Mathematics as Simulacra: Implications for a Critical ... · MATHEMATICS AS SIMULACRA 2 Abstract Mathematics education has failed to drag itself from the grips of humanist ideas about

Running head: MATHEMATICS AS SIMULACRA 1

Mathematics as Simulacra:

Implications for a Critical Mathematics Education

Brian R. Lawler

California State University San Marcos

MATHEMATICS AS SIMULACRA 2

Abstract

Mathematics education has failed to drag itself from the grips of humanist ideas about knowledge,

in particular an a priori ontological status of mathematics. The postmodern re-inscriptions of truth, what is

real, or what counts as real, have not impacted our ways of thinking with our core concern. In this paper I

use Baudrillard to rip mathematics free of this anchor, to cast mathematics itself as a simulacra,

references to a reference, constructions of a real that is not there. I argue that it is not until once freed of

the weight of reality that considering a non-fascist mathematics education is possible. I extend the

argument through the development of a possible four-point framework for thinking of a mathematics

education for social justice, then discuss a small sample of research on adolescents‘ mathematical

identity in order to begin to propose what might be a renewed direction for a research agenda toward a

critical mathematics education.

Keywords: critical mathematics education, social justice, mathematics, simulacra, Baudrillard,

personal epistemology, authority


Mathematics as Simulacra: Implications for a Critical Mathematics Education

―As soon as… a trait is labelled as a good thing in itself, all non-possessors of that trait

automatically are labeled as evil or worthless. Such an arbitrary labelling… is fascism‖

(Ellis, 1985, p. 9)

If the notion of fascism is based upon some arbitrary belief that those who possess certain traits

are intrinsically superior to others, and thus deserving of a higher status, or socio-political privilege, then

those who subscribe to a humanist mathematics education are intellectual fascists. That is to say, those

who subscribe to present day proposals for mathematics education are intellectual fascists.

When the patriarchal, European notions of mathematics, named as content or mathematical

practices (e.g. the U.S.‘s Common Core State Standards), are held up as desirable—necessary—ways of

knowing or thinking, the arbitrary belief that possessing this certain knowledge, or thinking in these

particular ways, is suggested to be more powerful, desirable. There is a simple reason that a belief in the

power of this mathematics, or in any particular way of knowing, is arbitrary: there is no objective evidence

to support it.1 The belief is based on value judgments that are definitional in character, they cannot be

empirically validated. There is no God‘s Book of Mathematics (a notion attributed to Paul Erdös) to

confirm one‘s mathematical knowing. Reality is unknowable2 and as such, any singular knowing

represents a viability rather than a truth. Knowledge, that is one‘s way of knowing, must fit reality, but

does not represent reality.

This is not to say that some other mathematics, some Ethnomathematics (Powell & Frankenstein,

1997), would be the better purpose for a mathematics education; replacing one truth regime (Foucault,

1980) with another solves no problem. It merely offers ―the same hearse with different license plates‖

(Ellis, 1985, p. 11).

Mathematics is a fabrication (Lawler, 2008, 2010, in press); it is an untruth. The postmodern turn

has provided the field of mathematics education a perspective on the spectre of truth, its complicity with

1 . As an example of such a discussion, see Skovsmose, 2009.

2 Vico (1968) stated, ―the true itself is made‖; see also Glasersfeld (1995), ―objectivity is the delusion that

observations could be made without an observer.‖


power. In the discourses of mathematics education, critical mathematics education, math education for

social justice, and others related to the learning of mathematics, we forget that there is not a mathematics

to be learned. This mathematics to which we refer is merely a simulacra.

In this paper, I use Jean Baudrillard‘s notion of simulacra to understand mathematics as a

hyperreality; more precisely, that student‘s constructed ways of mathematical knowing are references to a

referent, simulations of a real that isn't there. After reconstituting mathematics as simulacra, I discuss

potential implications for a critical mathematics education. I begin with a framework for a Mathematics

Education for Social Justice, and follow with further explication of the notion of authority withn=in this

skectch. Next I discuss the classroom experiences and personal epistemologies of two adolescent

mathematics students, toward considering what new curiosities and potentials can arise, ultimately

delineating several implications for the field of mathematics education, in particular potential research

trajectories to inform a critical mathematics education.

For the purpose of this work, I consider a critical mathematics education in the manner described

by Arthur Powell and Marylin Frankenstein (1997) as a mathematics education that emphasizes the social

nature of knowing, and takes seriously the powered position of the student in an educational setting. In

particular, this Freirian, ethnomathematical perspective of Powell and Frankenstein demands a need to

reconsider binaried categories common in academic thought. These include the dichotomy between

subjectivity and objectivity, action and reflection, teaching and learning, logic and intuition, practical and

theoretical knowledge. Skovsmose (1994, 2004) expands on this framework in important ways, namely

that within a critical mathematics education, students (and teachers) are attributed a critical competence.

This attribution rejects deficit models of knowing, sees the present ways of knowing of the learner as

central, beliefs learning builds up through critical interrogation of experience, and injects the learner with

the possibility to impact the world. It is a way to both re-understand the power of the mathematical learner

and a framework that empowers the mathematical learner.

Prior to beginning this explication, I remind us where mathematics education seems to live with

regards to the ontological status of knowledge. The language of knowledge construction (social or

otherwise) is prevalent. We consider the learner‘s mathematics to be no more (and no less) than a

construction of a human mind, or human minds, built up in interaction with others. It is a name we give to


help organize our knowing of the world, a world we seek to embue with consistency, knowability, truths.

We seek truths that allow us to feel a sort of comfort in our knowings, a stability, a reality, a place for our

selves, a meaning for our being, and a purpose for our existence. But Mathematics is a false god. Those

who proclaim to possess it or propose to educate others toward it are intellectual fascists.

Baudrillard and Simulacra

Jean Baudrillard was a 20th century French sociologist and philosopher, among other things; oft

associated with postmodernism and, in particular, post-structuralism. He followed upon the tradition of

sociologists like Levi-Strauss in making a link between sociology and semiotics,3 especially critiquing the

structural work of Swiss linguist de Saussure. Like Deleuze, Derrida, Foucault, Lacan and Lyotard, he

was among the postmodern French thinkers interested in semiotics who claimed meaning is created

through difference, what something is not. As such, one object‘s meaning is only understandable through

its relation to the meaning of other objects. It is a sort of self-referential view of meaning and knowing. As

such, the total or complete understanding of a word or sign, i.e. truth, is always already elusive. And one

is thus seduced by the power to know. Yet this is a false mistress, we are drawn toward a simulated

version of reality, one that is defined only through relation to other objects, which are defined by their

relations to other objects. Rather than coming to know a reality by making meaning through signs and

signifiers, we create knowing of a hyperreality. ―The distinction between true and false is blurred; as it is

Plato's cave: there are only images among images, opinions among other opinions, various sources of

information, but not ‗the Truth‘‖ (Solsona, 2010, paragraph 5). Baudrillard‘s hyperreality extended beyond

the inability to distinguish illusion from existence, but in a refusal of the suggested binary, sees

hyperreality as the illusion for a real that never did exist.

In a structuralist orientation toward semiotics, descriptive categories were real. Illusion has

existence as it‘s opposite, and both are grounded. But the notion that language signifies some pre-

existence for poststructuralism is unimaginable; rather language operates as a system of signs without

grounding. A poststructural semiotic approach to mathematics would suggest that mathematics is neither

real nor an illusion of some real; rather it is a hyperreality.

3 Semiotics is a field close to linguistics; it is comprised of a study of signs and sign processes, analogy, metaphor,

symbolism, and communication.


―The simulacrum is never that which conceals the truth—it is the truth which conceals that there is

none. The simulacrum is true‖ (Baudrillard, 1994, p. 1). Baudrillard refuses truth, to the extent to which he

condemned Foucault for falling short of a full refutation through the re-naming truth with the notion of

power relations; inasmuch Foucault could not release himself from a desire for truth (Baudrillard, 1987).

For Baudrillard, ―truth or reality lies only in the hope that crosses them and which becomes their great

productive power‖ (Solsona, 2010, paragraph 25). Baudrillard was not concerned about the quality of

relationship between the representation and reality, model and modeled, rather the always already

relations between model and (other) model. And in consideration of such relations, both models are taken

as real, to be the real. Rather than model and reality as polarized ideas, models (simulacra) coproduce

reality. Any truth, any meaning, is no more than references to referents (hyperreality).

Baudrillard believed that the relations between our lives and society were so steeped with

simulacra that all meaning had become meaningless. Meaning was infinitely mutable, always already

changing in its relations to other always already changing meaning. Baudrillard‘s (1994) theory launches

from a semiotic posture in which he names four stages to the sign order.

Stage 1: Sign operates as a faithful image or copy, it exists as a ―reflection of a profound

reality‖ (p. 6).

Stage 2: Sign operates as a perversion of reality, an unfaithful copy. Signs work to hint at the

existence of something real, but are unable to fully capture.

Stage 3: Sign operates as masking the absence of a reality; the sign pretends to be a faithful

copy. Yet there is no original for which the sign claims to represent. This suggestion

is the ―order of sorcery‖ (p. 6).

Stage 4: Sign operates as ―having no relation to reality whatsoever: it is its own pure

simulacrum‖ (p. 6). Signs merely are composed of the relationships among other

signs.

Baudrillard declares the phenomenon that truth could be no more than reference to referents as the

precession of the simulacra.

Simulation is to be considered as the current state of the simulacra. Prior to the era of

Enlightenment, the premodern of the Renaissance, the dominant state of the simulacra was that of the


counterfeit. The image clearly stood for a true referent, but the uniqueness and irreproducibility of the

object, such as royalty or holiness, could never be fully captured in the simulation. The sign operated

clearly as an artificial placeholder for some real item. Baudrillard considered this the first order of the

simulacra. Baudrillard attributed the second order to the turn of modernity, the Industrial Revolution. In the

second order, the dominant state of the simulacra is that which is produced. Distinctions between image

and reality break down due to the proliferation of mass production. What was conceived to be truth was

that which could be produced on some endless production line. The ability of the reproduction to imitate

the original threatened its reality.

Today, in the postmodern, the model itself serves is the simulacra; originality is a meaningless

concept—the model is a reproduction of an already reproduced reproduction. The model precedes reality;

the simulacra precedes the original. It is in this third order that originality becomes a meaningless idea,

and the real can no longer exist.

Baudrillard deconstructs Disneyland in order to describe the entangled orders of the simulacra. At

the first order, Disneyland‘s Main Street, the Frontier, Adventureland, the Pirates of the Caribbean, are all

present as a play of images, of illusions of what is or was real. Main Street is meant to represent

Anytown, USA. But in this illusion, it is masks, denatures, the profound reality of any one of these

Anytowns. In this way, the Disney Main Street operates on a second order.

However, ―Disneyland exists to hide that it is the ‗real‘ country, all of ‗real‘ America that is

Disneyland‖ (Baudrillard, 1994, p. 6)—in the sense that Disneyland must be considered too to be

America. Disneyland functions as a cover-up to what it itself is; it is in this way a simulation of the third

order. It is presented as imaginary, as a playground, so as to perpetuate the image of the rest of America

to be really the real.

Umberto Eco (1990) takes up a very similar deconstruction of Disneyland as the

precession of simulacra.

Disneyland's Main Street seems the first scene of the fiction whereas it is an extremely

shrewd commercial reality. Main Street—like the whole city, for that matter—is presented

as at once absolutely realistic and absolutely fantastic…. The houses of Disneyland are

full-size on the ground floor, and on a two-thirds scale on


ades are presented to us as

toy houses and invite us to enter them, but their interior is always a disguised

supermarket, where you buy obsessively, believing that you are still playing.

In this sense Disneyland is more hyperrealistic than the wax museum, precisely because

the latter still tries to make us believe that what we are seeing reproduces reality

absolutely, whereas Disneyland makes it clear that within its magic enclosure it is fantasy

that is absolutely reproduced. (pp. 23-24)

Eco conceived of the wax museum as a second-order simulacra, in that the American psyche, he argued,

follows obsessively toward a perfected reproduction. But that it is the fantasy that is reproduced positions

Disneyland as a third-order simulacra.

Disneyland not only produces illusion, but— in confessing it—stimulates the desire for it:

A real crocodile can be found in the zoo, and as a rule it is dozing or hiding, but

Disneyland tells us that faked nature corresponds much more to our daydream demands.

When, in the space of twenty-four hours, you go…from the wild river of Adventureland to

a trip on the Mississippi, where the captain of the paddle-wheel steamer says it is

possible to see alligators on the banks of the river, and then you don't see any, you risk

feeling homesick for Disneyland, where the wild animals don't have to be coaxed.

Disneyland tells us that technology can give us more reality than nature can. (p. 24).

The disappointment of the ―actual‖ riverboat and lapse into desire for the Adventureland version

exemplifies today's simulacrum society. It is dominated by the mere semblance of a reality. This

domination conceals the fact that it is only an apparition, and thereby diverts attention from the only

possible ―reality‖ or ―truth‖, which is in fact the simulacrum itself. Baudrillard says: ―The simulacrum is

never what hides the truth. It is truth that hides the fact that there is none. The simulacrum is true‖ (1994).

The simulacrum – when one knows what it is – does not lie; it is what it is. ―The lie occurs when an

attempt is made to pass off a simulacrum as truth; or in more radical terms; when we are told that there is

truth, and not a simulacrum.‖ (Solsona, 2010, paragraph 7).


Baudrillard is remarkably aligned with Lyotard‘s condemnation of the meta-narrative. It is this

metanarrative mathematics itself that I wish to interrogate in the remainder of paper, mathematics as

simulacra. Although the postmodern condition has created space for us to reposition what it is that

exists—what is a real, a truth—the dilemma I wish to present next is that people go on speaking about

mathematics as if it has some existence, as if there is a truth, or a nearer and nearer approximation of

this truth. Or even that we might model the mathematical knowing of another‘s mathematics, that which

we would recognize as an illusion. Or that these may even be considered as opposable ways of knowing.

When something is spoken about that seems real it is a simulacra, that is an illusion of a real that isn't

there. And, talking of an illusion is the recursively the same again too.

Wrestling ourselves from the language with which we think is difficult. ―We are always speaking

within the language of humanism, our mother tongue, a discourse that spawns structure after structure

after structure—binaries, categories, hierarchies, and other grids of regularity that are not only linguistic

but also very material‖ (St. Pierre & Willow, 2000, p. 4). In the next stage of this essay it will be my effort

to encourage the field of mathematics education to reimagine its work if there may truly not be a

mathematics to be learned. I press for a step backward from our humanist assumptions to recognize the

illusion/real binary so as to reconstitute the purpose and goals for mathematics education and to analyze

differently the ways in which mathematics education operates.

Is There Possibility for a Mathematics Education?

Reconstituting mathematics as a simulation without original, what becomes a notion for a critical

mathematics education? I begin consideration of such a question by naming a framework for a

mathematics education for social justice. As an example of where such a framework can inform research

in mathematics education, I present an analysis of the personal mathematical epistemologies of two

adolescents from prior research. And from the framework and examples, begin to pose challenges to and

questions for the field of mathematics education.

A Framework for Mathematics Education for Social Justice. Although there have been

multiple definitions for what it means to teach mathematics for social justice (Burton, 2003; Gutstein,

2006; Gutierrez, 2007; Keitel, 1998; Povey, 2002), here I suggest four cornerstones that characterize the

teaching of mathematics for social justice, and in particular each of the four cornerstones must be


considered: access, achievement, authority, and action. Social justice in mathematics education does not

end with attaining greater access to or achievement in mathematics, education, or the larger culture. The

notions of authority for knowing and both the confidence and compulsion to act are of equal or possibly

greater status.

I will not attend in this paper to the necessary cornerstone of equitable access,4 and will later

return to briefly consider what might be considered achievement5 in a age of simulacra. The lack of

attainment of these goals in mathematics education has been identified, decried, and deconstructed to a

point of exhaustion. The field of mathematics education has a long history documenting such iniquities

(Lawler, 2005), and there seems to be an adequate sense of what could be or ought to be done, or at

least a great many opinions and ideas—a ―discourse of moral judgments‖ (Atweh, Bland, and Ala‘i, 2011).

But rather than focusing on these iniquities, Gutierrez‘ gap-gazing (2008), I hope to elaborate on notions

of authority and action, especially in the context of a constructivist epistemology.

To set a stage, I draw upon constructivist tradition to recognize a children’s mathematics (Steffe,

2004), that which I as a teacher assume a student to have constructed; the mathematical ways of

knowing and ways of thinking that I attribute to the child. For the sake of the remainder of this paper, I will

refer to such mathematics as lower case (m)athematics when the specificity of meaning warrants

distinctions.

Continuing, mathematics for children refers to an adult‘s ways of mathematical knowing and

thinking (Steffe, 2004), drawn upon in order to hypothesize a zone of potential construction (Steffe &

Thompson, 2000) for directing interaction with a child. Although still always a constructed knowledge, we

as teachers treat this sort of mathematics, that which appears in textbooks and curriculum guides and

standards documents, as what is to be learned in the classroom. This particular mathematics, a

mathematics for children, will be referred to with an upper case (M)athematics. It is an idea similar to what

is often called School Math; however my characterization reflects the constructivist‘s ontological status of

knowledge, one which parallel‘s Baudrillard, positioning (M)athematics in the adult‘s ways of knowing.

4 Access as opportunity to learn, even if identified long ago, remains a significant gatekeeper (Moses & Cobb, 2001)

whether blatantly or in more nuanced forms (Nasir & Cobb, 2007). 5 Gutierrez (2002) calls for a fine-tuning of the concepts access and achievement as markers of a just mathematical

education. Acknowledging heterogeneity within and between groups of students, it is not evident that having all students reach the same goals represents justice for students‘ own desires or sense of self. Rather, she emphasizes a goal of being unable to predict patterns in achievement ―based solely on characteristics such as race, class, ethnicity, sex, beliefs and creeds, and proficiency in the dominant language‖ (p. 153).


Both (M)athematics and (m)athemetics are models for knowing, a model that we imagine some

omnipotent observer to be able to know. But that model is no more an illusion than the real to which it is

imagined to represent. Each are a version of the hyperreal; each are simulacra.

In a reconsideration of achievement as a goal of a socially just mathematics education, the (M) /

(m) distinction in itself poses a dilemma; it identifies, (re-)confirms, solidifies the notion of the

unknowability of a child‘s way of knowing (M)athematics. So the actual testing myth, one measure of

achievement, may be that any manner of assessment provides no access to a truth about the child‘s

mathematics. Yet this deconstruction of achievement only recognizes the second-order of the operation

of the simulacra. It treats the child‘s knowing as an illusion to some real that may actually be present. It is

Baudrillard that reminds us the referent of this name, (m)athematics, the child‘s mathematics, is always

already a referent itself. It is a notion that I imagine within my own unknowable mathematical way of

knowing; the models built for mathematical knowledge construction of the other are infinitely circular in

reference. What then can be made of this notion of achievement?

Reconstituting Mathematical Identity or Agency as Authority. The (M) / (m) distinction allows

for further discussion of achievement, as well as access, authority, and action. The notions of access and

achievement are posed fully in relation to (M)athematics. Ernest (2002) names this mathematical

empowerment. This privileged power/knowledge, an enlightenment era relic, remains a gateway (Moses

& Cobb, 2001) to the cultural capital that schools are directed to deliver. Gutstein (2006) noted that a

teaching goal for mathematics must embrace this potential to read the (M)athematical word, quite similar

to his teaching goal to succeed academically in the traditional sense. Gutstein extended this argument

that mathematics education should embrace the goal to read and write the world with mathematics;6

however he did not note the constructivist distinction among ways of referring to mathematics as I have

brought forth here.

To recognize that the child both writes the word of (m)athematics and writes the world with

(m)athematics (Freire, 2002) is fully imbuing the learner as an author of their experiential reality, the third

cornerstone for teaching mathematics for social justice. The child is an author of (m)athematics, and an

actor upon the world with her (m)athematics. To both attribute this authority to the child, as well as foster

6 Ernest‘s (2002) social empowerment.


the child‘s own awareness of this authority is the deference of power7 the constructivist epistemology

allows for. The child that sees oneself as the constructor of the knowledge guiding her way of knowing the

world, gains an ownership in the activity of living/interacting. This sort of self-concept in relation to

mathematics may be considered as a ―robust mathematics identity‖ (Martin & McGee, 2009, p. 233). It

reflects Ernest‘s (2002) epistemological empowerment, an ―individual‘s growth of confidence not only in

using mathematics, but also a personal sense of power over the creation and validation of knowledge‖ (p.

9).

Assuring confidence and competence in the learner‘s own ways of knowing and thinking are

essential, but incomplete without the coupling of an awareness of the perpetual incompleteness of these

ways of knowing. Taken to an extreme may render the mathematical learner overly egoist, unphased by

the knowing of others (Grieb & Easley, 1985; Lawler, 2008, 2010). Coming to value others‘ confidence

and competence in their knowing, and regarding that knowing of the other as not identical to one‘s own is

necessary (Lawler, 2005), what Boaler (2008) refers to as a relational equity. I consider this to be

conferring an independent existence on others. Regarding others‘ ways of knowing and thinking as not

identical to one‘s own always keeps in play possibility.

This shift in authority of knowledge, from the presumed guild of (M)athematicians, the

(M)athematics teacher, or (M)athematics textbook, to the constructing knower, justifies the subconscious

need to act upon the others that constitute one‘s social realm in a more just manner. This need to act, to

write the world, is the call for social action that underlies Gutstein‘s (2006) theory. That one does author

knowledge, mathematical or otherwise, places the knower at the foreground of the world that unfurls in

front of them. We know the world, the experiential world of constructivism, through our interactions with it.

Insomuch, we have a role in shaping that world. Through our own (m)athematics, we act upon the world.

To engage students in reflection, discussion, and decision on intentional acts and non-acts upon the

world engages them in the ethics of determining and enacting what is fair, a fundamental activity of social

justice. That children understand their role in authoring (writing) the world, and their decisions on how that

authoring shapes the world, speaks to the fourth component of social justice education, action.

7 ―Power is always present‖ (Foucault, 1997), and as such, ―there is freedom everywhere‖ (St. Pierre, 2000, p. 490).


The Generative Adolescent Mathematical Learner (GAML). My previous research has led me

to focus in particular on the personal epistemologies of high school mathematics students, in particular

students who seemed to have maintained a generative disposition toward mathematics. A generative

disposition is meant to characterize the learner as someone who operates mathematically in ways that

reflects an internal sense of authority for knowing and a constructive orientation to the knowledge they

come to know (Lawler, 2008). This sort of productive disposition seems to be a ―major goal in the current

educational reform movement, more generally, and in the reform movement in mathematics education, in

particular‖ (Yackel & Cobb, 1996, p. 473). In my research, I sought to better understand these personal

epistemologies, to examine the school structures that may have influenced the dispositions, and to gain

insight into how these adolescents seemed to maintain such a disposition so far into their schooling

(Lawler, 2008, 2010).

Previously, Grieb and Easley (1984) explored the social mechanism of primary schools regarding

the development of an independent attitude in mathematically creative children. They found that white,

middle-class males more often than females and minorities, can ―survive pressures toward conformity in

the early grades with their confidence in mathematical reasoning intact and thus preserve more courage

to tackle new types of problems throughout their schooling‖ (p. 318). It appeared to Grieb and Easley that

these boys have an advantage over girls and minorities of equal creativity because their teachers seemed

to not expect them to conform to social norms of arithmetic. Grieb and Easley hypothesized that socio-

cultural norms dictate affordances and constraints in the powered social interaction between teacher and

student. If the ―authoritarian‖ (p. 355) role of mathematics teachers were reduced, this would ―lead to

more pupil thinking and less straining to reproduce procedures‖ (p. 355). Hence, more students could

maintain a productive disposition in their mathematical learning.

I conducted this study in an urban high school of the American Northeast. With the help of the

mathematics faculty, I identified six Grade 11 students in two classrooms who had potential to be

characterized as GAMLs. For one month I co-taught these students in two classes led by two different

teachers. My interactions with the students were as both as teacher and researcher. Here I highlight a

few accounts of two subject‘s orientation to mathematics, the manners in which I perceived them to

operate mathematically, and the role of the teacher in their classroom.


One student, Fisk, described his sense of the existence of mathematics, ―It wasn‘t invented by

mathematicians because a mathematician was somebody who was good at that position. Somebody else

had to realize it at first, therefore they couldn‘t have been a master at that position.‖ He continued,

If something is discovered, that means it‘s already there. Invented is somebody‘s idea in

their own head, or an innovation on another, something to make it better. I can‘t say I

discovered sand. But I can say I did invent a sand castle.

For this student, sand existed prior to experience. Fisk considered his mathematical activity akin to

inventing the sand castle. He hadn‘t invented the materials with which he crafted meaning, but it was of

his own work to craft that meaning. In this way, his mathematical activity was inventing ways of doing a

mathematics. His ―sand castle‖ (mathematics) was not so much a generalized sand castle, but each

particular; hence the potential for invention.

Fisk‘s activity suggested he located authority internally. He was dissatisfied with mathematical

work until things made ―sense‖, in his mind. For example, as the class worked to convert standard-form

quadratics to vertex form, Fisk was wrestling with a problem, thinking aloud, ―It don‘t make no sense‖.

He did not ask for the teacher to confirm his accuracy or conclusions, but sought to share his

knowing with the teacher to either demonstrate that he was on track or to show off his understanding. If a

teacher or classmate disagreed, or hinted of an error through their tone, Fisk would think further.

Interestingly, ideas recorded to the whiteboard, either by the teacher or by other students, seemed to be

granted a heightened authority, as though sanctioned.

Fisk‘s teacher respected him for his mathematical thinking. In other mathematics classes, Fisk

shut down. The swings in perceived mathematical abilities by his teachers spoke to some need for ego

protection.

Jack‘s teacher also exhibited an openness to his ways of thinking and learning. Her simple belief

in him seemed to infuse a belief in himself, and thus affected his classroom activity. Jack wrote about this

impact, ―Since I been in the math class my teacher has confidence in me that I know what I am doing and

can help others.‖ Again later, ―My teacher also has a lot of confidence in me when I am doing math

because she know I can do math. She like[s] the way I do math that she wants me to share my ideas with

other students in the class.‖


Jack did much of his mathematical work in his head, never recorded to paper. Also, what

appeared to be detachment from the classroom would often and unforgettably conclude with a

surprisingly insightful connection to a topic being discussion. I asked him during an interview about what

prompted his breaks from intense mathematical work. He replied, ―I had to take a break ‗cause there was

wearin‘ on my mind.‖ Jack thought hard about mathematics. ―There‘s plenty of times when I been

daydreaming. Then like sometimes I daydream and I come right back in and I answer a question and then

I‘m right and I‘m like, how was I right?‖ Jack was aware that although he disconnected from what was

going on, he was not so disconnected; he provided solutions to questions asked. This quality surprised

him.

For Jack, the ideas of mathematics were invented, and then frequently rediscovered; invented

and discovered by people, including both mathematicians and himself. His initial dispositional survey and

follow-up interview bore out these findings. Jack‘s orientation toward the invented quality of mathematics

suggested that he would see himself as a mathematical author, and hence an authority for his knowing.

While interactions reported above support an internal locus of authority, other episodes suggested

contradictory evidence. For example, I asked him to share an idea with the class. His immediate response

was to ask me, ―Is it right?‖. However, considering Jack‘s frequent verbal participation in classroom

discussion, rather than seeking affirmation, this query more likely marked surprise to the possibility that

what he had offered in private to me was correct.

The students of this study indicated that adolescent mathematics learners can see mathematics

as activity, rather than as a static entity. The nature of mathematical activity and knowing was highly

social for the subjects of my study. And the knowledge lived in the interaction, never static, never

necessarily locked in place. And they seemed to not need it to, at least in the overwhelming ways adult

mathematics educators seem to treat the authority of an ontologically privileged mathematics knowledge.

While mathematics had an important social role, these subjects maintained a strong notion that it was for

them to determine a truth to the knowing, particularly evident in Fisk. The disagreement of another served

as a catalyst for further inquiry and drive to make meaning, rather than passive acceptance of proof from

external authority. These students drew heavily on interaction with others to confirm their own knowing, to

feel justified to speak of confirmed facts (Glasersfeld, 1995).


Occasional conflict between reported confidence in knowing and externalized artifacts of such

knowing had its roots in the need to communicate personal ways of thinking in conventional manners.

Both Fisk and Jack demonstrated moderate dysfunction in relation to classroom structures. The two

teachers I worked with in the study were open and flexible with students, but other teachers reported a

variety of trouble with these students. The subjects of my study could have been regarded as lazy,

uninterested in doing well in school, talkative, too confident in their ability, disrespectful of rules and

classroom norms, and even aggressively defiant of authority.

Implications for Research. As much as anything else, a poststructural view on knowledge, and

mathematics in particular, as no more (or no less) than simulacra, opens up new ways of considering

mathematical learners. What is a mathematical identity or mathematical agency? As I‘ve suggested, a

more appropriate approach may be to consider a question of authority while seeking to understand the

personal epistemologies of mathematical learners. How do we conceive of or describe the mathematical

activity, or the mathematical knowledge of students when all such description is that of an observer,

constructed via a self-analysis of ones own knowing of the observed, and then read and made sense of

by another, in essence the never-ending, self-referential cycle of Baudrillard‘s hyperreality? And if that

mathematical knowledge we wish to attribute to the subject is no more than simulacra, what is to be

counted as achievement?

Research efforts in mathematics education will need to redefine its methods and its means for

data collection, at minimum, if mathematical knowledge is not what it is thought to be. But further,

research itself must be reconstituted no longer as an apparatus for knowledge production, but for

something else. Such a reconstitution of the purpose of research would immediately reject the deficit-

minded work to determine what might be better than, or to promulgate the gap-gazing fetish.

And what is to be made of efforts to teach mathematics, when mathematics has been relegated

to a shadow of the revered knowledge that it cannot be. Although Fawcett (1938) conceived of

mathematics very differently at the time, he claimed that mathematics could be learned through

engagement in critical thinking. I suggest that through engaging students in such activity, a mathematics

would necessarily emerge (Lawler, 2008) and that in and of itself ought to be a worthy mathematical and

educational goal. Achievement would be relegated to simply being, thinking, interacting; notions that may


no longer be hierarch-able.8 And both learners and teachers would not be compelled to come to some

sort of external and particular knowledge, rather mathematical authority and the authority for knowing

more palpably resides in each knower.

Conclusion

Baudrillard (1987) pushes the postmodern view on knowledge well past that of power/knowledge

paradigm proposed by Foucault (1980) by suggesting that such an analogy makes reference to notions

that have an existence by insisting (or refusing to not insist) upon their non-existence, or ever-changing

nature. In this binaried act of the use of humanist language, quite possibly an activity that we cannot

avoid (St. Pierre & Willow, 2000), to examine the power relations of a regime of truth, the non-real of the

supposed real of knowledge is left in tact, an acceptance of a real that conceals that there is none.

Baudrillard throws out the real, leaving us only with simulacra with which to think; it is the simulacra that is

true.

I recognize that in this deconstruction of mathematics there are grave dangers, ones that I likely

cannot escape. As a product of my historicity, my subjectivity, my whiteness, my privilege, the ideas I

present here reflect some hierarchy of power (West, 1989, p. 91). It is clear through my citations and

ideological referents that the thinking here is unduly influenced by critical thought emergent from the

Frankfurt school, and virtually untouched by lines of thought such as the social theory of DuBois, or other

non-Western lines to which I am fully ignorant. Yet to fail to write in spite of such danger is dangerous too;

one voice silenced silences all. It remains my goal through my scholarship in mathematics education to

―stimulate, hasten, and enable alternative perceptions and practices by dislodging prevailing discourses

and powers‖ (West, 1985, p. 122).

It may be that only ―from the views of subordinated individuals and communities that we will learn

how to rethink mathematics education‖ (Gutierrez, 2010, p. 3). However, I do not believe this was the

intent of Gutierrez‘ statement, rather it was to say that no univocal approach to moving forward with a

reconstitution of mathematics can be adequate. There cannot be one voice, particular desirable

8 Turkle and Papert (1992) make a similar call for a different mathematics education. Their efforts to define

constructionism as an epistemological viewpoint intentionally disrupt epistemologies that may be described as hierarchical-centralized-distanced in order to create heterarchal-decentralized-personal conceptions of knowledge. They claim such a conception of knowledge ―reflects both the political/social and epistemological confrontations in the battle between curriculum-centered, teacher-driven forms of instruction, and student-centered developmental approaches to intellectual growth‖ (p. 15).


outcomes, arbitrary traits that are possessed. Such a goal, a system of standards, the belief that some

possess these idealized and necessary ways of knowings, is intellectual fascism.

Crippling the privileged way of knowing a (M)athematics, or in fact any mathematics, serves to

castrate the intellectual fascism of mathematics education. A decentralized (Turkle & Papert, 1992)

concept of mathematical knowledge, knowledge as simulacra, can allow for a new consideration of what

may constitute a critical mathematics education.

Azrael has come to claim mathematics, to lead it to its place amid the Procession of the

Disciplines toward the underworld of dead knowledge. It is among these rotting remains of modernity the

rhizome thrives.


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