Spring 2004 MATHEMATICS EDUCATION STUDENT ASSOCIATIONTHE UNIVERSITY OF GEORGIA

____ THE_____ MATHEMATICS ___

________ EDUCATOR _____Volume 14 Number 1

Editorial StaffEditorHolly Garrett Anthony

Associate EditorsGinger RhodesMargaret SloanErik Tillema

PublicationStephen BismarckLaurel BleichDennis Hembree

AdvisorsDenise S. MewbornNicholas OppongJames W. Wilson

MESA Officers2004-2005PresidentZelha Tunç-Pekkan

Vice-PresidentNatasha Brewley

SecretaryAmy J. Hackenberg

TreasurerGinger Rhodes

NCTMRepresentativeAngel Abney

UndergraduateRepresentativesErin BernsteinErin CainJessica Ivey

A Note from the EditorDear TME readers,

Along with the editorial team, I present the first of two issues to be produced during my brieftenure as editor of Volume 14 of The Mathematics Educator. This issue showcases the work of bothveteran and budding scholars in mathematics education. The articles range in topic and thus invite allthose vested in mathematics education to read on.

Both David Stinson and Amy Hackenberg direct our attention toward equity and social justice inmathematics education. Stinson discusses the “gatekeeping” status of mathematics, offers theoreticalperspectives he believes can change this, and motivates mathematics educators at all levels to rethinktheir roles in empowering students. Hackenberg’s review of Burton’s edited book, Which Way SocialJustice in Mathematics Education? is both critical and engaging. She artfully draws connections acrosschapters and applauds the picture of social justice painted by the diversity of voices therein.

Two invited pieces, one by Chandra Orrill and the other by Sybilla Beckmann, ask mathematicseducators to step outside themselves and reexamine features of PhD programs and elementarytextbooks. Orrill’s title question invites mathematics educators to consider what we value in classroomteaching, how we engage in and write about research on or with teachers, and what features of a PhDprogram can inform teacher education. Beckmann looks abroad to highlight simple diagrams used inSingapore elementary texts—that facilitate the development of students’ algebraic reasoning andproblem solving skills—and suggests that such representations are worthy of attention in the U. S.

Finally, Bharath Sriraman and Melissa Freiberg offer insights into the creativity ofmathematicians and the organization of rich experiences for preservice elementary teachers,respectively. Sriraman builds on creativity theory in his research to characterize the creative practicesof five well-published mathematicians in the production of mathematics. Freiberg reminds us of thedaily challenge of mathematics educators—providing preservice teachers rich classroomexperiences—and details the organization, coordination, and evaluation of Family Math Fun Nights inelementary schools.

It has been my goal thus far to entice you to read what follows, but I now want to focus yourattention on the work of TME. I invite and encourage TME readers to support our journal by gettinginvolved. Please consider submitting manuscripts, reviewing articles, and writing abstracts forpreviously published articles. It is through the efforts put forth by us all that TME continues to thrive.

Last I would like to comment that publication of Volume 14 Number 1 has been a rewardingprocess—at times challenging—but always worthwhile. I have grown as an editor, writer, and scholar.I appreciate the opportunity to work with authors and editors and look forward to continued work thisFall. I extend my thanks to all of the people who make TME possible: reviewers, authors, peers,faculty, and especially, the editors.

Holly Garrett Anthony105 Aderhold Hall tme@coe.uga.eduThe University of Georgia www.ugamesa.orgAthens, GA 30602-7124

About the coverCover artwork by Thomas E. Ricks. Fractal Worms I of the Seahorse Valley in the Mandelbrot Set, 2004.For questions or comments, contact: tomricks@uga.eduBenoit Mandelbrot was the pioneer of fractal mathematics, and the famous Mandelbrot set is his namesake. Based on a simple iterative equation applied to thecomplex number plane, the Mandelbrot set provides an infinitely intricate and varied landscape for exploration. Visual images of the set and surrounding points aremade by assigning a color to each point in the complex plane based on how fast the iterative equation’s value “escapes” toward infinity. The points that constitutethe actual Mandelbrot set, customarily colored black, are points producing a finite value. The Mandelbrot set is a fractal structure, and one can see self-similarforms within the larger set.Using computing software, anyone can delve within this intricate world and discover views never seen before. Modern computing power acts as a microscopeallowing extraordinary magnification of the set’s detail.The fanciful drawing Fractal Worms I is based on the structure of spirals residing in the commonly called “Seahorse Valley” of the Mandelbrot Set. Using alightboard, Thomas Ricks drew the fractal worms on a sheet of art paper laid over a computer printout of the Seahorse spirals. With the light shining through bothsheets of paper, he drew the various fractal worms following the general curve of the spirals. The printout was produced by a Mandelbrot set explorer softwarepackage called “Xaos”, developed by Jan Hubicka and Thomas Marsh and available at: http://xaos.theory.org/

This publication is supported by the College of Education at The University of Georgia.

____________THE___________________________ MATHEMATICS________

______________ EDUCATOR ____________An Official Publication of

The Mathematics Education Student AssociationThe University of Georgia

Spring 2004 Volume 14 Number 1

Table of Contents

2 Guest Editorial… Do You Need a PhD to Teach K–8 Mathematics in WaysRespected by the Mathematics Education Community?CHANDRA HAWLEY ORRILL

8 Mathematics as “Gate-Keeper” (?): Three Theoretical Perspectives that AimToward Empowering All Children With a Key to the GateDAVID W. STINSON

19 The Characteristics of Mathematical CreativityBHARATH SRIRAMAN

35 Getting Everyone Involved in Family MathMELISSA R. FREIBERG

42 In Focus… Solving Algebra and Other Story Problems with Simple Diagrams: aMethod Demonstrated in Grade 4–6 Texts Used in SingaporeSYBILLA BECKMANN

47 Book Review… Diverse Voices Call for Rethinking and Refining Notions of EquityAMY J. HACKENBERG

52 Upcoming Conferences53 Subscription Form54 Submissions Information

© 2004 Mathematics Education Student Association.All Rights Reserved

The Mathematics Educator2004, Vol. 14, No. 1, 2–7

2 Do You Need a PhD?

Guest Editorial…Do You Need a PhD to Teach K–8 Mathematics in WaysRespected by the Mathematics Education Community?

Chandra Hawley Orrill

The genesis of this editorial was a conversationabout an article in which Ball (1991) provideddescriptions of three teachers’ approaches to workingwith their students. In Ball’s article, teachers withoutPhDs in mathematics or mathematics educationstruggled to engage their students in developingmeaningful concepts of mathematics. They could notp r o v i d e m u l t i p l e i n t e r p r e t a t i o n s o fconcepts—particularly representations that providedconcrete explanations or tie-ins to the real world. Theydemonstrated only stepwise approaches to doingmathematics, clinging tightly to procedures andalgorithms, and provided no evidence that they had adeeper understanding of the mathematics. In starkcontrast, the same Ball article offered a vignette ofLampert’s teaching that illustrated a rich mathematicalexperience for students. Lampert provided multipleperspectives, introduced multiple representations, anddemonstrated a deep understanding of bothmathematics and student learning throughout theepisode described.

Given the number of articles in the literaturepainting the ‘typical’ mathematical experience as onethat is impoverished, and the growing body ofliterature written by PhD researcher-teachers, Iwondered, “Do you need a PhD to teach elementary

and middle school mathematics in ways thatmathematics educators would value?” After all, theBalls, Lamperts, and McClains1 in the literature offerhigh-quality mathematics instruction, attend to studentthinking, provide opportunities for knowledgeconstruction, and introduce students to a variety oftools they can use later (e.g., visual representations andproblem solving strategies). Further, these researcher-teachers seem to have a gift for promoting studentthinking and moving an entire class forward byscaffolding lessons, questioning students, and creatinga classroom community where learners consider eachother’s work critically and interact meaningfully. Thereality, however, is that not all mathematics teachershave PhDs and it is unlikely that most ever will.2

In working through this question both with thegraduate students with whom I work and in preparationfor this editorial, I have developed some ideas bothabout researcher-teachers as a “special” group andabout why having a PhD might matter. Based on mythoughts I would like to propose two conjectures aboutresearcher-teacher efforts. First, I conjecture that weshould consider the way we think about researcher-teachers versus research on/with teachers. Second, Ipropose that certain features of PhD programs can beapplied to teacher professional development and/orundergraduate education to support all teachers increating richer mathematics learning experiences fortheir students. This editorial explores these twoconjectures in more detail.

Researcher-Teachers as a Special GroupIn order to understand some of the unique qualities

of the teaching exemplified by researcher-teachers, it isworthwhile to consider why they do what they do sowell. There are a variety of factors that impact both theway these people teach and the way we, as consumersof research, read about their teaching. First, researcher-teachers teach well because they have significantknowledge of mathematics and how children learnmathematics. There is no doubt that teachers, with orwithout PhDs, who have strong pedagogicalknowledge and strong content knowledge, create richer

Chandra Hawley Orrill is a Research Scientist in the Learningand Performance Support Laboratory at the University ofGeorgia. Her research interest is teacher professionaldevelopment with an emphasis on teaching in the midst ofchange. She is also interested in how professional developmentimpacts the opportunities teachers create for student learning.AcknowledgementsThe research reported here came from a variety of projectsspanning six years. These projects were supported by grants fromthe Andrew W. Mellon Foundation and the Russell SageFoundation, the National Science Foundation, Georgia’sTeacher Quality Program (formerly Georgia’s EisenhowerHigher Education Program), and the Office of the Vice Presidentfor Research at the University of Georgia. Opinions expressedhere are my own and do not necessarily reflect those of thegranting agencies. My thanks to Holly Anthony, Ernise Singleton,Peter Rich, Craig Shepherd, Laurel Bleich, and Drew Polly fortheir ongoing discussions with me about whether a teacher needsa PhD to teach K-8 mathematics.

Chandra Hawley Orrill 3

learning experiences for their students (e.g., Ball,Lubienski, & Mewborn, 2001).

Further, in the process of earning a PhD,researcher-teachers presumably develop reflectivedispositions, grapple with their own epistemologicalbeliefs, and define their visions of learning andteaching. This produces teachers who criticallyexamine the world around them and who areintrospective in ways that are productive for achievingthe classroom environment valued by mathematicseducation researchers and described in the NCTMStandards (NCTM, 2000). By developing thisdisposition, researcher-teachers are in a unique positionto make critical changes to the classroom environmentas needs are identified. Too often, regular classroomteachers do not have the time or skills to analyzeformal or informal data about their students and theirteaching. In fact, many classroom teachers have onlybeen exposed to the most basic concepts of studentlearning theory and research. As a result, even if theytried to make sense of the data presented in theirclassroom, they would be ill-equipped to makeimportant changes based on those data.

In addition, researcher-teachers have somepragmatic luxuries that typical teachers do not have.For example, they usually only teach one subject toone class per day, while a typical elementary teachermight teach four subjects to one class, and a middleschool teacher might teach one or two subjects to fouror five classes each day. This provides the researcher-teacher with more time for reflection and refinement.To be fair, researcher-teachers typically do have otherwork responsibilities – they do not simply teach for 50minutes and “call it a day.” However, their situation isvery different from that of a typical classroom teacher.Researcher-teachers have support with the reflectionprocess from others studying the classroom, and oftenhave no additional responsibilities such as conductingparent conferences, developing individualized plans forcertain students, and attending the team meetingscommon in many teachers’ daily experience. Whilethis difference should not be viewed or used as anexcuse for classroom teachers to avoid improving theirpractice, it is undeniable that a researcher-teacher’s jobis fundamentally different from that of the typicalclassroom teacher.

In addition to teaching expertise and workload,researcher-teachers have some advantages overteachers when participating in others’ studies. Unlikemost “typical” teachers, researcher-teachers are, bydefinition, philosophically aligned with and invested inthe goals of the research. They already have agreement

with the researcher about what good teaching andlearning look like – after all, they are typically eitherthe researcher (e.g., Ball, 1990a and Lampert, 2001) orthey are a full member of the research team (e.g.,McClain in Bowers, Cobb, & McClain, 1999). Theimportance of this is profound. A researcher-teacherwants the same (not negotiated or compromised)outcomes as the researcher, because she either is theresearcher or is a member of the research team. Theresearcher-teacher, therefore, attends to those issuesand aspects of the classrooms and student learning thatare the focus of the research. Further, the researcher-teacher provides unlimited, or nearly unlimited, richaccess to her thinking for the research effort because,again, she has a vested interest in capturing thatthinking. Thus, teacher and researcher alignment interms of goals, values, and expectations is important.

One potential disadvantage for researcher-teachersworth noting is the potential for bias to confound theresearch. After all, the researcher-teacher has a biasedview of the teaching being studied because it is herown. Further, because she is invested in the researchand because she is a member of the research team, it ispossible that her teaching is biased to make theresearch work. That is, if the researcher is looking forparticular aspects of teaching, such as student-teacherinteractions, the researcher-teacher may attend to thoseinteractions more in the course of instruction than shewould under other circumstances. Clearly, the impactof this on the research is determined by both theresearch questions and the data collection and analysistechniques used.

Research On/With TeachersIn order to understand the differences between

researcher-teacher research and research on or withfull-time teachers, it is necessary to explore some ofthe issues involved with doing research on/withteachers. Research in regular classrooms differs insome significant ways from the researcher-teacherwork alluded to in this editorial. To highlight some ofthese differences, I offer examples from my ownexperience in working with middle grades mathematicsteachers.

One major difference I alluded to is the values ateacher holds. In the course of my career, I have beenfortunate to work with several “good” teachers.However, the ways in which they were “good” weredirect reflections of their own values and the values ofthe system within which they were working.Sometimes, they were good in the eyes of theadministrators with whom they worked because they

4 Do You Need a PhD?

kept their students under control. Sometimes they weregood for my research in that their practice had theelements I was interested in, thus making it easier forme to find the kinds of interactions I was looking for intheir classrooms. Sometimes they were good in thatthey were predisposed to reflective practice allowingme, as a researcher, easier access to their ideas throughobservation and interviews. The quality of the teachers,though, depended on what measure they were held upagainst and what measures they, personally, felt theywere trying to align with.

Another important aspect of working with teachersis a lack of access to certain aspects of their thinking.For example, I have never been able to analyze a dataset without thinking, at some point, “I wonder what shewas thinking when she did that?” or “Did she notunderstand what that student was asking?”Acknowledging this lack of access to a teacher’sthinking requires researchers to be careful in theiranalysis of the teacher’s actions and beliefs and toexplain how thinking and actions are interpreted.Further, at times, such limitations require researchersto analyze situations from their own perspectives aswell as from the teacher’s perspective to understand asituation.

As a practical example of the influence ofresearcher and teacher alignment issues, I offer twosituations from my own work: one addressing the“good” teacher issue and the other addressing the needto understand the situation from the teacher’sperspective. My goal in presenting these two examplesis to highlight issues that arise in research with teacherswho are not members of the research team. In onestudy (Orrill, 2001), I worked with two middle schoolteachers (one mathematics and one science) in NewYork City to understand how to structure professionaldevelopment to support uses of computer-basedsimulations. My goal for the professional developmentwas to enhance teachers’ attention to student problem-solving skills in the context of computer-based,workplace simulations. The mathematics teacher wasconsidered to be “good” by her principal and otherteachers. In my observations of her classroom, I foundthat she taught mathematics in much the same way asthe “typical” teachers we read about in case study aftercase study. She offered many procedures but providedinadequate opportunities for students to interact withthe content in ways that would allow them to developdeep understanding of the mathematical conceptsunderlying those procedures. However, this teacherhad remarkable skill in classroom management, whichwas highly valued in her school. Further, she had

developed techniques that supported her students inachieving acceptable scores on the New Yorkstandardized tests. By these standards, she wasconsidered “good.” When she used the simulations Iwas researching, she maintained the same kinds ofapproaches, particularly early in the study. She keptstudents on task and directed them to work moreefficiently. Given my goal of understanding how topromote problem solving, her interactions with thestudents were inadequate and impoverished. Shetypically did not ask the students questions thatprovided insight into their thinking and she did notallow them to struggle with a problem. Instead, shedirected them to an efficient approach for solving theproblem they were working on, which effectively keptthem on task and motivated them to move forward.While this presented a challenge to me as theresearcher, it would not be fair for me to “accuse” herof being less than a good teacher when she was clearlymeeting the expectations of the system within whichshe worked. This is clearly a case in which there was amismatch between what I, the researcher, valued andwhat the teacher and system valued. Had I beenresearching my own practice or the practice of aresearch team member, this tension would have beenremoved.

As a second example, a teacher I have worked withmore recently proved a perplexing puzzle for my teamas we considered her teaching. A point of particularinterest was the teacher’s frustration with poor studentperformance on tests – regardless of what students didin class, a significant number failed her tests. In myanalysis of this case, I recognized that this teacher’sbeliefs about teaching and learning significantlydiffered from my own. Until I realized this, I wasunable to understand the magnitude of the barrier theteacher felt she was facing. At the simplest level, shebelieved that her role as a good teacher was to presentnew material and provide an opportunity for studentsto practice that material. The students’ job, in her view,was to engage in that practice and develop anunderstanding from it. Therefore, when students werenot succeeding, she became extremely frustrated sinceshe had presented information and providedopportunities for practice. In her worldview, studentsuccess was out of her hands – she had already donewhat she could to support them. As the researcher inthat setting, it was difficult to understand herfrustration because I was working from a constructivistperspective. Specifically, I was looking for anenvironment in which the teacher provided studentsopportunities to develop their own thinking via an

Chandra Hawley Orrill 5

assortment of models, experiences, and collaborativeexchanges. Student test failure, for me, was anindicator that learning was not complete and thatstudents needed different opportunities to build andconnect knowledge. It took considerable analysis forme, as a researcher with a different perspective anddifferent goals, to understand how the teacherunderstood her role and how she enacted her beliefsabout her role in the classroom.

My point in these two examples is that in muchresearch there are significant and important differencesin the worldviews of the participants and theresearcher. These differences can lead to frustrations indata collection, hurdles in data analysis, and, in theworst cases, assessments of the teachers that are simplynot fair. For example, in the early 1990’s there weremany articles written about the implementation of thestandards in California (e.g., Ball, 1990b; Cohen, 1990;Wilson, 1990). In many of these cases, the teachersstruggled to implement a set of standards that werewritten from a particular perspective that they did notfully understand. This led to implementations that werefar from ideal in the eyes of the researchers whounderstood the initial intent of the standards. Toooften, teachers were presented by researchers ashopeless or inadequate—in contrast, the teachersreportedly perceived themselves as adhering to thesenew standards. Likely, if the researchers and teachershad philosophical alignment afforded by theresearcher-teacher approach the findings would havebeen tremendously different. After all, had thesestudies focused on researcher-teachers, the teacher andthe researcher would have had a shared understandingof the intent of the standards and had a shared vision ofwhat their implementation should look like.

PhD Program Features That Could Be Useful InTeacher Development

While not all people who hold PhDs are goodteachers, certain habits of mind are developed as partof the process of earning a PhD that can significantlyimpact the learning environment a teacher designs.Given the high-quality of teaching exhibited by theresearcher-teachers referred to in this article, it seemslikely that there are aspects of the PhD program thatcould be adapted for teacher professional development.

First, the researcher-teacher typically hasdeveloped solid pedagogical knowledge, contentknowledge, and pedagogical content knowledge. Thiscomes from having time and encouragement to readabout different practices in a focused way,participating in shared discourse with colleagues,

conducting research in others’ classrooms, and havingother similar experiences. This is in stark contrast tothe elementary or middle grades teacher who hastypically had four years of college—with coursesspread across the curriculum—and only limited “lifeexperience” to relate to in the courses that help developthese knowledge areas. Second, one of the mostpowerful outcomes of earning a PhD is thedevelopment of a concrete picture of a desired learningenvironment that looks beyond issues of classroommanagement and logistics to focus on the kinds oflearning and teaching that will take place. Third, PhDsdevelop a rich, precise vocabulary aligned with that ofthe standards-writers and the researchers. In becominga researcher, the holder of the PhD becomes active inthe conversation of the field—meaning that person hasdeveloped a refined vocabulary and vision that isshared, in some way, by the field. This is not to saythat there is a definitive definition of K-8 mathematicseducation that is shared across the field of mathematicseducation, rather that there is a shared way ofdiscussing and thinking about mathematics educationthat allows a more consistent enactment of standardsand practices.

Finally, many researcher-teachers implement ordevelop a “special” curriculum. In the case of Lampert(2001), the teacher was creating open-ended problemseach day to support mathematical topics. In othercases, the research team has developed materials forthe researcher-teachers to implement. Often, thesematerials are far richer than traditional mathematicstextbooks. While there may not be a single dispositionthat could be pulled from the process of earning a PhDthat allows researcher-teachers to be successfulimplementers of non-traditional materials, it is clearthat there is something different between PhD-holdingresearcher-teachers and other teachers. Likely, part ofthis ability is related to the knowledge constructs theresearcher-teachers have that allow them to implementthose materials. In my own work, I have found thatteachers who are not well-versed in the curricula, wholack conceptual knowledge, or who lack thepedagogical content knowledge to see connectionsbetween various mathematical ideas do not know howto utilize these kinds of materials to make theexperiences mathematically rich for their students.Clearly, some attention to the aspects of earning a PhDthat relate to these dispositions would benefitpreservice and inservice teachers.

6 Do You Need a PhD?

Teacher DevelopmentWhile it may not be feasible, or even reasonable, to

expect teachers to pursue doctoral degrees, there maybe some characteristics of doctoral education that areworthwhile for consideration as components or foci ofprofessional development and undergraduate programs.To frame this section, I want to draw on the work ofCohen and Ball (1999) who have argued that thelearning environment is shaped by the interactions ofthree critical elements: teachers, students, andmaterials/content. This model assumes that for eachelement a variety of beliefs, values, and backgroundswork together to create each unique learningenvironment. Considering the classroom from thisperspective is critical to understanding why thesolutions to the problems highlighted in research onand with teachers are complex.

What We See NowA quick overview of my definition of the “typical”

classroom may be warranted at this point. Based on theclassrooms described in the literature and those I workin, the typical mathematics classroom remains focusedon teachers’ delivering information to the students,typically by working sample problems on the board.Students are responsible for using this information towork problems on worksheets or in their books.Students are asked to do things like name the fractionalportion of a circle that is colored in or to work 20addition or multiplication problems. Many teachers usemanipulatives or drawn representations to introducenew ideas to their students. However, their intent is toprovide a concrete example and move the students tothe abstract activities of arithmetic as quickly aspossible or to use the manipulative to motivate thestudents to want to do the arithmetic. Mathematicslearning in these classrooms is more about developingefficient means for working problems than developingrich understandings of why those methods work.Referring back to the Cohen and Ball triangle ofinteractions, the interactions in these classrooms couldbest be characterized by what follows. The teacherinterprets the materials/content and delivers thatinterpretation to the students. The students look toteachers as holders of all information. Teachers are toprovide guidance when students are unable to solve aproblem, to provide feedback about the “rightness” ofstudent work, and to find the errors students have madein their work. The students interact with the materialsby working problems. The students may or may notinteract with the concepts at a meaningful level – thatdepends on the teacher and the activity. In these

classrooms, success is measured in the number ofproblems students can answer correctly, often within aspecific amount of time.

How Features of PhD Programs May Change ThisTo enhance the interactions among teachers,

students, and materials/content there are a number ofelements from doctoral training that may be worthpursuing. First, teachers can use guided reflection as ameans to step out of the teaching moment to considercritical aspects of the teaching and learningenvironment. Through reflection, teachers have theopportunity to align their beliefs and practices (e.g.Wedman, Espinosa, & Laffey, 1998) and to make theirintent more explicit rather than relying on tacit “gutinstinct” (e.g., Richardson, 1990). The reflectivepractitioner can learn to look at a learning environmentas a whole by considering how students and materialsare interacting, looking for evidence of conceptualdevelopment, and thinking about ways to improve theirown role in the classroom. The researcher-teachers(Ball, Lampert, and McClain) cited in this article allreported using reflection regularly as part of theirpractice.

Another element of the PhD experience worthconsideration is the development of solid content andpedagogical knowledge. Teachers who do notunderstand mathematics cannot be as effective as thosewho do. For example, teachers who do not know howto use representations to model multiplication offractions cannot use that pedagogical strategy in theirclassrooms. Teachers who lack adequate content orpedagogical knowledge cannot know what to do whena student suggests an approach to solving a problemthat does not work—too often the only approach theteacher has is to point out errors to the student anddemonstrate “one more time” the “right” way to workthe problem. I assert that combining teacherdevelopment of content knowledge and pedagogicalknowledge with the development of a reflectivedisposition will lead to the emergence of pedagogicalcontent knowledge. By pedagogical contentknowledge, I refer to knowledge that is a combinationof knowing what content can be learned/taught withwhich pedagogies and knowing when to use each ofthese approaches to teach students.

Some of the habits of mind developed in a doctoralprogram in education translate directly into practicewithout focusing on the entire teacher-student-materials interaction triad. For example, onepotentially powerful factor to address is the teacher-student interaction. PhD programs in education offer

Chandra Hawley Orrill 7

tremendous opportunities for thinking about thisrelationship in meaningful ways, and in the researcher-teacher work, attention to this interaction is ubiquitous.It is absolutely critical to support teachers in learningto listen to students and respond to them in meaningfulways. Further, given the poor grounding most teachershave in learning theory, it may be that developing atheoretical understanding of how people learn shouldbe a part of this (this is supported in recent researchsuch as Philipp, Clement, Thanheiser, Schappelle, &Sowder, 2003). Finally, focusing professionaldevelopment on techniques for questioning that allowthe teacher to access student understanding willprovide teachers with ways to access student thinking.

ConclusionWhile it is not realistic to expect that all classroom

teachers will earn doctoral degrees, there are elementsthat go into the attainment of a PhD that can lead toimproved classroom teaching. Therefore, it seemsreasonable to capitalize on what we know about theprocess of getting and having a doctorate versus moretraditional routes to becoming a teacher.

Granted, there are aspects of researcher-teachers'activities that are not addressed simply by consideringtheir educational background or their role in theresearch team. For example, high quality materials areextremely important. Further, it is vital that teachersare supported in learning how to interact with thosematerials (and the content they are trying to convey) ifwe want to raise the bar on teaching and learning. Noone can create rich learning experiences aroundmaterials they do not understand. On the other hand,researcher-teachers have been able to find ways tocapitalize on even the weakest of materials. Forexample, Lampert (2001) discusses how she was ableto use the topic ideas from the traditional textbook herschool used to develop rich problems that allowedstudents prolonged and repeated exposure to criticalmathematics content—it is clear that the typical teacheris unable to capitalize on materials in these ways.Certainly, there is an appropriate place in professionaldevelopment efforts to support teachers’ use ofmaterials.

While this article has only begun to explore thedifferences between a typical classroom teacher’senvironment and that of a researcher-teacher, it appearsthat researcher-teachers have some advantages overother teachers. They are better able to understand andaddress what is going on in the classroom, as well asthe material they are expected to work with.Researcher-teachers are also better able to

communicate with others in the field and to understandinput from the research. Unfortunately, it is notpractical to expect most teachers to earn a doctoraldegree. The question then becomes, “What elementscan we take from earning an advanced degree that willhelp teachers in the classroom?” By incorporatingthese elements into teacher education and professionaldevelopment programs, we can greatly improveclassroom instruction.

REFERENCESBall, D. L. (1990a). Halves, pieces, and twoths: Constructing

representational concepts in teaching fractions. East Lansing, MI:National Center for Research on Teacher Education.

Ball, D. L. (1990b). Reflection and deflections of policy: The case of CarolTurner. Educational Evaluation and Policy Analysis, 12(3), 247–259.

Ball, D. L. (1991). Research on teaching mathematics: Making subject-matter knowledge part of the equation. In J. Brophy (Ed.), Advancesin research on teaching (Vol. 3, pp. 1–48). Greenwich, CT: JAI Press.

Ball, D. L., Lubienski, S. T., & Mewborn, D. S. (2001). Research onteaching mathematics: The unsolved problem of teachers'mathematical knowledge. In V. Richardson (Ed.), Handbook ofresearch on teaching (4th ed.). Washington, DC: AmericanEducational Research Association.

Bowers, J., Cobb, P., & McClain, K. (1999). The evolution of mathematicalpractices: A case study. Cognition and Instruction, 17(1), 25–64.

Cohen, D. (1990). A revolution in one classroom: The case of Mrs. Oublier.Educational Evaluation and Policy Analysis, 12(3), 327–345.

Cohen, D., & Ball, D. B. (1999). Instruction, capacity, & improvement (No.CPRE-RR-43). Philadelphia, PA: Consortium for Policy Research inEducation.

Lampert, M. (2001). Teaching problems and the problems of teaching. NewHave, CT: Yale University Press.

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

Orrill, C. H. (2001). Building technology-based learning-centeredclassrooms: The evolution of a professional development framework.Educational Technology Research and Development, 49(1), 15–34.

Philipp, R. A., Clement, L., Thanheiser, E., Schappelle, B., & Sowder, J. T.(2003). Integrating mathematics and pedagogy: An investigation ofthe effects on elementary preservice teachers' beliefs and learning ofmathematics. Paper presented at the Research Presession of the 81stAnnual Meeting of the National Council of Teachers of Mathematics,San Antonio, TX. Available online:http://www.sci.sdsu.edu/CRMSE/IMAP/pubs.html.

Richardson, V. (1990). Significant and worthwhile change in teachingpractice. Educational Researcher, 19(7), 10–18.

Wedman, J. M., Espinosa, L. M., & Laffey, J. M. (1998). A process forunderstanding how a field-based course influences teachers' beliefsand practices, Paper presented at the Annual Meeting of the AmericanEducational Research Association, San Diego, CA.

Wilson, S. M. (1990). A conflict of interests: The case of Mark Black.Educational Evaluation and Policy Analysis, 12(3), 309–326.

1 I cite examples of each of these researcher-teachers’ work throughout thiseditorial. This list is not exhaustive.2 Reasons why I believe this is true range from the lack of incentives relativeto the effort required to earn a PhD to the mismatch between the intent ofPhD programs and what teachers do in their everyday lives. This is not toassert that earning a PhD is not helpful for a teacher, rather that it is notlikely in the current educational system.

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.

David W. Stinson 9

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

David W. Stinson 11

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

David W. Stinson 13

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.

The Mathematics Educator2004, Vol. 14, No. 1, 19–34

Bharath Sriraman 19

The Characteristics of Mathematical CreativityBharath Sriraman

Mathematical creativity ensures the growth of mathematics as a whole. However, the source of this growth, thecreativity of the mathematician, is a relatively unexplored area in mathematics and mathematics education. Inorder to investigate how mathematicians create mathematics, a qualitative study involving five creativemathematicians was conducted. The mathematicians in this study verbally reflected on the thought processesinvolved in creating mathematics. Analytic induction was used to analyze the qualitative data in the interviewtranscripts and to verify the theory driven hypotheses. The results indicate that, in general, the mathematicians’creative processes followed the four-stage Gestalt model of preparation-incubation-illumination-verification. Itwas found that social interaction, imagery, heuristics, intuition, and proof were the common characteristics ofmathematical creativity. Additionally, contemporary models of creativity from psychology were reviewed andused to interpret the characteristics of mathematical creativity.

Mathematical creativity ensures the growth of thefield of mathematics as a whole. The constant increasein the number of journals devoted to mathematicsresearch bears evidence to the growth of mathematics.Yet what lies at the essence of this growth, thecreativity of the mathematician, has not been thesubject of much research. It is usually the case thatmost mathematicians are uninterested in analyzing thethought processes that result in mathematical creation(Ervynck, 1991). The earliest known attempt to studymathematical creativity was an extensive questionnairepublished in the French periodical L'EnseigementMathematique (1902). This questionnaire and a lectureon creativity given by the renowned 20th centurymathematician Henri Poincaré to the Societé dePsychologie inspired his colleague Jacques Hadamard,another prominent 20th century mathematician, toinvestigate the psychology of mathematical creativity(Hadamard, 1945). Hadamard (1945) undertook aninformal inquiry among prominent mathematicians andscientists in America, including George Birkhoff,George Polya, and Albert Einstein, about the mentalimages used in doing mathematics. Hadamard (1945),influenced by the Gestalt psychology of his time,theorized that mathematicians’ creative processesfollowed the four-stage Gestalt model (Wallas, 1926)of preparation-incubation-illumination-verification.As we will see, the four-stage Gestalt model is acharacterization of the mathematician's creativeprocess, but it does not define creativity per se. How

does one define creativity? In particular what exactly ismathematical creativity? Is it the discovery of a newtheorem by a research mathematician? Does studentdiscovery of a hitherto known result also constitutecreativity? These are among the areas of exploration inthis paper.

The Problem Of Defining CreativityMathematical creativity has been simply described

as discernment, or choice (Poincaré, 1948). Accordingto Poincaré (1948), to create consists precisely in notmaking useless combinations and in making thosewhich are useful and which are only a small minority.Poincaré is referring to the fact that the “proper”combination of only a small minority of ideas results ina creative insight whereas a majority of suchcombinations does not result in a creative outcome.This may seem like a vague characterization ofmathematical creativity. One can interpret Poincaré's"choice" metaphor to mean the ability of themathematician to choose carefully between questions(or problems) that bear fruition, as opposed to thosethat lead to nothing new. But this interpretation doesnot resolve the fact that Poincaré’s definition ofcreativity overlooks the problem of novelty. In otherwords, characterizing mathematical creativity as theability to choose between useful and uselesscombinations is akin to characterizing the art ofsculpting as a process of cutting away the unnecessary!

Poincaré's (1948) definition of creativity was aresult of the circumstances under which he stumbledupon deep results in Fuchsian functions. The first stagein creativity consists of working hard to get an insightinto the problem at hand. Poincaré (1948) called thisthe preliminary period of conscious work. This periodis also referred to as the preparatory stage (Hadamard,

Bharath Sriraman is an assistant professor of mathematics andmathematics education at the University of Montana. Hispublications and research interests are in the areas of cognition,foundational issues, mathematical creativity, problem-solving,proof, and gifted education.

20 Mathematical Creativity

1945). In the second, or incubatory, stage (Hadamard,1945), the problem is put aside for a period of time andthe mind is occupied with other problems. In the thirdstage the solution suddenly appears while themathematician is perhaps engaged in other unrelatedactivities. "This appearance of sudden illumination is amanifest sign of long, unconscious prior work"(Poincaré, 1948). Hadamard (1945) referred to this asthe illuminatory stage. However, the creative processdoes not end here. There is a fourth and final stage,which consists of expressing the results in language orwriting. At this stage, one verifies the result, makes itprecise, and looks for possible extensions throughutilization of the result. The “Gestalt model” has someshortcomings. First, the model mainly applies toproblems that have been posed a priori bymathematicians, thereby ignoring the fascinatingprocess by which the actual questions evolve.Additionally, the model attributes a large portion ofwhat “happens” in the incubatory and illuminatoryphases to subconscious drives. The first of theseshortcomings, the problem of how questions aredeveloped, is partially addressed by Ervynck (1991) inhis three-stage model.

Ervynck (1991) described mathematical creativityin terms of three stages. The first stage (Stage 0) isreferred to as the preliminary technical stage, whichconsists of "some kind of technical or practicalapplication of mathematical rules and procedures,without the user having any awareness of thetheoretical foundation" (p. 42). The second stage(Stage 1) is that of algorithmic activity, which consistsprimarily of performing mathematical techniques, suchas explicitly applying an algorithm repeatedly. Thethird stage (Stage 2) is referred to as crea t i ve(conceptual, constructive) activity. This is the stage inwhich true mathematical creativity occurs and consistsof non-algorithmic decision making. "The decisionsthat have to be taken may be of a widely divergentnature and always involve a choice" (p. 43). AlthoughErvynck (1991) tries to describe the process by which amathematician arrives at the questions through hischaracterizations of Stage 0 and Stage 1, hisdescription of mathematical creativity is very similar tothose of Poincaré and Hadamard. In particular his useof the term “non-algorithmic decision making” isanalogous to Poincaré’s use of the “choice” metaphor.

The mathematics education literature indicates thatvery few attempts have been made to explicitly definemathematical creativity. There are references made tocreativity by the Soviet researcher Krutetskii (1976) inthe context of students’ abilities to abstract and

generalize mathematical content. There is also anoutstanding example of a mathematician (GeorgePolya) attempting to give heuristics to tackle problemsin a manner akin to the methods used by trainedmathematicians. Polya (1954) observed that in "tryingto solve a problem, we consider different aspects of itin turn, we roll it over and over in our minds; variationof the problem is essential to our work." Polya (1954)emphasized the use of a variety of heuristics forsolving mathematical problems of varying complexity.In examining the plausibility of a mathematicalconjecture, mathematicians use a variety of strategies.In looking for conspicuous patterns, mathematiciansuse such heuristics as (1) verifying consequences, (2)successively verifying several consequences, (3)verifying an improbable consequence, (4) inferringfrom analogy, and (5) deepening the analogy.

As is evident in the preceding paragraphs, theproblem of defining creativity is by no means an easyone. However, psychologists’ renewed interest in thephenomenon of creativity has resulted in literature thatattempts to define and operationalize the word“creativity.” Recently psychologists have attempted tolink creativity to measures of intelligence (Sternberg,1985) and to the ability to abstract, generalize(Sternberg, 1985), and solve complex problems(Frensch & Sternberg, 1992). Sternberg and Lubart(2000) define creativity as the ability to produceunexpected original work that is useful and adaptive.Mathematicians would raise several arguments withthis definition, simply because the results of creativework may not always have implications that are“useful” in terms of applicability in the contemporaryworld. A recent example that comes to mind is AndrewWiles’ proof of Fermat’s Last Theorem. Themathematical community views his work as creative. Itwas unexpected and original but had no applicability inthe sense Sternberg and Lubart (2000) suggest. Hence,I think it is sufficient to define creativity as the abilityto produce novel or original work, which is compatiblewith my personal definition of mathematical creativityas the process that results in unusual and insightfulsolutions to a given problem, irrespective of the levelof complexity. In the context of this study involvingprofessional mathematicians, mathematical creativity isdefined as the publication of original results inprominent mathematics research journals.

The Motivation For Studying CreativityThe lack of recent mathematics education literature

on the subject of creativity was one of the motivationsfor conducting this study. Fifteen years ago Muir

Bharath Sriraman 21

(1988) invited mathematicians to complete a modifiedand updated version of the survey that appeared inL'Enseigement Mathematique (1902) but the results ofthis endeavor are as yet unknown. The purpose of thisstudy was to gain insight into the nature ofmathematical creativity. I was interested in distillingcommon attributes of the creative process to see ifthere were any underlying themes that characterizedmathematical creativity. The specific questions ofexploration in this study were:

Is the Gestalt model of mathematical creativity stillapplicable today?

What are the characteristics of the creative processin mathematics?

Does the study of mathematical creativity have anyimplications for the classroom?

Literature ReviewAny study on the nature of mathematical creativity

begs the question as to whether the mathematiciandiscovers or invents mathematics. Therefore, thisreview begins with a brief description of the four mostpopular viewpoints on the nature of mathematics. Thisis followed by a comprehensive review ofcontemporary models of creativity from psychology.

The Nature of MathematicsMathematicians actively involved in research have

certain beliefs about the ontological nature ofmathematics that influence their approach to research(Davis & Hersh, 1981; Sriraman, 2004a). The Platonistviewpoint is that mathematical objects exist prior totheir discovery and that “any meaningful questionabout a mathematical object has a definite answer,whether we are able to determine it or not” (Davis &Hersh, 1981). According to this view, mathematiciansdo not invent or create mathematics - they discovermathematics. Logicists hold that “all concepts ofmathematics can ultimately be reduced to logicalconcepts” which implies that “all mathematical truthscan be proved from the axioms and rules of inferenceand logic alone” (Ernest, 1991). Formalists do notbelieve that mathematics is discovered; they believemathematics is simply a game, created bymathematicians, based on strings of symbols that haveno meaning (Davis & Hersh, 1981).

Constructivism (incorporating Intuitionism) is oneof the major schools of thought (besides Platonism,Logicism and Formalism) that arose due to thecontradictions that emerged in the development of thetheory of sets and the theory of functions during theearly part of the 20th century. The constructivist

(intuitionist) viewpoint is that “human mathematicalactivity is fundamental in the creation of newknowledge and that both mathematical truths and theexistence of mathematical objects must be establishedby constructive methods" (Ernest, 1991, p. 29).Contradictions like Russell’s Paradox were a majorblow to the absolutist view of mathematicalknowledge, for if mathematics is certain and all itstheorems are certain, how can there be contradictionsamong its theorems? The early constructivists inmathematics were the intuitionists Brouwer andHeyting. Constructivists claim that both mathematicaltruths and the existence of mathematical objects mustbe established by constructivist methods.

The question then is how does a mathematician goabout conducting mathematics research? Do thequestions appear out of the blue, or is there a mode ofthinking or inquiry that leads to meaningful questionsand to the methodology for tackling these questions? Icontend that the types of questions asked aredetermined to a large extent by the culture in which themathematician lives and works. Simply put, it isimpossible for an individual to acquire knowledge ofthe external world without social interaction.According to Ernest (1994) there is no underlyingmetaphor for the wholly isolated individual mind.Instead, the underlying metaphor is that of persons inconversation, persons who participate in meaningfullinguistic interaction and dialogue (Ernest, 1994).Language is the shaper, as well as being the“summative” product, of individual minds(Wittgenstein, 1978). The recent literature inpsychology acknowledges these social dimensions ofhuman activity as being instrumental in the creativeprocess.The Notion of Creativity in Psychology

As stated earlier, research on creativity has been onthe fringes of psychology, educational psychology, andmathematics education. It is only in the last twenty-fiveyears that there has been a renewed interest in thephenomenon of creativity in the psychologycommunity. The Handbook of Creativity (Sternberg,2000), which contains a comprehensive review of allresearch then available in the field of creativity,suggests that most of the approaches used in the studyof creativity can be subsumed under six categories:mystical, pragmatic, psychodynamic, psychometric,cognitive, and social-personality. Each of theseapproaches is briefly reviewed.

22 Mathematical Creativity

The mystical approachThe mystical approach to studying creativity

suggests that creativity is the result of divineinspiration or is a spiritual process. In the history ofmathematics, Blaise Pascal claimed that many of hismathematical insights came directly from God. Therenowned 19th century algebraist Leopold Kroneckersaid that “God made the integers, all the rest is thework of man” (Gallian, 1994). Kronecker believed thatall other numbers, being the work of man, were to beavoided; and although his radical beliefs did not attractmany supporters, the intuitionists advocated his beliefsabout constructive proofs many years after his death.There have been attempts to explore possiblerelationships between mathematicians’ beliefs aboutthe nature of mathematics and their creativity (Davisand Hersh, 1981; Hadamard, 1945; Poincaré, 1948;Sriraman, 2004a). These studies indicate that such arelationship does exist. It is commonly believed thatthe neo-Platonist view is helpful to the researchmathematician because of the innate belief that thesought after result/relationship already exists.

The pragmatic approachThe pragmatic approach entails “being concerned

primarily with developing creativity” (Sternberg, 2000,p. 5), as opposed to understanding it. Polya’s (1954)emphasis on the use of a variety of heuristics forsolving mathematical problems of varying complexityis an example of a pragmatic approach. Thus,heuristics can be viewed as a decision-makingmechanism which leads the mathematician down acertain path, the outcome of which may or may not befruitful. The popular technique of brainstorming, oftenused in corporate or other business settings, is anotherexample of inducing creativity by seeking as manyideas or solutions as possible in a non-critical setting.

The psychodynamic approachThe psychodynamic approach to studying

creativity is based on the idea that creativity arisesfrom the tension between conscious reality andunconscious drives (Hadamard, 1945; Poincaré, 1948,Sternberg, 2000, Wallas, 1926; Wertheimer, 1945).The four-step Gestalt model (preparation-incubation-illumination-verification) is an example of the use of apsychodynamic approach to studying creativity. Itshould be noted that the gestalt model has served askindling for many contemporary problem-solvingmodels (Polya, 1945; Schoenfeld, 1985; Lester, 1985).Early psychodynamic approaches to creativity wereused to construct case studies of eminent creators such

as Albert Einstein, but the behaviorists criticized thisapproach because of the difficulty in measuringproposed theoretical constructs.

The psychometric approachThe psychometric approach to studying creativity

entails quantifying the notion of creativity with the aidof paper and pencil tasks. An example of this would bethe Torrance Tests of Creative Thinking, developed byTorrance (1974), that are used by many giftedprograms in middle and high schools to identifystudents that are gifted/creative. These tests consist ofseveral verbal and figural tasks that call for problem-solving skills and divergent thinking. The test is scoredfor fluency, flexibility, originality (the statistical rarityof a response), and elaboration (Sternberg, 2000).Sternberg (2000) states that there are positive andnegative sides to the psychometric approach. On thepositive side, these tests allow for research with non-eminent people, are easy to administer, and objectivelyscored. The negative side is that numerical scores failto capture the concept of creativity because they arebased on brief paper and pencil tests. Researchers callfor using more significant productions such as writingsamples, drawings, etc., subjectively evaluated by apanel of experts, instead of simply relying on anumerical measure.

The cognitive approachThe cognitive approach to the study of creativity

focuses on understanding the “mental representationsand processes underlying human thought” (Sternberg,2000, p. 7). Weisberg (1993) suggests that creativityentails the use of ordinary cognitive processes andresults in original and extraordinary products. Theseproducts are the result of cognitive processes acting onthe knowledge already stored in the memory of theindividual. There is a significant amount of literature inthe area of information processing (Birkhoff, 1969;Minsky, 1985) that attempts to isolate and explaincognitive processes in terms of machine metaphors.

The social-personality approachThe social-personality approach to studying

creativity focuses on personality and motivationalvariables as well as the socio-cultural environment assources of creativity. Sternberg (2000) states thatnumerous studies conducted at the societal levelindicate that “eminent levels of creativity over largespans of time are statistically linked to variables suchas cultural diversity, war, availability of role models,availability of financial support, and competitors in adomain” (p. 9).

Bharath Sriraman 23

Most of the recent literature on creativity(Csikszentmihalyi, 1988, 2000; Gruber & Wallace,2000; Sternberg & Lubart, 1996) suggests thatcreativity is the result of a confluence of one or moreof the factors from these six aforementionedcategories. The “confluence” approach to the study ofcreativity has gained credibility, and the researchliterature has numerous confluence theories for betterunderstanding the process of creativity. A review of themost commonly cited confluence theories of creativityand a description of the methodology employed fordata collection and data analysis in this study follow.

Confluence Theories of CreativityThe three most commonly cited “confluence”

approaches to the study of creativity are the “systemsapproach” (Csikszentmihalyi, 1988, 2000); “the casestudy as evolving systems approach” (Gruber &Wallace, 2000), and the “investment theory approach”(Sternberg & Lubart, 1996). The case study as anevolving system has the following components. First, itviews creative work as multi-faceted. So, inconstructing a case study of a creative work, one mustdistill the facets that are relevant and construct the casestudy based on the chosen facets. Some facets that canbe used to construct an evolving system case study are:(1) uniqueness of the work; (2) a narrative of what thecreator achieved; (3) systems of belief; (4) multipletime-scales (construct the time-scales involved in theproduction of the creative work); (5) problem solving;and (6) contextual frame such as family, schooling, andteacher’s influences (Gruber & Wallace, 2000). Insummary, constructing a case study of a creative workas an evolving system entails incorporating the manyfacets suggested by Gruber & Wallace (2000). Onecould also evaluate a case study involving creativework by looking for the above mentioned facets.

The systems approachThe systems approach takes into account the social

and cultural dimensions of creativity instead of simplyviewing creativity as an individualistic psychologicalprocess. The systems approach studies the interactionbetween the individual, domain, and field. The fieldconsists of people who have influence over a domain.For example, editors of mathematics research journalshave influence over the domain of mathematics. Thedomain is in a sense a cultural organism that preservesand transmits creative products to individuals in thefield. The systems model suggests that creativity is aprocess that is observable at the “intersection whereindividuals, domains and fields interact”

(Csikzentmihalyi, 2000). These three components -individual, domain, and field - are necessary becausethe individual operates from a cultural or symbolic(domain) aspect as well as a social (field) aspect.

“The domain is a necessary component ofcreativity because it is impossible to introduce avariation without reference to an existing pattern. Newis meaningful only in reference to the old”(Csikzentmihalyi, 2000). Thus, creativity occurs whenan individual proposes a change in a given domain,which is then transmitted by the field through time.The personal background of an individual and hisposition in a domain naturally influence the likelihoodof his making a contribution. For example, amathematician working at a research university is morelikely to produce research papers because of the timeavailable for “thinking” as well as the creativeinfluence of being immersed in a culture where ideasflourish. It is no coincidence that in the history ofscience, there are significant contributions fromclergymen such as Pascal and Mendel because theyhad the means and the leisure to “think.”Csikszentmihalyi (2000) argues that novel ideas, whichcould result in significant changes, are unlikely to beadopted unless they are sanctioned by the experts.These “gatekeepers” (experts) constitute the field. Forexample, in mathematics, the opinion of a very smallnumber of leading researchers was enough to certifythe validity of Andrew Wiles’ proof of Fermat’s LastTheorem.

There are numerous examples in the history ofmathematics that fall within the systems model. Forinstance, the Bourbaki, a group of mostly Frenchmathematicians who began meeting in the 1930s,aimed to write a thorough unified account of allmathematics. The Bourbaki were essentially a group ofexpert mathematicians that tried to unify all ofmathematics and become the gatekeepers of the field,so to speak, by setting the standard for rigor. Althoughthe Bourbakists failed in their attempt, students of theBourbakists, who are editors of certain prominentjournals, to this day demand a very high degree of rigorin submitted articles, thereby serving as gatekeepers ofthe field.

A different example is that of the role of proof.Proof is the social process through which themathematical community validates the mathematician'screative work (Hanna, 1991). The Russian logicianManin (1977) said "A proof becomes a proof after thesocial act of accepting it as a proof. This is true ofmathematics as it is of physics, linguistics, andbiology."

24 Mathematical Creativity

In summary, the systems model of creativitysuggests that for creativity to occur, a set of rules andpractices must be transmitted from the domain to theindividual. The individual then must produce a novelvariation in the content of the domain, and thisvariation must be selected by the field for inclusion inthe domain.

Gruber and Wallace’s case study as evolvingsystems approachIn contrast to Csikszentmihalyi’s (2000) argument

calling for a focus on communities in which creativitymanifests itself, Gruber and Wallace (2000) propose amodel that treats each individual as a unique evolvingsystem of creativity and ideas; and, therefore, eachindividual’s creative work must be studied on its own.This viewpoint of Gruber and Wallace (2000) is abelated victory of sorts for the Gestaltists, whoessentially proclaimed the same thing almost a centuryago. Gruber and Wallace’s (2000) use of terminologythat jibes with current trends in psychology seems tomake their ideas more acceptable. They propose amodel that calls for “detailed analytic and sometimesnarrative descriptions of each case and efforts tounderstand each case as a unique functioning system(Gruber & Wallace, 2000, p. 93). It is important to notethat the emphasis of this model is not to explain theorigins of creativity, nor is it the personality of thecreative individual, but on “how creative work works”(p. 94). The questions of concern to Gruber andWallace are: (1) What do creative people do when theyare being creative? and (2) How do creative peopledeploy available resources to accomplish somethingunique? In this model creative work is defined as thatwhich is novel and has value. This definition isconsistent with that used by current researchers increativity (Csikszentmihalyi, 2000; Sternberg &Lubart, 2000). Gruber and Wallace (2000) also claimthat creative work is always the result of purposefulbehavior and that creative work is usually a longundertaking “reckoned in months, years and decades”(p. 94).

I do not agree with the claim that creative work isalways the result of purposeful behavior. Onecounterexample that comes to mind is the discovery ofpenicillin. The discovery of penicillin could beattributed purely to chance. On the other hand, thereare numerous examples that support the claim thatcreative work sometimes entails work that spans years,and in mathematical folklore there are numerousexamples of such creative work. For example, Kepler’slaws of planetary motion were the result of twenty

years of numerical calculations. Andrew Wiles’ proofof Fermat’s Last Theorem was a seven-yearundertaking. The Riemann hypothesis states that theroots of the zeta function (complex numbers z, atwhich the zeta function equals zero) lie on the lineparallel to the imaginary axis and half a unit to theright of it. This is perhaps the most outstandingunproved conjecture in mathematics with numerousimplications. The analyst Levinson undertook adetermined calculation on his deathbed that increasedthe credibility of the Riemann-hypothesis. This isanother example of creative work that falls withinGruber and Wallace's (2000) model.

The investment theory approachAccording to the investment theory model, creative

people are like good investors; that is, they buy lowand sell high (Sternberg & Lubart, 1996). The contexthere is naturally in the realm of ideas. Creative peopleconjure up ideas that are either unpopular ordisrespected and invest considerable time convincingother people about the intrinsic worth of these ideas(Sternberg & Lubart, 1996). They sell high in the sensethat they let other people pursue their ideas while theymove on to the next idea. Investment theory claims thatthe convergence of six elements constitutes creativity.The six elements are intelligence, knowledge, thinkingstyles, personality, motivation, and environment. It isimportant that the reader not mistake the wordintelligence for an IQ score. On the contrary, Sternberg(1985) suggests a triarchic theory of intelligence thatconsists of synthetic (ability to generate novel, taskappropriate ideas), analytic, and practical abilities.Knowledge is defined as knowing enough about aparticular field to move it forward. Thinking styles aredefined as a preference for thinking in original ways ofone’s choosing, the ability to think globally as well aslocally, and the ability to distinguish questions ofimportance from those that are not important.Personality attributes that foster creative functioningare the willingness to take risks, overcome obstacles,and tolerate ambiguity. Finally, motivation and anenvironment that is supportive and rewarding areessential elements of creativity (Sternberg, 1985).

In investment theory, creativity involves theinteraction between a person, task, and environment.This is, in a sense, a particular case of the systemsmodel (Csikszentmihalyi, 2000). The implication ofviewing creativity as the interaction between person,task, and environment is that what is considered novelor original may vary from one person, task, andenvironment to another. The investment theory model

Bharath Sriraman 25

suggests that creativity is more than a simple sum ofthe attained level of functioning in each of the sixelements. Regardless of the functioning levels in otherelements, a certain level or threshold of knowledge isrequired without which creativity is impossible. Highlevels of intelligence and motivation can positivelyenhance creativity, and compensations can occur tocounteract weaknesses in other elements. For example,one could be in an environment that is non-supportiveof creative efforts, but a high level of motivation couldpossibly overcome this and encourage the pursuit ofcreative endeavors.

This concludes the review of three commonly citedprototypical confluence theories of creativity, namelythe systems approach (Csikszentmihalyi, 2000), whichsuggests that creativity is a sociocultural processinvolving the interaction between the individual,domain, and field; Gruber & Wallace’s (2000) modelthat treats each individual case study as a uniqueevolving system of creativity; and investment theory(Sternberg & Lubart, 1996), which suggests thatcreativity is the result of the convergence of sixelements (intelligence, knowledge, thinking styles,personality, motivation, and environment).

Having reviewed the research literature oncreativity, the focus is shifted to the methodologyemployed for studying mathematical creativity.

MethodologyThe Interview Instrument

The purpose of this study was to gain an insightinto the nature of mathematical creativity. In an effortto determine some of the characteristics of the creativeprocess, I was interested in distilling commonattributes in the ways mathematicians createmathematics. Additionally, I was interested in testingthe applicability of the Gestalt model. Because themain focus of the study was to ascertain qualitativeaspects of creativity, a formal interview methodologywas selected as the primary method of data collection.The interview instrument (Appendix A) was developedby modifying questions from questionnaires inL’Enseigement Mathematique (1902) and Muir (1988).The rationale behind using this modified questionnairewas to allow the mathematicians to express themselvesfreely while responding to questions of a generalnature and to enable me to test the applicability of thefour-stage Gestalt model of creativity. Therefore, theexisting instruments were modified to operationalizethe Gestalt theory and to encourage the natural flow ofideas, thereby forming the basis of a thesis that wouldemerge from this exploration.

Background of the SubjectsFive mathematicians from the mathematical

sciences faculty at a large Ph.D. granting mid-westernuniversity were selected. These mathematicians werechosen based on their accomplishments and thediversity of the mathematical areas in which theyworked, measured by counting the number ofpublished papers in prominent journals, as well asnoting the variety of mathematical domains in whichthey conducted research. Four of the mathematicianswere tenured full professors, each of whom had beenprofessional mathematicians for more than 30 years.One of the mathematicians was considerably youngerbut was a tenured associate professor. All interviewswere conducted formally, in a closed door setting, ineach mathematician’s office. The interviews wereaudiotaped and transcribed verbatim.

Data AnalysisSince creativity is an extremely complex construct

involving a wide range of interacting behaviors, Ibelieve it should be studied holistically. The principleof analytic induction (Patton, 2002) was applied to theinterview transcripts to discover dominant themes thatdescribed the behavior under study. According toPatton (2002), "analytic induction, in contrast togrounded theory, begins with an analyst's deducedpropositions or theory-derived hypotheses and is aprocedure for verifying theories and propositions basedon qualitative data” (Taylor and Bogdan, 1984, p. 127).Following the principles of analytic induction, the datawas carefully analyzed in order to extract commonstrands. These strands were then compared totheoretical constructs in the existing literature with theexplicit purpose of verifying whether the Gestalt modelwas applicable to this qualitative data as well as toextract themes that characterized the mathematician’screative process. If an emerging theme could not beclassified or named because I was unable to grasp itsproperties or significance, then theoretical comparisonswere made. Corbin and Strauss (1998) state that “usingcomparisons brings out properties, which in turn can beused to examine the incident or object in the data. Thespecific incidents, objects, or actions that we use whenmaking theoretical comparisons can be derived fromthe literature and experience. It is not that we useexperience or literature as data “but rather that we usethe properties and dimensions derived from thecomparative incidents to examine the data in front ofus” (p. 80). Themes that emerged were socialinteraction, preparation, use of heuristics, imagery,incubation, illumination, verification, intuition, and

26 Mathematical Creativity

proof. Excerpts from interviews that highlight thesecharacteristics are reconstructed in the next sectionalong with commentaries that incorporate the widerconversation, and a continuous discussion ofconnections to the existing literature.

Results, Commentaries & DiscussionThe mathematicians in this study worked in

academic environments and regularly fulfilled teachingand committee duties. The mathematicians were free tochoose their areas of research and the problems onwhich they focused. Four of the five mathematicianshad worked and published as individuals and asmembers of occasional joint ventures withmathematicians from other universities. Only one ofthe mathematicians had done extensive collaborativework. All but one of the mathematicians were unableto formally structure their time for research, primarilydue to family commitments and teachingresponsibilities during the regular school year. All themathematicians found it easier to concentrate onresearch in the summers because of lighter or non-existent teaching responsibilities during that time. Twoof the mathematicians showed a pre-dispositiontowards mathematics at the early secondary schoollevel. The others became interested in mathematicslater, during their university education. Themathematicians who participated in this study did notreport any immediate family influence that was ofprimary importance in their mathematicaldevelopment. Four of the mathematicians recalledbeing influenced by particular teachers, and onereported being influenced by a textbook. The threemathematicians who worked primarily in analysismade a conscious effort to obtain a broad overview ofmathematics not necessarily of immediate relevance totheir main interests. The two algebraists expressedinterest in other areas of mathematics but wereprimarily active in their chosen field.

Supervision Of Research & Social InteractionAs noted earlier, all the mathematicians in this

study were tenured professors in a research university.In addition to teaching, conducting research, andfulfilling committee obligations, many mathematiciansplay a big role in mentoring graduate studentsinterested in their areas of research. Researchsupervision is an aspect of creativity because anyinteraction between human beings is an ideal settingfor the exchange of ideas. During this interaction themathematician is exposed to different perspectives onthe subject, and all of the mathematicians in this study

valued the interaction they had with their graduatestudents. Excerpts of individual responses follow.1

Excerpt 1

A. I've had only one graduate student per semesterand she is just finishing up her PhD right now,and I'd say it has been a very good interactionto see somebody else get interested in thesubject and come up with new ideas, andexploring those ideas with her.

B. I have had a couple of students who have sortof started but who haven't continued on to aPhD, so I really can't speak to that. But theinteraction was positive.

C. Of course, I have a lot of collaborators, theseare my former students you know…I amalways all the time working with students, thisis normal situation.

D. That is difficult to answer (silence)…it ispositive because it is good to interact withother people. It is negative because it can takea lot of time. As you get older your braindoesn't work as well as it used toand…younger people by and large their mindsare more open, there is less garbage in therealready. So, it is exciting to work with youngerpeople who are in their most creative time.When you are older, you have moreexperience, when you are younger your mindworks faster …not as fettered.

E. Oh…it is a positive factor I think, because itcontinues to stimulate ideas …talking aboutthings and it also reviews things for you in theprocess, puts things in perspective, and keepthe big picture. It is helpful really in your ownresearch to supervise students.

Commentary on Excerpt 1The responses of the mathematicians in the

preceding excerpt are focused on research supervision;however, all of the mathematicians acknowledged therole of social interaction in general as an importantaspect that stimulated creative work. Many of themathematicians mentioned the advantages of beingable to e-mail colleagues and going to researchconferences and other professional meetings. This isfurther explored in the following section, whichfocuses on preparation.

Bharath Sriraman 27

Preparation and the Use of HeuristicsWhen mathematicians are about to investigate a

new topic, there is usually a body of existing researchin the area of the new topic. One of goals of this studywas to find out how creative mathematiciansapproached a new topic or a problem. Did they try theirown approach, or did they first attempt to assimilatewhat was already known about that topic? Did themathematicians make use of computers to gain insightinto the problem? What were the various modes ofapproaching a new topic or problem? The responsesindicate that a variety of approaches were used.

Excerpt 2

A. Talk to people who have been doing this topic.Learn the types of questions that come up.Then I do basic research on the main ideas. Ifind that talking to people helps a lot more thanreading because you get more of a feel forwhat the motivation is beneath everything.

B. What might happen for me, is that I may startreading something, and, if feel I can do a betterjob, then I would strike off on my own. But forthe most part I would like to not have toreinvent a lot that is already there. So, a lot ofwhat has motivated my research has been thedesire to understand an area. So, if somebodyhas already laid the groundwork then it'shelpful. Still I think a large part of doingresearch is to read the work that other peoplehave done.

C. It is connected with one thing that simply…mystyle was that I worked very much and I evenwork when I could not work. Simply theproblems that I solve attract me so much, thatthe quest ion was who wil l diefirst…mathematics or me? It was never clearwho would die.

D. Try and find out what is known. I won't sayassimilate…try and find out what's known andget an overview, and try and let the problemspeak…mostly by reading because you don'thave that much immediate contact with otherpeople in the field. But I find that I get morefrom listening to talks that other people aregiving than reading.

E. Well! I have been taught to be a good scholar.A good scholar attempts to find out what isfirst known about something or other beforethey spend their time simply going it on their

own. That doesn't mean that I don'tsimultaneously try to work on something.

Commentary on Excerpt 2These responses indicate that the mathematician

spends a considerable amount of time researching thecontext of the problem. This is primarily done byreading the existing literature and by talking to othermathematicians in the new area. This finding isconsistent with the systems model, which suggests thatcreativity is a dynamic process involving theinteraction between the individual, domain, and field(Csikzentmihalyi, 2000). At this stage, it is reasonableto ask whether a mathematician works on a singleproblem until a breakthrough occurs or does amathematician work on several problems concurrently?It was found that each of the mathematicians workedon several problems concurrently, using a back andforth approach.

Excerpt 3

A. I work on several different problems for aprotracted period of time… there have beentimes when I have felt, yes, I should be able toprove this result, then I would concentrate onthat thing for a while but they tend to beseveral different things that I was thinkingabout a particular stage.

B. I probably tend to work on several problems atthe same time. There are several differentques t i ons t ha t I am work ingon…mm…probably the real question is howoften do you change the focus? Do I work ontwo different problems on the same day? Andthat is probably up to whatever comes to mindin that particular time frame. I might startworking on one rather than the other. But Iwould tend to focus on one particular problemfor a period of weeks, then you switch tosomething else. Probably what happens is thatI work on something and I reach a dead endthen I may shift gears and work on a differentproblem for a while, reach a dead end thereand come back to the original problem, so it’sback and forth.

C . I must simply think on one thing and notswitch so much.

D. I find that I probably work on one. Theremight be a couple of things floating around butI am working on one and if I am not getting

28 Mathematical Creativity

anywhere, then I might work on the other andthen go back.

E. I usually have couple of things going. When Iget stale on one, then I will pick up the other,and bounce back and forth. Usually I have onethat is primarily my focus at a given time, andI will spend time on it over another; but it isnot uncommon for me to have a couple ofproblems going at a given time. Sometimeswhen I am looking for an example that is notcoming, instead of spending my time beatingmy head against the wall, looking for thatexample is not a very good use of time.Working on another helps to generate ideasthat I can bring back to the other problem.

Commentary on Excerpt 3The preceding excerpt indicates that

mathematicians tend to work on more than oneproblem at a given time. Do mathematicians switchback and forth between problems in a completelyrandom manner, or do they employ and exhaust asystematic train of thought about a problem beforeswitching to a different problem? Many of themathematicians reported using heuristic reasoning,trying to prove something one day and disprove it thenext day, looking for both examples andcounterexamples, the use of "manipulations" (Polya,1954) to gain an insight into the problem. Thisindicates that mathematicians do employ some of theheuristics made explicit by Polya. It was unclearwhether the mathematicians made use of computers togain an experimental or computational insight into theproblem. I was also interested in knowing the types ofimagery used by mathematicians in their work. Themathematicians in this study were queried about this,and the following excerpt gives us an insight into thataspect of mathematical creativity.

ImageryThe mathematicians in this study were asked about

the kinds of imagery they used to think aboutmathematical objects. Their responses are reportedhere to give the reader a glimpse of the waysmathematicians think of mathematical objects. Theirresponses also highlight the difficulty of explicitlydescribing imagery.

Excerpt 4Yes I do, yes I do, I tend to draw a lot of pictures

when I am doing research, I tend to manipulate thingsin the air, you know to try to figure out how things

work. I have a very geometrically based intuition anduhh…so very definitely I do a lot of manipulations.

A. That is a problem because of the particulararea I am in. I can't draw any diagrams, thingsare infinite, so I would love to be able to getsome kind of a computer diagram to show thecomplexity for a particular ring… to havesometh ing l ike the Ju l i a se t sor…mmm…fractal images, things which areinfinite but you can focus in closer and closerto see possible relationships. I have thoughtabout that with possibilities on the computer.To think about the most basic ring, you wouldhave to think of the ring of integers and all ofthe relationships for divisibility, so how do yousomehow describe this tree of divisibility forintegers…it is infinite.

B . Science is language, you think throughlanguage. But it is language simply; you puttogether theorems by logic. You first see thetheorem in nature…you must see thatsomewhat is reasonable and then you go andbegin and then of course there is big, big, bigwork to just come to some theorem in non-linear elliptic equations…

C. A lot of mathematics, whether we are teachingor doing, is attaching meaning to what we aredoing and this is going back to the earlierquestion when you talked about how do you doit, what kind of heuristics do you use? Whatkind of images do you have that you are using?A lot of doing mathematics is creating theseabstract images that connect things and thenmaking sense of them but that doesn't appearin proofs either.

D. Pictorial, linguistic, kinesthetic...any of them isthe point right! Sometimes you think of one,sometimes another. It really depends on theproblem you are looking at, they are verymuch…often I think of functions as verykinesthetic, moving things from here to there.Other approaches you are talking about isgoing to vary from problem to problem, oreven day to day. Sometimes when I amworking on research, I try to view things in asmany different ways as possible, to see what isreally happening. So there are a variety ofapproaches.

Bharath Sriraman 29

Commentary on Excerpt 4Besides revealing the difficulty of describing

mental imagery, all the mathematicians reported thatthey did not use computers in their work. Thischaracteristic of the pure mathematician's work isechoed in Poincaré's (1948) use of the “choice”metaphor and Ervynck's (1991) use of the term “non-algorithmic decision making.” The doubts expressedby the mathematicians about the incapability ofmachines to do their work brings to mind the reportedwords of Garrett Birkhoff, one of the great appliedmathematicians of our time. In his retirementpresidential address to the Society for Industrial andApplied Mathematics, Birkhoff (1969) addressed therole of machines in human creative endeavors. Inparticular, part of this address was devoted todiscussing the psychology of the mathematicians (andhence of mathematics). Birkhoff (1969) said:

The remarkable recent achievements of computershave partially fulfilled an old dream. Theseachievements have led some people to speculatethat tomorrow's computers will be even more"intelligent" than humans, especially in theirpowers of mathematical reasoning...the ability ofgood mathematicians to sense the significant and toavoid undue repetition seems, however, hard tocomputerize; without it, the computer has to pursuemillions of fruitless paths avoided by experiencedhuman mathematicians. (pp. 430-438)

Incubation and IlluminationHaving reported on the role of research supervision

and social interaction, the use of heuristics andimagery, all of which can be viewed as aspects of thepreparatory stage of mathematical creativity, it isnatural to ask what occurs next. As the literaturesuggests, after the mathematician works hard to gaininsight into a problem, there is usually a transitionperiod (conscious work on the problem ceases andunconscious work begins), during which the problem isput aside before the breakthrough occurs. Themathematicians in this study reported experiences thatare consistent with the existing literature (Hadamard,1945; Poincaré, 1948).

Excerpt 5B. One of the problems is first one does some

preparatory work, that has to be the left side[of the brain], and then you let it sit. I don'tthink you get ideas out of nowhere, you haveto do the groundwork first, okay. This is whypeople will say, now we have worked on thisproblem, so let us sleep on it. So you do the

preparation, so that the sub-conscious orintuitive side may work on it and the answercomes back but you can't really tell when. Youhave to be open to this, lay the groundwork,think about it and then these flashes ofintuition come and they represent the otherside of the brain communicating with you atwhatever odd time.

D. I am not sure you can really separate thembecause they are somewhat connected. Youspend a lot of time working on something andyou are not getting anywhere with it…with thedeliberate effort, then I think your mindcontinues to work and organize. And maybewhen the pressure is off the idea comes…butthe idea comes because of the hard work.

E. Usually they come after I have worked veryhard on something or another, but they maycome at an odd moment. They may come intomy head before I go to bed …What do I do atthat point? Yes I write it down (laughing).Sometimes when I am walking somewhere, themind flows back to it (the problem) and sayswhat about that, why don't you try that. Thatsort of thing happens. One of the best ideas Ihad was when I was working on my thesis…Saturday night, having worked on it quite abit, sitting back and saying why don't I thinkabout it again…and ping! There it was…Iknew what it was, I could do that. Often ideasare handed to you from the outside, but theydon't come until you have worked on it longenough.

Commentary on Excerpt 5As is evident in the preceding excerpt, three out of

the five mathematicians reported experiencesconsistent with the Gestalt model. Mathematician Cattributed his breakthroughs on problems to hisunflinching will to never give up and to divineinspiration, echoing the voice of Pascal in a sense.However, Mathematician A attributed breakthroughs tochance. In other words, making the appropriate(psychological) connections by pure chance whicheventually result in the sought after result.

I think it is necessary to comment about theunusual view of mathematician A. Chance plays animportant role in mathematical creativity. Great ideasand insights may be the result of chance such as thediscovery of penicillin. Ulam (1976) estimated thatthere is a yearly output of 200,000 theorems in

30 Mathematical Creativity

mathematics. Chance plays a role in what is consideredimportant in mathematical research since only ahandful of results and techniques survive out of thevolumes of published research. I wish to draw adistinction between chance in the "Darwinian" sense(as to what survives), and chance in the psychologicalsense (which results in discovery/invention). The roleof chance is addressed by Muir (1988) as follows.

The act of creation of new entities has two aspects:the generation of new possibilities, for which wemight attempt a stochastic description, and theselection of what is valuable from among them.However the importation of biological metaphorsto explain cultural evolution is dubious…bothcreation and selection are acts of design within asocial context. (p. 33)

Thus, Muir (1988) rejects the Darwinianexplanation. On the other hand, Nicolle (1932) inBiologie de L'Invention does not acknowledge the roleof unconsciously present prior work in the creativeprocess. He attributes breakthroughs to pure chance.

By a streak of lightning, the hitherto obscureproblem, which no ordinary feeble lamp wouldhave revealed, is at once flooded in light. It is like acreation. Contrary to progressive acquirements,such an act owes nothing to logic or to reason. Theact of discovery is an accident. (Hadamard, 1945)

Nicolle's Darwinian explanation was rejected byHadamard on the grounds that to claim creation occursby pure chance is equivalent to asserting that there areeffects without causes. Hadamard further argued thatalthough Poincaré attributed his particularbreakthrough in Fuchsian functions to chance, Poincarédid acknowledge that there was a considerable amountof previous conscious effort, followed by a period ofunconscious work. Hadamard (1945) further arguedthat even if Poincaré's breakthrough was the result ofchance alone, chance alone was insufficient to explainthe considerable body of creative work credited toPoincaré in almost every area of mathematics. Thequestion then is how does (psychological) chancework?

It is my conjecture that the mind throws outfragments (ideas) that are products of past experience.Some of these fragments can be juxtaposed andcombined in a meaningful way. For example, if onereads a complicated proof consisting of a thousandsteps, a thousand random fragments may not be enoughto construct a meaningful proof. However the mindchooses relevant fragments from these randomfragments and links them into something meaningful.Wedderburn's Theorem, that a finite division ring is a

field, is one instance of a unification of apparentlyrandom fragments because the proof involves algebra,complex analysis, and number theory.

Polya (1954) addresses the role of chance in aprobabilistic sense. It often occurs in mathematics thata series of mathematical trials (involving computation)generate numbers that are close to a Platonic ideal. Theclassic example is Euler's investigation of the infiniteseries 1 + 1/4 + 1/9 + 1/16 +…+ 1/n2 +…. Eulerobtained an approximate numerical value for the sumof the series using various transformations of theseries. The numerical approximation was 1.644934.Euler confidently guessed the sum of the series to beπ2/6. Although the numerical value obtained by Eulerand the value of π2/6 coincided up to seven decimalplaces, such a coincidence could be attributed tochance. However, a simple calculation shows that theprobability of seven digits coinciding is one in tenmillion! Hence, Euler did not attribute this coincidenceto chance but boldly conjectured that the sum of thisseries was indeed π2/6 and later proved his conjectureto be true (Polya, 1954, pp. 95-96).

Intuition, Verification and ProofOnce illumination has occurred, whether through

sheer chance, incubation, or divine intervention,mathematicians usually try to verify that theirintuitions were correct with the construction of a proof.The following section discusses how thesemathematicians went about the business of verifyingtheir intuitions and the role of formal proof in thecreative process. They were asked whether they reliedon repeatedly checking a formal proof, used multipleconverging partial proofs, looked first for coherencewith other results in the area, or looked at applications.Most of the mathematicians in this study mentionedthat the last thing they looked at was a formal proof.This is consistent with the literature on the role offormal proof in mathematics (Polya, 1954; Usiskin,1987). Most of the mathematicians mentioned the needfor coherence with other results in the area. Themathematician’s responses to the posed questionfollow.

Excerpt 6

B. I think I would go for repeated checking of theformal proof…but I don't think that that isreally enough. All of the others have to also betaken into account. I mean, you can believethat something is true although you may notfully understand it. This is the point that wasmade in the lecture by … of … University on

Bharath Sriraman 31

Dirichlet series. He was saying that we havehad a formal proof for some time, but that isnot to say that it is really understood, and whatdid he mean by that? Not that the proof wasn'tunderstood, but it was the implications of theresult that are not understood, theirconnections with other results, applicationsand why things really work. But probably thefirst thing that I would really want to do ischeck the formal proof to my satisfaction, sothat I believe that it is correct although at thatpoint I really do not understand itsimplications… it is safe to say that it is mysurest guide.

C. First you must see it in the nature, something,first you must see that this theoremcorresponds to something in nature, then if youhave this impression, it is something relativelyreasonable, then you go to proofs…and ofcourse I have also several theorems and proofsthat are wrong, but the major amount of proofsand theorems are right.

D. The last thing that comes is the formal proof. Ilook for analogies with other things… Howyour results that you think might be true wouldilluminate other things and would fit in thegeneral structure.

E. Since I work in an area of basic research, it isusually coherence with other things, that isprobably more than anything else. Yes, onecould go back and check the proof and that sortof thing but usually the applications are yet tocome, they aren't there already. Usually whatguides the choice of the problem is thepotential for application, part of whatrepresents good problems is their potential foruse. So, you certainly look to see if it makessense in the big picture…that is a coherencephenomenon. Among those you've given me,that’s probably the most that fits.

Commentary on Excerpt 6This excerpt indicates that for mathematicians,

valid proofs have varied degrees of rigor. “Amongmathematicians, rigor varies depending on time andcircumstance, and few proofs in mathematics journalsmeet the criteria used by secondary school geometryteachers (each statement of proof is backed byreasons). Generally one increases rigor only when theresult does not seem to be correct” (Usiskin, 1987).Proofs are in most cases the final step in this testing

process. “Mathematics in the making resembles anyother human knowledge in the making. The result ofthe mathematician’s creative work is demonstrativereasoning, a proof; but the proof is discovered byplausible reasoning, by guessing” (Polya, 1954). Howmathematicians approached proof in this study wasvery different from the logical approach found in proofin most textbooks. The logical approach is an artificialreconstruction of discoveries that are being forced intoa deductive system, and in this process the intuitionthat guided the discovery process gets lost.

ConclusionsThe goal of this study was to gain an insight into

mathematical creativity. As suggested by the literaturereview, the existing literature on mathematicalcreativity is relatively sparse. In trying to betterunderstand the process of creativity, I find that theGestalt model proposed by Hadamard (1945) is stillapplicable today. This study has attempted to add somedetail to the preparation-incubation-illumination-verification model of Gestalt by taking into account therole of imagery, the role of intuition, the role of socialinteraction, the use of heuristics, and the necessity ofproof in the creative process.

The mathematicians worked in a setting that wasconducive to prolonged research. There was aconvergence of intelligence, knowledge, thinkingstyles, personality, motivation and environment thatenabled them to work creatively (Sternberg, 2000;Sternberg & Lubart, 1996, 2000). The preparatorystage of mathematical creativity consists of variousapproaches used by the mathematician to lay thegroundwork. These include reading the existingliterature, talking to other mathematicians in theparticular mathematical domain (Csikzentmihalyi,1988; 2000), trying a variety of heuristics (Polya,1954), and using a back-and-forth approach ofplausible guessing. One of the mathematicians said thathe first looked to see if the sought after relationshipscorresponded to natural phenomenon.

All of the mathematicians in this study worked onmore than one problem at a given moment. This isconsistent with the investment theory view of creativity(Sternberg & Lubart, 1996). The mathematiciansinvested an optimal amount of time on a givenproblem, but switched to a different problem if nobreakthrough was forthcoming. All the mathematiciansin this study considered this as the most important anddifficult stage of creativity. The prolonged hard workwas followed by a period of incubation where theproblem was put aside, often while the preparatory

32 Mathematical Creativity

stage is repeated for a different problem; and thus,there is a transition in the mind from conscious tounconscious work on the problem. One mathematiciancited this as the stage at which the "problem begins totalk to you." Another offered that the intuitive side ofthe brain begins communicating with the logical side atthis stage and conjectured that this communication wasnot possible at a conscious level.

The transition from incubation to illuminationoften occurred when least expected. Many reported thebreakthrough occurring as they were going to bed, orwalking, or sometimes as a result of speaking tosomeone else about the problem. One mathematicianillustrated this transition with the following: "You talkto somebody and they say just something that mighthave been very ordinary a month before but if they sayit when you are ready for it, and Oh yeah, I can do itthat way, can’t I! But you have to be ready for it.Opportunity knocks but you have to be able to answerthe door."

Illumination is followed by the mathematician’sverifying the result. In this study, most of themathematicians looked for coherence of the result withother existing results in the area of research. If theresult cohered with other results and fit the generalstructure of the area, only then did the mathematiciantry to construct a formal proof. In terms of themathematician’s beliefs about the nature ofmathematics and its influence on their research, thestudy revealed that four of the mathematicians leanedtowards Platonism, in contrast to the popular notionthat Platonism is an exception today. A detaileddiscussion of this aspect of the research is beyond thescope of this paper; however, I have found that beliefsregarding the nature of mathematics not onlyinfluenced how these mathematicians conductedresearch but also were deeply connected to theirtheological beliefs (Sriraman, 2004a).

The mathematicians hoped that the results of theircreative work would be sanctioned by a group ofexperts in order to get the work included in the domain(Csikzentmihalyi, 1988, 2000), primarily in the form ofpublication in a prominent journal. However, theacceptance of a mathematical result, the end product ofcreation, does not ensure its survival in the Darwiniansense (Muir, 1988). The mathematical result may ormay not be picked up by other mathematicians. If themathematical community picks it up as a viable result,then it is likely to undergo mutations and lead to newmathematics. This, however, is determined by chance!

ImplicationsIt is in the best interest of the field of mathematics

education that we identify and nurture creative talent inthe mathematics classroom. "Between the work of astudent who tries to solve a difficult problem inmathematics and a work of invention (creation)…thereis only a difference of degree" (Polya, 1954).Creativity as a feature of mathematical thinking is not apatent of the mathematician! (Krutetskii, 1976); andalthough most studies on creativity have focused oneminent individuals (Arnheim, 1962; Gardner, 1993,1997; Gruber, 1981), I suggest that contemporarymodels from creativity research can be adapted forstudying samples of creativity such as are produced byhigh school students. Such studies would reveal moreabout creativity in the classroom to the mathematicseducation research community. Educators couldconsider how often mathematical creativity ismanifested in the school classroom and how teachersmight identify creative work. One plausible way toapproach these concerns is to reconstruct and evaluatestudent work as a unique evolving system of creativity(Gruber & Wallace, 2000) or to incorporate some ofthe facets suggested by Gruber & Wallace (2000). Thisnecessitates the need to find suitable problems at theappropriate levels to stimulate student creativity.

A common trait among mathematicians is thereliance on particular cases, isomorphic reformulations,or analogous problems that simulate the originalproblem situations in their search for a solution (Polya,1954; Skemp, 1986). Creating original mathematicsrequires a very high level of motivation, persistence,and reflection, all of which are considered indicators ofcreativity (Amabile, 1983; Policastro & Gardner, 2000;Gardner, 1993). The literature suggests that mostcreative individuals tend to be attracted to complexity,of which most school mathematics curricula has verylittle to offer. Classroom practices and math curriculararely use problems with the sort of underlyingmathematical structure that would necessitate students’having a prolonged period of engagement and theindependence to formulate solutions. It is myconjecture that in order for mathematical creativity tomanifest itself in the classroom, students should begiven the opportunity to tackle non-routine problemswith complexity and structure - problems whichrequire not only motivation and persistence but alsoconsiderable reflection. This implies that educatorsshould recognize the value of allowing students toreflect on previously solved problems to drawcomparisons between various isomorphic problems(English, 1991, 1993; Hung, 2000; Maher & Kiczek,

Bharath Sriraman 33

2000; Maher & Martino, 1997; Maher & Speiser,1996; Sriraman, 2003; Sriraman, 2004b). In addition,encouraging students to look for similarities in a classof problems fosters "mathematical" behavior (Polya,1954), leading some students to discover sophisticatedmathematical structures and principles in a mannerakin to the creative processes of professionalmathematicians.

REFERENCESAmabile, T. M. (1983). Social psychology of creativity: A

componential conceptualization. Journal of Personality andSocial Psychology, 45, 357−376.

Arnheim, R. (1962). Picasso’s guernica. Berkeley: University ofCalifornia Press.

Birkhoff, G. (1969). Mathematics and psychology. SIAM Review,11, 429−469.

Corbin, J., & Strauss, A. (1998). Basics of qualitative research.Thousand Oaks, CA: Sage.

Csikszentmihalyi, M. (1988). Society, culture, and person: Asystems view of creativity. In R. J. Sternberg (Ed.), The natureof creativity: Contemporary psychological perspectives (pp.325−339). Cambridge UK: Cambridge University Press.

Csikszentmihalyi, M. (2000). Implications of a systems perspectivefor the study of creativity. In R. J. Sternberg (Ed.), Handbookof creativity (pp. 313−338). Cambridge UK: CambridgeUniversity Press.

Davis, P. J., & Hersh, R. (1981). The mathematical experience.New York: Houghton Mifflin.

English, L. D. (1991). Young children's combinatoric strategies.Educational Studies in Mathematics, 22, 451−474.

English, L. D. (1993). Children's strategies in solving two- andthree-dimensional combinatorial problems. Journal forResearch in Mathematics Education, 24(3), 255−273.

Ernest, P. (1991). The philosophy of mathematics education,Briston, PA: Falmer.

Ernest, P. (1994). Conversation as a metaphor for mathematics andlearning. Proceedings of the British Society for Research intoLearning Mathematics Day Conference, ManchesterMetropolitan University (pp. 58−63). Nottingham: BSRLM.

Ervynck, G. (1991). Mathematical creativity. In D. Tall (Ed.),Advanced mathematical thinking (pp. 42−53). Dordrecht:Kluwer.

Frensch, P., & Sternberg, R. (1992). Complex problem solving:Principles and mechanisms. New Jersey: Erlbaum.

Gallian, J. A. (1994). Contemporary abstract algebra. Lexington,MA: Heath.

Gardner, H. (1997). Extraordinary minds. New York: Basic Books.Gardner, H. (1993). Frames of mind. New York: Basic Books.Gruber, H. E. (1981). Darwin on man. Chicago: University of

Chicago Press.Gruber, H. E., & Wallace, D. B. (2000). The case study method and

evolving systems approach for understanding unique creativepeople at work. In R. J. Sternberg (Ed.), Handbook ofcreativity (pp. 93-115). Cambridge UK: Cambridge UniversityPress.

Hadamard, J. (1945). Essay on the psychology of invention in themathematical field. Princeton, NJ: Princeton University Press.

Hanna, G. (1991). Mathematical proof. In D. Tall (Ed.). Advancedmathematical thinking (pp. 54−60). Dordrecht: Kluwer.

Hung, D. (2000). Some insights into the generalizations ofmathematical meanings. Journal of Mathematical Behavior,19, 63–82.

Krutetskii, V. A. (1976). The psychology of mathematical abilitiesin school children. (J. Kilpatrick & I. Wirszup, Eds.; J. Teller,Trans.). Chicago: University of Chicago Press. (Original workpublished 1968)

L'Enseigement Mathematique. (1902), 4, 208–211.L'Enseigement Mathematique. (1904), 6, 376.Lester, F. K. (1985). Methodological considerations in research on

mathematical problem solving. In E. A. Silver (Ed.), Teachingand learning mathematical problem solving: Multipleresearch perspectives (pp. 41–70). Hillsdale, NJ: Erlbaum.

Maher, C. A., & Kiczek R. D. (2000). Long term building ofmathematical ideas related to proof making. Contributions toPaolo Boero, G. Harel, C. Maher, M. Miyasaki. (organizers)Proof and Proving in Mathematics Education. Paperdistributed at ICME9 -TSG 12. Tokyo/Makuhari, Japan.

Maher, C. A., & Speiser M. (1997). How far can you go with blocktowers? Stephanie's intellectual development. Journal ofMathematical Behavior, 16(2), 125−132.

Maher, C. A., & Martino A. M. (1996). The development of theidea of mathematical proof: A 5-year case study. Journal forResearch in Mathematics Education, 27(2), 194−214.

Manin, Y. I. (1977). A course in mathematical logic. New York:Springer-Verlag.

Minsky, M. (1985). The society of mind. New York: Simon &Schuster.

Muir, A. (1988). The psychology of mathematical creativity.Mathematical Intelligencer, 10(1), 33−37.

Nicolle, C. (1932). Biologie de l'invention, Paris: Alcan.Patton, M. Q. (2002). Qualitative research and evaluation methods.

Thousand Oaks, CA: Sage.Policastro, E., & Gardner, H. (2000). From case studies to robust

generalizations: An approach to the study of creativity. In R. J.Sternberg (Ed.), Handbook of creativity (pp. 213−225).Cambridge, UK: Cambridge University Press.

Poincaré, H. (1948). Science and method. New York: Dover.Polya, G. (1945). How to solve it. Princeton, NJ: Princeton

University Press.Polya, G. (1954). Mathematics and plausible reasoning: Induction

and analogy in mathematics (Vol. II). Princeton, NJ: PrincetonUniversity Press.

Schoenfeld, A. H. (1985). Mathematical problem solving. NewYork: Academic Press.

Skemp, R. (1986). The psychology of learning mathematics.Middlesex, UK: Penguin Books.

Sriraman, B. (2003). Mathematical giftedness, problem solving,and the ability to formulate generalizations. The Journal ofSecondary Gifted Education. XIV(3), 151−165.

Sriraman, B. (2004a). The influence of Platonism on mathematicsresearch and theological beliefs. Theology and Science, 2(1),131−147.

34 Mathematical Creativity

Sriraman, B. (2004b). Discovering a mathematical principle: Thecase of Matt. Mathematics in School (UK), 3(2), 25−31.

Sternberg, R. J. (1985). Human abilities: An informationprocessing approach. New York: W. H. Freeman.

Sternberg, R. J. (2000). Handbook of creativity. Cambridge, UK:Cambridge University Press.

Sternberg, R. J., & Lubart, T. I. (1996). Investing in creativity.American Psychologist, 51, 677−688.

Sternberg, R. J., & Lubart, T. I. (2000). The concept of creativity:Prospects and paradigms. In R. J. Sternberg (Ed.), Handbookof creativity (pp. 93−115). Cambridge, UK: CambridgeUniversity Press.

Taylor, S. J., & Bogdan, R. (1984). Introduction to qualitativeresearch methods: The search for meanings. New York: JohnWiley & Sons.

Torrance, E. P. (1974). Torrance tests of creative thinking: Norms-technical manual. Lexington, MA: Ginn.

Ulam, S. (1976). Adventures of a mathematician. New York:Scribners.

Usiskin, Z. P. (1987). Resolving the continuing dilemmas in schoolgeometry. In M. M. Lindquist, & A. P. Shulte (Eds.), Learningand teaching geometry, K-12: 1987 yearbook (pp. 17−31).Reston, VA: National Council of Teachers of Mathematics.

Wallas, G. (1926). The art of thought. New York: Harcourt, Brace& Jovanovich.

Weisberg, R. W. (1993). Creativity: Beyond the myth of genius.New York: Freeman.

Wertheimer, M. (1945). Productive thinking. New York: Harper.Wittgenstein, L. (1978). Remarks on the foundations of

mathematics (Rev. Ed.).Cambridge: Massachusetts Institute ofTechnology Press.

APPENDIX A: Interview ProtocolThe interview instrument was developed by modifying questionsfrom questionnaires in L’Enseigement Mathematique (1902) andMuir (1988).1. Describe your place of work and your role within it.2. Are you free to choose the mathematical problems you tackle

or are they determined by your work place?3. Do you work and publish mainly as an individual or as part of

a group?4. Is supervision of research a positive or negative factor in your

work?5. Do you structure your time for mathematics?6. What are your favorite leisure activities apart from

mathematics?7. Do you recall any immediate family influences, teachers,

colleagues or texts, of primary importance in yourmathematical development?

8. In which areas were you initially self-educated? In whichareas do you work now? If different, what have been thereasons for changing?

9. Do you strive to obtain a broad overview of mathematics notof immediate relevance to your area of research?

10. Do you make a distinction between thought processes inlearning and research?

11. When you are about to begin a new topic, do you prefer toassimilate what is known first or do you try your ownapproach?

12. Do you concentrate on one problem for a protracted period oftime or on several problems at the same time?

13. Have your best ideas been the result of prolonged deliberateeffort or have they occurred when you were engaged in otherunrelated tasks?

14. How do you form an intuition about the truth of a proposition?15. Do computers play a role in your creative work (mathematical

thinking)?16. What types of mental imagery do you use when thinking about

mathematical objects?

Note: Questions regarding foundational and theologicalissues have been omitted in this protocol. The discussionresulting from these questions are reported in Sriraman(2004a).

The Mathematics Educator2004, Vol. 14, No. 1, 35–41

Melissa R. Freiberg 35

Getting Everyone Involved in Family MathMelissa R. Freiberg

Teachers from the departments of Mathematics and Computer Science, and Curriculum and Instruction at theUniversity of Wisconsin-Whitewater collaboratively developed and implemented an evening math event,Family Math Fun Night, at local elementary schools. As an assignment, preservice elementary teachersdeveloped hands-on mathematical activities, adaptable for different ages and abilities, to engage children andparents. The pre-service elementary teachers presented a variety of activities at each school site to small groupsof families and school personnel. This paper outlines the purpose, structure, and benefits of the project for all itsparticipants.

In an age when we continually hear about thenecessity of parent awareness and involvement inschools, there are still limited connections amongschools, parents, and higher education institutions. It isespecially important for parents and teachers to beaware of the premises and types of activities thatsupport effective mathematics learning as advocated bythe National Council of Teachers of Mathematics(NCTM, 2000). However, many parents did not growup learning in ways the NCTM advocates; they seehands-on activities as a fun “waste of time” rather thanan avenue for providing conceptual underpinnings formathematics. Teachers must realize that fun hands-onactivities, though motivating for students, must alsohave mathematical integrity in order to be included inthe curriculum. To facilitate both parents and teachersreaching these goals, our university presents what wecall Family Math Fun Night (FMFN) at areaelementary schools.

Numerous schools and districts report using somevariation of Family Math to help parents understandtheir children’s mathematics curricula better (Wood,1991, 1992; Carlson, 1991; Pagni, 2002; Kyle,McIntyre, & Moore, 2001). Our program is a variationof Stenmark, Thompson, and Cossey’s (1986) FamilyMath. In contrast to their Family Math, we choose tohave our preservice teachers present activities atelementary schools. This provides our preserviceteachers with an opportunity to have a positive, earlyexperience in schools and allows them to test ideasabout mathematics education they have learned in theiruniversity classes. Also, FMFNs provide an

entertaining family experience centered on academicswith very little expense to or preparation by the school.Finally, FMFNs offer a unique opportunity forprofessional interaction among university and schoolfaculties and staff.

At the University of Wisconsin-Whitewater, werequire the FMFN project for students enrolled in theMath for Elementary Teachers content courses andprovide it as an optional project for students enrolled inthe elementary mathematics methods course. Sincestudents take the Math for Elementary Teacherscourses in their freshman or sophomore year, FMFN isa good way to get preservice teachers thinking aboutthe content they are going to teach. Also, theexperience supports the developmental view ofmathematics learning presented in the content courseand provides an experiential background for students inthe methods courses. The preservice teachers useactivities from the Family Math books we keep onreserve, and we encourage students to devise or findactivities from other sources. The preservice teachersin the methods courses are especially encouraged toexamine professional journals and databases inpreparation for their projects.

Parent-teacher groups at schools provide a smallamount of funding (usually about $25) to purchasestickers, pencils, erasers, etc. for prizes; though somepre-service teachers buy their own, and many pre-service teacher groups do not give out prizes at all. Thelack of prizes does not seem to affect the popularity ofthe activities for most children. For past FMFNs, wehave received small grants from NASA to deviseactivities that have a space theme. We have notdesignated a theme for the event since, but have foundthat a theme often emerges. For example, we have hadFMFNs whose activities revolve around sports andFMFNs whose activities relate to voting.

Reflecting on our version of FMFN raises pointsof interest that are worth sharing: (1) the types of

Melissa Freiberg is an associate professor in the Department ofCurriculum and Instruction at the University of Whitewater-Wisconsin. She has a PhD in Urban Education with an emphasisin teacher education. Her research interests are teacherinduction and hands-on learning.

36 Family Math

activities that are presented at the events and whatdetermines their quality, (2) what considerations arenecessary for coordinating a FMFN, and (3) what canbe learned as a result of the experiences. In thefollowing sections, I will attend to each of thesecategories.

Types of ActivitiesFor each school site, the university organizers

provide two activities in addition to those the pre-service teachers present. The first activity uses jarscontaining snacks that are taken to the school a weekprior to the FMFN. Jars of varying shape are used fordifferent age levels. Each class within an age levelestimates the number of snacks in the jar and recordsits estimate. During the FMFN, individual students orparents can make estimates and enter them for aparticular class. The class with the closest estimatereceives the snacks. This activity serves two purposes.The first purpose is to generate interest in andawareness of the event and encourage participation.The second purpose is to support NCTM’s efforts(NCTM, 2000) by emphasizing estimation skills. Thesecond activity provided by the university requires aschool representative to greet children and parents atthe door and ask them to add a sticker to his or herbirth month on a pre-designed bar graph. This helpstake attendance for the evening and also helps childrensee the process of data collection and how a graphevolves from the process.

Preservice teachers design all other activities, andtheir activities must involve mathematics conceptscovered in their math classes (Math For theElementary Teachers I—numeration, whole numberand fraction operations, problem solving; Math For theElementary Teacher II—geometry, measurement,probability, and statistics). The types of projects thepreservice teachers choose to present usually fall intothe categories of drill and practice, problem solving, orestimation. I will discuss types of activities that fallinto each category and then discuss two exceptionalactivities that do not fall into any of the threecategories.

Drill and PracticeAlthough students are charged (and monitored) to

do more than BINGO or flash cards as the essence ofthe activity, drill and practice may be part of theactivity. Pre-service teachers’ initial attempts atcreating these activities are generally weak but withcoaching or feedback, they develop more thought-provoking activities. Rich activities designed to

incorporate drill and practice are usually presented inthe context of a game. For example, one student groupused a plastic bowling set to practice: addition andsubtraction facts with younger children, how to keep arunning total with slightly older children, and how toidentify fractions and percents for upper elementarychildren. Other examples of drill and practice activitiesare educational video games in which correct responseshelp students reach a goal (fuel for the spacecraft,money to buy souvenirs, moving closer to a target,etc.). These activities allow children to pick thedifficulty of the task and move through different levelsof calculation, building their self-confidence andknowledge. We encourage preservice teachers tobroaden their activities to include topics such asgeometry, estimation, logic, patterns, graphinterpretation, and computation since all are importantto review. Board games are yet another way to supportdrill and practice activity. The board is laid out on thefloor so that students walk around it landing on spaces.When a student is on a space, he or she is asked amathematics question that varies depending on the ageof the student.

Problem Solving Examples of problem solving activities are games

from which preservice teachers create adaptations.Preservice teachers like to challenge themselves withgames that incorporate mathematical ideas and skillsand then adapt them to the skill level of the children.Adaptations of games such as Yahtzee® Equations® or24® help children plan and carry out differentstrategies. Memory games, similar to Concentration®

are used to match fractions to decimals, operations toresults, or various representations of numbers. Thesegames1 are inexpensive to produce, easy to explain,and easily adaptable for different ages and grade levels.

A second example of a problem solving activity isasking children to identify or copy patterns in beads,pictures, tessellations, or shapes. Bead stringing iscommonly used to demonstrate patterns. The youngestchildren describe and extend simple patterns whilesomewhat older children choose a preset pattern andstring beads to illustrate the pattern. The oldest groupof students designs bead strings that contain multiplepatterns such as combining patterns of color withpatterns of shape or size. This activity is moreexpensive because children keep the materials they useto make the bracelets or necklaces.

Melissa R. Frieberg 37

EstimationIn addition to the introductory snack estimation

activity, almost every FMFN has at least onepreservice teacher designed activity that asks childrento estimate capacity, weight, area, and/or quantity. Onepopular activity requires children to estimate throughthe use of indirect measurement. In this activity, thereare approximately 15 objects to measure and thecharacteristics of objects vary in difficulty according tochildren’s differing abilities.

In recent years, we have seen a growing number ofactivities that use estimation to help students developprobability concepts. These activities illustrate ourpreservice teachers’ increased awareness of theimportance of estimation and probability as well astheir increased confidence in students’ abilities to dosuch activities. In these activities, children are askedhow frequently an event happens or how close anestimated answer is to the correct solution.Exceptional Activities

Two exceptional activities from the past do not fallinto any of the above categories. They are exceptionalbecause they are unique and demonstrate the creativityof the preservice teachers who made them. The firstwas presented in one of the first FMFNs we ran.Preservice teachers, with the help of the students, usedmath symbols to represent letters of each child's nameon a nametag. Children were then told to see if theycould figure out other people’s names by equating theletter of a name with a math symbol. For example,Anne's name might be + φ ≠ = (add, null set, not equal,equal) and she would then know the math symbol thatcorresponded to the letter “a”, “n”, and “e” and coulduse this to deduce the names of other people.

The second exceptional activity had three picturesmade up of geometric shapes. Children were given apaper shape and asked to match their paper shape withthe shape in one of the pictures. The youngest childrenhad shapes that were congruent to shapes in thepicture, while older students were asked to find shapessimilar to their shape but that differed in size, color, ororientation. The preservice teachers prompted childrento name the shape and describe its attributes. Thisactivity proved quite challenging for children but wasextremely popular.

Coordinating an FMFNIn organizing FMFNs, we have discovered that

communication among all the parties involved isessential. We have developed guidelines and a timelineto facilitate communication, to give schools and

preservice teachers a clear understanding ofexpectations, and to detail past problems we havefaced. Since incorporating FMFN into our curriculum,we have identified objectives and assessments assuringthat FMFN activities are mathematically sound (seeAppendix A and Form A). The most frequent problemswe encounter revolve around logistics such ascoordinating transportation to schools, advertising theevent in the community, and setting up the schoolspace. The following steps are used to conduct ourFMFN events and might be helpful for those who wantto organize similar work:

1. Contact schools that might be interested in hostingthe event. We contact school districts through directmailings or use various connections our department hasto area schools. After several years of conducting threeFMFNs each semester, most schools contact us toschedule the event.

2. Information about FMFN is given to our preserviceteachers with their class syllabus. The preserviceteachers are allowed to choose the topic around whichthey will make their activity (within guidelinesmentioned earlier). Groups may be made up of studentsfrom different classes requiring FMFN or from classesthat offer it as an optional activity.

3. The preservice teachers turn in a description of theiractivities (see Form A and Evaluation Form) indicatinghow it will be adjusted for various ages/grades, howparents will be involved, and how they will assess thesuccess of their activities. This allows the faculty toassess the activities for mathematical integrity andavoid redundancies in activities. It also gives students afoundation for writing their reflections on the event(See number 8).

4. We assign our preservice teachers to specific datesand schools based on preferences and class schedules.Groups are usually made up of three to four people andabout twenty groups are assigned to each school.

5. We confirm who is assigned to each school andallow groups to indicate special set-up needs (see FormB). The preservice teachers indicate if they are able toprovide transportation to the schools so car pools canbe established.

6. A faculty member or preservice teacher visits eachschool to determine space and resource availability, todiscuss the role of the school staff, and to givesuggestions for advertising the event. We suggest theschool connect FMFN with a regularly scheduled PTAmeeting. Sending reminders home with school

38 Family Math

children, having the event on the school calendar, andwriting an article in the school or local newsletterexplaining the event are ways that have been effectivein bringing FMFN to the attention of parents.

7. The night of the event, university and schoolpersonnel monitor the preservice teacher groups andthe families attending. At the close of the event, weannounce the winning class for the estimation exerciseand leave activity kits at the school for classroom use.

8. Each university preservice teacher group turns in awritten reflection of impressions of the event. Thisreport is not only helpful in assessing the universitystudents' learning, but also helps us identify problemsthat might need to be addressed in the future. Thisreport focuses on the content and success of theactivity, how students handled problems and questionsthat arose, how students interacted with parents andteachers, and how they collaborated with their groups.

9. An individual report is also required from eachpreservice teacher. This report is focused on how thestudent felt the group process worked, what waslearned about mathematics, and a self-reflection aboutone's ability as a teacher.

Conclusions In the introduction, we stated that we found this

activity to be beneficial to university preserviceteachers, university faculty and staff, school staff,parents, and especially children. Although this paper isnot intended to present a research study on FMFN, webelieve that we have seen beneficial results for thoseinvolved.

The university students have consistently, andalmost unanimously, responded positively to theirparticipation in FMFN both in their reports and in classdiscussions. Even students who described themselvesas poor math students found the experience to beenjoyable and uplifting. They appreciated the chance towork with a small group of elementary students. Asone student said, "I found that helping them [thechildren] out with solving a problem was an interestingand rewarding experience...this is what teaching is allabout." Many university students were especiallysurprised and buoyed by the fact that they were able toadjust questions, offer hints and assistance, or explainmathematical ideas more easily than they anticipated.They also learned how to share responsibilities, ask forhelp, and make changes to their activities as needed.Too often preservice teachers believe these things are asign of weakness rather than a sign of collaboration.FMFN helps change that perception.

One of the most rewarding results of thisexperience for the university faculty and staff is theopportunity to work collaboratively across departmentsand colleges. College of Education faculty/staff whoteach the elementary mathematics methods coursesassist faculty and staff from the MathematicsDepartment in planning, implementing, assessing, andrevising the program. Additionally, the experienceprovides an opportunity for the MathematicsDepartment members to visit local elementary schoolswith teachers and children. Education faculty and staffwho do regular supervision of student teachers inschools get to see students' abilities to teach to avariety of ages and abilities, which requires flexibilityand instant adaptations that might be missed in singlegrade level settings.

Teachers, administrators, and parents are effusivein their praise for the event. The university studentsmention that they often have classroom teacherswaiting “like vultures” to pick up the activity at the endof the night. Alternately, classroom teachers giveuniversity students ideas for improving or adapting theprojects for different children’s needs or abilities.Administrators find that the turnout for this event ishigher than for other school sponsored programs and,interestingly, draws more fathers. We average about200 participants at each event, even in schools wherethere are fewer than 300 students.

Parents have a varied level of involvement inactivities from merely standing and waiting to sittingdown and participating with their children in theactivity. Many times parents mention that they aresurprised at how well their children performed on agiven task or how well they thought through aproblem. In rare instances parents appear to beimpatient or negative with respect to their children’sefforts, and the university students get their first chanceto try out their mediating skills. Although not a benefitto children, parent outbursts do give university studentsan opportunity to see how parents influence children’slearning.

Most importantly, it appears that the elementarychildren who attend FMFN come away satisfied.University and school faculty have observed thatstudents almost universally leave the event feelingsuccessful and empowered in math. Certainly childrenfearful in math are less likely to attend, but we havewatched children start out very tentatively and soonfind themselves immersed in an activity. Virtually allthe children at each event try every activity, but theyreturn to certain activities—and these are rarely theeasiest activities. This behavior indicates that students

Melissa R. Frieberg 39

are motivated by activities that challenge them andmake them think rather than simple mastery.

In conclusion, we have found that all of us havegained from the experiences. As university instructorswe continually need to listen to our students in order toadapt and refine the expectations and requirements forFMFN. As prospective mathematics teachers, ourstudents have the chance to devise and carry outactivities in a low-stress, supportive atmosphere.

Schools and teachers are provided with examplesof activities that complement classroom instruction.Parents see how their children’s active involvement inactivities enhances their learning, and parents maycome away with a better understanding of themathematics curriculum. Finally, children always seemto walk away feeling successful and eager to move onto the next level in mathematics.

REFERENCESCarlson, C. G. (1991). Getting parents involved in their children’s

education. Education Digest, 57(10), 10–12.Kyle, D. W., McIntyre, E., & Moore, G. H. (2001). Connecting

mathematics instruction with the families of young children.Teaching Children Mathematics, 8, 80–86.

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

Pagni, D. (2002). Mathematics outside of schools. TeachingChildren Mathematics, 9, 75–78.

Stenmark, J. K., Thompson, V., & Cossey, R. (1986). Family math.Berkeley: University of California-Berkeley,

Wood, J. (Ed.). (1992). Variations on a theme: Family math night.Curriculum review, 32(2), 10. Retrieved May 17, 2004, fromGalileo database (ISSN 0147-2453; No. 9705276559).

Wood, J. (Ed.). (1991). “Family math” teaches English as well asmath. Curriculum review, 31(1), 21. Retrieved May 17, 2004,from Galileo database (ISSN 0147-2453; No. 9705223330).

Internet site for student information on FMFN:http://facstaff.uww.edu/whitmorr/whitmore/FMFN.html

1Equations® is a game in which a specific number of cards aredrawn. The cards have whole numbers on them, and students are toarrange the cards and determine operations that will create anequation. 24® is a similar game in which each card has four wholenumbers that, when using different operations on the numbers, willequal 24. Concentration® is a game in which a set of cards is placedface down in an array, and players take turns turning up two cardsat a time looking for pairs. In commercially made games these areusually identical pictures; however, in educational games these maybe two equivalent numbers using different symbols orrepresentations.

Family Math Fun Night Project RequirementsYour grade for this project is based on 80 points. The numbers

following the due dates below indicate the points that can be earnedon each portion of the project.Jan 31 (5 points) Form A – Group Membership and Activity IdeaHand in one copy of Form A to each instructor of members of yourgroup. Your Group ID Code will be assigned when returned.February 17 for District #1 (School A), February 19 for all others(20 points) Activity DescriptionTyped descriptions of your FMFN project should include:Names of group members with leader indicated, Group ID Code,name of activity, date and location of presentation.Procedures and/or instructions you will be giving for the activity.What the child is to do and learn from your activity? Includesample problems and activities for each level.If adults accompany children at the event, how will the adultparticipate in your activity?If prizes are used, how will they be awarded? Who will supply theprizes?How will you evaluate different aspects of your activity? Refer tothe attached evaluation sheet used by faculty and questions listedbelow.Feb 24 (School A), Feb 26 (School B), March 5 (School C), March12 (School D) Form B – Needs ListHand in one copy of Form B to each instructor of members of yourgroup.Mar 7 (School A), March 17 (School B), March 31 (School C),April 7 (School D) (5 pts) FMFN Evaluation FormHand in two copies of FMFN Evaluation Form to your groupleader’s instructor with answers completed for the questions on theright side of the form.Mar 11 (School A), March 20 (School B), April 3 (School C), April10 (School D) (30 points) FMFN EventRun your activity (6:30-8:00 p.m. in School A & B, 6:00 to 7:30 inSchool D) and have fun.Arrive at school 30 to 60 minutes prior to start.Set up your activity.Try to find time to visit and play the activities of other groupsduring the evening.Mar 19 (School A), Apr. 4 (School B), April 11 (School C), April16 (School D) (20 points) Individual EvaluationSorry, no evaluations will be returned until all evaluations havebeen graded.Your individual evaluation of the group learning activity (2 to 3pages) should include:Your name, Group ID Code, your activity name, school attended,group members’ names and their instructor, if other than yourinstructor, and method(s) used to evaluate your activity.Did your group work well together? Why or why not? How welldid you work within your group? What part of the project did youdo?Briefly state what the math concepts were that you were integratinginto your activity. Was this activity an effective means to conveythese concepts to the student? How could your activity be adaptedfor use in a classroom?What strategies did you see students use? What strategies did youuse to help them succeed?

40 Family Math

Did things go as planned during FMFN? What did you notanticipate?How did you modify/adjust your activity during the evening tomeet the needs of the students/parents? Include specific examplesof difficulties and adaptations.What would you do differently if you did a similar activity again?What did you learn about yourself and the grade(s) you areplanning to teach? Is teaching at this level still your goal? Why orwhy not?Grammar and other English mechanics will count.

Form AGroup Membership and Activity Idea (Spring 2003)Due: Friday, January 31, 2003Value: 5 pointAssigned Group ID Code:Please turn in one copy of this form to each teacher of a member ofyour group. (Group ID Code will be assigned after you submitForm A. Use it on all subsequent submissions.)Materials will be returned the group via the leader.Group Leader’s Name, Phone, Email Address, Course/Section,Teacher's Name, Other Members’ Names:Brief Activity Description:Indicate your choice for FMFN presentation. Consider eveningclasses, sports schedules, previous commitments, and workschedules of all members of the group in making your selections. Ifyour group requires a particular time, please explain thecircumstances. You will not be allowed to switch assignments afterthey have been made unless you can find a group able to exchangewith you.___Our group has no preference of night presentation; any nightwill work for us.___Our group would prefer the following nights: (Please circle firstand second choices, and give reasons in space to the right.)Does your group have transportation for FMFN? yes noCould your group provide transportation for others theevening of FMFN? yes no

Form BFamily Math Fun Night: Needs ListInstructor(s) Group ID Code Due:Please turn in one copy of this form to each teacher of a member ofyour group.Name of Activity:Brief description of activity:Group Leader:Other group members:Things you may need for your activity:Tables - Limit your project to one table. These may be lunch tableswith attached benchesChairs - remember most elementary teachers do not sit downTape, scissors, pencils, paper, scrap paper, markers, etc - pleasebring your own !!Our group will need to have (please indicate how many)Table (zero or one): Chairs:

If the table has attached benches, our group will need only __additional chairs.Our group would also like the following to be supplied by the hostschool:Our group would prefer to be located (please check one and giveyour reasoning in the space to the right)

___so we can hang things on a wall behind us___in a corner of the room___in the center of the room___it doesn't matter___near a power source

Our group (please check one)___doesn't plan to use prizes___will supply its own prizes___is counting on having the school supply prizes

Appendix AThis semester you will work with elementary students and their

parents/guardians in a project called Family Math Fun Night(FMFN). This project is designed to show children and parents thatmathematics is an essential part of their everyday life and can beFUN!! Most importantly, it provides the opportunity for you to beinvolved with elementary children as they do mathematics inenjoyable problem solving activities.

As a member of a group, you will be presenting an activity forFamily Math Fun Night (FMFN) at one of four elementary schools:School A (PK-5, 300 students); School B (K-5, 280 students; andSchool C (K-3, 400 students). All children from these schools andtheir families will be invited to attend from 6:30 - 8:00 p.m. (6:00 to7:30 in one school). The fourth elementary presentation is from 1:30to 3:00 at School D. We will run all events like a carnival havingbooths (tables) set up with various activities. There will not be awhole group presentation. You should plan to be at your school atleast one half hour early. This will allow you time to set up youractivity, and to visit and enjoy the activities of other groups beforethe children and parents arrive. You should be cleaned-up and out ofthe school 30 minutes after the closing time.

FORMING GROUPS. Who will design the activities for thiscarnival? Your group will select, make, and present your activity atFMFN. Form a group of four; a group with 3 or 5 students must beapproved by your instructor(s). Group members may be from anysection of the course you are taking. As you are selecting groups,think about class and work schedules for all of the members of yourgroup: work on this project will be done outside of class. Also, besure that each member of your group can be at the school to presentthe activity. You may indicate your group's preference for eveningof presentation. VERY IMPORTANT: After groups have beenassigned an evening, you will not be able to change assignmentsunless you can find a group willing to switch with you.

SELECTING AN ACTIVITY. Your group should select anactivity that is accessible and meaningful to the full range ofstudents in attendance. If you are unsure what is taught at variousgrade levels, do some research in the LMC on the lower level of thelibrary. Your activity should be fun and challenging for studentsand parents and need not be competitive. It should involve problemsolving, not merely mechanics or facts. Flash card type drill is notusually fun, and is not appropriate for a FMFN activity. Be sure toinvolve parents in your activity; parents should be doing not justwatching. “Helping by giving hints and encouragement” is notsufficient adult involvement. Your activity will need to be plannedwith space limitations in mind. Plan on setting up on one six toeight-foot table. Please also realize there will be about twentyactivities in a gym-sized room; consider how your activity and its

Melissa R. Frieberg 41

sounds and lights will affect others. You are not to present anactivity with music, popping balloons or other distractions forneighboring groups. Be aware of copyright laws! For example, thelatest cartoon characters may attract elementary students, but maybe an infringement of copyright. Invent a clone! Be creative! Don'tjust take an activity from a book or off a shelf; put something ofyourself into it. Don’t just use the activity you, or a friend, used lastsemester. Math 148 and 149 students should develop an activitythat involves math topics they will be covering in class. Realizingthat there are many connections between the mathematics in thetwo courses, this does not exclude presenting a topic from yourcourse in an activity that also uses a topic from the other course.Take this opportunity to develop an activity you could use in yourfuture classroom. Please do not use TWISTER activities.

The book Family Math has been placed on reserve (2 hour, noovernight) in the library. You will need to ask for it by name at themain circulation desk. This book has over 100 Family Mathactivities. You may wish to use one of these, combine a couple,modify one, or come up with an idea on your own. You could alsocheck Teaching Children Mathematics, other periodicals, and theInternet for ideas. Make this a fun learning experience for you!

WHAT YOU WILL NEED. Your group must have a sign withthe name of your activity. You may need to make some equipmentto be used at your booth such as markers, counters, game board,etc. Other things such as pencils, scissors, ruler, scrap paper, andmanipulatives are also useful. The LMC has some equipment thatcan be checked out. If they cannot meet your needs, your instructormay have some ideas. You may also want to have copies ofhandouts, problems, or puzzles available for parents/teachers totake home. Remember these are activities for the children andparents, so make sure they have plenty to DO.

If you feel that prizes would be appropriate for your activity,please indicate this on Form B that is due 2 weeks before yourFMFN. The PTO's of the various schools have given us somemoney with which to purchase small prizes - pencils, erasers,stickers, etc. These will be divided among the groups requestingthem. There will not be a large number of prizes per group. Pleaselimit the candy your group plans to use; not all children are allowedcandy, especially after supper. Many groups in the past havepresented very successful activities without prizes. Do not spend alot of money purchasing prizes. The students should be having fundoing math -- NOT seeing who can accumulate the most/bestprizes!

EVALUATION. Three-quarters of your grade will be assignedthrough group work. If your group contains members from morethan one class, some written work must be submitted to eachinstructor involved. Your group will supply two copies of theFMFN Evaluation Form a few days prior to your activity night. Acopy of this form is attached. On the night of your presentation,faculty attending FMFN will evaluate your project. A week afteryour FMFN, a typed individual reflective evaluation is to besubmitted. Select a method to help you evaluate your activity. Youmay get written evaluations from students and parents; keep ajournal of student/parent reactions during the evening, etc.

March 11, the first FMFN, is only SEVEN weeks away! It istime to get started selecting a group and an activity NOW. Thedeadline for forming groups and selecting an activity is January31st.

Evaluation FormSubmit two copies to your group leader’s instructorActivity Date: Instructor(s):Activity Name:Group Members:Faculty evaluators will use the following portion (and rate between1 and 5).Math content: (Problem solving, concept development, more thanmechanics)Adaptability of project: (Grade level, special needs, mental, writtenand manipulative capabilities)Materials: (Quality, durability and economy of materials)Appeal and Creativity: (Attract and retain participants)Interaction: (With students and adults, where possible)Professionalism: (Dress, group demeanor, setup on time,enthusiasm)Total Points (out of 30):Average number of points:(Based on evaluations)Groups are to provide the following information in the spaceprovided:Describe your activity’s math content and how you emphasized it.How did you adapt your activity to meet all students’ capabilities?Describe the quality, durability and economy of yourmaterials.

The Mathematics Educator2004, Vol. 14, No. 1, 42–46

Sybilla Beckman 42

Solving Algebra and Other Story Problems with SimpleDiagrams: a Method Demonstrated in

Grade 4–6 Texts Used in SingaporeSybilla Beckmann

Out of the 38 nations studied in the 1999 Trends in International Mathematics and Science Study (TIMSS),children in Singapore scored highest in mathematics (National Center for Education Statistics, NCES, 2003).Why do Singapore’s children do so well in mathematics? The reasons are undoubtedly complex and involvesocial aspects. However, the mathematics texts used in Singapore present some interesting, accessible problem-solving methods, which help children solve problems in ways that are sensible and intuitive. Could the textsused in Singapore be a significant factor in children’s mathematics achievement? There are some reasons tobelieve so. In this article, I give reasons for studying the way mathematics is presented in the elementarymathematics texts used in Singapore; show some of the mathematics problems presented in these texts and thesimple diagrams that accompany these problems as sense-making aids; and present data from TIMSS indicatingthat children in Singapore are proficient problem solvers who far outperform U.S. children in problem-solving.

Why Study the Methods of Singapore’sMathematics Texts?

What is special about the elementary mathematicstexts used in Singapore? These texts look verydifferent from major elementary school mathematicstexts used in the U.S. The presentation of mathematicsin Singapore’s elementary texts is direct and brief.Words are used sparingly, but even so, problemssometimes have complex sentence structures. The pagelayout is clean and uncluttered. Perhaps the moststriking feature is the heavy use of pictures anddiagrams to present material succinctly—althoughpictures are never used for embellishment. Simplepictures and diagrams accompany many problems, andthe same types of pictures and diagrams are usedrepeatedly, as supports for different types of problems,and across grade levels. These simple pictures anddiagrams are not mere procedural aids designed to helpchildren produce speedy solutions withoutunderstanding. Rather, the pictures and diagramsappear to be designed to help children make sense ofproblems and to use solution strategies that can bejustified on solid conceptual grounds. Because of thispictorial, sense-making approach, the elementary texts

used in Singapore can include problems that are quitecomplex and advanced. Children can reasonably beexpected to solve these problems given the problem-solving and sense-making tools they have beenexposed to.

Thus the strong performance of Singapore’schildren in mathematics may be due in part to the waymathematics is presented in their textbooks, includingthe way simple pictures and diagrams are used tocommunicate mathematical ideas and to provide sense-making aids for solving problems. If so, then teachers,mathematics educators, and instructional designers inthe U.S. will benefit from studying the presentation ofmathematics in Singapore’s textbooks, so that they canhelp children in the U.S. improve their understandingof mathematics and their ability to solve problems.

Using Strip Diagrams to Solve StoryProblems

One of the most interesting aspects of theelementary school mathematics texts and workbooksused in Singapore (Curriculum Planning andDevelopment Division, Ministry of Education,Singapore, 1999, hereafter referred to as PrimaryMathematics and Primary Mathematics Workbook) isthe repeated use of a few simple types of diagrams toaid in solving problems. Starting in volume 3A, whichis used in the first half of 3rd grade, simple “stripdiagrams” accompany a variety of story problems.Consider the following 3rd grade subtraction storyproblem:

Sybilla Beckmann is a mathematician at the University ofGeorgia who has a strong interest in education. She hasdeveloped three mathematics content courses for prospectiveelementary teachers and has written a textbook, Mathematics forElementary Teachers, published by Addison-Wesley, for use insuch courses. In the 2004/2005 academic year, she will teach aclass of 6th grade mathematics daily at a local public middleschool.

Sybilla Beckman 43

Mary made 686 biscuits. She sold some of them. If298 were left over, how many biscuits did she sell?(Primary Mathematics volume 3A, page 20,problem 4)

The problem is accompanied by a strip diagram likethe one shown in Figure 1.

Figure 1: How Many Biscuits Were Sold?

On the next page in volume 3A is the followingproblem:

Meilin saved $184. She saved $63 more than Betty.How much did Betty save? (Primary Mathematicsvolume 3A, page 21, problem 7)

This problem is accompanied by a strip diagram likethe one in Figure 2.

Figure 2: How Much Did Betty Save?

These two problems are examples of some of themore difficult types of subtraction story problems forchildren. The first problem is difficult because we musttake an unknown number of biscuits away from theinitial number of biscuits. This problem is of the typechange-take-from, unknown change (see Fuson, 2003,for a discussion of the classification of addition andsubtraction story problems). The second problem isdifficult because it includes the phrase “$63 morethan,” which may prompt children to add $63 ratherthan subtract it. This problem is of type compare,inconsistent (see Fuson, 2003). The term inconsistent isused because the phrase “more than” is inconsistentwith the required subtraction. Other linguisticallydifficult problems, including those that involve amultiplicative comparison with a phrase such as “Ntimes as many as”, are common in P r i m a r yMathematics and are often supported with a stripdiagram. Consider the following 3rd grade problem,which is supported with a diagram like the one inFigure 3:

A farmer has 7 ducks. He has 5 times as manychickens as ducks….How many more chickensthan ducks does he have? (Primary Mathematicsvolume 3A, page 46, problem 4)

(Note: The first part of the problem asks how manychickens there are in all, hence the question mark aboutall the chickens in Figure 3 below.)

Figure 3: How Many More Chickens Than Ducks?

Although the strip diagrams will not always helpchildren carry out the required calculations (forexample, we don’t see how to carry out the subtraction$184 – $63 from Figure 2), they are clearly designed tohelp children decide which operations to use. Insteadof relying on superficial and unreliable clues like keywords, the simple visual diagram can help childrenunderstand why the appropriate operations make sense.The diagram prompts children to choose theappropriate operations on solid conceptual grounds.

From volume 3A onward, strip diagrams regularlyaccompany some of the addition, subtraction,multiplication, division, fraction, and decimal storyproblems. Other problems that could be solved with theaid of a strip diagram do not have an accompanyingdiagram and do not mention drawing a diagram.Fraction problems, such as the following 4th gradeproblem, are naturally modeled with strip diagramssuch as the accompanying diagram in Figure 4:

David spent 2/5 of his money on a storybook. Thestorybook cost $20. How much money did he haveat first? (Primary Mathematics volume 4A, page62, problem 11)

Without a diagram, the problem becomes muchmore difficult to solve. We could formulate it with theequation (2/5)x = 20 where x stands for David’soriginal amount of money, which we can solve bydividing 20 by 2/5. Notice that the diagram can help ussee why we should divide fractions by multiplying bythe reciprocal of the divisor. When we solve theproblem with the aid of the diagram, we first divide$20 by 2, and then we multiply the result by 5. In otherwords, we multiply $20 by 5/2, the reciprocal of 2/5.

44 Solving Problems with Simple Diagrams

Figure 4: How Much Money Did David Have?

The problems presented previously are arithmeticproblems, even though we could also formulate andsolve these problems algebraically with equations. Butstarting with volume 4A, which is used in the first halfof 4th grade, algebra story problems begin to appear.Consider the following problems:

1. 300 children are divided into two groups. Thereare 50 more children in the first group than in thesecond group. How many children are there in thesecond group? (Primary Mathematics volume 4A,page 40, problem 8)

2. The difference between two numbers is 2184. Ifthe bigger number is 3 times the smaller number,find the sum of the two numbers. (P r i m a r yMathematics volume 4A, page 40, problem 9)

3. 3000 exercise books are arranged into 3 piles.The fist pile has 10 more books than the secondpile. The number of books in the second pile istwice the number of books in the third pile. Howmany books are there in the third pile? (PrimaryMathematics volume 4A, page 41, problem 10)

These problems are readily formulated and solvedalgebraically with equations, but since the text has notintroduced equations with variables, the children arepresumably expected to draw diagrams to help themsolve these problems. Notice that from an algebraicpoint of view, the second problem is most naturallyformulated with two linear equations in two unknowns,and yet 4th graders can solve this problem.

The 5th grade Primary Mathematics texts andworkbooks include many algebra story problems whichare to be solved with the aid of strip diagrams. Somedo not have accompanying diagrams, but others do,and some include a number of prompts, such as adiagram like the one in Figure 5 which accompaniesthe following problem:

Raju and Samy shared $410 between them. Rajureceived $100 more than Samy. How much moneydid Samy receive? (Primary Mathematics volume5A, page 23, problem 1)

Figure 5: Raju and Samy Split Some Money

Notice that the manipulations we perform withstrip diagrams usually correspond to the algebraicmanipulations we perform in solving the problemalgebraically. For example, to solve the previous Rajuand Samy problem, we could let S be Samy’s initialamount of money. Then,

2S + 100 = 410as we also see in Figure 5. When we solve the problemalgebraically, we subtract 100 from 410 and thendivide the resulting 310 by 2, just as we do when wesolve the problem with the aid of the strip diagram.

Strip diagrams make it possible for children whohave not studied algebra to attempt remarkablycomplex problems, such as the following two, whichare accompanied by diagrams like the ones in Figure 6and Figure 7 respectively:

Encik Hassan gave 2/5 of his money to his wifeand spent 1/2 of the remainder. If he had $300 left,how much money did he have at first? (PrimaryMathematics volume 5A, page 59, problem 6)

Raju had 3 times as much money as Gopal. AfterRaju spent $60 and Gopal spent $10, they each hadan equal amount of money left. How much moneydid Raju have at first? (Primary Mathematicsvolume 6B, page 67, problem 1)

Sybilla Beckman 45

Figure 6: How Much Money Did Encik Hassan Have atFirst?

Figure 7: How Much Did Raju Have at First?

Performance of 8th Graders on TIMSSIn light of the complex problems that children in

Singapore are taught how to solve in elementaryschool, the strong performance of Singapore’s 8thgraders on the TIMSS assessment is not surprising.Among the released TIMSS 8th grade assessmentitems in the content domain “Fractions and NumberSense” classified as “Investigating and SolvingProblems,” Singapore 8th graders scored higher thanU.S. 8th graders on all items. These released itemsincluded the following problems (see NCES, 2003):

Laura had $240. She spent 5/8 of it. How muchmoney did she have left? (Problem R14, page 29.Overall percent correct, Singapore: 78%, UnitedStates: 25%).

Penny had a bag of marbles. She gave one-third ofthem to Rebecca, and then one-fourth of theremaining marbles to John. Penny then had 24marbles left in the bag. How many marbles were inthe bag to start with?

A. 36

B. 48

C. 60

D. 96

(Problem N16, page 19. Overall percent correct,Singapore: 81%, United States: 41%)

These problems are similar to problems in PrimaryMathematics. The strong performance of Singapore 8thgraders on these problems indicates that the instructionchildren receive in solving these kinds of problems iseffective. Similarly, among the released TIMSS 8thgrade assessment items in the content domain“Algebra” classified as “Investigating and SolvingProblems,” Singapore 8th graders scored higher thanU.S. 8th graders on all items.

But the strong problem-solving abilities ofSingapore’s 8th graders in fractions and number senseand in algebra does not necessarily result in factualknowledge in other mathematical domains in which thechildren have not had instruction. For example, U.S.8th graders scored higher than Singapore 8th graderson the following item in the content domain “DataRepresentation, Analysis and Probability” classified as“Knowing”:

If a fair coin is tossed, the probability that it willland heads up is 1/2. In four successive tosses, afair coin lands heads up each time. What is likelyto happen when the coin is tossed a fifth time?

A. It is more likely to land tails up than heads up.

B. It is more likely to land heads up than tails up.

C. It is equally likely to land heads up or tails up.

D. More information is needed to answer thequestion.

(Problem F08, page 74. Overall percent correct,United States: 62%, Singapore: 48%)

The mathematics texts used in Singapore through8th grade do not address probability. Thus thedifference in performance in fraction, number sense,and algebra problem-solving versus knowledge aboutprobability can reasonably be attributed to effectiveinstruction.

46 Solving Problems with Simple Diagrams

ConclusionThe mathematics textbooks used in elementary

schools in Singapore show how to represent quantitieswith drawings of strips. With the aid of these simplestrip diagrams, children can use straightforwardreasoning to solve many challenging story problemsconceptually. The TIMSS 8th grade assessment showsthat 8th graders in Singapore are effective problemsolvers and are much better problem solvers than U.S.8th graders. Although cultural factors probably alsoaffect the strong mathematics performance of childrenin Singapore, children in the U.S. could probablystrengthen their problem-solving abilities by learningSingapore’s methods and by being exposed to morechallenging and linguistically complex story problemsearly in their mathematics education.

REFERENCESCurriculum Planning and Development Division, Ministry of

Education, Singapore (1999, 2000). Primary Mathematics (3rded.) volumes 1A–6B. Singapore: Times Media PrivateLimited. Note: additional copyright dates listed on books inthis series are 1981, 1982, 1983, 1984, 1985, 1992, 1993,1994, 1995, 1996, 1997, thus 8th graders who took the 1999TIMSS assessment used an edition of these books.

Curriculum Planning and Development Division, Ministry ofEducation, Singapore (1999, 2000). Primary MathematicsWorkbook (3rd ed.) volumes 1A–6B. Singapore: Times MediaPrivate Limited.

Fuson, K. C. (2003). Developing Mathematical Power in WholeNumber Operations. In J. Kilpatrick, W. G. Martin, and D.Schifter, (Eds.), A Research companion to principles andstandards for school mathematics (pp. 68–94). Reston,VA:National Council of Teachers of Mathematics.

National Center for Education Statistics (2003). Trends ininternational mathematics and science study. Retrieved May3, 2004, from http://nces.ed.gov/timss/results.asp and fromhttp://nces.ed.gov/timss/educators.asp

The Mathematics Educator2004, Vol. 14, No. 1, 47–51

47 Book Review

Amy Hackenberg is at work on her doctoral dissertation on theemergence of sixth graders’ algebraic reasoning from theirquantitative reasoning in the context of mathematically caringteacher-student relations. In addition to her fascination withmathematical learning and the orchestration of it, she iscompelled by issues of social justice, the nature andconsequences of social interaction, and the relationship betweenthe “social” and the “psychological” in mathematics education.

Book Review…Diverse Voices Call for Rethinking and Refining Notions of

EquityAmy J. Hackenberg

Burton, L. (Ed.). (2003). Which way social justice in mathematics education? Westport, CT:Praeger. 344 pp. ISBN 1-56750-680-1 (hb). $69.95.

Editor Leone Burton remarks that the title of thisbook reflects a “shift in focus from equity to a moreinclusive perspective that embraces social justice as acontested area of investigation within mathematicseducation” (p. xv). What’s interesting is that thequestion in the title lacks a verb—is the question“which are ways to social justice in mathematicseducation?” Or more tentatively, “which ways mightbring about social justice in mathematics education?”Or perhaps the focus is more on research, either up tonow or in the future: “which ways have research onsocial justice in mathematics education taken? Or“which ways could (should?) research on social justicein mathematics education take?” Each of the thirteenchapters in the volume addresses at least one of thosefour questions. Overall, this book responds to its titlequestion through diverse voices that call for expandingwork on gender issues into broader sociocultural,political, and technological contexts; rethinking andrefining key notions such as equity, citizenship, anddifference; and considering how to conduct studies thatreach beyond school and university boundaries towardfamilies, communities, and policy-makers.

The collection is the third volume in theInternational Perspectives on Mathematics Educationseries for which Burton has served as series editor.1 Inher introduction she describes the origin of the book inthe activities of the International Organization ofWomen in Mathematics Education (IOWME) at theNinth International Congress of Mathematics

Education (ICME9) in Tokyo, Japan, in 2000. Perhapsthis context explains why approximately half of thechapters focus primarily on gender, while otherchapters include issues related to differences in race,class, language, and thinking styles. Burton notes thatthis book, as the fourth publication of IOWME,“reflects the development of the group’s interests thathave evolved over 16 years from a sharp focus ongender issues to its present wider interest in socialjustice” (p. xiii).

In the introduction Burton also outlines the processby which the book developed. After a general call forpapers, an international review panel of mathematicseducators reviewed submissions. Chapter authors werethen paired to give feedback to each other on theirwork in order to promote dialogue as well as “cross-referencing possibilities” (p. xv). As perhaps is alwaysthe case in an edited book without summary pieces tohighlight connections between chapters, the cross-referencing of concepts in this volume could beexpanded. Burton does a nice job of drawing someconnections in her introduction, but otherwise suchresonance is largely left to the reader. Fortunately, as Ihope to demonstrate in this review, there is ampleopportunity to draw connections between chapters (andalso occasionally to wish that an author had heededanother author’s points or ideas!)

Organization of the BookThe thirteen chapters in the book are organized

into three sections. The four chapters in the firstsection focus on definitional work, conceptualframeworks, and reviews of and recommendations forresearch, thereby “setting the scene” (p. 1). The authorsof this section are from Australia (Brew), Germany(Jungwirth), the United Kingdom (Povey), and theUnited States (Hart). The second section consists ofseven chapters primarily about studies that take placein classrooms and address the question “what does

48 Book Review

social justice mean in classrooms?” (p. 101). Theauthors of this section are from Australia (Forgasz,Leder, and Thomas; Zevenbergen), Germany (Ferriand Kaiser), Malawi (Chamdimba), the United Statesand Peru (Secada, Cueto, and Andrade), and the UnitedKingdom (Mendick; Wiliam). The last section includestwo chapters focused specifically on “computers andmathematics learning” (p. 261) with regard to socialjustice. The authors (Wood, Viskic, and Petocz; Vale)come from Australia and Eastern Europe, but all nowpractice mathematics education in Australia.

The placement of chapters within this organizationis a little puzzling. Wiliam’s illuminating chapter onthe construction of statistical differences and itsimplications is included in the second section onclassroom studies, but since it grapples with definitionsand conceptual ideas (and is not a classroom study), itmight have been better placed in the first moretheoretically-oriented section. Brew’s chapter, a studyabout reasons that mothers return to studymathematics, is included in the first section but seemsto fit better in the second, despite the fact that the studydoes not take place in mathematics classrooms.Support for changing the placement of Brew’s chapteris provided by the position of Mendick’s: Her report ofyoung British men’s choices to study mathematicsbeyond compulsory schooling is only peripherallylocated in classrooms and was still placed in the secondsection.

The other weak organizational aspect of the bookis the inclusion of only two chapters in the third sectionon computers and mathematics learning. One wondersif there were intentions for a more substantial sectionbut some papers did not make the publication deadline.In any case, because both chapters in this section reporton studies set in classrooms, it seems that they couldhave been included in the second section—or thatperhaps two sections about studies might have beenwarranted, one that focused directly on studies inmathematics classrooms and one that included researchon mathematics education outside of immediateclassroom contexts.

Conceptually-Oriented Chapters: What Is Equity?What Is Social Justice?

Organizational difficulties aside, I focus first onthe more conceptually-oriented chapters, which arecontained in the first three chapters of the first sectionof the book as well as in Wiliam’s chapter from thesecond section. These authors engage in definitionaland conceptual work that forms a foundation forresearch on social justice. All four authors ponder the

nature of equity and justice within different contexts: atypology of gender-sensitive teaching, previous andcurrent research on equity and justice in mathematicseducation, citizenship education in the UnitedKingdom, and statistical analyses of gender differencesin mathematics education.

Jungwirth describes a typology of gender-sensitiveteaching that consists of three types distinguished bymodifications made according to gender, the degree towhich gender groups are identified and treated asmonolithic, and corresponding conceptions of equity.In Type I teaching, teachers are “gender-blind” andmake no modifications according to gender since theybelieve that boys and girls can do math equally well. InType II teaching, teachers adjust practices based ongender but tend to treat students of a single gender asmonolithic (i.e., tend to essentialize.) Jungwirthbelieves that in the third (and implicitly mostadvanced) type, the concept of equity “no longerapplies…Equity here refers to the individual, withrespect to learning arrangements and, somewhatqualified, to outcomes” (p. 16). Teachers engaging inType III teaching attend to individual differenceswithin gender groups and tailor teaching to individuals.

Although Jungwirth’s typology offers a conceptualframework for examining the equitable implications ofteachers’ orientations toward mathematics teaching andmathematics classrooms, her dichotomizing of groupsand individuals is problematic. For example, in theirattention to individuals, might not Type III teacherscreate classrooms in which mathematics could bedevoid of women, which Jungwirth sees asconsiderably less evolved than even Type I teaching?The problem seems to be in characterizing equitybased on group-individual dichotomies—to adhere toostrongly to group identities can result in essentializing,while to focus primarily on the individual can leave outtrends and broad characteristics of groups that areimportant considerations in work toward equity andsocial justice (cf. Lubienski, 2003).

These issues are reflected in Hart’s review ofscholarship on equity and justice in mathematicseducation over the last 25 years. Her chapter is notablefor explicit discussion about different ways researchershave used equity and justice (and equality); for herclearly stated choice to use equity to mean justice; andfor her formulation of calls for future research. Inparticular, she calls for research on pedagogies thatcontribute to justice; self-study of educators’ ownpractices; and more research that explores studentmotivation, socialization, identity, and agency withrespect to mathematics. Hart highlights Martin’s

Amy J. Hackenberg 49

(2000) study on factors contributing to failure andsuccess of African American students in mathematicsas an exemplar for future research because of itsmultilevel framework for analyzing mathematicssocialization and identity. Although her points abouthis work are well taken, the considerable space shegives to this relatively recent study seems odd givenher aims to review 25 years of research.

Povey continues Jungwirth’s and Hart’sdefinitional work by considering the complex andcontested notion of citizenship in relation to socialjustice and mathematics education. She describes howrecent mandates for citizenship education in Englandreinforce a conservative perspective by focusing onpolitical and legal citizenship (the right to vote, forexample), without questioning the nature and characterof social citizenship, let alone its connections to “the(mathematics) education of future citizens” (p. 52).Povey believes that for citizenship to be a usefulconcept in democratizing mathematics classrooms theconcept “will have to be more plural, more active, andmore concerned with participation in the here andnow” (p. 56).

Perhaps the strongest chapter of these four (andone of the strongest in the collection) is Wiliam’s onthe construction of statistical differences inmathematical assessments. He demonstrates that ingender research in mathematics education, effect sizesof standardized differences between male and femaletest scores are relatively small, and the variabilitywithin a gender is greater than between genders. Basedon this analysis, Wiliam concludes that differencesbetween genders depend on what counts asmathematics on assessments. In particular, what countsas mathematics may be maintained because it supportspatriarchal hegemony.

As an implication of his argument, Wiliamproposes “random justice” (p. 202) to produce equityin selection based on test scores. Wiliam calls thepercentage of the population that reaches a certainstandard (for, say, entrance to medical school) arecruitment population. Usually, selecting from arecruitment population (i.e., creating a selectionpopulation) involves choosing a small top percentageof it. This mode of selection perpetuates selecting moremales than females, largely because males showgreater variability in their test scores compared tofemales (males produce more highs and lows.) Wiliamproposes that a random sample of the recruitmentpopulation that sustains the gender (or racial, class,etc.) make-up of it is “the only fair way” (p. 204) ofcreating a selection population. Although this proposal

may seem counterintuitive (and certainly differs fromtypical U.S. selection processes!), Wiliam makes acompelling argument that is worth reading.

Chapters on Studies in or SurroundingMathematics Classrooms

In these chapters—Brew’s chapter from the firstsection as well as the other 8 chapters in the book—thediverse voices in the volume become quite apparent,not only because of the different geographical locationsor ethnic heritages of the authors but because of thediverse ways in which the authors focus on issues ofsocial justice in relation to mathematics classrooms andmathematical study. These nine chapters can also beloosely grouped as exemplifying, supporting,informing, or aligning with the more conceptually-oriented chapters.

In particular, two chapters that focus specificallyon teaching practices in relation to social justice mayexemplify and inform Jungwith’s typology. Theauthors of these chapters attend to how teachersapproach students who belong to disadvantagedgroups. Chamdimba, whose research took place in thesouthern African country of Malawi, studied the year11 students of a Malawian teacher who agreed to usecooperative learning to potentially promote a “learner-friendly classroom climate” (p. 156) for girls. As aresearcher, Chamdimba might exemplify a Type IIorientation out of her concerns over Malawian girls’lack of representation and achievement in mathematicsand subsequent Malawian women’s lack of bargainingpower as a group for social and economic resources inthe country. Chamdimba’s conclusion that femalestudents experienced largely positive effects mighthelp Jungwirth refine her typology so that recognizingstudents as part of disenfranchised groups and actingon that recognition to address the group is seen aslegitimate and useful (i.e., not necessarily less evolvedthan Type III teaching.) However, Chamdimba’s studyis also subject to scrutiny over whether a particularclassroom structure can bring about improvements inall Malawian females’ educational, social, andeconomic status.

Perhaps a better example of the subtlety involvedin the group-individual distinctions with regard tosocial justice is found in Zevenbergen’s study.Zevenbergen used Bourdieu’s tools as a frame forunderstanding teachers’ beliefs about students fromsocially disadvantaged backgrounds in the South-EastQueensland region of Australia. Eight of the 9 teachersinterviewed expressed views of students as deficientdue to poverty and cultural practices. Stretching

50 Book Review

Jungwirth’s typology beyond gender-sensitivity, theninth teacher had more of a Type III orientation in herrespect for these students as individuals. However, byexpressing an understanding of how parents’ lack ofcultural capital prevented them from challenging theways in which schools (under)served their children,this teacher did not ignore these students as belongingto a disadvantaged group. This teacher’s ability tounderstand and value students as both individuals andpart of a group might allow Jungwirth to amplify andfurther articulate her typology.

These two chapters and three others exhibit workthat aligns with Hart’s call for research on pedagogiesthat contribute to social justice and on one’s ownteaching in relation to social justice. Vale’s two casestudies of computer-intensive mathematics learning intwo junior secondary mathematics classrooms focus onhow teachers’ practices with technology impede (butmight facilitate) more just classroom environments.Vale’s work is complemented by the three universityclassroom studies presented by Wood, Viskic, andPetocz. In studying their own computer-intensiveteaching of differential equations, statistics, andpreparatory mathematics classes, these threeresearchers found positive attitudes toward the use oftechnology across gender. Finally, Ferri and Kaiser’scomparative case study on the styles of mathematicalthinking of year 9 and 10 students (ages 15-16) hasimplications for developing pedagogies that recognizedifferences other than due to gender, race, or class, andthat thereby contribute to justice and diversity inclassrooms.

However, Secada, Cueto, and Andrade’s large-scale, comprehensive study of the conditions ofschooling for fourth and fifth-grade children who speakAymara, Quechua, and Spanish in Peru may be thestrongest example of work toward Hart’srecommendation of multilevel frameworks in researchon social justice. These researchers intended to create a“policy-relevant study” (p. 106). To do so theyarticulated their conceptions of equity as distributivesocial justice (opportunity to learn mathematics is asocial good and should not be related to accidents ofbirth) and socially enlightened self-interest (it is ineveryone’s interest for everyone to do well so as not tocause great cost to society). In addition, the researcherstook as a premise that equity must come with both highquality and equality (i.e., lowering the bar does notfoster equity). Thus they contribute to definitionalwork while formulating “practical” conclusions andrecommendations for Peruvian governmental policy.

Finally, the remaining three chapters in the bookconnect with Povey’s chapter in exploring a particularcontested and complex concept or relate to Wiliam’swork on considering the construction of difference.Brew’s study entails rethinking aspects of the complexconcept of mothering in the context of mathematicallearning of both mothers and their children. Byincluding voices of the children in the study, Brew isable to show the fluid roles of care-taking betweenstudying mothers and their children (e.g., childrensometimes acted as carers for their mothers) and “thepivotal role that children can play…in providing notonly a consistent motivating factor but also enhancingtheir mother’s intellectual development” (p. 94).

What Povey does for citizenship and Brew does formothering, Mendick does for masculinity in thecontext of doing mathematics. In a very strong andthoughtful chapter, she describes stories of three youngBritish men who have opted to study mathematics intheir A-levels even though they do not enjoy it.Mendick’s smart use of a poststructuralist perspectivethat deconstructs the classic opposition betweenstructure and agency allows her to argue that taking upmathematics is a way for the men to “do masculinity”in a variety of ways: to prove their intelligence toemployers and others as well as to secure a future inlabor market. The stories of the three males prompt thequestion: “why is maths a more powerful proof ofability than other subjects?” (p. 182). To respond,Mendick contrasts the men’s stories with youngwomen’s stories (part of her larger research project.)

This artful move is not intended to drawdichotomies between how men and women “do maths”differently—Mendick cautions against such simplisticconclusions and notes that some females usemathematics the way these three males do. Instead thecontrast allows her to demonstrate and deepen hertheorizing of masculinity as a relational configurationof a practice, as well as to argue for more complexityin gender reform work. Thus for her, “maths andgender are mutually constitutive; maths reform work isgender reform work” (p. 184). By examining gender inthis way, like Wiliam, she calls into questiondifferences between males and females in relation tomathematics and supports his contention that whatcounts as mathematics (and, Mendick would add, asmasculine and feminine) is the basis for thesedifferences.

Differences between males and females are alsothe subject of the chapter by Fogasz, Leder, andThomas. They used a new survey instrument to capturethe beliefs of over 800 grade 7–10 Australian students

Amy J. Hackenberg 51

regarding gender stereotyping of mathematics. Theirfindings revealed interesting reversals of expected(stereotyped) beliefs. For example, their participantsbelieved that boys are more likely than girls to give upwhen they find a problem too difficult, and that girlsare more likely than boys to like math and find itinteresting. However, through an examination ofparticipation rates and achievement levels of male andfemale grade 12 mathematics students from 1994 to1999 in Victoria, Australia, the researchers refuterecent, media-hyped contentions (see, e.g., Conlin,2003; Weaver-Hightower, 2003) that males are nowdisadvantaged in mathematics. Frankly, Fogasz andcolleagues might have benefited from Wiliam’s adviceon examining effect size—it is hard to know how muchsignificance to give to the differences they found.Nevertheless, their work supports the notion thatmathematics may be maintained as a male domaindespite certain advances of females.

Overall, I agree with Burton that the chapters inthis volume achieve the goal of providing “anintroduction for new researchers as well as stimulationfor those seeking to develop their thinking in new orunfamiliar directions” (p. xiii). Although theorganization is a bit puzzling and some chapters areclearly stronger than others, the book is a useful readfor researchers in mathematics education. Moreimportant, the diversity of voices—and the connectionsthat readers can draw among this diversity—gives acomplex and layered picture of how resources,sociocultural contexts, governmental policy, teacherand student practices, human preferences andexpectations, and researchers’ theorizing andinterpretations, all contribute to “…who does, and whodoes not, become a learner of mathematics” (p. xviii).

REFERENCESConlin, M. (2003, May 26). The new gender gap. Business Week

online. Retrieved September 1, 2003, fromhttp://www.businessweek.com

Lubienski, S. T. (2003). Celebrating diversity and denyingdisparities: A critical assessment. Educational Researcher,32(8), 30–38.

Martin, D. B. (2000). Mathematics success and failure amongAfrican-American youth: The roles of sociohistorical context,community forces, school influence, and individual agency.Mahwah, NJ: Lawrence Erlbaum.

Weaver-Hightower, M. (2003). The “boy turn” in research ongender and education. Review of Educational Research, 73(4).471–498.

1 The first volume was Multiple Perspectives on MathematicsTeaching and Learning (2000) edited by Jo Boaler; the secondvolume was Researching Mathematics Classrooms: A CriticalExamination of Methodology (2002) edited by Simon Goodchildand Lyn English.

52 Book Review

CONFERENCES 2004…CMESG/GCEDM Universite Laval May 28–June 1Canadian Mathematics Education Study Group Quebec, Canadahttp://plato.acadiau.ca/courses/educ/reid/cmesg/cmesg.html

HIC Honolulu, Hawaii June 9–12The 3rd Annual Hawaii International Conference onStatistics, Mathematics and Related Fieldshttp://www.hicstatistics.org/index.htm

EDGE Symposium Atlanta, Georgia June 25–26Graduate School Experience for Women in Mathematics:From Assessment to Actionhttp://www.edgeforwomen.org/symposium.html

AMESA Potchefstroom, July 1–4Tenth Annual National Congress South Africahttp://www.sun.ac.za/MATHED/AMESA/AMESA2004/Index.htm

ICOTS7 Salvador, Brazil July 2–7International Conference on Teaching Statisticshttp://www.maths.otago.ac.nz/icots7/layout.php

ICME – 10 Copenhagen, Denmark July 4–11The 10th International Congress on Mathematics Educationhttp://www.icme-10.dk

HPM Uppsala, Sweden July 12–17History & Pedagogy of Mathematics Conferencehttp://www-conference.slu.se/hpm/about/

PME-28 Bergen, Norway July 14–18International Group for the Psychology of Mathematics Educationhttp://home.hia.no/~annebf/pme28/

JSM of the ASA Toronto, Canada August 8–12Joint Statistical Meetings of the American Statistical Associationhttp://www.amstat.org/meetings

CABRI 2004 Rome, Italy September 9–12Third CabriGeometry International Conferencehttp://italia2004.cabriworld.com/redazione/cabrieng2004

GCTM Rock Eagle, Georgia October 14–16GCTM Annual Conferencehttp://www.gctm.org/georgia_mathematics_conference.htm

PME-NA Toronto, Canada October 21–24North American ChapterInternational Group for the Psychology of Mathematics Educationhttp://www.pmena.org

SSMA College Park, Georgia October 21–23School Science and Mathematics Associationhttp://www.ssma.org

AAMT 2005 Sydney, Australia January 17–20Australian Association of Mathematics Teachers 2005http://www.aamt.edu.au/mmv

53

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In this Issue,

Guest Editorial… Do You Need a PhD to Teach K–8 Mathematics in Ways Respected bythe Mathematics Education Community?CHANDRA HAWLEY ORRILL

Mathematics as “Gate-Keeper” (?): Three Theoretical Perspectives that Aim TowardEmpowering All Children With a Key to the GateDAVID W. STINSON

The Characteristics of Mathematical CreativityBHARATH SRIRAMAN

Getting Everyone Involved in Family MathMELISSA R. FREIBERG

In Focus… Solving Algebra and Other Story Problems with Simple Diagrams: a MethodDemonstrated in Grade 4–6 Texts Used in SingaporeSYBILLA BECKMANN

Book Review… Diverse Voices Call for Rethinking and Refining Notions of EquityAMY J. HACKENBERG