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The Mathematics Educator2004, Vol. 14, No. 1, 8–18

8 Mathematics as “Gate-Keeper” (?)

Mathematics as “Gate-Keeper” (?): Three TheoreticalPerspectives that Aim Toward Empowering All Children With

a Key to the GateDavid W. Stinson

In this article, the author’s intent is to begin a conversation centered on the question: How might mathematicseducators ensure that gatekeeping mathematics becomes an inclusive instrument for empowerment rather thanan exclusive instrument for stratification? In the first part of the discussion, the author provides a historicalperspective of the concept of “gatekeeper” in mathematics education. After substantiating mathematics as agatekeeper, the author proceeds to provide a definition of empowering mathematics within a Freirian frame, anddescribes three theoretical perspectives of mathematics education that aim toward empowering all children witha key to the gate: the situated perspective, the culturally relevant perspective, and the critical perspective. Last,within a Foucauldian frame, the author concludes the article by asking the reader to think differently.

My graduate assistantship in The Department ofMathematics Education at The University of Georgiafor the 2002–2003 academic year was to assist with afour-year Spencer-funded qualitative research projectentitled “Learning to Teach Elementary Mathematics.”This assistantship presented the opportunity to conductresearch at elementary schools in two suburbancounties—a new experience for me since my priorprofessional experience in education had been withinthe context of secondary mathematics education. Myresearch duties consisted of organizing, coding,analyzing, and writing-up existing data, as well ascollecting new data. This new data includedtranscribed interviews of preservice and noviceelementary school teachers and fieldnotes fromclassroom observations.

By January 2003, I had conducted fiveobservations in 1st, 2nd, and 3rd grade classrooms attwo elementary schools with diverse populations. I wasimpressed with the preservice and novice elementaryteachers’ mathematics pedagogy and ability to interactwith their students. Given that my research interest isequity and social justice in education, I was mindful ofthe “racial,” ethnic, gender, and class make-up of theclassroom and how these attributes might help meexplain the teacher-student interactions I observed. My

experiences as a secondary mathematics teacher,preservice-teacher supervisor, and researcher supportedOakes’s (1985) assertions that often students aredistributed into “ability” groups based on their race,gender, and class. Nonetheless, my perception afterfive observations was that ability grouping accordingto these attributes was diminishing—at least in theseelementary schools. In other words, the student make-up of each mathematics lesson that I observedappeared to be representative of the demographics ofthe school.

However, on my sixth observation, at anelementary school with 35.8 % Black, 12.8 % Asian,5.3 % Hispanic, 3.5 % Multi-racial, and 0.5 %American Indian1 children, I observed a 3rd grademathematics lesson that was 94.4% White (at least itwas 50% female). The make-up of the classroom wasnot initially unrepresentative of the school’sracial/ethnic demographics, but became so shortlybefore the start of the mathematics lesson as somestudents left the classroom while others entered. WhenI questioned why the students were exchanged betweenclassrooms, I was informed that the mathematicslesson was for the “advanced” third graders. Becauseof my experience in secondary mathematics education,I am aware that academic tracking is a nationallypracticed education policy, and that it even occurs inmany districts and schools as early as 5th grade—butthese were eight-year-old children! Has the structure ofpublic education begun to decide who is and who is not“capable” mathematically in the 3rd grade? Has thestructure of public education begun to decide who willbe proletariat and who will be bourgeoisie in the 3rdgrade—with eight-year-old children? How did school

David W. Stinson is a doctoral candidate in The Department ofMathematics Education at The University of Georgia. In the fallof 2004, he will join the faculty of the Middle-SecondaryEducation and Instructional Technology Department at GeorgiaState University. His research interests are the sociopolitical andcultural aspects of mathematics and mathematics teaching andlearning with an emphasis on equity and social justice inmathematics education and education in general.

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mathematics begin to (re)produce and regulate racial,ethnic, gender, and class divisions, becoming a“gatekeeper”? And (if) school mathematics is agatekeeper, how might mathematics educators ensurethat gatekeeping mathematics becomes an inclusiveinstrument for empowerment rather than an exclusiveinstrument for stratification?

This article provides a two-part discussion centeredon the last question. The first part of the discussionprovides a historical perspective of the concept ofgatekeeper in mathematics education, verifying thatmathematics is an exclusive instrument forstratification, effectively nullifying the if. The intent ofthis historical perspective is not to debate whethermathematics should be a gatekeeper but to provide aperspective that reveals existence of mathematics as agatekeeper (and instrument for stratification) in thecurrent education structure of the United States. In thediscussion, I state why I believe all students are notprovided with a key to the gate.

After arguing that mathematics is a gatekeeper andinequities are present in the structure of education, Iproceed to the second part of the discussion: howmight mathematics educators ensure that gatekeepingmathematics becomes an inclusive instrument forempowerment? In this discussion, I first defineempowerment and empowering mathematics. Then, Imake note of the “social turn” in mathematicseducation research, which provides a framework forthe situated, culturally relevant, and criticalperspectives of mathematics education that arepresented. Finally, I argue that these theoreticalperspectives replace characteristics of exclusion andstratification (of gatekeeping mathematics) withcharacteristics of inclusion and empowerment. Iconclude the article by challenging the reader to thinkdifferently.

Mathematics a Gatekeeper: A HistoricalPerspective

Discourse regarding the “gatekeeper” concept inmathematics can be traced back over 2300 years ago toPlato’s (trans. 1996) dialogue, The Republic. In thefictitious dialogue between Socrates and Glauconregarding education, Plato argued that mathematicswas “virtually the first thing everyone has tolearn…common to all arts, science, and forms ofthought” (p. 216). Although Plato believed that allstudents needed to learn arithmetic—”the trivialbusiness of being able to identify one, two, and three”(p. 216)—he reserved advanced mathematics for thosethat would serve as philosopher guardians2 of the city.

He wrote:We shall persuade those who are to perform highfunctions in the city to undertake calculation, butnot as amateurs. They should persist in their studiesuntil they reach the level of pure thought, wherethey will be able to contemplate the very nature ofnumber. The objects of study ought not to bebuying and selling, as if they were preparing to bemerchants or brokers. Instead, it should serve thepurposes of war and lead the soul away from theworld of appearances toward essence and reality.(p. 219)

Although Plato believed that mathematics was ofvalue for all people in everyday transactions, the studyof mathematics that would lead some men from“Hades to the halls of the gods” (p. 215) should bereserved for those that were “naturally skilled incalculation” (p. 220); hence, the birth of mathematicsas the privileged discipline or gatekeeper.

This view of mathematics as a gatekeeper haspersisted through time and manifested itself in earlyresearch in the field of mathematics education in theUnited States. In Stanic’s (1986) review ofmathematics education of the late 19th and early 20thcenturies, he identified the 1890s as establishing“mathematics education as a separate and distinctprofessional area” (p. 190), and the 1930s asdeveloping the “crisis” (p. 191) in mathematicseducation. This crisis—a crisis for mathematicseducators—was the projected extinction ofmathematics as a required subject in the secondaryschool curriculum. Drawing on the work of Kliebard(c.f., Kliebard, 1995), Stanic provided a summary ofcurriculum interest groups that influenced the positionof mathematics in the school curriculum: (a) thehumanists, who emphasized the traditional disciplinesof study found in Western philosophy; (b) thedevelopmentalists, who emphasized the “natural”development of the child; (c) the social efficiencyeducators, who emphasized a “scientific” approach thatled to the natural development of social stratification;and (d) the social meliorists, who emphasizededucation as a means of working toward social justice.

Stanic (1986) noted that mathematics educators, ingeneral, sided with the humanists, claiming:“mathematics should be an important part of the schoolcurriculum” (p. 193). He also argued that thedevelopment of the National Council of Teachers ofMathematics (NCTM) in 1920 was partly in responseto the debate that surrounded the position ofmathematics within the school curriculum.

10 Mathematics as “Gate-Keeper” (?)

The founders of the Council wrote:Mathematics courses have been assailed on everyhand. So-called educational reformers havetinkered with the courses, and they, not knowingthe subject and its values, in many cases havethrown out mathematics altogether or made itentirely elective. …To help remedy the existingsituation the National Council of Teachers ofMathematics was organized. (C. M. Austin asquoted in Stanic, 1986, p. 198)

The backdrop to the mathematics education crisiswas the tremendous growth in school population thatoccurred between 1890 and 1940—a growth of nearly20 times (Stanic, 1986). This dramatic increase in thestudent population yielded the belief that the overallintellectual capabilities of students had decreased;consequently, students became characterized as the“army of incapables” (G. S. Hall as quoted in Stanic,1986, p. 194). Stanic presented the results of thisprevailing belief by citing the 1933 National Survey ofSecondary Education, which concluded that less thanhalf of the secondary schools required algebra andplane geometry. And, he illustrated mathematicsteachers’ perspectives by providing George Counts’1926 survey of 416 secondary school teachers.Eighteen of the 48 mathematics teachers thought thatfewer pupils should take mathematics, providing acontrast to teachers of other academic disciplines whobelieved that “their own subjects should be morelargely patronized” (G. S. Counts as quoted in Stanic,p. 196). Even so, the issues of how mathematics shouldbe positioned in the school curriculum and who shouldtake advanced mathematics courses was not a majornational concern until the 1950s.

During the 1950s, mathematics education in U.S.schools began to be attacked from many segments ofsociety: the business sector and military for graduatingstudents who lacked computational skills, colleges forfailing to prepare entering students with mathematicsknowledge adequate for college work, and the publicfor having “watered down” the mathematicscurriculum as a response to progressivism (Kilpatrick,1992). The launching of Sputnik in 1957 furtherexacerbated these attacks leading to a national demandfor rigorous mathematics in secondary schools. Thisdemand spurred a variety of attempts to reformmathematics education: “the ‘new’ math of the 1960s,the ‘back-to-basic’ programs of the 1970s, and the‘problem-solving’ focus of the 1980s” (Johnston,1997). Within these programs of reform, the questionswere not only what mathematics should be taught and

how, but more importantly, who should be taughtmathematics.

The question of who should be taught mathematicsinitially appeared in the debates of the 1920s andcentered on “ascertaining who was prepared for thestudy of algebra” (Kilpatrick, 1992, p. 21). Thesedebates led to an increase in grouping studentsaccording to their presumed mathematics ability. This“ability” grouping often resulted in excluding femalestudents, poor students, and students of color from theopportunity to enroll in advanced mathematics courses(Oakes, 1985; Oakes, Ormseth, Bell, & Camp, 1990).Sixty years after the beginning of the debates, therecognition of this unjust exclusion from advancedmathematics courses spurred the NCTM to publish theCurriculum and Evaluation Standards for SchoolMathema t i c s (Standards, 1989) that includedstatements similar to the following:

The social injustices of past schooling practices canno longer be tolerated. Current statistics indicatethat those who study advanced mathematics aremost often white males. …Creating a just societyin which women and various ethnic groups enjoyequal opportunities and equitable treatment is nolonger an issue. Mathematics has become a criticalfilter for employment and full participation in oursociety. We cannot afford to have the majority ofour population mathematically illiterate: Equity hasbecome an economic necessity. (p. 4)

In the Standards the NCTM contrasted societalneeds of the industrial age with those of theinformation age, concluding that the educational goalsof the industrial age no longer met the needs of theinformation age. They characterized the informationage as a dramatic shift in the use of technology whichhad “changed the nature of the physical, life, and socialsciences; business; industry; and government” (p. 3).The Council contended, “The impact of thistechnological shift is no longer an intellectualabstraction. It has become an economic reality” (p. 3).

The NCTM (1989) believed this shift demandednew societal goals for mathematics education: (a)mathematically literate workers, (b) lifelong learning,(c) opportunity for all, and (d) an informed electorate.They argued, “Implicit in these goals is a schoolsystem organized to serve as an important resource forall citizens throughout their lives” (p. 3). These goalsrequired those responsible for mathematics educationto strip mathematics from its traditional notions ofexclusion and basic computation and develop it into adynamic form of an inclusive literacy, particularlygiven that mathematics had become a critical filter for

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full employment and participation within a democraticsociety. Countless other education scholars(Frankenstein, 1995; Moses & Cobb, 2001; Secada,1995; Skovsmose, 1994; Tate, 1995) have madesimilar arguments as they recognize the need for allstudents to be provided the opportunity to enroll inadvanced mathematics courses, arguing that a dynamicmathematics literacy is a gatekeeper for economicaccess, full citizenship, and higher education. In theparagraphs that follow, I highlight quantitative andqualitative studies that substantiate mathematics as agatekeeper.

The claims that mathematics is a “critical filter” orgatekeeper to economic access, full citizenship, andhigher education are quantitatively substantiated bytwo reports by the U. S. government: the 1997 WhitePaper entitled Mathematics Equals Opportunity andthe 1999 follow-up summary of the 1988 NationalEducation Longitudinal Study (NELS: 88) entitled DoGatekeeper Courses Expand Education Options? TheU. S. Department of Education prepared both reportsbased on data from the NELS: 88 samples of 24,599eighth graders from 1,052 schools, and the 1992follow-up study of 12,053 students.

In Mathematics Equals Opportunity, the followingstatements were made:

In the United States today, mastering mathematicshas become more important than ever. Studentswith a strong grasp of mathematics have anadvantage in academics and in the job market. The8th grade is a critical point in mathematicseducation. Achievement at that stage clears theway for students to take rigorous high schoolmathematics and science courses—keys to collegeentrance and success in the labor force.

Students who take rigorous mathematics andscience courses are much more likely to go tocollege than those who do not.

Algebra is the “gateway” to advanced mathematicsand science in high school, yet most students donot take it in middle school.

Taking rigorous mathematics and science coursesin high school appears to be especially importantfor low-income students.

Despite the importance of low-income studentstaking rigorous mathematics and science courses,these students are less likely to take them. (U. S.Department of Education, 1997, pp. 5–6)

This report, based on statistical analyses, explicitlystated that algebra was the “gateway” or gatekeeper toadvanced (i.e., rigorous) mathematics courses and that

advanced mathematics provided an advantage inacademics and in the job market—the same argumentprovided by the NCTM and education scholars.

The statistical analyses in the report entitled, DoGatekeeper Courses Expand Educational Options? (U.S. Department of Education, 1999) presented thefollowing findings:

Students who enrolled in algebra as eighth-graderswere more likely to reach advanced math courses(e.g., algebra 3, trigonometry, or calculus, etc.) inhigh school than students who did not enroll inalgebra as eighth-graders.

Students who enrolled in algebra as eighth-graders,and completed an advanced math course duringhigh school, were more likely to apply to a four-year college than those eighth-grade students whodid not enroll in algebra as eighth-graders, but whoalso completed an advanced math course duringhigh school. (pp. 1–2)

The summary concluded that not all students whotook advanced mathematics courses in high schoolenrolled in a four-year postsecondary school, althoughthey were more likely to do so—again confirmingmathematics as a gatekeeper.

Nicholas Lemann’s (1999) book The Big Test: TheSecret History of the American Meritocracy provides aqualitative substantiation that mathematics is agatekeeper to economic access, full citizenship, andhigher education. In Parts I and II of his book, Lemannpresented a detailed historical narrative of the mergerbetween the Educational Testing Service with theCollege Board. Leman argued this merger establishedhow mathematics would directly and indirectlycategorize Americans—becoming a gatekeeper—forthe remainder of the 20th and beginning of the 21stcenturies. During World War I, the United States WarDepartment (currently known as the Department ofDefense) categorized people using an adapted versionof Binet’s Intelligence Quotient test to determine theentering rank and duties of servicemen. Thiscategorization evolved into ranking people by“aptitude” through administering standardized tests incontemporary U. S. education.

In Part III of his book, Lemann presented a case-study characterization of contemporary Platonicguardians, individuals who unjustly (or not) benefitedfrom the concept of aptitude testing and the ideal ofAmerican meritocracy. Lemann argued that because oftheir ability to demonstrate mathematics proficiency(among other disciplines) on standardized tests, theseindividuals found themselves passing through the gates

12 Mathematics as “Gate-Keeper” (?)

to economic access, full citizenship, and highereducation.

The concept of mathematics as providing the keyfor passing through the gates to economic access, fullcitizenship, and higher education is located in the coreof Western philosophy. In the United States, schoolmathematics evolved from a discipline in “crisis” intoone that would provide the means of “sorting”students. As student enrollment in public schoolsincreased, the opportunity to enroll in advancedmathematics courses (the key) was limited becausesome students were characterized as “incapable.”Female students, poor students, and students of colorwere offered a limited access to quality advancedmathematics education. This limited access was amotivating factor behind the Standards, and thesubsequent NCTM documents.3

NCTM and education scholars’ argument thatmathematics had and continues to have a gatekeepingstatus has been confirmed both quantitatively andqualitatively. Given this status, I pose two questions:(a) Why does U.S. education not provide all studentsaccess to a quality, advanced (mathematics) educationthat would empower them with economic access andfull citizenship? and (b) How can we as mathematicseducators transform the status quo in the mathematicsclassroom?

To fully engage in the first question demands adeconstruction of the concepts of democratic publicschooling and American meritocracy and an analysis ofthe morals and ethics of capitalism. To provide such adeconstruction and analysis is beyond the scope of thisarticle. Nonetheless, I believe that Bowles’s(1971/1977) argument provides a comprehensive, yetcondensed response to the question of why U. S.education remains unequal without oversimplifying thecomplexities of the question. Through a historicalanalysis of schooling he revealed four components ofU. S. education: (a) schools evolved not in pursuit ofequality, but in response to the developing needs ofcapitalism (e.g., a skilled and educated work force); (b)as the importance of a skilled and educated work forcegrew within capitalism so did the importance ofmaintaining educational inequality in order toreproduce the class structure; (c) from the 1920s to1970s the class structure in schools showed no signs ofdiminishment (the same argument can be made for the1970s to 2000s); and (d) the inequality in educationhad “its root in the very class structures which it servesto legitimize and reproduce” (p. 137). He concluded bywriting: “Inequalities in education are thus seen as part

of the web of capitalist society, and likely to persist aslong as capitalism survives” (p. 137).

Although Bowles’s statements imply that only theoverthrow of capitalism will emancipate educationfrom its inequalities, I believe that developingmathematics classrooms that are empowering to allstudents might contribute to educational experiencesthat are more equitable and just. This development mayalso assist in the deconstruction of capitalism so that itmight be reconstructed to be more equitable and just.The following discussion presents three theoreticalperspectives that I have identified as empoweringstudents. These perspectives aim to assist in moreequitable and just educative experiences for allstudents: the situated perspective, the culturallyrelevant perspective, and the critical perspective. Ibelieve these perspectives provide a plausible answerto the second question asked above: How do we asmathematics educators transform the status quo in themathematics classroom?

An Inclusive Empowering Mathematics EducationTo frame the discussion that follows, I provide a

definition of e m p o w e r m e n t and empoweringmathematics. Freire (1970/2000) framed the notion ofempowerment within the concept of conscientização,defined as “learning to perceive social, political andeconomic contradictions, and to take action against theoppressive elements of reality” (p. 35). He argued thatconscientização leads people not to “destructivefanaticism” but makes it possible “for people to enterthe historical process as responsible Subjects4” (p. 36),enrolling them in a search for self-affirmation.Similarly, Lather (1991) defined empowerment as theability to perform a critical analysis regarding thecauses of powerlessness, the ability to identify thestructures of oppression, and the ability to act as asingle subject, group, or both to effect change towardsocial justice. She claimed that empowerment is alearning process one undertakes for oneself; “it is notsomething done ‘to’ or ‘for’ someone” (Lather, 1991,p. 4). In effect, empowerment provides the subject withthe skills and knowledge to make sociopoliticalcritiques about her or his surroundings and to takeaction (or not) against the oppressive elements of thosesurroundings. The emphasis in both definitions is self-empowerment with an aim toward sociopoliticalcritique. With this emphasis in mind, I next defineempowering mathematics.

Ernest (2002) provided three domains ofempowering mathematics—mathematical, social, andepistemological—that assist in organizing how I define

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empower ing mathemat ics . Mathemat ica lempowerment relates to “gaining the power over thelanguage, skills and practices of using mathematics”(section 1, ¶ 3) (e.g., school mathematics). Socialempowerment involves using mathematics as a tool forsociopolitical critique, gaining power over the socialdomains—“the worlds of work, life and social affairs”(section 1, ¶ 4). And, epistemological empowermentconcerns the “individual’s growth of confidence in notonly using mathematics, but also a personal sense ofpower over the creation and validation ofknowledge”(section 1, ¶ 5). Ernest argued, and I agree,that all students gain confidence in their mathematicsskills and abilities through the use of mathematics inroutine and nonroutine ways and that this confidencewill logically lead to higher levels of mathematicsattainment. All students achieving higher levels ofattainment will assist in leveling the racial, gender, andclass imbalances that currently persist in advancedmathematics courses. Effectively, Ernest’s definition ofempowering mathematics echoes the definition ofempowerment stated earlier.

Using Ernest’s three domains of empoweringmathematics as a starting point, I selected threeempowering mathematics perspectives. In doing so, Ikept in mind Stanic’s (1989) challenge to mathematicseducators: “If mathematics educators take seriously thegoal of equity, they must question not just the commonview of school mathematics but also their own taken-for-granted assumptions about its nature and worth” (p.58). I believe that the situated perspective, culturallyrelevant perspective, and critical perspective, invarying degrees, motivate such questioning andresonate with the definition I have given ofempowering mathematics. These configurations arecomplex theoretical perspectives derived from multiplescholars who sometimes have conflicting workingdefinitions. These perspectives, located in the “socialturn” (Lerman, 2000, p. 23) of mathematics educationresearch, originate outside the realm of “traditional”mathematics education theory, in that they are rootedin anthropology, cultural psychology, sociology, andsociopolitical critique. In the discussion that follows, Iprovide sketches of each theoretical perspective bybriefly summarizing the viewpoints of key scholarsworking within the perspective. I then explain howeach perspective holds possibilities in transforminggatekeeping mathematics from an exclusive instrumentfor stratification into an inclusive instrument forempowerment.

The Situated PerspectiveThe situated perspective is the coupling of

scholarship from cultural anthropology and culturalpsychology. In the situated perspective, learningbecomes a process of changing participation inchanging communities of practice in which anindividual’s resulting knowledge becomes a functionof the environment in which she or he operates.Consequently, in the situated perspective, the dualismsof mind and world are viewed as artificial constructs(Boaler, 2000b). Moreover, the situated perspective, incontrast to constructivist perspectives, emphasizesinteractive systems that are larger in scope than thebehavioral and cognitive processes of the individualstudent.

Mathematics knowledge in the situated perspectiveis understood as being co-constituted in a communitywithin a context. It is the community and context inwhich the student learns the mathematics thatsignificantly impacts how the student uses andunderstands the mathematics (Boaler, 2000b). Boaler(1993) suggested that learning mathematics in contextsassists in providing student motivation and interest andenhances transference of skills by linking classroommathematics with real-world mathematics. She argued,however, that learning mathematics in contexts doesnot mean learning mathematics ideas and proceduresby inserting them into “real-world” textbook problemsor by extending mathematics to larger real-world classprojects. Rather, she suggested that the classroom itselfbecomes the context in which mathematics is learnedand understood: “If the students’ social and culturalvalues are encouraged and supported in themathematics classroom, through the use of contexts orthrough an acknowledgement of personal routes anddirection, then their learning will have more meaning”(p. 17).

The situated perspective offers different notions ofwhat it means to have mathematics ability, changingthe concept from “either one has mathematics ability ornot” to an analysis of how the environment co-constitutes the mathematics knowledge that is learned(Boaler, 2000a). Boaler argued that this change in howmathematics ability is assessed in the situatedperspective could “move mathematics education awayfrom the discriminatory practices that produce morefailures than successes toward something considerablymore equitable and supportive of social justice” (p.118).

14 Mathematics as “Gate-Keeper” (?)

The Culturally Relevant PerspectiveWorking toward social justice is also a component

of the culturally relevant perspective. Ladson-Billings(1994) developed the “culturally relevant” (p. 17)perspective as she studied teachers who weresuccessful with African-American children. Thisperspective is derived from the work of culturalanthropologists who studied the cultural disconnectsbetween (White) teachers and students of color andmade suggestions about how teachers could “matchtheir teaching styles to the culture and homebackgrounds of their students” (Ladson-Billings, 2001,p. 75). Ladson-Billings defined the culturally relevantperspective as promoting student achievement andsuccess through cultural competence (teachers assiststudents in developing a positive identification withtheir home culture) and through sociopoliticalconsciousness (teachers help students develop a civicand social awareness in order to work toward equityand social justice).

Teachers working from a culturally relevantperspective (a) demonstrate a belief that children canbe competent regardless of race or social class, (b)provide students with scaffolding between what theyknow and what they do not know, (c) focus oninstruction during class rather than busy-work orbehavior management, (d) extend students’ thinkingbeyond what they already know, and (e) exhibit in-depth knowledge of students as well as subject matter(Ladson-Billings, 1995). Ladson-Billings argued thatall children “can be successful in mathematics whentheir understanding of it is linked to meaningfulcultural referents, and when the instruction assumesthat all students are capable of mastering the subjectmatter” (p. 141).

Mathematics knowledge in the culturally relevantperspective is viewed as a version ofethnomathematics—ethno defined as all culturallyidentifiable groups with their jargons, codes, symbols,myths, and even specific ways of reasoning andinferring; mathema defined as categories of analysis;and t ics defined as methods or techniques (D’Ambrosio, 1985/1997, 1997). In the culturally relevantmathematics classroom, the teacher builds from thestudents’ ethno or informal mathematics and orientsthe lesson toward their culture and experiences, whiledeveloping the students’ critical thinking skills(Gutstein, Lipman, Hernandez, & de los Reyes, 1997).The positive results of teaching from a culturallyrelevant perspective are realized when studentsdevelop mathematics empowerment: deducingmathematical generalizations and constructing creative

solution methods to nonroutine problems, andperceiving mathematics as a tool for sociopoliticalcritique (Gutstein, 2003).

The Critical PerspectivePerceiving mathematics as a tool for sociopolitical

critique is also a feature of the critical perspective. Thisperspective is rooted in the social and political critiqueof the Frankfurt School (circa 1920) whosemembership included but was not limited to MaxHorkheimer, Theodor Adorno, Leo Lowenthal, andFranz Neumann. The critical perspective ischaracterized as (a) providing an investigation into thesources of knowledge, (b) identifying social problemsand plausible solutions, and (c) reacting to socialinjustices. In providing these most general andunifying characteristics of a critical education,Skovsmose (1994) noted, “A critical education cannotbe a simple prolongation of existing socialrelationships. It cannot be an apparatus for prevailinginequalities in society. To be critical, education mustreact to social contradictions” (p. 38).

Skovsmose (1994), drawing from Freire’s(1970/2000) popularization of the conceptc o n s c i e n t i z a ç ã o and his work in literacyempowerment, derived the term “mathemacy” (p. 48).Skovsmose claimed that since modern society is highlytechnological and the core of all modern-daytechnology is mathematics that mathemacy is a meansof empowerment. He stated, “If mathemacy has a roleto play in education, similar to but not identical to therole of literacy, then mathemacy must be seen ascomposed of different competences: a mathematical, atechnological, and a reflective” (p. 48).

In the critical perspective, mathematics knowledgeis seen as demonstrating these three competencies(Skovsmose, 1994). Mathematical competence isdemonstrating proficiency in the normally understoodskills of school mathematics, reproducing andmastering various theorems, proofs, and algorithms.Technological competence demonstrates proficiency inapplying mathematics in model building, usingmathematics in pursuit of different technological aims.And, reflective competence achieves mathematics’critical dimension, reflecting upon and evaluating thejust and unjust uses of mathematics. Skovsmosecontended that mathemacy is a necessary condition fora politically informed citizenry and efficient laborforce, claiming that mathemacy provides a means forempowerment in organizing and reorganizing socialand political institutions and their accompanyingtraditions.

David W. Stinson 15

Transforming Gatekeeping MathematicsThe preceding sketches demonstrate that these

three theoretical perspectives approach mathematicsand mathematics teaching and learning differently thantraditional perspectives. All three perspectives, invarying degrees, question the taken-for-grantedassumptions about mathematics and its nature andworth, locate the formation of mathematics knowledgewithin the social community, and argue thatmathematics is an indispensable instrument used insociopolitical critique. In the following paragraphs Iexplicate the degrees to which the three perspectivesaddress these issues.

The situated perspective negates the assumptionthat mathematics is a contextually free discipline,contending that it is the context in which mathematicsis learned that determines how it will be used andunderstood. The culturally relevant perspective negatesthe assumption that mathematics is a culturally freediscipline, recognizing mathematics is not separatefrom culture but is a product of culture. The criticalperspective redefines the worth of mathematicsthrough an acknowledgment and critical examinationof the just and, often overlooked, unjust uses ofmathematics.

The situated perspective locates mathematicsknowledge in the social community. In thisperspective, mathematics is not learned from amathematics textbook and then applied to real-worldcontexts, but is negotiated in communities that exist inreal-world contexts. The culturally relevant perspectivealso locates mathematics knowledge in the socialcommunity. This perspective argues teachers shouldbegin to build on the collective mathematicsknowledge present in the classroom communities,moving toward mathematics found in textbooks. Thecritical perspective does not locate mathematicsknowledge in the social community but is orientedtowards using mathematics to critique and transformthe social and political communities in whichmathematics exists and has its origins.

The situated perspective posits that students willbegin to understand mathematics as a discipline that islearned in the context of communities. It is in this waythat students may learn how mathematics can beapplied in uncovering the inequities and injusticespresent in communities or can be used forsociopolitical critique. Similarly, one of the two tenetsof the culturally relevant perspective is for the teacherto assist students in developing a sociopoliticalconsciousness. Finally, using mathematics as a meansfor sociopolitical critique is essential to the critical

perspective, as mathematics is understood as a tool thatcan be used for critique.

How do the three aspects of mathematics andmathematics teaching and learning relate to each otherin these perspectives and how does this relationshipaddress the three domains of empoweringmathematics? First, mathematics empowerment isachieved because each perspective questions theassumptions that are often taken-for-granted about thenature and worth of mathematics. Although all threeperspectives see value in the study of mathematics,including “academic”5 mathematics, they differ fromtraditional perspectives in that academic mathematicsitself is troubled6 with regards to its contextualexistence, its cultural connectedness, and its criticalutility. Second, students achieve social empowermentbecause all three perspectives argue that studentsshould engage in mathematics contextually andculturally; and, therefore students have the opportunityto gain confidence in using mathematics in routine andnonroutine problems. The advocates for these threeperspectives argue that as students expand the use ofmathematics into nonroutine problems, they becomecognizant of how mathematics can be used as a tool forsociopolitical critique. Finally students achieveepistemological empowerment because all threeperspectives trouble academic mathematics that in turnmay lead students to understand that the concept of a“true” or “politically-free” mathematics is a fiction.Students will hopefully understand that mathematicsknowledge is (and always has been) a contextually andculturally (and politically) constructed humanendeavor. If students achieve this perspective ofmathematics, they will better understand their role asproducers of mathematics knowledge, not justconsumers. Hence, the three domains of empoweringmathemat ics—mathemat ica l , soc ia l , andepistemological—are achieved in each perspective orthrough various combinations of the three perspectives.

The chief aim of an empowering mathematics is totransform gatekeeping mathematics from a disciplineof oppressive exclusion into a discipline ofempowering inclusion. (This aim is inclusive ofmathematics educators and education researchers.)Empowering inclusion is achieved when students (andteachers of mathematics) are presented with theopportunity to learn that the foundations ofmathematics can be troubled. This troubling ofmathematics’ foundations transforms the discourse inthe mathematics classroom from a discourse oftransmitting mathematics to a “chosen” few students,into a discourse of exploring mathematics with all

16 Mathematics as “Gate-Keeper” (?)

students. Empowering inclusion is achieved whenstudents (and teachers of mathematics) are presentedwith the opportunity to learn that, similar to literacy,mathemacy is a tool that can be used to reword worlds.This rewording of worlds (Freire, 1970/2000) withmathematics transforms mathematics from a tool usedby a few students in “mathematical” pursuits, into atool used by all students in sociopolitical pursuits.Finally, empowering inclusion is achieved whenstudents (and teachers of mathematics) are presentedwith the opportunity to learn that mathematicsknowledge is constructed human knowledge. Thisreturning to the origins of mathematics knowledgetransforms mathematics from an Ideal of the godsreproduced by a few students, into a human endeavorproduced by all students.

Concluding ThoughtsThe concept of mathematics as gatekeeper has a

very long and disturbing history. There have beeneducators satisfied with the gatekeeping status ofmathematics and those that have questioned not onlyits gatekeeping status but also its nature and worth. Inmy thinking about mathematics as a gatekeeper and thepossibility of transforming mathematics education, Ioften reflect on Foucault’s challenge. He challenged usto think the un-thought, to think: “how is it that oneparticular statement appeared rather than another?”(Foucault, 1969/1972, p. 27). With Foucault’schallenge in mind, I often think what if Plato had said,

We shall persuade those who are to perform highfunctions in the city to undertake ________, butnot as amateurs. They should persist in their studiesuntil they reach the level of pure thought, wherethey will be able to contemplate the very nature of________…. it should serve the purposes of warand lead the soul away from the world ofappearances toward essence and reality. (trans.1996, p. 219)

In the preceding blanks, I insert different humanpursuits, such as writing, speaking, painting, sculpting,dancing, and so on, asking: does mathematics reallylead the soul away from the world of appearancestoward essence and reality?7 Or could dancing, forexample, achieve the same result? While rethinkingPlato’s centuries old comment, I rethink the privilegedstatus of mathematics as a gatekeeper (and as aninstrument of stratification). But rather than askingwhat is school mathematics as gatekeeper or what doesit mean, I ask different questions: How does schoolmathematics as gatekeeper function? Where is schoolmathematics as gatekeeper to be found? How does

school mathematics as gatekeeper get produced andregulated? How does school mathematics asgatekeeper exist? (Bové, 1995). These questionstransform the discussions around gatekeepermathematics from discussions that attempt to findmeaning in gatekeeper mathematics to discussions thatexamine the ethics of gatekeeper mathematics. Implicitin this examination is an analysis of how the structureof schools and those responsible for that structure areimplicated (or not) in reproducing the unethical effectsof gatekeeping mathematics.

Will asking the questions noted above transformgatekeeping mathematics from an exclusive instrumentfor stratification into an inclusive instrument forempowerment? Will asking these questions stop the“ability” sorting of eight-year-old children? Willasking these questions encourage mathematics teachers(and educators) to adopt the situated, culturallyrelevant, or critical perspectives, perspectives that aimtoward empowering all children with a key? AlthoughI believe that there are no definitive answers to thesequestions, I do believe that critically examining (andimplementing) the different possibilities formathematics teaching and learning found in thetheoretical perspectives explained in this articleprovides a sensible beginning to transformingmathematics education. In closing, I fervently proclaimthe way we use mathematics today in our nation’sschools must stop! Mathematics should not be used asan instrument for stratification but rather an instrumentfor empowerment!

REFERENCESBoaler, J. (2000a). Exploring situated insights into research and

learning. Journal for Research in Mathematics Education,31(1), 113–119.

Boaler, J. (2000b). Mathematics from another world: Traditionalcommunities and the alienation of learners. Journal ofMathematical Behavior, 18(4), 379–397.

Boaler, J. (1993). The role of context in the mathematicsclassroom: Do they make mathematics more “real”? For theLearning of Mathematics, 13(2), 12–17.

Bové, P. A. (1995). Discourse. In F. Lentricchia & T. McLaughlin(Eds.), Critical terms for literary study (pp. 50–65). Chicago:University of Chicago Press.

Bowles, S. (1977). Unequal education and the reproduction of thesocial division of labor. In J. Karabel & A. H. Halsey (Eds.),Power and ideology in education (pp. 137–153). New York:Oxford University Press. (Original work published in 1971)

D’Ambrosio, U. (1997). Ethnomathematics and its place in thehistory and pedagogy of mathematics. In A. B. Powell & M.Frankenstein (Eds.), Ethnomathematics: ChallengingEurocentrism in mathematics education (pp. 13–24). Albany:State University of New York Press. (Original work publishedin 1985)

David W. Stinson 17

D’Ambrosio, U. (1997). Foreword. In A. B. Powell & M.Frankenstein (Eds.), Ethnomathematics: ChallengingEurocentrism in mathematics education (pp. xv–xxi). Albany:State University of New York Press.

Derrida, J. (1997). Of grammatology (Corrected ed.). Baltimore:Johns Hopkins University Press. (Original work published1974)

Ernest, P. (2002). Empowerment in mathematics education.Philosophy of Mathematics Education, 15. Retrieved January26, 2004, fromhttp://www.ex.ac.uk/~PErnest/pome15/empowerment.htm

Foucault, M. (1972). The archaeology of knowledge (1st Americaned.). New York: Pantheon Books. (Original work published1969)

Frankenstein, M. (1995). Equity in mathematics education: Class inthe world outside the class. In W. G. Secada, E. Fennema, &L. Byrd (Eds.), New directions for equity in mathematicseducation (pp. 165–190). Cambridge: Cambridge UniversityPress.

Freire, P. (2000). Pedagogy of the oppressed (30th anniversaryed.). New York: Continuum. (Original work published 1970)

Gutstein, E. (2003). Teaching and learning mathematics for socialjustice in an urban, Latino school. Journal for Research inMathematics Education, 34(1), 37–73.

Gutstein, E., Lipman, P., Hernandez, P., & de los Reyes, R. (1997).Culturally relevant mathematics teaching in a MexicanAmerican context. Journal for Research in MathematicsEducation, 28(6), 709–737.

Johnston, H. (1997). Teaching mathematics for understanding:Strategies and activities (Unpublished manuscript). Atlanta:Georgia State University.

Kilpatrick, J. (1992). A history of research in mathematicseducation. In D. A. Grouws (Ed.), Handbook of research onmathematics teaching and learning (pp. 3–38). New York:Macmillan.

Kliebard, H. M. (1995). The struggle for the American curriculum.New York: Routledge.

Ladson-Billings, G. (2001). The power of pedagogy: Does teachingmatter? In W. H. Watkins, J. H. Lewis, & V. Chou (Eds.),Race and education: The roles of history and society ineducating African American students (pp. 73–88). Boston:Allyn & Bacon.

Ladson-Billings, G. (1995). Making mathematics meaningful in amulticultural context. In W. G. Secada, E. Fennema, & L.Byrd (Eds.), New directions for equity in mathematicseducation (pp. 126–145). Cambridge: Cambridge UniversityPress.

Ladson-Billings, G. (1994). The Dreamkeepers: Successfulteachers of African American children. San Francisco: Jossey-Bass.

Lather, P. (1991). Getting smart: Feminist research and pedagogywith/in the postmodern. New York: Routledge.

Lemann, N. (1999). The big test: The secret history of theAmerican meritocracy (1st ed.). New York: Farrar Straus andGiroux.

Lerman, S. (2000). The social turn in mathematics educationresearch. In J. Boaler (Ed.), International perspectives onmathematics education, (pp. 19–44). Westport, CT: Ablex.

Moses, R. P., & Cobb, C. E. (2001). Radical equations: Mathliteracy and civil rights. Boston: Beacon Press.

National Council of Teachers of Mathematics. (2000). Principlesand standards for school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics. (1995). Assessmentstandards for school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics. (1991). Professionalstandards for teaching mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics. (1989). Curriculumand evaluation standards for school mathematics. Reston,VA: Author.

Oakes, J. (1985). Keeping track: How schools structure inequality.New Haven: Yale University Press.

Oakes, J., Ormseth, T., Bell, R., & Camp, P. (1990). Multiplyinginequalities: The effects of race, social class, and tracking onopportunities to learn mathematics and science. SantaMonica, CA: Rand Corporation.

Plato. (trans. 1996). The republic (R. W. Sterling & W. C. Scott,Trans.) (Paperback ed.). New York: Norton.

Secada, W. G. (1995). Social and critical dimensions for equity inmathematics education. In W. G. Secada, E. Fennema, & L. B.Adajian (Eds.), New directions for equity in mathematicseducation (pp. 146–164). Cambridge: Cambridge UniversityPress.

Skovsmose, O. (1994). Towards a critical mathematics education.Educational Studies in Mathematics, 27, 35–57.

Spivak, G. C. (1997). Translator’s preface. In J. Derrida (Ed.), Ofgrammatology (Corrected ed., pp. ix - lxxxvii). Baltimore:Johns Hopkins University Press. (Original work published1974)

Stanic, G. M. A. (1989). Social inequality, cultural discontinuity,and equity in school mathematics. Peabody Journal ofEducation, 66(2), 57–71.

Stanic, G. M. A. (1986). The growing crisis in mathematicseducation in the early twentieth century. Journal for Researchin Mathematics Education, 17(3), 190–205.

Tate, W. F. (1995). Economics, equity, and the nationalmathematics assessment: Are we creating a national toll road?In W. G. Secada, E. Fennema, & L. Byrd (Eds.), Newdirections for equity in mathematics education (pp. 191–206).Cambridge: Cambridge University Press.

U.S. Department of Education. (1997). Mathematics equalsopportunity. White Paper prepared for U.S. Secretary ofEducation Richard W. Riley. Retrieved January 26, 2004,from http://www.ed.gov/pubs/math/mathemat.pdf

U.S. Department of Education. (1999). Do gatekeeper coursesexpand education options? National Center for EducationStatistics. Retrieved January 26, 2004, fromhttp://nces.ed.gov/pubsearch/pubsinfo.asp?pubid=1999303

1 The student racial/ethnic data were based on the 2001-2002Georgia Public Education Report Card; the racial/ethnicclassifications were the State of Georgia’s not this author’s. Fordetails of racial/ethnic data on all schools in the State of Georgiasee: http://techservices.doe.k12.ga.us/reportcard/2 Plato (trans. 1996) in establishing his utopian Republic imaginedthat the philosopher guardians of the city, identified as the

18 Mathematics as “Gate-Keeper” (?)

aristocracy, would be children taken from their parents at an earlyage and educated at the academy until of age when they woulddutifully rule as public servants and not for personal gain. Platobelieved that these children would be from all classes: “it maysometimes happen that a silver child will be born of a goldenparent, a golden child from a silver parent and so on” (p. 113); andfrom both sexes: “we must conclude that sex cannot be the criterionin appointments to government positions…there should be nodifferentiation” (pp. 146-147). However, Plato’s concept ofaristocracy has been greatly misinterpreted within Westernideology. The concept has historically and consistently favored thesocial positionality of the White heterosexual Christian male ofbourgeois privilege.3 Throughout the remainder of this article the term NCTMdocuments designates the Professional Standards for TeachingM a t h e m a t i c s (1991), Assessment Standards for SchoolMathematics (1995), Principles and Standards for SchoolMathematics (2000), and the Curriculum and Evaluation Standardsfor School Mathematics (1989).4 Freire (1970/2000) defined the term Subjects, with a capital S, as“those who know and act, in contrast to objects, which are knownand acted upon” (p. 36).5 I define the term “academic” mathematics as D`Ambrosio(1997) defined the term: mathematics that is taught and learned inschools, differentiated from ethnomathematics.6 In this context, I use the term trouble to place academicmathematics under erasure. Spivak (1974/1997) explainedDerrida’s (1974/1997) sous rature , that is, under erasure, aslearning “to use and erase our language at the same time” (p. xviii).She claimed that Derrida is “acutely aware… [of] the strategy ofusing the only available language while not subscribing to itspremises, or ‘operat[ing] according to the vocabulary of the verything that one delimits’ (MP 18, SP 147)” (p. xviii). In other words,I argue that these three perspectives, while purporting the teachingof the procedures and concepts of academic mathematics (i.e., thelanguage of mathematics), also place it sous rature so as not tolimit the mathematics creativity and engagement of all students.7 Even though I trouble Plato’s remark regarding “essence andreality,” the purpose of this article is not to engage in thatargument, an argument that I believe will be my life’s work.