of mathematics, fractal geometry provides a rich site for wonderingabout what it is that mathematicians do, how mathematical research pro-ceeds, how mathematics itself evolves, and how our formal mathematicsshapes and reshapes our more unformulated perceptions and involvementsin the world. For the mathematics teacher, for example, fractal geometryhas the capacity to interrupt common sense, as it provides a set of imagesand metaphors that might be used to examine some of the transparentbackdrop of conventional educational practice. The Euclidean notions thatunderpin linear lesson plans, hierarchical curricula, grid-like classroomarrangements, and rigid boundaries of subjects become more apparent inthe context of forms that are nonlinear and dynamic, and whose bound-aries slip between dimensions. In the same way that cultural idiosyncrasiesand worldviews are revealed when a society’s customs are cast againstthose of other cultures, so the usually invisible aspects of one mathemat-ical system become more apparent when that system is juxtaposed withanother.

The second reason for developing the basic irony around fractal geom-etry was more local: As a topic that was relatively new to the methodsstudents, fractal geometry provided an opportunity to monitor our ownprocesses of mathematics learning. In terms of the issue guiding the


Figure 1. A Fractal Card. (The photograph is courtesy of Barbara Budd of the VancouverSchool District.)

inquiry, I felt it critical to be immersed in mathematical activity whilewondering about the nature of mathematics. A tenet of interpretive inquiryis that one cannot stand outside the object of one’s investigation, but must,rather, consider how one is shaped while participating in shaping thatobject.

The basic irony was introduced through the construction of FractalCards, an activity that involves establishing and recursively applying asequence of folds and cuts on paper in order to create a three-dimensionalrepresentation of a fractal image (see Simmt & Davis, 1998). It beganwith little formal introduction. Several completed cards were presentedand some terms that pertained to their construction were defined before theclass was launched into the guided construction of the example pictured inFigure 1. Within a few minutes, all present had generated a card, and agood portion of the class had begun experimenting with other cuts, folds,and combinations.

Most of the introductory three-hour time block was devoted to playingwith the activity: exploring variations, attempting to recreate completedcards, noticing relationships, generalizing patterns, and identifying topicsin mandated curriculum documents that might be addressed through thisactivity. As it turns out, virtually every topic in our provincial curriculumwas at least touched on.

30 BRENT DAVIS

UNCOVERING ASSUMPTIONS

Prior to the second class into the topic, I had assembled some questionsto assist with orienting the discussion of the fractal cards activity withthe intention of getting at the issue of the nature of mathematics: How isthis activity mathematical? How does it problematize the earlier defini-tions? What does it say about the process of mathematical research? Is thisdomain of inquiry about discovery? Creation? Both? Neither?

The discussion went in quite a different direction, however. The firstand, as it turned out, the most persistent issue to arise had to do withfitting the activity into the established school curriculum. Other topics ofconcern included questions around structuring the activity to reduce ambi-guity, ensuring that all necessary background topics had been covered, andevaluating student performances.

In other words, there was nothing ironic about the discussion: Theintended ground of the task became the stubborn figure of student con-cern. The fractal cards activity, rather than serving as the backdrop of aninterrogation of what tends to be taken for granted in mathematics teach-ing, was treated like any other classroom activity. It became an item tobe dissected, located, and catalogued within the matrix of the mandatedcurriculum.

Rather than despairing of my plans and giving in to the desire to controlthe direction of the learning – which would have amounted to a fallingin to the same mindset that I was seeking to avoid – I chose to let thediscussion continue around these issues for the entire first half of the class.As it turned out, several important considerations did come up, includingmaking teaching more attentive and responsive to learner actions and mov-ing the teaching focus away from isolated concepts to contexts wherein anarray of ideas arise. These served as the starting place of the post-breakportion of the session.

In that second half, I more deliberately attempted to prompt an inver-sion of the figure and ground of the earlier discussion, seeking to usefractal geometry as a vehicle for experiencing some of the preoccupationsabout teaching that had just been identified. The first point of examinationwas the unquestioned assumption that an official curriculum documentprovided the final word on which topics were to be addressed and inwhat order they should be covered. I began by recounting some of thedetails of the emergence of fractal geometry as a domain of mathemat-ical inquiry. The fits-and-starts nature of its evolution, its tumultuousstatus within a more formalist culture, and its reliance on electronic tech-nologies and trial-and-error are qualities that suggest the development offractal geometry is better thought of in terms of its own images than in


terms of the linear logic that is more typically used to describe traditionalmathematics.

There was no disagreement with this idea, and a few students suggestedthat the analogy could be taken a step further: The activities of a singlemathematician might be seen as a similar phenomenon to the more generalrealm of mathematics research. Notions of non-linearity and recursivityseemed to be appropriate in describing both. Additionally, qualities of thecomplex relationship between mathematician and mathematics were nicelyhighlighted by the fractal image. The whole (of mathematics) is not a mereassemblage of the parts (mathematicians); rather the whole seems to unfoldfrom the part and to be enfolded in the part.

The discussion then turned to a comparison of the fractal cards activityto the dominant structures of mathematics lessons in the students’ owneducational histories. The lack of rigidly specified learning goals, the rangeof mathematical notions addressed, the participatory role of the teacher,the unpredictable but sophisticated directions of inquiry, and the mannerin which the classroom investigation seemed to reflect many aspects ofresearch in mathematics were all cited as features that separated the activ-ity from prior experiences. After these and other points were discussed, Iventured to suggest that, like the structure of mathematical inquiry, mathe-matics learning and school curricula might be better thought of in terms ofthe nonlinear imagery and recursive structures offered by fractal geometrythan in terms of the lines, grids, and neatly distinguished areas aligned withmore Euclidean notions. That is, fractal imagery was useful for describ-ing phenomena that ranged from mathematics research to the activities ofindividuals in the mathematics class.

This suggestion was readily accepted with regard to individual cogni-tion and mathematics research. Such phenomena, it was agreed, are far toocomplex to be understood in terms of simple trajectories or linear causes-and-effects. The analogy, however, met with resistance on the issue offormal mathematics curricula. Despite the emergent structure of the fractalcards activity, it was argued that a formal curriculum represents an attemptto order existence. Unlike the complex and unpredictable nature of theresearch process, schooling is intended to tame rather than to embrace thechaotic backdrop of life. And this assertion is supported by virtually allof the textbooks, the teacher’s manuals, and the programs of study thatline classroom bookshelves. The purpose of mathematics teaching, it wasargued, is to equip learners with the basics for living, not to alert learnersto the obvious complexities of existence.

32 BRENT DAVIS

RECONCEPTUALIZING THE NOTION OF BASICS

The termbasics, in fact, arose with a surprising frequency. Suspecting thatthere was consensus on neither the meaning of the term nor on the empha-sis that should be afforded the basics in the mathematics classroom, I endedthe second class by assigning the task of preparing a brief description ofwhat each person meant by the wordbasics.

By the following week, most of the students noticed that they had beenusing the term somewhat ambiguously. For the most part, their use wasin reference both to those concepts that are considered fundamental toother concepts, e.g., in the way that counting is basic to addition, and tothose competencies that have been deemed necessary for an adult citizenin our society. It was further noted that, in both cases, what is basic isclearly neither pre-given nor stable, but dependent on particular interestsand social circumstances.

Despite the apparent variations in meaning, however, the similarity ofboth senses was quickly noted by students. Each seems to derive froma conception of knowledge as rooted in fundamentals that can be spec-ified and upon which more sophisticated understandings can be erected.Hardly confined to discussions of school mathematics, this conception ofknowledge is deeply inscribed in Western academic traditions. The mod-ern desires “to get to the bottom of things”, “to identify root causes”, “toreduce to first principles”, and – ironically, as one student noted with regardto this exercise in irony – “to interrogate the ground” of particular activitiesare affiliated with this conception to some extent.

Modern mathematics serves as the principal, although not exclusive,model for the conception of knowledge that suggests that all valid claimscan be traced to a few self-evident premises. The teaching of mathemat-ics has borrowed this assumed structure – which is to say, mathematicspowerfully informs its own teaching as it offers not just a collection oftopics to be studied but a relational structure that has been interpretedas a somewhat prescriptive learning sequence. Although the myth of thatpristine and unambiguous structure has been punctured by mathematicsitself, the rational argument is sobasic to Western mindsets that it con-tinues as the dominant image after which the teaching of mathematics ispatterned.

In this context, I suggested an alternative interpretation of basics, viz.,as the knowledge that has slipped into unformulated activity, as opposed toa list of concepts or competencies. Basics, in this sense, are the tacit groundof activity and perception. This idea was discussed by Grumet (1995) whocontrasted it to more popular interpretations:


The basics [as popularly conceived] are drawn from a fantasy of communion, and weproject this wish on to history, mistaking the rounded edges of the past for a perfect circleof consensus. When the fantasy shifts location from historical sentimentality to a currentcurriculum, it becomes an agenda of control imposed on a community whose diversitysplinters the steady rhythms of shared lives. Rather than describing an existing consensus,the basics are deployed to create an arbitrary compact. (p. 15)

Whether the basics are regarded as a narrow band of testable conceptsor a more hazily defined collection of social competencies, they remainarbitrary compacts, intended to deny the complexities of existence.

In other words, both sides of the ever-present back-to-basics debatemight be making the same error, which amounts to an ironic failure toattend to the basics – that is, to the predominant worldview that is tacitly,but pervasively, enacted in our schools. This point was highlighted in arecent exchange between Lynne Cheney and Thomas Romberg in the edi-torial pages ofThe New York Times(1997, August 11, p. A13). Cheney’sfamiliar argument is summed up in her final sentence: “If we want ourchildren to be mathematically competent and creative, we must give thema base of knowledge [read: a mastery of particular procedures] upon whichthey can build.” Unlike Cheney, Romberg attempted to interrupt some ofthe uninterrogated ground of these sorts of debates, delving into the natureof mathematics and rationales for teaching it. However, he ended up inalmost the same place: “Unless we reform math education so that ourchildren can be prepared for the immense technological changes alreadyoccurring, our nation will lose – and so will our children.”

Granted, these two discussants argue for slightly different sets of com-petencies – Cheney for what is more classically considered as basic,Romberg for the abilities to “communicate, reason, compute, generalizeand formalize 20th century experience, and to serve 21st century goals.”But both amount to a desire to articulate a set of universal basics and, fromthere, to ensure that those basics are learned. It is not surprising, then, thattheir closing sentences sound suspiciously similar. Neither writer seemsto have gotten at what is being taken for granted, at those assumptionsthat support “the steady rhythms of shared lives” (Grumet, 1995, p. 16),at those basics that we don’t need to go to school to learn as they “arethreaded through our body knowledge” (ibid.).

Both arguments suggest a conception of life processes as analogous tothe structure of the linear, logical argument whereby, in order to progresstoward predicted ends, one must begin with solid foundations. The implicitcommitment to direct and linear conceptions of human existence aredemonstrated most obviously in the desire to manage the future through abasiceducation today. The underlying view of history and progress is thatof logical, linear, and predictable unfoldings. As demonstrated in the brief

34 BRENT DAVIS

history of fractal geometry undertaken in class, this Newtonian conceptionof the universe has been losing ground in much of contemporary academicinquiry in which there has been a broad shift in metaphor away from thelanguage of causal connections and formalist structures and toward moreorganic notions (see Brockman, 1995). The same progression of movingaway from employing mathematics or physics as the model of knowledgetoward taking up biological notions to describe human knowing has beensweeping across most domains of psychological and social inquiry, includ-ing mathematics education. This shift inbasics is powerfully revealedin the constructivist-oriented investigations that now dominate mathemat-ics education research – but which, it must be admitted, have not muchaffected standard school curricula.

Such points made for some lively class discussions. In the end, however,my students were reluctant to set aside the linear curriculum guides and theeven more linear lesson plans that had been so fundamental to their livesin formal educational settings. As they cited needs for effectiveness, effi-ciency, uniformity, testability, and control, our third class period devotedto this exercise in irony ended with a general agreement that the notionof basics as embodied or enacted suggests a fractal-like character thatcompels consideration of both agent and collective as similar, dynamic,and nested. However, the formulated basics of curriculum guides, howeverartificial, were argued to provide a necessary structure for formal educa-tion, a perspective grounded in the preservice teachers’ perceived needsfor predictable outcomes. It bears mentioning that there was also consider-able agreement on the suggestion that mathematics teaching would likelyfollow the evolutions in mindset that are more evident in other culturaldomains. Put differently, participants agreed that mathematics will likelycontinue to inform its own teaching through the assumed structure of itsown knowledge claims. As postmodern sensibilities are taken up insideand outside of mathematics, new possibilities for curriculum and teachingwill likely arise.

There was no great consensus, however, on the issue of the mathematicsteacher’s role in effecting such changes. But the variations in educationalphilosophy were not split between those who argued that teachers mustseek to preserve the status quo through working to transmit culture andthose who felt teachers must be critical or radical instigators of socialtransformation. On the contrary, it seemed that everyone agreed that teach-ers were inevitably agents of social change. The difference in opinion layin whether that agency is enacted as an inevitable part of a larger cul-tural evolution, or whether it should be taken up more deliberately andconscientiously.


EXPLORING INDIVIDUAL COGNITION

Our basic irony, over the course of just a few classes, had thus taken usinto discussions of the nature and evolution of knowledge, the processesof social change, varied philosophies of education, and cognition. As ateacher, I was pleased with the way students were beginning to interrogatewhat was being taken for granted and to propose alternatives for thinkingand acting. However, feeling the need to bring a little focus to the discus-sions, and wanting to address issues of learning more directly, I thoughtthat it might be a good time to look at theories of cognition.

Given the complexity of current discussions, cognition is a topic that Ihave had trouble presenting in an accessible manner to preservice teachers.Framing the topic in terms of fractal geometry, however, proved to be aneffective introduction. In brief, I began by noting some of the most com-mon foci of conventional educational research, citing issues of learning,the joint production of knowledge, and the role of society in determiningthe character of the individual. As had already been taken up in class,these phenomena are hardly distinct, but seem to be intertwined, in muchthe same way that mathematician and mathematics are caught up in oneanother. In particular, the way that each seems to be as complex a phenom-enon as the others is analogous to the way that the intricateness of fractalimages does not vary with scale.

The lesson was decidedly more structured and teacher-centered, as Ielected a principally lecture format to present some background infor-mation on theories of cognition. The presentation was to be interspersedwith moments of discussion, intended to provide students with an oppor-tunity to articulate, reformulate, and re-present understandings, as wellas to point to possibilities for further investigation. The following repre-sents a brief summary of the key points of the lecture and the ensuingdiscussion.

The differences and similarities of the varied constructivisms, the issuesthey address, and their objects of inquiry have been discussed in sev-eral contexts (e.g., Bereiter, 1994; Davis, 1996; Davis & Sumara, 1997;Spivey, 1997; Steffe & Gale, 1995) as researchers have looked for sharedassumptions, similar processes, common logics, and parallel products.These constructivisms share, for example, the goal of questioning ourcommon sense about thinking. All of them tend to highlight uncriticalmachine-based (e.g., mind as computer), ocular-centric (e.g., understand-ing as seeing), and corporatist (e.g., knowledge as capital) metaphors.Each questions the ways that cognition has been cut up and located.Mind/body, self/other, individual/collective, nature/nurture are some of thedichotomies that have been rendered problematic by interpretations of

36 BRENT DAVIS

cognition that embrace the inseparability of activity, understanding, andidentity.

Given their prominence in current mathematics education research, Ibegan with an introduction of those constructivist discourses that focuson issues of individual cognition. A central thesis of these discourses,viz., that the cognizing agent’s basis of meaning is found in her or hisdirect experience with a dynamic and responsive world, was illustratedthrough reference to an unanticipated outcome of the fractal cards activ-ity. Earlier in the term, several students had reported that they had “seenfractals everywhere” – in trees, in salad bars, in landscapes, and so on –after they had been introduced to the new category of geometric forms.Conception and perception had become conflated for these students inthis event. In this way, the event served as support for the constructivistassertion that cognition is a process of maintaining an adequate fit withone’s ever-changing circumstances, as opposed to progressing toward anoptimal internal representation of an external world.

A more sensual, embodied attitude toward cognition was also readilyappreciated in the context of our fractal cards activity. Construction ofthe cards provided an opportunity to attend to the kinesthetic dimensionsof understanding. The importance of attending to the physical ground ofone’s understandings proved to be a point of tremendous interest to thepre-service teachers. The balance of this class was spent in small group dis-cussion of the sorts of physical experiences that might underlie or influencemathematical competencies.

EXPLORING COLLECTIVE COGNITION

I began the next class with the reminder that subject-centered construc-tivisms not only posited a more sensual cognition, but a cognition that wasnot strictly internal. That is, not only do other parts of the body partic-ipate in thought, knowing is also distributed across the objects of one’sworld. Although a departure from the commonsensical notion that thoughtand memory reside in the brain, the stretching of the cognition beyondneural processes and physical activity to include various artifacts seemedto be appreciated. One student’s example of the necessity of a pencil andpaper to think through mathematical notions was particularly helpful indeveloping the idea.

For the purpose of stimulating discussion, I suggested that one’s cogni-tion should not only include one’s physical experiences and the artifactsof one’s world, but other agents in one’s world. There was immedi-ate resistance to the notion, expressed in the acceptance that others


might influence your thinking, but thoughts are ultimately personal andbounded.

Not wanting to launch into an extended debate, I proceeded with apresentation of a second category of constructivist discourses, social con-structivisms, which focus more on small groups, e.g., pairs of students,a teacher and a pupil, or a classroom, as learners build shared under-standings. This perspective on cognition is thus much less focused onthe individual than more subject-centered constructivisms and tends to beconcerned with conversation patterns, relational dynamics, and collectivetraits.

The fractal cards activity and our subsequent discussions served as anillustration of the manner in which individual understandings are caughtup in the movement of the collective. In each activity, for example, par-ticular strands of inquiry and interest arose and spread, while others werepassed over. The directions of investigation, more often than not, were notdeliberate. Rather, a simple question, a chance remark, a surprise happen-ing would occasion the collective interest to follow one path instead ofanother of an infinite range of possibilities. That is, the character of thecollective activity was similar (in the mathematical sense) to the characterof individual cognition.

Despite these noted similarities, the suggestion that cognitive processesmight not be strictly individual was at first rejected by most class mem-bers. The fractal geometric notion of self-similarity, however, proved quitehelpful in this regard. In the same way that a part of a fractal imageresembles the whole – but is not identical to the whole – so the cognitiveprocesses of the individual might be thought to resemble the dynamicsof a group. Within these nested dynamics, as with the fractal image, thepart, that is, the individual, can be seen as a whole unto itself, with itsown particular integrity. Of course, the phenomenon of knowledge, con-sidered on the level of the individual sense-maker, is something differentthan knowledge considered on the level of social grouping. But, in thecontext of the fractal cards activity, it was obvious that these were tightlyrelated and interdependent. Personal understandings were clearly inextri-cable from the emergent foci of the collective over the preceding classes.Within our discussion, it became evident that the relationship betweenindividual cognition and collective cognition was not a simple matter ofa back-and-forth or dialectical relationship. Rather, individual and collec-tive cognition appeared to be knitted together fractally. That is, as with afractal card, there seemed to be a self-similarity between smaller and largerelements.

38 BRENT DAVIS

In keeping with the emphasis on irony, a large portion of this discussionfocused on the contrast between these fractal-informed notions and morepopular conceptions of cognition. In particular, the dependence of morepopular theories on rigid divisions, well-bounded regions, hierarchies, andlinear processes helped to uncover some of the mathematized sensibilitiesthat underpin and infuse much of current thinking about thinking.

In terms of theoretical discussions, as was highlighted in the ensuingclass discussion, this self-similarity is demonstrated in the shared use ofevolutionary and ecological metaphors – and, in particular, in the centralityof the notion of adequatefit with prevailing circumstances as the measureof individual or collective knowledge. The principal point of departure ofsocial constructivist theories from subject-centered constructivist theories,then, is not on matters of process or product, but on the phenomenological-and-biological order of the object of inquiry, i.e., collective rather thanindividual activity.

EXPLORING CULTURAL COGNITION

My impression was that the fractal imagery was essential to drawing atten-tion to the similarities of these two categories of constructivism and toenabling an understanding of cognition as a much broader and more com-plex phenomenon than had been previously assumed. Further, the nexttopic of discussion, critical and sociocultural theories and their contribu-tions to thinking about thinking, was supported by the fractal image. Thetopic was broached through the question, “What happens if you pull thecamera back even further on this nested interpretation of individual andcollective cognition?”

Unlike subject-centered constructivist discourses, which tend to bemost interested in how the individual shapes an understanding of the world,cultural constructivists are generally more interested in how the worldshapes the understanding of the individual. Rooted in critical and inter-pretive philosophic traditions, these discourses began their inquiries intothe complex characters of culture and identity long before ComplexityTheory came together as a field of study. Moreover, although predatingfractal geometry and non-linear dynamics, critical theorists have long beenarguing for very fractal-like notions to trouble the rigidly logical and linearsensibilities that permeate Western thought.

With reference to mathematics education, some of the topics dis-cussed by cultural constructivists have included the hidden agendas ofthe classroom, such as the establishment and maintenance of gender andracial norms, the enactment of tacit social contracts among educators


and learners, and the privileging of mathematics and mathematized sys-tems of knowledge. The actual in-class discussion of these issues tookup more than two sessions, as we examined postmodernist, feminist, andneo-Marxist critiques of education, generally, and mathematics education,specifically.

Notably, these discussions were characterized by what I interpreted tobe a deep appreciation of the complex intertwinings of individual and cul-ture. This understanding was highlighted in a part of the discussion whereseveral students, making figurative use of the notion of self-similarity,pointed to the centrality of the notion of body in subject-centered, social,and cultural constructivisms alike. The figure of the body is different ineach case – with subject-centered constructivisms focusing on the bodybiologic, social constructivisms focusing on the body epistemic, and cul-tural constructivisms focusing on the body politic – and these intertwiningbodies might be considered in terms of the different orders of phenomenathat serve as the foci of the varied constructivisms.

This metaphoric commitment to the body across the interpretive frame-works proved vital to understanding their shared logics. If one imaginesthe notion of body as analogous to a region of a fractal figure – that is,as an element which can be regarded as an object unto itself, as a fractalcomponent of a larger object, or as an assemblage of smaller objects –the relationships among the varied constructivisms become more apparent.Further, the body metaphor recalls the fact that one is always, at best,studying only a part of the whole. But that part, since it is a fractal andnot a mere fragment, has the capacity to point beyond itself.

I ended the multi-session study of constructivisms and cognition byasking, “Where does cognition happen?” The question served to highlightthree common themes of constructivist discourses: First, as just mentioned,each focuses on a body (biologic, epistemic, and/or politic) as the site ofcontestation. Second, they all draw on evolutionary theory to describe thedynamics of their subjects/objects of inquiry, whether the individual, thesocial group, or a culture. Third, cognition is not seen as locatedin a body,but as a means of describing the relationships that make that body cohere orthat enable that body to maintain its viability and integrity within a larger,similarly dynamic and responsive context.

The question also served as a reminder of the ironic nature of the dis-cussion. The assertion that cognition could not be unambiguously locatedprompted a brief discussion of our discomfort with phenomena that haveno tidy edges, that refuse linear characterizations, that disallow reductivedescriptions and explanations – in brief, that do not fit with concept, forms,or analytic tools of classical mathematics.

40 BRENT DAVIS

RECONCEPTUALIZING COGNITION

In terms of the theories of cognition and their implications for mathematicsteaching, my original intention was to limit examinations to subject-centered, social, and cultural constructivisms. The discussions, however,spilled beyond these bounds – in part, ironically, because of the frac-tal analogy used to draw attention to the common ground of the variedinterpretive frameworks.

The fractal logic developed above suggests that the tendency to limitdiscussions of cognition to the domain of human sociality is troublesome.This point was, in fact, made by several of the students in the context ofour discussion of cultural constructivisms. If cognition is fractal-like, itwas argued, we should be able to extend the analysis in two directions:both to sub-human processes and beyond humanity. From my position asinstructor, this was an important moment in the basic irony, as it seemedthat the fractal notion had moved beyond its intended role of an illustra-tive analogy and allowed, and perhaps even compelled, us to examine themulti-tiered dynamics of cognition. That is, the possibility of a furtheriteration of the analysis was made evident and was encouraged throughthe emergent understandings of fractal geometry.

In response to students’ questioning the possibility and relevance ofextending these discussions of cognition, I elected to use a small portionof class time to present some details on other domains of inquiry that havetaken up similar strands of thought. On the matter of sub-human processesthat that might be aligned with this fractal logic, for example, I pointed totwo areas of inquiry, viz., studies of brain structure and research into theimmune system.

On the former, recent research has demonstrated that the brainseems to be fractally structured and that the activities at each level oforganization resemble those at every other level (see Calvin, 1996). Suchanalysis renders the Euclidean model of a pyramid-shaped hierarchy,with individual neurons at the base and the functioning brain at the apex,inadequate, because each level of functioning seems to have its ownparticular autonomy and integrity. That is, cognition is not merely a globalprocess that emerges in the amalgamated activities of neurons, but aprocess that is embodied in each element, be it neuron, minicolumn, orhemisphere. In a related manner, recent HIV/AIDS-prompted researchhas demonstrated that the immune system learns, forgets, hypothesizes,errs, and recovers in a complex dance with other bodily systems(Varela, 1989). Neither fully autonomous nor a mere mechanical compo-nent of a larger whole, then, it seems that a person’s immune system is


related to the person in the same way that the individual is related to thecollective.

In the other direction, current work in global ecology has demonstratedthe devastating consequences of a conceptual separation of humanity fromnature. Perhaps finding its clearest articulation in the Gaia Hypothesis (seeLovelock, 1979), and gaining in popularity as a way of understandinghumanity’s role in the biosphere, this conception of life on Earth positsthat our species is a mere sub-system of a larger organismic relationality– a grander body of which we are part. The implications for formal edu-cation are immediate in this regard: Knowledge is not merely knowledgeof the world, but knowledgein the world, wholly complicit in shaping theconditions and realities of the planet.

Notably, in our discussions, these extensions were not merely matters ofstretching an analogy. Rather, the fractal cards activity itself, like so manyfractal images, prompted us toward this way of thinking as many of thecards that had been generated bore striking and unexpected resemblancesto objects of the natural world: trees, snails, faces, mountains, an arteryseparating into capillaries and then rejoining into a vein, to name a few.As each of these appeared, the ideal realm of mathematics was pulled tothe ground of our physical engagement with the not-human part of theworld. This, in fact, might have been the most significant aspect of theexercise in irony. The habit of thinking of mathematics as existing on anideal Platonic plane (or, more recently, on a social plane) was interruptedwith the realization that our mathematical knowledge does not separate usfrom but knits us together with the rest of the biosphere. As Kline (1980)has suggested,

Unexpected . . . uses of mathematical theories arise because the theories are physicallygrounded to start with and are by no means due to the prophetic insight of all-wise math-ematicians who wrestle solely with their souls. The continuing successful use of thesecreations is by no means fortuitous. (p. 295)

Kline might be interpreted as suggesting a sort of fusing of subject-centered, social, and cultural constructivisms. Beyond that, he is clearly sit-uating humanity, through mathematics, in conversation with a responsiveand dynamic universe.

With the same sort of insight, our basic irony into the nature of mathe-matics moved beyond questions of personal, social, and cultural activityinto wondering about existence itself. That is, there was a recognitionof the inextricability of issues of epistemology and ontology. This samesensibility is part of recent ecological theories of cognition, such as enac-tivism (Varela, Thompson & Rosch, 1991). Briefly, such theories mightbe described as further iterations of constructivisms. They simultaneously

42 BRENT DAVIS

apply the same sorts of logics, metaphors, and images at phenomenal andbiological levels that range from the sub-cellular to the planetary, neces-sarily collapsing questions of cognition with questions of life. The idea isconcisely captured in aphorisms offered by Maturana and Varela (1987):Knowing is doing; knowing is being. The attentiveness to the biologicalrepresents an important interruption of the conventional emphases on inter-preted experience and formulated knowledge. The role of bodily needs andphysical drives – phenomena that, within those discourses that are boundedby human sociality, tend to be disregarded, reduced to social constructions,or seen as base instincts to be overcome – thus play a renewed and vitalrole in ecological thinking about thinking.

The import of these matters with regard to the original focus of the basicirony (i.e., the nature of mathematics) was not lost in our discussions. Withthe recognition of the self-similar intertwinings of subjective knowing andcollective knowledge, mathematics, whether considered in terms of a man-ner of inquiry or a body of knowledge, came to be seen as inseparable fromthe activities of the agents who are enacting mathematized sensibilities.A return to the question, “What is mathematics?” thus prompted us toconsider how we position ourselves in relation to the perceived-to-be not-human part of the world and, within that stance, to ask how we make senseof our preoccupations and motivations.

Ensuing discussions focused on the relevance of understanding cur-riculum as a complex form, rather than an artificial structure imposed tomanage complexity. In other words, we examined the potential for think-ing of curriculum as another iterative layer, lodged in the fractal form thatnow included individual, collective, culture, and biosphere. This imaginingprompted the conversation toward, for example, the need for more fluidunderstandings of learning goals, lesson plans, and teaching approaches– a dramatic shift in thinking from the earlier noted resistance to morecomplexified understandings of curriculum.

The balance of the course was more focused on what it might mean toenact such a complex curriculum, with a significant portion of our timedevoted to examining the role of the teacher. As I have developed theseissues elsewhere (e.g., Davis, 1994, 1996, 1997), I will limit my remarksto describing the consequent attitude toward pedagogy as more participa-tory, more mindful, more attentive, and more tentative than the controllingmanner of mathematics teaching that is now widely critiqued.


STUDENT RESPONSE

The final assignment in this course involved the crafting of a position paperon mathematics teaching. Students were asked to take up, among othermatters, issues of the nature of mathematics and processes of cognitionin a discussion of their emerging conceptions of what it means to teachmathematics. In this section, I draw from some of this written work in aneffort to represent how, at the course’s end, the issues addressed in theexercise in irony were being taken up by these preservice teachers.

On the issue of the nature of mathematics, students now demonstratedconsiderable appreciation for the relevance of this topic, in their responsesto the question, “What is mathematics?” The following comment wastypical:

When we started, I knew what math was, and I couldn’t figure out why you would ask usthat. Now [I] realize that I don’t know the answer. The weird [ironic] thing is that I alsorealize mynot knowing is going to make me a better math teacher.

Such comments were generally followed by attempts to take up a fractalanalogy to discuss the nature of mathematics. Interestingly, for most stu-dents, the analogy compelled them to also talk about individual cognizingagents. A second person had the following to say:

The more I think about it, the more I realize that the question, “What is mathematics?”,isn’t really the right one to be asking. Phrased that way, it makes me think that mathematicsmust be something, out there, separate from us. A better question, at least for those of uswho will be teaching the subject, seems to be “What does mathematics do?”

This student went on to examine how mathematical sensibilities participatein shaping perceptions and world views. Her paper ended with a examina-tion of how a discussion of the nature of mathematics is really a discussionof ourselves. This conflation of the body of mathematical knowledge withthe bodies of knowers, she pointed out, is a conclusion that must be drawnif one thinks in terms of fractal geometry rather than use the more tidilybounded regions of Euclidean figures to “cut up the world.”

In fact, and to my surprise, the idea of understanding the relationshipbetween collective knowledge and individual knowing in terms of onebeing nested in the other was perhaps the most common theme in thestudents’ papers. It was certainly more prominently represented in theirwritings than it was as a topic of class discussion. In almost every case,similarities were noted between the complex and unpredictable ways thatmathematics has evolved and the ways that individual mathematical under-standings emerge. Although few students made explicit use of ecologicaland evolutionary notions in their examinations of mathematics and cogni-

44 BRENT DAVIS

tion, for the most part they demonstrated well developed appreciations ofthe core principles of the varied constructivisms.

They also demonstrated that they were able to bring these under-standings to bear on their thinking about teaching. As one student putit,

Instead of thinking of the classroom as a collection of discrete units, it’s interesting to thinkof it in terms of a single amoeba-like body that’s made up of many smaller, distinguishable,but not really separable amoeba-like bodies.

The struggle for an alternative metaphoric frame is evident in this state-ment. My own interpretation is that this student was attempting to artic-ulate a conception of personal knowing and collective knowledge thatabandons the language of classical physics and adopts the language ofbiology. The shift might also be represented in terms of leaving behindEuclidean notions and taking up a more fractal geometric frame. Thisinterpretation was borne out in his paper, in which he critiqued a cause-and-effect mentality that he saw as underpinning the desires to predictoutcomes and control behaviors. In its place, metaphors drawn from lifeprocesses and other complex phenomena were used. His closing paragraphhighlighted this transition:

When I think about my own mathematics learning, most of that happened when therewasn’t a teacher anywhere near me. But that didn’t mean that teachers didn’t matter. Theydid. Profoundly. It’s just that they nevercausedme to learn what I learned. So what itcomes down to for me is this: remembering that my students’ learning will depend on whatI do, but will never be determined by it. As a teacher, then, I’m a participant in students’learning in pretty much the same way that, as a group, my mathematics classroom becomespart of the body of mathematics.

Although not explicitly stated, once again the concepts of nestedness, eco-logical intertwinings, and complex evolutions are intuited. I read thesestatements as demonstrations of the students’ abilities to be ironic – that is,as noted earlier, to deliberately interrogate the ground of one’s assumptionsby turning the language and the logic of one’s thinking onto itself.

BACK TO BASICS

In retrospect, the course that I taught was not so much guided by an exam-ination of the nature of mathematics, as I had originally intended, as bythe sort of thinking about thinking announced by McCulloch (1963) whenhe asked, “What is number that man may know it, and a man that he mayknow number?” (p. 1).


In posing this reflexive question, McCulloch was identifying mathe-matics as a humanity, as the potential site of anthropological inquiry. Andit is precisely that attitude that I have sought to bring to my own teaching.For me, the study of mathematics is a study of ourselves. Understand-ing mathematics, particularly for teachers, involves an appreciation of themanner in which mathematized notions are woven through our bodies, thatis, the ways that mathematics is continuously enacted beneath the surfaceof conscious awareness.

For the prospective mathematics teacher, the sort of teaching emphasisrepresented by a basic irony has a manifold purpose: compelling exam-ination of the subject matter, its nature, its contribution to perception,thought, and activity; prompting interrogation of the place of schoolingas part of a mathematized culture and mathematics as part of a culture ofeducation; occasioning an awareness of our complicity, through our math-ematics, in the conditions of the planet; fostering a mindfulness towardthe ways knowledge is enacted and the manners in which mathematicsteaching, like the fractal image, always points beyond what is immediatelypresent.

There are many readily available sites for such investigations. Besideswidely debated issues around notions of the basics, one might examinethe pervasive presence of number, the primacy of rationality over othermodes of knowing, the ubiquitous presence of the line (in time lines,story lines, lines of reason), the mathematized notions that infuse ourlanguage and underpin cultural ideals (e.g., equality, independence, auton-omy, order), and the new infusion of mathematized sensibilities that derivefrom computer use and computer-based metaphors. Or one might engagein a mathematical anthropology, borrowing from Sumara’s (1996) notionof “literary anthropology” (p. 231). This involves an analysis of a commonartifact or activity for what it reveals about sensibilities that have fadedinto transparency, that prejudice perceptions, that permeate identities. Vir-tually any article can serve as the basis for such an inquiry, as each of thethingsof our world “is modeled to the human action which it serves. Eachone spreads round it an atmosphere of humanity” (Merleau-Ponty, 1962,p. 347). Further, any topic in a grade school curriculum will do as the focalpoint of a basic irony or a mathematical anthropology – as will, for thatmatter, the actual forms in which curricula, textbooks, concepts, problems,and evaluation schemes are presented. Or one might simply examine one’ssurroundings and their shaping influences on perception. As Abram (1996)suggests:

The superstraight lines and right angles of our . . . architecture . . . make our animal senseswither even as they support the abstract intellect; the wild, earth-born nature of the materi-

46 BRENT DAVIS

als – the woods, clays, metals, and stones that went into the building – are readily forgottenbehind the abstract and calculable form. (p. 64)

Indeed, in our study of fractal images, one of the most engaging tasksinvolved stepping out-of-doors to identify and examine the self-similarityof trees, rivulets of water, flowers, and other natural forms. Most studentswondered why they had been unable to see this self-similarity prior to ourinquiry, agreeing that the Euclidean world that has been erected aroundus had likely played an important role in dulling perceptions of theseforms.

In terms of using a basic irony as a pedagogical device, a critical ele-ment is an engagement in some aspect of mathematical inquiry because,by definition, an exercise in irony relies on turning a mode of thinkingonto itself. Possible topics for such inquiry include fuzzy logic, non-lineardynamics, and knot theory. For me, fractal geometry has proven very use-ful, even in such contexts as afternoon workshops where there is littleopportunity for sustained engagement. Its utility, I believe, arises from twofactors: the ease with which its core principles can be illustrated and theready body of natural forms that are fractal-like but that are not popularlyperceived as geometric. Combined, these qualities remind us of the roleof knowledge in perception and thus open the door to examinations of theways that mathematics infuses world views and mind sets – that is, to theways that our knowing also involves forgetting, an allowing of metaphorsand analogies to slip into literalness (Rorty, 1989).

It is this forgetting that most prompts my interest in basic ironies. Con-ceived as a deliberate re-cognizing of the fractional dimensions of knowingand knowledge, a basic irony is a rethinking of what it means to knowand to teach mathematics. Re-iterating what others have already suggested(e.g., Skovsmose’s (1985) “mathematical archaeology” and Frankenstein’s(1983) “critical mathematics”), and pushing the central worry beyondhumanity’s technologies and power structures, this pedagogical empha-sis reminds us of the very ground of our activity, that is, of the basics.Following Grumet (1995),

We don’t need to go to school to learn these basics. They are threaded through body-knowledge, and no amount of resolve can make them disappear. What is basic to educationis neither the system that surrounds us nor the situation of each individual’s livedexperience. What is basic to education is the relation between the two. (p. 16)

With this assertion, I find myself, ironically, where I and my methods stu-dents began in the quest to better understand the nature of mathematics:with a concern for relations.

But the re-emergence of the idea is not a simplerecurrence. Rather,having undertaken the basic irony, the notion of relation is now a very


different one. And so its return is arecursiveevent; it underscores thatan irony is not a matter to be resolved or concluded, but an occasion forturning thinking and acting back onto themselves.

REFERENCES

Abram, D. (1996).The spell of the sensuous: Perception and language in a more-than-human world. New York: Pantheon Books.

Bereiter, C. (1994). Constructivism, socioculturalism, and Popper’s world.EducationalResearcher, 23(7), 21–23.

Brockman, J. (Ed.) (1995).The third culture: Beyond the scientific revolution. New York:Touchstone.

Calvin, W.H. (1996).How brains think: Evolving intelligence, then and now. New York:Basic Books.

Cheney, L. (1997 August 11). Once again, basic skills fall prey to a fad.The New YorkTimes, p. A13.

Davis, B. (1994). Mathematics teaching: Moving from telling to listening.Journal ofCurriculum and Supervision, 9, 267–283.

Davis, B. (1996).Teaching mathematics: Toward a sound alternative. New York: Garland.Davis, B. (1997). Listening for differences: An evolving conception of mathematics

teaching.Journal for Research in Mathematics Education, 28, 355–376.Davis, B. & Sumara, D.J. (1997). Cognition, complexity, and teacher education.Harvard

Educational Review, 67, 105–125.Frankenstein, M. (1983). Critical mathematics education: An application of Paulo Freire’s

epistemology.Journal of Education, 163, 315–339.Gödel, K. (1962[1931]).On formally undecidable propositions of Principia Mathematica

and related systems. New York: BasicBooks.Grumet, M.R. (1995). The curriculum: What are the basics and are we teaching them?

In J.L. Kincheloe & S.R. Steinberg (Eds.),Thirteen questions: Reframing education’sconversation(2nd ed., 15–21). New York: Peter Lang.

Kline, M. (1980). Mathematics: The loss of certainty. New York: Oxford UniversityPress.

Lovelock, J. (1979).Gaia, a new look at life on earth. New York: Oxford UniversityPress.

Maturana, H. & Varela, F. (1987).The tree of knowledge: The biological roots of humanunderstanding. Boston, MA: Shambhala.

McCulloch, W. (1963).Embodiments of mind. Cambridge, MA: The MIT Press.Merleau-Ponty, M. (1962).Phenomenology of perception. London: Routledge.Romberg, T. (1997 August 11). Mediocre is not enough.The New York Times, p. A13.Rorty, R. (1989).Contingency, irony, solidarity. New York: Cambridge University Press.Simmt, E. & Davis, B. (1998). Fractal cards: A space for exploration in geometry and

discrete mathematics.Mathematics Teacher, 91 (February), 102–108.Skovsmose, O. (1985). Mathematical education versus critical education.Educational

Studies in Mathematics, 16, 337–354.Spivey, N.N. (1997).The constructivist metaphor: Reading, writing, and the making of

meaning. San Diego, CA: Academic Press.Steffe, L. & Gale, J. (Eds.) (1995).Constructivism in education. Hillsdale, NJ: Erlbaum.

48 BRENT DAVIS

Sumara, D.J. (1996).Private readings in public: Schooling the literary imagination. NewYork: Peter Lang.

Varela, F. (1989).Principles of biological autonomy. New York: Elsevier North Holland.Varela, F., Thompson, E. & Rosch, E. (1991).The embodied mind: Cognitive science and

human experience. Cambridge, MA: The MIT Press.

Faculty of Education,York University,4700 Keele Street,Toronto, Ontario M3J 1P3,[email protected]

Documents

Basic Irony: Examining the Foundations of School Mathematics With Preservice Teachers