v14n1

____ THE _____ MATHEMATICS ___ ________ EDUCATOR _____ Volume 14 Number 1

Spring 2004

MATHEMATICS EDUCATION STUDENT ASSOCIATION THE UNIVERSITY OF GEORGIA

Editorial Staff

A Note from the Editor

Editor Holly Garrett Anthony

Dear TME readers, Along with the editorial team, I present the first of two issues to be produced during my brief tenure as editor of Volume 14 of The Mathematics Educator. This issue showcases the work of both veteran and budding scholars in mathematics education. The articles range in topic and thus invite all those vested in mathematics education to read on.

Both David Stinson and Amy Hackenberg direct our attention toward equity and social justice in mathematics education. Stinson discusses the “gatekeeping” status of mathematics, offers theoretical perspectives he believes can change this, and motivates mathematics educators at all levels to rethink their roles in empowering students. Hackenberg’s review of Burton’s edited book, Which Way Social Justice in Mathematics Education? is both critical and engaging. She artfully draws connections across chapters 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 mathematics educators to step outside themselves and reexamine features of PhD programs and elementary textbooks. Orrill’s title question invites mathematics educators to consider what we value in classroom teaching, how we engage in and write about research on or with teachers, and what features of a PhD program can inform teacher education. Beckmann looks abroad to highlight simple diagrams used in Singapore elementary texts—that facilitate the development of students’ algebraic reasoning and problem 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 of mathematicians and the organization of rich experiences for preservice elementary teachers, respectively. Sriraman builds on creativity theory in his research to characterize the creative practices of five well-published mathematicians in the production of mathematics. Freiberg reminds us of the daily challenge of mathematics educators—providing preservice teachers rich classroom experiences—and details the organization, coordination, and evaluation of Family Math Fun Nights in elementary schools.

It has been my goal thus far to entice you to read what follows, but I now want to focus your attention on the work of TME. I invite and encourage TME readers to support our journal by getting involved. Please consider submitting manuscripts, reviewing articles, and writing abstracts for previously 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 rewarding process—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 this Fall. I extend my thanks to all of the people who make TME possible: reviewers, authors, peers, faculty, and especially, the editors.

Associate Editors Ginger Rhodes Margaret Sloan Erik Tillema Publication Stephen Bismarck Laurel Bleich Dennis Hembree Advisors Denise S. Mewborn Nicholas Oppong James W. Wilson

MESA Officers 2004-2005 President Zelha Tunç-Pekkan Vice-President Natasha Brewley Secretary Amy J. Hackenberg Treasurer Ginger Rhodes NCTM Representative Angel Abney

Holly Garrett Anthony

Undergraduate Representatives Erin Bernstein Erin Cain Jessica Ivey

105 Aderhold Hall The University of Georgia Athens, GA 30602-7124

tme@coe.uga.edu www.ugamesa.org

About the cover Cover artwork by Thomas E. Ricks. Fractal Worms I of the Seahorse Valley in the Mandelbrot Set, 2004.

For questions or comments, contact: tomricks@uga.edu Benoit Mandelbrot was the pioneer of fractal mathematics, and the famous Mandelbrot set is his namesake. Based on a simple iterative equation applied to the complex number plane, the Mandelbrot set provides an infinitely intricate and varied landscape for exploration. Visual images of the set and surrounding points are made 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 constitute the actual Mandelbrot set, customarily colored black, are points producing a finite value. The Mandelbrot set is a fractal structure, and one can see self-similar forms 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 microscope allowing 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 a lightboard, 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 both sheets of paper, he drew the various fractal worms following the general curve of the spirals. The printout was produced by a Mandelbrot set explorer software package 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 Association The 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 Ways Respected by the Mathematics Education Community?

CHANDRA HAWLEY ORRILL

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

19 The Characteristics of Mathematical Creativity BHARATH SRIRAMAN 35 Getting Everyone Involved in Family Math MELISSA R. FREIBERG 42 In Focus… Solving Algebra and Other Story Problems with Simple Diagrams: a Method Demonstrated in Grade 4–6 Texts Used in Singapore SYBILLA BECKMANN 47 Book Review… Diverse Voices Call for Rethinking and Refining Notions of Equity AMY J. HACKENBERG 52 Upcoming Conferences 53 Subscription Form 54 Submissions Information © 2004 Mathematics Education Student Association.

All Rights Reserved

The Mathematics Educator 2004, Vol. 14, No. 1, 2–7

Guest Editorial…

Do You Need a PhD to Teach K–8 Mathematics in Ways Respected by the Mathematics Education Community?

Chandra Hawley Orrill The genesis of this editorial was a conversation about an article in which Ball (1991) provided descriptions of three teachers’ approaches to working with their students. In Ball’s article, teachers without PhDs in mathematics or mathematics education struggled to engage their students in developing meaningful concepts of mathematics. They could not provide multiple interpretations of concepts—particularly representations that provided concrete explanations or tie-ins to the real world. They demonstrated only stepwise approaches to doing mathematics, clinging tightly to procedures and algorithms, and provided no evidence that they had a deeper understanding of the mathematics. In stark contrast, the same Ball article offered a vignette of Lampert’s teaching that illustrated a rich mathematical experience for students. Lampert provided multiple perspectives, introduced multiple representations, and demonstrated a deep understanding of both mathematics and student learning throughout the episode described.

Given the number of articles in the literature painting the ‘typical’ mathematical experience as one that is impoverished, and the growing body of literature written by PhD researcher-teachers, I wondered, “Do you need a PhD to teach elementary Chandra Hawley Orrill is a Research Scientist in the Learning and Performance Support Laboratory at the University of Georgia. Her research interest is teacher professional development with an emphasis on teaching in the midst of change. She is also interested in how professional development impacts the opportunities teachers create for student learning.

Acknowledgements The research reported here came from a variety of projects spanning six years. These projects were supported by grants from the Andrew W. Mellon Foundation and the Russell Sage Foundation, the National Science Foundation, Georgia’s Teacher Quality Program (formerly Georgia’s Eisenhower Higher Education Program), and the Office of the Vice President for Research at the University of Georgia. Opinions expressed here are my own and do not necessarily reflect those of the granting agencies. My thanks to Holly Anthony, Ernise Singleton, Peter Rich, Craig Shepherd, Laurel Bleich, and Drew Polly for their ongoing discussions with me about whether a teacher needs a PhD to teach K-8 mathematics.

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and middle school mathematics in ways that mathematics educators would value?” After all, the Balls, Lamperts, and McClains1 in the literature offer high-quality mathematics instruction, attend to student thinking, provide opportunities for knowledge construction, and introduce students to a variety of tools they can use later (e.g., visual representations and problem solving strategies). Further, these researcherteachers seem to have a gift for promoting student thinking and moving an entire class forward by scaffolding lessons, questioning students, and creating a classroom community where learners consider each other’s work critically and interact meaningfully. The reality, however, is that not all mathematics teachers have PhDs and it is unlikely that most ever will.2 In working through this question both with the graduate students with whom I work and in preparation for this editorial, I have developed some ideas both about researcher-teachers as a “special” group and about why having a PhD might matter. Based on my thoughts I would like to propose two conjectures about researcher-teacher efforts. First, I conjecture that we should consider the way we think about researcherteachers versus research on/with teachers. Second, I propose that certain features of PhD programs can be applied to teacher professional development and/or undergraduate education to support all teachers in creating richer mathematics learning experiences for their students. This editorial explores these two conjectures in more detail.

Researcher-Teachers as a Special Group In order to understand some of the unique qualities of the teaching exemplified by researcher-teachers, it is worthwhile to consider why they do what they do so well. There are a variety of factors that impact both the way these people teach and the way we, as consumers of research, read about their teaching. First, researcherteachers teach well because they have significant knowledge of mathematics and how children learn mathematics. There is no doubt that teachers, with or without PhDs, who have strong pedagogical knowledge and strong content knowledge, create richer Do You Need a PhD?

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

Further, in the process of earning a PhD, researcher-teachers presumably develop reflective dispositions, grapple with their own epistemological beliefs, and define their visions of learning and teaching. This produces teachers who critically examine the world around them and who are introspective in ways that are productive for achieving the classroom environment valued by mathematics education researchers and described in the NCTM Standards (NCTM, 2000). By developing this disposition, researcher-teachers are in a unique position to make critical changes to the classroom environment as needs are identified. Too often, regular classroom teachers do not have the time or skills to analyze formal or informal data about their students and their teaching. In fact, many classroom teachers have only been exposed to the most basic concepts of student learning theory and research. As a result, even if they tried to make sense of the data presented in their classroom, they would be ill-equipped to make important changes based on those data.

In addition, researcher-teachers have some pragmatic luxuries that typical teachers do not have.

For example, they usually only teach one subject to one class per day, while a typical elementary teacher might teach four subjects to one class, and a middle school teacher might teach one or two subjects to four or five classes each day. This provides the researcherteacher with more time for reflection and refinement.

To be fair, researcher-teachers typically do have other work responsibilities – they do not simply teach for 50 minutes and “call it a day.” However, their situation is very different from that of a typical classroom teacher.

Researcher-teachers have support with the reflection process from others studying the classroom, and often have no additional responsibilities such as conducting parent conferences, developing individualized plans for certain students, and attending the team meetings common in many teachers’ daily experience. While this difference should not be viewed or used as an excuse for classroom teachers to avoid improving their practice, it is undeniable that a researcher-teacher’s job is fundamentally different from that of the typical classroom teacher.

In addition to teaching expertise and workload, researcher-teachers have some advantages over teachers when participating in others’ studies. Unlike most “typical” teachers, researcher-teachers are, by definition, philosophically aligned with and invested in the goals of the research. They already have agreement Chandra Hawley Orrill

with the researcher about what good teaching and learning look like – after all, they are typically either the researcher (e.g., Ball, 1990a and Lampert, 2001) or they are a full member of the research team (e.g., McClain in Bowers, Cobb, & McClain, 1999). The importance of this is profound. A researcher-teacher wants the s a m e (not negotiated or compromised) outcomes as the researcher, because she either is the researcher or is a member of the research team. The researcher-teacher, therefore, attends to those issues and aspects of the classrooms and student learning that are the focus of the research. Further, the researcherteacher provides unlimited, or nearly unlimited, rich access to her thinking for the research effort because, again, she has a vested interest in capturing that thinking. Thus, teacher and researcher alignment in terms of goals, values, and expectations is important.

One potential disadvantage for researcher-teachers worth noting is the potential for bias to confound the research. After all, the researcher-teacher has a biased view of the teaching being studied because it is her own. Further, because she is invested in the research and because she is a member of the research team, it is possible that her teaching is biased to make the research work. That is, if the researcher is looking for particular aspects of teaching, such as student-teacher interactions, the researcher-teacher may attend to those interactions more in the course of instruction than she would under other circumstances. Clearly, the impact of this on the research is determined by both the research questions and the data collection and analysis techniques used.

Research On/With Teachers In order to understand the differences between researcher-teacher research and research on or with full-time teachers, it is necessary to explore some of the issues involved with doing research on/with teachers. Research in regular classrooms differs in some significant ways from the researcher-teacher work alluded to in this editorial. To highlight some of these differences, I offer examples from my own experience in working with middle grades mathematics teachers.

One major difference I alluded to is the values a teacher holds. In the course of my career, I have been fortunate to work with several “good” teachers.

However, the ways in which they were “good” were direct reflections of their own values and the values of the system within which they were working.

Sometimes, they were good in the eyes of the administrators with whom they worked because they 3

kept their students under control. Sometimes they were good for my research in that their practice had the elements I was interested in, thus making it easier for me to find the kinds of interactions I was looking for in their classrooms. Sometimes they were good in that they were predisposed to reflective practice allowing me, as a researcher, easier access to their ideas through observation and interviews. The quality of the teachers, though, depended on what measure they were held up against and what measures they, personally, felt they were trying to align with.

Another important aspect of working with teachers is a lack of access to certain aspects of their thinking.

For example, I have never been able to analyze a data set without thinking, at some point, “I wonder what she was thinking when she did that?” or “Did she not understand what that student was asking?”

Acknowledging this lack of access to a teacher’s thinking requires researchers to be careful in their analysis of the teacher’s actions and beliefs and to explain how thinking and actions are interpreted.

Further, at times, such limitations require researchers to analyze situations from their own perspectives as well as from the teacher’s perspective to understand a situation.

As a practical example of the influence of researcher and teacher alignment issues, I offer two situations from my own work: one addressing the “good” teacher issue and the other addressing the need to understand the situation from the teacher’s perspective. My goal in presenting these two examples is to highlight issues that arise in research with teachers who are not members of the research team. In one study (Orrill, 2001), I worked with two middle school teachers (one mathematics and one science) in New York City to understand how to structure professional development to support uses of computer-based simulations. My goal for the professional development was to enhance teachers’ attention to student problemsolving skills in the context of computer-based, workplace simulations. The mathematics teacher was considered to be “good” by her principal and other teachers. In my observations of her classroom, I found that she taught mathematics in much the same way as the “typical” teachers we read about in case study after case study. She offered many procedures but provided inadequate opportunities for students to interact with the content in ways that would allow them to develop deep understanding of the mathematical concepts underlying those procedures. However, this teacher had remarkable skill in classroom management, which was highly valued in her school. Further, she had 4

developed techniques that supported her students in achieving acceptable scores on the New York standardized tests. By these standards, she was considered “good.” When she used the simulations I was researching, she maintained the same kinds of approaches, particularly early in the study. She kept students on task and directed them to work more efficiently. Given my goal of understanding how to promote problem solving, her interactions with the students were inadequate and impoverished. She typically did not ask the students questions that provided insight into their thinking and she did not allow them to struggle with a problem. Instead, she directed them to an efficient approach for solving the problem they were working on, which effectively kept them on task and motivated them to move forward.

While this presented a challenge to me as the researcher, it would not be fair for me to “accuse” her of being less than a good teacher when she was clearly meeting the expectations of the system within which she worked. This is clearly a case in which there was a mismatch between what I, the researcher, valued and what the teacher and system valued. Had I been researching my own practice or the practice of a research team member, this tension would have been removed.

As a second example, a teacher I have worked with more recently proved a perplexing puzzle for my team as we considered her teaching. A point of particular interest was the teacher’s frustration with poor student performance on tests – regardless of what students did in class, a significant number failed her tests. In my analysis of this case, I recognized that this teacher’s beliefs about teaching and learning significantly differed from my own. Until I realized this, I was unable to understand the magnitude of the barrier the teacher felt she was facing. At the simplest level, she believed that her role as a good teacher was to present new material and provide an opportunity for students to practice that material. The students’ job, in her view, was to engage in that practice and develop an understanding from it. Therefore, when students were not succeeding, she became extremely frustrated since she had presented information and provided opportunities for practice. In her worldview, student success was out of her hands – she had already done what she could to support them. As the researcher in that setting, it was difficult to understand her frustration because I was working from a constructivist perspective. Specifically, I was looking for an environment in which the teacher provided students opportunities to develop their own thinking via an Do You Need a PhD?

assortment of models, experiences, and collaborative exchanges. Student test failure, for me, was an indicator that learning was not complete and that students needed different opportunities to build and connect knowledge. It took considerable analysis for me, as a researcher with a different perspective and different goals, to understand how the teacher understood her role and how she enacted her beliefs about her role in the classroom.

My point in these two examples is that in much research there are significant and important differences in the worldviews of the participants and the researcher. These differences can lead to frustrations in data collection, hurdles in data analysis, and, in the worst cases, assessments of the teachers that are simply not fair. For example, in the early 1990’s there were many articles written about the implementation of the standards in California (e.g., Ball, 1990b; Cohen, 1990; Wilson, 1990). In many of these cases, the teachers struggled to implement a set of standards that were written from a particular perspective that they did not fully understand. This led to implementations that were far from ideal in the eyes of the researchers who understood the initial intent of the standards. Too often, teachers were presented by researchers as hopeless or inadequate—in contrast, the teachers reportedly perceived themselves as adhering to these new standards. Likely, if the researchers and teachers had philosophical alignment afforded by the researcher-teacher approach the findings would have been tremendously different. After all, had these studies focused on researcher-teachers, the teacher and the researcher would have had a shared understanding of the intent of the standards and had a shared vision of what their implementation should look like.

PhD Program Features That Could Be Useful In Teacher Development While not all people who hold PhDs are good teachers, certain habits of mind are developed as part of the process of earning a PhD that can significantly impact the learning environment a teacher designs.

Given the high-quality of teaching exhibited by the researcher-teachers referred to in this article, it seems likely that there are aspects of the PhD program that could be adapted for teacher professional development.

First, the researcher-teacher typically has developed solid pedagogical knowledge, content knowledge, and pedagogical content knowledge. This comes from having time and encouragement to read about different practices in a focused way, participating in shared discourse with colleagues, Chandra Hawley Orrill

conducting research in others’ classrooms, and having other similar experiences. This is in stark contrast to the elementary or middle grades teacher who has typically had four years of college—with courses spread across the curriculum—and only limited “life experience” to relate to in the courses that help develop these knowledge areas. Second, one of the most powerful outcomes of earning a PhD is the development of a concrete picture of a desired learning environment that looks beyond issues of classroom management and logistics to focus on the kinds of learning and teaching that will take place. Third, PhDs develop a rich, precise vocabulary aligned with that of the standards-writers and the researchers. In becoming a researcher, the holder of the PhD becomes active in the conversation of the field—meaning that person has developed a refined vocabulary and vision that is shared, in some way, by the field. This is not to say that there is a definitive definition of K-8 mathematics education that is shared across the field of mathematics education, rather that there is a shared way of discussing and thinking about mathematics education that allows a more consistent enactment of standards and practices.

Finally, many researcher-teachers implement or develop a “special” curriculum. In the case of Lampert (2001), the teacher was creating open-ended problems each day to support mathematical topics. In other cases, the research team has developed materials for the researcher-teachers to implement. Often, these materials are far richer than traditional mathematics textbooks. While there may not be a single disposition that could be pulled from the process of earning a PhD that allows researcher-teachers to be successful implementers of non-traditional materials, it is clear that there is something different between PhD-holding researcher-teachers and other teachers. Likely, part of this ability is related to the knowledge constructs the researcher-teachers have that allow them to implement those materials. In my own work, I have found that teachers who are not well-versed in the curricula, who lack conceptual knowledge, or who lack the pedagogical content knowledge to see connections between various mathematical ideas do not know how to utilize these kinds of materials to make the experiences mathematically rich for their students.

Clearly, some attention to the aspects of earning a PhD that relate to these dispositions would benefit preservice and inservice teachers.

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Teacher Development While it may not be feasible, or even reasonable, to expect teachers to pursue doctoral degrees, there may be some characteristics of doctoral education that are worthwhile for consideration as components or foci of professional development and undergraduate programs.

To frame this section, I want to draw on the work of Cohen and Ball (1999) who have argued that the learning environment is shaped by the interactions of three critical elements: teachers, students, and materials/content. This model assumes that for each element a variety of beliefs, values, and backgrounds work together to create each unique learning environment. Considering the classroom from this perspective is critical to understanding why the solutions to the problems highlighted in research on and with teachers are complex.

What We See Now A quick overview of my definition of the “typical” classroom may be warranted at this point. Based on the classrooms described in the literature and those I work in, the typical mathematics classroom remains focused on teachers’ delivering information to the students, typically by working sample problems on the board.

Students are responsible for using this information to work problems on worksheets or in their books.

Students are asked to do things like name the fractional portion of a circle that is colored in or to work 20 addition or multiplication problems. Many teachers use manipulatives or drawn representations to introduce new ideas to their students. However, their intent is to provide a concrete example and move the students to the abstract activities of arithmetic as quickly as possible or to use the manipulative to motivate the students to want to do the arithmetic. Mathematics learning in these classrooms is more about developing efficient means for working problems than developing rich understandings of why those methods work.

Referring back to the Cohen and Ball triangle of interactions, the interactions in these classrooms could best be characterized by what follows. The teacher interprets the materials/content and delivers that interpretation to the students. The students look to teachers as holders of all information. Teachers are to provide guidance when students are unable to solve a problem, to provide feedback about the “rightness” of student work, and to find the errors students have made in their work. The students interact with the materials by working problems. The students may or may not interact with the concepts at a meaningful level – that depends on the teacher and the activity. In these 6

classrooms, success is measured in the number of problems students can answer correctly, often within a specific amount of time.

How Features of PhD Programs May Change This To enhance the interactions among teachers, students, and materials/content there are a number of elements from doctoral training that may be worth pursuing. First, teachers can use guided reflection as a means to step out of the teaching moment to consider critical aspects of the teaching and learning environment. Through reflection, teachers have the opportunity to align their beliefs and practices (e.g.

Wedman, Espinosa, & Laffey, 1998) and to make their intent more explicit rather than relying on tacit “gut instinct” (e.g., Richardson, 1990). The reflective practitioner can learn to look at a learning environment as a whole by considering how students and materials are interacting, looking for evidence of conceptual development, and thinking about ways to improve their own role in the classroom. The researcher-teachers (Ball, Lampert, and McClain) cited in this article all reported using reflection regularly as part of their practice.

Another element of the PhD experience worth consideration is the development of solid content and pedagogical knowledge. Teachers who do not understand mathematics cannot be as effective as those who do. For example, teachers who do not know how to use representations to model multiplication of fractions cannot use that pedagogical strategy in their classrooms. Teachers who lack adequate content or pedagogical knowledge cannot know what to do when a student suggests an approach to solving a problem that does not work—too often the only approach the teacher has is to point out errors to the student and demonstrate “one more time” the “right” way to work the problem. I assert that combining teacher development of content knowledge and pedagogical knowledge with the development of a reflective disposition will lead to the emergence of pedagogical content knowledge. By pedagogical content knowledge, I refer to knowledge that is a combination of knowing what content can be learned/taught with which pedagogies and knowing when to use each of these approaches to teach students.

Some of the habits of mind developed in a doctoral program in education translate directly into practice without focusing on the entire teacher-studentmaterials interaction triad. For example, one potentially powerful factor to address is the teacherstudent interaction. PhD programs in education offer Do You Need a PhD?

tremendous opportunities for thinking about this relationship in meaningful ways, and in the researcherteacher work, attention to this interaction is ubiquitous.

It is absolutely critical to support teachers in learning to listen to students and respond to them in meaningful ways. Further, given the poor grounding most teachers have in learning theory, it may be that developing a theoretical understanding of how people learn should be a part of this (this is supported in recent research such as Philipp, Clement, Thanheiser, Schappelle, & Sowder, 2003). Finally, focusing professional development on techniques for questioning that allow the teacher to access student understanding will provide teachers with ways to access student thinking.

Conclusion While it is not realistic to expect that all classroom teachers will earn doctoral degrees, there are elements that go into the attainment of a PhD that can lead to improved classroom teaching. Therefore, it seems reasonable to capitalize on what we know about the process of getting and having a doctorate versus more traditional routes to becoming a teacher.

Granted, there are aspects of researcher-teachers' activities that are not addressed simply by considering their educational background or their role in the research team. For example, high quality materials are extremely important. Further, it is vital that teachers are supported in learning how to interact with those materials (and the content they are trying to convey) if we want to raise the bar on teaching and learning. No one can create rich learning experiences around materials they do not understand. On the other hand, researcher-teachers have been able to find ways to capitalize on even the weakest of materials. For example, Lampert (2001) discusses how she was able to use the topic ideas from the traditional textbook her school used to develop rich problems that allowed students prolonged and repeated exposure to critical mathematics content—it is clear that the typical teacher is unable to capitalize on materials in these ways.

Certainly, there is an appropriate place in professional development efforts to support teachers’ use of materials.

While this article has only begun to explore the differences between a typical classroom teacher’s environment and that of a researcher-teacher, it appears that researcher-teachers have some advantages over other teachers. They are better able to understand and address what is going on in the classroom, as well as the material they are expected to work with.

Researcher-teachers are also better able to Chandra Hawley Orrill

communicate with others in the field and to understand input from the research. Unfortunately, it is not practical to expect most teachers to earn a doctoral degree. The question then becomes, “What elements can we take from earning an advanced degree that will help teachers in the classroom?” By incorporating these elements into teacher education and professional development programs, we can greatly improve classroom instruction.

REFERENCES Ball, 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 Carol Turner. Educational Evaluation and Policy Analysis, 12(3), 247–259.

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

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

Bowers, J., Cobb, P., & McClain, K. (1999). The evolution of mathematical practices: 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 in Education.

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

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

Orrill, C. H. (2001). Building technology-based learning-centered classrooms: 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 of the effects on elementary preservice teachers' beliefs and learning of mathematics. Paper presented at the Research Presession of the 81st Annual 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 teaching practice. Educational Researcher, 19(7), 10–18.

Wedman, J. M., Espinosa, L. M., & Laffey, J. M. (1998). A process for understanding how a field-based course influences teachers' beliefs and practices, Paper presented at the Annual Meeting of the American Educational 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.

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I cite examples of each of these researcher-teachers’ work throughout this editorial. This list is not exhaustive.

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Reasons why I believe this is true range from the lack of incentives relative to the effort required to earn a PhD to the mismatch between the intent of PhD programs and what teachers do in their everyday lives. This is not to assert that earning a PhD is not helpful for a teacher, rather that it is not likely in the current educational system.

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The Mathematics Educator 2004, Vol. 14, No. 1, 8–18

Mathematics as “Gate-Keeper” (?): Three Theoretical Perspectives that Aim Toward Empowering All Children With a Key to the Gate David W. Stinson In this article, the author’s intent is to begin a conversation centered on the question: How might mathematics educators ensure that gatekeeping mathematics becomes an inclusive instrument for empowerment rather than an exclusive instrument for stratification? In the first part of the discussion, the author provides a historical perspective of the concept of “gatekeeper” in mathematics education. After substantiating mathematics as a gatekeeper, the author proceeds to provide a definition of empowering mathematics within a Freirian frame, and describes three theoretical perspectives of mathematics education that aim toward empowering all children with a 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 of Mathematics Education at The University of Georgia for the 2002–2003 academic year was to assist with a four-year Spencer-funded qualitative research project entitled “Learning to Teach Elementary Mathematics.”

This assistantship presented the opportunity to conduct research at elementary schools in two suburban counties—a new experience for me since my prior professional experience in education had been within the context of secondary mathematics education. My research duties consisted of organizing, coding, analyzing, and writing-up existing data, as well as collecting new data. This new data included transcribed interviews of preservice and novice elementary school teachers and fieldnotes from classroom observations.

By January 2003, I had conducted five observations in 1st, 2nd, and 3rd grade classrooms at two elementary schools with diverse populations. I was impressed with the preservice and novice elementary teachers’ mathematics pedagogy and ability to interact with their students. Given that my research interest is equity and social justice in education, I was mindful of the “racial,” ethnic, gender, and class make-up of the classroom and how these attributes might help me explain the teacher-student interactions I observed. My David W. Stinson is a doctoral candidate in The Department of Mathematics Education at The University of Georgia. In the fall of 2004, he will join the faculty of the Middle-Secondary Education and Instructional Technology Department at Georgia State University. His research interests are the sociopolitical and cultural aspects of mathematics and mathematics teaching and learning with an emphasis on equity and social justice in mathematics education and education in general.

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experiences as a secondary mathematics teacher, preservice-teacher supervisor, and researcher supported Oakes’s (1985) assertions that often students are distributed into “ability” groups based on their race, gender, and class. Nonetheless, my perception after five observations was that ability grouping according to these attributes was diminishing—at least in these elementary schools. In other words, the student makeup of each mathematics lesson that I observed appeared to be representative of the demographics of the school.

However, on my sixth observation, at an elementary 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 grade mathematics lesson that was 94.4% White (at least it was 50% female). The make-up of the classroom was not initially unrepresentative of the school’s racial/ethnic demographics, but became so shortly before the start of the mathematics lesson as some students left the classroom while others entered. When I questioned why the students were exchanged between classrooms, I was informed that the mathematics lesson was for the “advanced” third graders. Because of my experience in secondary mathematics education, I am aware that academic tracking is a nationally practiced education policy, and that it even occurs in many districts and schools as early as 5th grade—but these were eight-year-old children! Has the structure of public education begun to decide who is and who is not “capable” mathematically in the 3rd grade? Has the structure of public education begun to decide who will be proletariat and who will be bourgeoisie in the 3rd grade—with eight-year-old children? How did school Mathematics as “Gate-Keeper” (?)

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

This article provides a two-part discussion centered on the last question. The first part of the discussion provides a historical perspective of the concept of gatekeeper in mathematics education, verifying that mathematics is an exclusive instrument for stratification, effectively nullifying the if. The intent of this historical perspective is not to debate whether mathematics should be a gatekeeper but to provide a perspective that reveals existence of mathematics as a gatekeeper (and instrument for stratification) in the current education structure of the United States. In the discussion, I state why I believe all students are not provided with a key to the gate.

After arguing that mathematics is a gatekeeper and inequities are present in the structure of education, I proceed to the second part of the discussion: how might mathematics educators ensure that gatekeeping mathematics becomes an inclusive instrument for empowerment? In this discussion, I first define empowerment and empowering mathematics. Then, I make note of the “social turn” in mathematics education research, which provides a framework for the situated, culturally relevant, and critical perspectives of mathematics education that are presented. Finally, I argue that these theoretical perspectives replace characteristics of exclusion and stratification (of gatekeeping mathematics) with characteristics of inclusion and empowerment. I conclude the article by challenging the reader to think differently.

Mathematics a Gatekeeper: A Historical Perspective Discourse regarding the “gatekeeper” concept in mathematics can be traced back over 2300 years ago to Plato’s (trans. 1996) dialogue, The Republic. In the fictitious dialogue between Socrates and Glaucon regarding education, Plato argued that mathematics was “virtually the first thing everyone has to learn…common to all arts, science, and forms of thought” (p. 216). Although Plato believed that all students needed to learn arithmetic—”the trivial business of being able to identify one, two, and three” (p. 216)—he reserved advanced mathematics for those that would serve as philosopher guardians2 of the city.

David W. Stinson

He wrote: We shall persuade those who are to perform high functions in the city to undertake calculation, but not as amateurs. They should persist in their studies until they reach the level of pure thought, where they will be able to contemplate the very nature of number. The objects of study ought not to be buying and selling, as if they were preparing to be merchants or brokers. Instead, it should serve the purposes of war and lead the soul away from the world of appearances toward essence and reality. (p. 219)

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

This view of mathematics as a gatekeeper has persisted through time and manifested itself in early research in the field of mathematics education in the United States. In Stanic’s (1986) review of mathematics education of the late 19th and early 20th centuries, he identified the 1890s as establishing “mathematics education as a separate and distinct professional area” (p. 190), and the 1930s as developing the “crisis” (p. 191) in mathematics education. This crisis—a crisis for mathematics educators—was the projected extinction of mathematics as a required subject in the secondary school curriculum. Drawing on the work of Kliebard (c.f., Kliebard, 1995), Stanic provided a summary of curriculum interest groups that influenced the position of mathematics in the school curriculum: (a) the humanists, who emphasized the traditional disciplines of study found in Western philosophy; (b) the developmentalists, who emphasized the “natural” development of the child; (c) the social efficiency educators, who emphasized a “scientific” approach that led to the natural development of social stratification; and (d) the social meliorists, who emphasized education as a means of working toward social justice.

Stanic (1986) noted that mathematics educators, in general, sided with the humanists, claiming: “mathematics should be an important part of the school curriculum” (p. 193). He also argued that the development of the National Council of Teachers of Mathematics (NCTM) in 1920 was partly in response to the debate that surrounded the position of mathematics within the school curriculum.

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The founders of the Council wrote: Mathematics courses have been assailed on every hand. So-called educational reformers have tinkered with the courses, and they, not knowing the subject and its values, in many cases have thrown out mathematics altogether or made it entirely elective. …To help remedy the existing situation the National Council of Teachers of Mathematics was organized. (C. M. Austin as quoted in Stanic, 1986, p. 198)

The backdrop to the mathematics education crisis was the tremendous growth in school population that occurred between 1890 and 1940—a growth of nearly 20 times (Stanic, 1986). This dramatic increase in the student population yielded the belief that the overall intellectual 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 this prevailing belief by citing the 1933 National Survey of Secondary Education, which concluded that less than half of the secondary schools required algebra and plane geometry. And, he illustrated mathematics teachers’ perspectives by providing George Counts’ 1926 survey of 416 secondary school teachers.

Eighteen of the 48 mathematics teachers thought that fewer pupils should take mathematics, providing a contrast to teachers of other academic disciplines who believed that “their own subjects should be more largely patronized” (G. S. Counts as quoted in Stanic, p. 196). Even so, the issues of how mathematics should be positioned in the school curriculum and who should take advanced mathematics courses was not a major national concern until the 1950s.

During the 1950s, mathematics education in U.S. schools began to be attacked from many segments of society: the business sector and military for graduating students who lacked computational skills, colleges for failing to prepare entering students with mathematics knowledge adequate for college work, and the public for having “watered down” the mathematics curriculum as a response to progressivism (Kilpatrick, 1992). The launching of Sputnik in 1957 further exacerbated these attacks leading to a national demand for rigorous mathematics in secondary schools. This demand spurred a variety of attempts to reform mathematics 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 questions were not only what mathematics should be taught and

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how, but more importantly, who should be taught mathematics.

The question of who should be taught mathematics initially appeared in the debates of the 1920s and centered on “ascertaining who was prepared for the study of algebra” (Kilpatrick, 1992, p. 21). These debates led to an increase in grouping students according to their presumed mathematics ability. This “ability” grouping often resulted in excluding female students, poor students, and students of color from the opportunity to enroll in advanced mathematics courses (Oakes, 1985; Oakes, Ormseth, Bell, & Camp, 1990).

Sixty years after the beginning of the debates, the recognition of this unjust exclusion from advanced mathematics courses spurred the NCTM to publish the Curriculum and Evaluation Standards for School M a t h e m a t i c s (Standards, 1989) that included statements similar to the following: The social injustices of past schooling practices can no longer be tolerated. Current statistics indicate that those who study advanced mathematics are most often white males. …Creating a just society in which women and various ethnic groups enjoy equal opportunities and equitable treatment is no longer an issue. Mathematics has become a critical filter for employment and full participation in our society. We cannot afford to have the majority of our population mathematically illiterate: Equity has become an economic necessity. (p. 4)

In the Standards the NCTM contrasted societal needs of the industrial age with those of the information age, concluding that the educational goals of the industrial age no longer met the needs of the information age. They characterized the information age as a dramatic shift in the use of technology which had “changed the nature of the physical, life, and social sciences; business; industry; and government” (p. 3).

The Council contended, “The impact of this technological shift is no longer an intellectual abstraction. It has become an economic reality” (p. 3).

The NCTM (1989) believed this shift demanded new 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 school system organized to serve as an important resource for all citizens throughout their lives” (p. 3). These goals required those responsible for mathematics education to strip mathematics from its traditional notions of exclusion and basic computation and develop it into a dynamic form of an inclusive literacy, particularly given that mathematics had become a critical filter for Mathematics as “Gate-Keeper” (?)

full employment and participation within a democratic society. Countless other education scholars (Frankenstein, 1995; Moses & Cobb, 2001; Secada, 1995; Skovsmose, 1994; Tate, 1995) have made similar arguments as they recognize the need for all students to be provided the opportunity to enroll in advanced mathematics courses, arguing that a dynamic mathematics literacy is a gatekeeper for economic access, full citizenship, and higher education. In the paragraphs that follow, I highlight quantitative and qualitative studies that substantiate mathematics as a gatekeeper.

The claims that mathematics is a “critical filter” or gatekeeper to economic access, full citizenship, and higher education are quantitatively substantiated by two reports by the U. S. government: the 1997 White Paper entitled Mathematics Equals Opportunity and the 1999 follow-up summary of the 1988 National Education Longitudinal Study (NELS: 88) entitled Do Gatekeeper Courses Expand Education Options? The U. S. Department of Education prepared both reports based on data from the NELS: 88 samples of 24,599 eighth graders from 1,052 schools, and the 1992 follow-up study of 12,053 students.

In Mathematics Equals Opportunity, the following statements were made: In the United States today, mastering mathematics has become more important than ever. Students with a strong grasp of mathematics have an advantage in academics and in the job market. The 8th grade is a critical point in mathematics education. Achievement at that stage clears the way for students to take rigorous high school mathematics and science courses—keys to college entrance and success in the labor force.

Students who take rigorous mathematics and science courses are much more likely to go to college than those who do not.

Algebra is the “gateway” to advanced mathematics and science in high school, yet most students do not take it in middle school.

Taking rigorous mathematics and science courses in high school appears to be especially important for low-income students.

Despite the importance of low-income students taking 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, explicitly stated that algebra was the “gateway” or gatekeeper to advanced (i.e., rigorous) mathematics courses and that David W. Stinson

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

The statistical analyses in the report entitled, D o Gatekeeper Courses Expand Educational Options? (U. S. Department of Education, 1999) presented the following findings: Students who enrolled in algebra as eighth-graders were more likely to reach advanced math courses (e.g., algebra 3, trigonometry, or calculus, etc.) in high school than students who did not enroll in algebra as eighth-graders.

Students who enrolled in algebra as eighth-graders, and completed an advanced math course during high school, were more likely to apply to a fouryear college than those eighth-grade students who did not enroll in algebra as eighth-graders, but who also completed an advanced math course during high school. (pp. 1–2)

The summary concluded that not all students who took advanced mathematics courses in high school enrolled in a four-year postsecondary school, although they were more likely to do so—again confirming mathematics as a gatekeeper.

Nicholas Lemann’s (1999) book The Big Test: The Secret History of the American Meritocracy provides a qualitative substantiation that mathematics is a gatekeeper to economic access, full citizenship, and higher education. In Parts I and II of his book, Lemann presented a detailed historical narrative of the merger between the Educational Testing Service with the College Board. Leman argued this merger established how mathematics would directly and indirectly categorize Americans—becoming a gatekeeper—for the remainder of the 20th and beginning of the 21st centuries. During World War I, the United States War Department (currently known as the Department of Defense) categorized people using an adapted version of Binet’s Intelligence Quotient test to determine the entering rank and duties of servicemen. This categorization evolved into ranking people by “aptitude” through administering standardized tests in contemporary U. S. education.

In Part III of his book, Lemann presented a casestudy characterization of contemporary Platonic guardians, individuals who unjustly (or not) benefited from the concept of aptitude testing and the ideal of American meritocracy. Lemann argued that because of their ability to demonstrate mathematics proficiency (among other disciplines) on standardized tests, these individuals found themselves passing through the gates

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to economic access, full citizenship, and higher education.

The concept of mathematics as providing the key for passing through the gates to economic access, full citizenship, and higher education is located in the core of Western philosophy. In the United States, school mathematics evolved from a discipline in “crisis” into one that would provide the means of “sorting” students. As student enrollment in public schools increased, the opportunity to enroll in advanced mathematics courses (the key) was limited because some students were characterized as “incapable.”

Female students, poor students, and students of color were offered a limited access to quality advanced mathematics education. This limited access was a motivating factor behind the Standards, and the subsequent NCTM documents.3 NCTM and education scholars’ argument that mathematics had and continues to have a gatekeeping status has been confirmed both quantitatively and qualitatively. Given this status, I pose two questions: (a) Why does U.S. education not provide all students access to a quality, advanced (mathematics) education that would empower them with economic access and full citizenship? and (b) How can we as mathematics educators transform the status quo in the mathematics classroom?

To fully engage in the first question demands a deconstruction of the concepts of democratic public schooling and American meritocracy and an analysis of the morals and ethics of capitalism. To provide such a deconstruction and analysis is beyond the scope of this article. Nonetheless, I believe that Bowles’s (1971/1977) argument provides a comprehensive, yet condensed response to the question of why U. S. education remains unequal without oversimplifying the complexities of the question. Through a historical analysis of schooling he revealed four components of U. S. education: (a) schools evolved not in pursuit of equality, but in response to the developing needs of capitalism (e.g., a skilled and educated work force); (b) as the importance of a skilled and educated work force grew within capitalism so did the importance of maintaining educational inequality in order to reproduce the class structure; (c) from the 1920s to 1970s the class structure in schools showed no signs of diminishment (the same argument can be made for the 1970s to 2000s); and (d) the inequality in education had “its root in the very class structures which it serves to legitimize and reproduce” (p. 137). He concluded by writing: “Inequalities in education are thus seen as part

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

Although Bowles’s statements imply that only the overthrow of capitalism will emancipate education from its inequalities, I believe that developing mathematics classrooms that are empowering to all students might contribute to educational experiences that are more equitable and just. This development may also assist in the deconstruction of capitalism so that it might be reconstructed to be more equitable and just.

The following discussion presents three theoretical perspectives that I have identified as empowering students. These perspectives aim to assist in more equitable and just educative experiences for all students: the situated perspective, the culturally relevant perspective, and the critical perspective. I believe these perspectives provide a plausible answer to the second question asked above: How do we as mathematics educators transform the status quo in the mathematics classroom?

An Inclusive Empowering Mathematics Education To frame the discussion that follows, I provide a definition of e m p o w e r m e n t and empowering mathematics. Freire (1970/2000) framed the notion of empowerment within the concept of conscientização, defined as “learning to perceive social, political and economic contradictions, and to take action against the oppressive elements of reality” (p. 35). He argued that conscientização leads people not to “destructive fanaticism” but makes it possible “for people to enter the historical process as responsible Subjects4” (p. 36), enrolling them in a search for self-affirmation.

Similarly, Lather (1991) defined empowerment as the ability to perform a critical analysis regarding the causes of powerlessness, the ability to identify the structures of oppression, and the ability to act as a single subject, group, or both to effect change toward social justice. She claimed that empowerment is a learning process one undertakes for oneself; “it is not something done ‘to’ or ‘for’ someone” (Lather, 1991, p. 4). In effect, empowerment provides the subject with the skills and knowledge to make sociopolitical critiques about her or his surroundings and to take action (or not) against the oppressive elements of those surroundings. The emphasis in both definitions is selfempowerment with an aim toward sociopolitical critique. With this emphasis in mind, I next define empowering mathematics.

Ernest (2002) provided three domains of empowering mathematics—mathematical, social, and epistemological—that assist in organizing how I define Mathematics as “Gate-Keeper” (?)

empowering mathematics.

Mathematical empowerment relates to “gaining the power over the language, skills and practices of using mathematics” (section 1, ¶ 3) (e.g., school mathematics). Social empowerment involves using mathematics as a tool for sociopolitical critique, gaining power over the social domains—“the worlds of work, life and social affairs” (section 1, ¶ 4). And, epistemological empowerment concerns the “individual’s growth of confidence in not only using mathematics, but also a personal sense of power over the creation and validation of knowledge”(section 1, ¶ 5). Ernest argued, and I agree, that all students gain confidence in their mathematics skills and abilities through the use of mathematics in routine and nonroutine ways and that this confidence will logically lead to higher levels of mathematics attainment. All students achieving higher levels of attainment will assist in leveling the racial, gender, and class imbalances that currently persist in advanced mathematics courses. Effectively, Ernest’s definition of empowering mathematics echoes the definition of empowerment stated earlier.

Using Ernest’s three domains of empowering mathematics as a starting point, I selected three empowering mathematics perspectives. In doing so, I kept in mind Stanic’s (1989) challenge to mathematics educators: “If mathematics educators take seriously the goal of equity, they must question not just the common view of school mathematics but also their own takenfor-granted assumptions about its nature and worth” (p.

58). I believe that the situated perspective, culturally relevant perspective, and critical perspective, in varying degrees, motivate such questioning and resonate with the definition I have given of empowering mathematics. These configurations are complex theoretical perspectives derived from multiple scholars who sometimes have conflicting working definitions. These perspectives, located in the “social turn” (Lerman, 2000, p. 23) of mathematics education research, originate outside the realm of “traditional” mathematics education theory, in that they are rooted in anthropology, cultural psychology, sociology, and sociopolitical critique. In the discussion that follows, I provide sketches of each theoretical perspective by briefly summarizing the viewpoints of key scholars working within the perspective. I then explain how each perspective holds possibilities in transforming gatekeeping mathematics from an exclusive instrument for stratification into an inclusive instrument for empowerment.

David W. Stinson

The Situated Perspective The situated perspective is the coupling of scholarship from cultural anthropology and cultural psychology. In the situated perspective, learning becomes a process of changing participation in changing communities of practice in which an individual’s resulting knowledge becomes a function of the environment in which she or he operates.

Consequently, in the situated perspective, the dualisms of mind and world are viewed as artificial constructs (Boaler, 2000b). Moreover, the situated perspective, in contrast to constructivist perspectives, emphasizes interactive systems that are larger in scope than the behavioral and cognitive processes of the individual student.

Mathematics knowledge in the situated perspective is understood as being co-constituted in a community within a context. It is the community and context in which the student learns the mathematics that significantly impacts how the student uses and understands the mathematics (Boaler, 2000b). Boaler (1993) suggested that learning mathematics in contexts assists in providing student motivation and interest and enhances transference of skills by linking classroom mathematics with real-world mathematics. She argued, however, that learning mathematics in contexts does not mean learning mathematics ideas and procedures by inserting them into “real-world” textbook problems or by extending mathematics to larger real-world class projects. Rather, she suggested that the classroom itself becomes the context in which mathematics is learned and understood: “If the students’ social and cultural values are encouraged and supported in the mathematics classroom, through the use of contexts or through an acknowledgement of personal routes and direction, then their learning will have more meaning” (p. 17).

The situated perspective offers different notions of what it means to have mathematics ability, changing the concept from “either one has mathematics ability or not” to an analysis of how the environment coconstitutes the mathematics knowledge that is learned (Boaler, 2000a). Boaler argued that this change in how mathematics ability is assessed in the situated perspective could “move mathematics education away from the discriminatory practices that produce more failures than successes toward something considerably more equitable and supportive of social justice” (p.

118).

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The Culturally Relevant Perspective Working 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 were successful with African-American children. This perspective is derived from the work of cultural anthropologists who studied the cultural disconnects between (White) teachers and students of color and made suggestions about how teachers could “match their teaching styles to the culture and home backgrounds of their students” (Ladson-Billings, 2001, p. 75). Ladson-Billings defined the culturally relevant perspective as promoting student achievement and success through cultural competence (teachers assist students in developing a positive identification with their home culture) and through sociopolitical consciousness (teachers help students develop a civic and social awareness in order to work toward equity and social justice).

Teachers working from a culturally relevant perspective (a) demonstrate a belief that children can be competent regardless of race or social class, (b) provide students with scaffolding between what they know and what they do not know, (c) focus on instruction during class rather than busy-work or behavior management, (d) extend students’ thinking beyond what they already know, and (e) exhibit indepth knowledge of students as well as subject matter (Ladson-Billings, 1995). Ladson-Billings argued that all children “can be successful in mathematics when their understanding of it is linked to meaningful cultural referents, and when the instruction assumes that all students are capable of mastering the subject matter” (p. 141).

Mathematics knowledge in the culturally relevant perspective is viewed as a version of ethnomathematics— ethno defined as all culturally identifiable groups with their jargons, codes, symbols, myths, and even specific ways of reasoning and inferring; mathema defined as categories of analysis; and tics defined as methods or techniques (D’ Ambrosio, 1985/1997, 1997). In the culturally relevant mathematics classroom, the teacher builds from the students’ ethno or informal mathematics and orients the lesson toward their culture and experiences, while developing the students’ critical thinking skills (Gutstein, Lipman, Hernandez, & de los Reyes, 1997).

The positive results of teaching from a culturally relevant perspective are realized when students develop mathematics empowerment: deducing mathematical generalizations and constructing creative 14

solution methods to nonroutine problems, and perceiving mathematics as a tool for sociopolitical critique (Gutstein, 2003).

The Critical Perspective Perceiving mathematics as a tool for sociopolitical critique is also a feature of the critical perspective. This perspective is rooted in the social and political critique of the Frankfurt School (circa 1920) whose membership included but was not limited to Max Horkheimer, Theodor Adorno, Leo Lowenthal, and Franz Neumann. The critical perspective is characterized as (a) providing an investigation into the sources of knowledge, (b) identifying social problems and plausible solutions, and (c) reacting to social injustices. In providing these most general and unifying characteristics of a critical education, Skovsmose (1994) noted, “A critical education cannot be a simple prolongation of existing social relationships. It cannot be an apparatus for prevailing inequalities in society. To be critical, education must react to social contradictions” (p. 38).

Skovsmose (1994), drawing from Freire’s (1970/2000) popularization of the concept c o n s c i e n t i z a ç ã o and his work in literacy empowerment, derived the term “mathemacy” (p. 48).

Skovsmose claimed that since modern society is highly technological and the core of all modern-day technology is mathematics that mathemacy is a means of empowerment. He stated, “If mathemacy has a role to play in education, similar to but not identical to the role of literacy, then mathemacy must be seen as composed of different competences: a mathematical, a technological, and a reflective” (p. 48).

In the critical perspective, mathematics knowledge is seen as demonstrating these three competencies (Skovsmose, 1994). Mathematical competence is demonstrating proficiency in the normally understood skills of school mathematics, reproducing and mastering various theorems, proofs, and algorithms.

Technological competence demonstrates proficiency in applying mathematics in model building, using mathematics in pursuit of different technological aims.

And, reflective competence achieves mathematics’ critical dimension, reflecting upon and evaluating the just and unjust uses of mathematics. Skovsmose contended that mathemacy is a necessary condition for a politically informed citizenry and efficient labor force, claiming that mathemacy provides a means for empowerment in organizing and reorganizing social and political institutions and their accompanying traditions.

Mathematics as “Gate-Keeper” (?)

Transforming Gatekeeping Mathematics The preceding sketches demonstrate that these three theoretical perspectives approach mathematics and mathematics teaching and learning differently than traditional perspectives. All three perspectives, in varying degrees, question the taken-for-granted assumptions about mathematics and its nature and worth, locate the formation of mathematics knowledge within the social community, and argue that mathematics is an indispensable instrument used in sociopolitical critique. In the following paragraphs I explicate the degrees to which the three perspectives address these issues.

The situated perspective negates the assumption that mathematics is a contextually free discipline, contending that it is the context in which mathematics is learned that determines how it will be used and understood. The culturally relevant perspective negates the assumption that mathematics is a culturally free discipline, recognizing mathematics is not separate from culture but is a product of culture. The critical perspective redefines the worth of mathematics through an acknowledgment and critical examination of the just and, often overlooked, unjust uses of mathematics.

The situated perspective locates mathematics knowledge in the social community. In this perspective, mathematics is not learned from a mathematics textbook and then applied to real-world contexts, but is negotiated in communities that exist in real-world contexts. The culturally relevant perspective also locates mathematics knowledge in the social community. This perspective argues teachers should begin to build on the collective mathematics knowledge present in the classroom communities, moving toward mathematics found in textbooks. The critical perspective does not locate mathematics knowledge in the social community but is oriented towards using mathematics to critique and transform the social and political communities in which mathematics exists and has its origins.

The situated perspective posits that students will begin to understand mathematics as a discipline that is learned in the context of communities. It is in this way that students may learn how mathematics can be applied in uncovering the inequities and injustices present in communities or can be used for sociopolitical critique. Similarly, one of the two tenets of the culturally relevant perspective is for the teacher to assist students in developing a sociopolitical consciousness. Finally, using mathematics as a means for sociopolitical critique is essential to the critical David W. Stinson

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

How do the three aspects of mathematics and mathematics teaching and learning relate to each other in these perspectives and how does this relationship address the three domains of empowering mathematics? First, mathematics empowerment is achieved because each perspective questions the assumptions that are often taken-for-granted about the nature and worth of mathematics. Although all three perspectives see value in the study of mathematics, including “academic”5 mathematics, they differ from traditional perspectives in that academic mathematics itself is troubled6 with regards to its contextual existence, its cultural connectedness, and its critical utility. Second, students achieve social empowerment because all three perspectives argue that students should engage in mathematics contextually and culturally; and, therefore students have the opportunity to gain confidence in using mathematics in routine and nonroutine problems. The advocates for these three perspectives argue that as students expand the use of mathematics into nonroutine problems, they become cognizant of how mathematics can be used as a tool for sociopolitical critique. Finally students achieve epistemological empowerment because all three perspectives trouble academic mathematics that in turn may lead students to understand that the concept of a “true” or “politically-free” mathematics is a fiction.

Students will hopefully understand that mathematics knowledge is (and always has been) a contextually and culturally (and politically) constructed human endeavor. If students achieve this perspective of mathematics, they will better understand their role as producers of mathematics knowledge, not just consumers. Hence, the three domains of empowering mathematics—mathematical, social, and epistemological—are achieved in each perspective or through various combinations of the three perspectives.

The chief aim of an empowering mathematics is to transform gatekeeping mathematics from a discipline of oppressive exclusion into a discipline of empowering inclusion. (This aim is inclusive of mathematics educators and education researchers.)

Empowering inclusion is achieved when students (and teachers of mathematics) are presented with the opportunity to learn that the foundations of mathematics can be troubled. This troubling of mathematics’ foundations transforms the discourse in the mathematics classroom from a discourse of transmitting mathematics to a “chosen” few students, into a discourse of exploring mathematics with all 15

students. Empowering inclusion is achieved when students (and teachers of mathematics) are presented with 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) with mathematics transforms mathematics from a tool used by a few students in “mathematical” pursuits, into a tool used by all students in sociopolitical pursuits.

Finally, empowering inclusion is achieved when students (and teachers of mathematics) are presented with the opportunity to learn that mathematics knowledge is constructed human knowledge. This returning to the origins of mathematics knowledge transforms mathematics from an Ideal of the gods reproduced by a few students, into a human endeavor produced by all students.

Concluding Thoughts The concept of mathematics as gatekeeper has a very long and disturbing history. There have been educators satisfied with the gatekeeping status of mathematics and those that have questioned not only its gatekeeping status but also its nature and worth. In my thinking about mathematics as a gatekeeper and the possibility of transforming mathematics education, I often reflect on Foucault’s challenge. He challenged us to think the un-thought, to think: “how is it that one particular statement appeared rather than another?”

(Foucault, 1969/1972, p. 27). With Foucault’s challenge in mind, I often think what if Plato had said, We shall persuade those who are to perform high functions in the city to undertake ________, but not as amateurs. They should persist in their studies until they reach the level of pure thought, where they will be able to contemplate the very nature of ________…. it should serve the purposes of war and lead the soul away from the world of appearances toward essence and reality. (trans.

1996, p. 219)

In the preceding blanks, I insert different human pursuits, such as writing, speaking, painting, sculpting, dancing, and so on, asking: does mathematics really lead the soul away from the world of appearances toward essence and reality?7 Or could dancing, for example, achieve the same result? While rethinking Plato’s centuries old comment, I rethink the privileged status of mathematics as a gatekeeper (and as an instrument of stratification). But rather than asking what is school mathematics as gatekeeper or what does it mean, I ask different questions: How does school mathematics as gatekeeper function? Where is school mathematics as gatekeeper to be found? How does 16

school mathematics as gatekeeper get produced and regulated? How does school mathematics as gatekeeper exist? (Bové, 1995). These questions transform the discussions around gatekeeper mathematics from discussions that attempt to find meaning in gatekeeper mathematics to discussions that examine the ethics of gatekeeper mathematics. Implicit in this examination is an analysis of how the structure of schools and those responsible for that structure are implicated (or not) in reproducing the unethical effects of gatekeeping mathematics.

Will asking the questions noted above transform gatekeeping mathematics from an exclusive instrument for stratification into an inclusive instrument for empowerment? Will asking these questions stop the “ability” sorting of eight-year-old children? Will asking these questions encourage mathematics teachers (and educators) to adopt the situated, culturally relevant, or critical perspectives, perspectives that aim toward empowering all children with a key? Although I believe that there are no definitive answers to these questions, I do believe that critically examining (and implementing) the different possibilities for mathematics teaching and learning found in the theoretical perspectives explained in this article provides a sensible beginning to transforming mathematics education. In closing, I fervently proclaim the way we use mathematics today in our nation’s schools must stop! Mathematics should not be used as an instrument for stratification but rather an instrument for empowerment!

REFERENCES Boaler, 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: Traditional communities and the alienation of learners. Journal of Mathematical Behavior, 18(4), 379–397.

Boaler, J. (1993). The role of context in the mathematics classroom: Do they make mathematics more “real”? For the Learning 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 the social 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 the history and pedagogy of mathematics. In A. B. Powell & M. Frankenstein (Eds.), Ethnomathematics: Challenging Eurocentrism in mathematics education (pp. 13–24). Albany: State University of New York Press. (Original work published in 1985) Mathematics as “Gate-Keeper” (?)

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

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

Derrida, J. (1997). Of grammatology (Corrected ed.). Baltimore: Johns Hopkins University Press. (Original work published 1974)

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

Ernest, P. (2002). Empowerment in mathematics education.

Philosophy of Mathematics Education, 15. Retrieved January 26, 2004, from http://www.ex.ac.uk/~PErnest/pome15/empowerment.htm Foucault, M. (1972). The archaeology of knowledge (1st American ed.). New York: Pantheon Books. (Original work published 1969) Frankenstein, M. (1995). Equity in mathematics education: Class in the world outside the class. In W. G. Secada, E. Fennema, & L. Byrd (Eds.), New directions for equity in mathematics education (pp. 165–190). Cambridge: Cambridge University Press.

Freire, P. (2000). Pedagogy of the oppressed (30th anniversary ed.). New York: Continuum. (Original work published 1970) Gutstein, E. (2003). Teaching and learning mathematics for social justice in an urban, Latino school. Journal for Research in Mathematics Education, 34(1), 37–73.

Gutstein, E., Lipman, P., Hernandez, P., & de los Reyes, R. (1997).

Culturally relevant mathematics teaching in a Mexican American context. Journal for Research in Mathematics Education, 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 mathematics education. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 3–38). New York: Macmillan.

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

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

National Council of Teachers of Mathematics. (1989). Curriculum and 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). Multiplying inequalities: The effects of race, social class, and tracking on opportunities to learn mathematics and science. Santa Monica, 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 in mathematics education. In W. G. Secada, E. Fennema, & L. B. Adajian (Eds.), New directions for equity in mathematics education (pp. 146–164). Cambridge: Cambridge University Press.

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.), Of grammatology (Corrected ed., pp. ix - lxxxvii). Baltimore: Johns Hopkins University Press. (Original work published 1974) Stanic, G. M. A. (1989). Social inequality, cultural discontinuity, and equity in school mathematics. Peabody Journal of Education, 66(2), 57–71.

Kliebard, H. M. (1995). The struggle for the American curriculum.

New York: Routledge.

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

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

Tate, W. F. (1995). Economics, equity, and the national mathematics assessment: Are we creating a national toll road?

In W. G. Secada, E. Fennema, & L. Byrd (Eds.), New directions for equity in mathematics education (pp. 191–206).

Cambridge: Cambridge University Press.

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

Ladson-Billings, G. (1994). The Dreamkeepers: Successful teachers of African American children. San Francisco: JosseyBass.

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

Lemann, N. (1999). The big test: The secret history of the American meritocracy (1st ed.). New York: Farrar Straus and Giroux.

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

David W. Stinson

U.S. Department of Education. (1997). Mathematics equals opportunity. White Paper prepared for U.S. Secretary of Education Richard W. Riley. Retrieved January 26, 2004, from http://www.ed.gov/pubs/math/mathemat.pdf U.S. Department of Education. (1999). Do gatekeeper courses expand education options? National Center for Education Statistics. Retrieved January 26, 2004, from http://nces.ed.gov/pubsearch/pubsinfo.asp?pubid=1999303 1

The student racial/ethnic data were based on the 2001-2002 Georgia Public Education Report Card; the racial/ethnic classifications were the State of Georgia’s not this author’s. For details of racial/ethnic data on all schools in the State of Georgia see: http://techservices.doe.k12.ga.us/reportcard/ 2

Plato (trans. 1996) in establishing his utopian Republic imagined that the philosopher guardians of the city, identified as the

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aristocracy, would be children taken from their parents at an early age and educated at the academy until of age when they would dutifully rule as public servants and not for personal gain. Plato believed that these children would be from all classes: “it may sometimes happen that a silver child will be born of a golden parent, a golden child from a silver parent and so on” (p. 113); and from both sexes: “we must conclude that sex cannot be the criterion in appointments to government positions…there should be no differentiation” (pp. 146-147). However, Plato’s concept of aristocracy has been greatly misinterpreted within Western ideology. The concept has historically and consistently favored the social positionality of the White heterosexual Christian male of bourgeois privilege.

3

Throughout the remainder of this article the term NCTM documents designates the Professional Standards for Teaching M a t h e m a t i c s (1991), Assessment Standards for School Mathematics (1995), Principles and Standards for School Mathematics (2000), and the Curriculum and Evaluation Standards for 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 known and acted upon” (p. 36).

5

I define the term “academic” mathematics as D`Ambrosio (1997) defined the term: mathematics that is taught and learned in schools, differentiated from ethnomathematics.

6

In this context, I use the term trouble to place academic mathematics under erasure. Spivak (1974/1997) explained Derrida’s (1974/1997) sous rature, that is, under erasure, as learning “to use and erase our language at the same time” (p. xviii).

She claimed that Derrida is “acutely aware… [of] the strategy of using the only available language while not subscribing to its premises, or ‘operat[ing] according to the vocabulary of the very thing that one delimits’ (MP 18, SP 147)” (p. xviii). In other words, I argue that these three perspectives, while purporting the teaching of the procedures and concepts of academic mathematics (i.e., the language of mathematics), also place it sous rature so as not to limit the mathematics creativity and engagement of all students.

7

Even though I trouble Plato’s remark regarding “essence and reality,” the purpose of this article is not to engage in that argument, an argument that I believe will be my life’s work.

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Mathematics as “Gate-Keeper” (?)

The Mathematics Educator 2004, Vol. 14, No. 1, 19–34

The Characteristics of Mathematical Creativity Bharath Sriraman Mathematical creativity ensures the growth of mathematics as a whole. However, the source of this growth, the creativity of the mathematician, is a relatively unexplored area in mathematics and mathematics education. In order to investigate how mathematicians create mathematics, a qualitative study involving five creative mathematicians was conducted. The mathematicians in this study verbally reflected on the thought processes involved in creating mathematics. Analytic induction was used to analyze the qualitative data in the interview transcripts 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. It was found that social interaction, imagery, heuristics, intuition, and proof were the common characteristics of mathematical creativity. Additionally, contemporary models of creativity from psychology were reviewed and used to interpret the characteristics of mathematical creativity .

Mathematical creativity ensures the growth of the field of mathematics as a whole. The constant increase in the number of journals devoted to mathematics research bears evidence to the growth of mathematics.

Yet what lies at the essence of this growth, the creativity of the mathematician, has not been the subject of much research. It is usually the case that most mathematicians are uninterested in analyzing the thought processes that result in mathematical creation (Ervynck, 1991). The earliest known attempt to study mathematical creativity was an extensive questionnaire published in the French periodical L'Enseigement Mathematique (1902). This questionnaire and a lecture on creativity given by the renowned 20th century mathematician Henri Poincaré to the Societé de Psychologie inspired his colleague Jacques Hadamard, another prominent 20th century mathematician, to investigate the psychology of mathematical creativity (Hadamard, 1945). Hadamard (1945) undertook an informal inquiry among prominent mathematicians and scientists in America, including George Birkhoff, George Polya, and Albert Einstein, about the mental images used in doing mathematics. Hadamard (1945), influenced by the Gestalt psychology of his time, theorized that mathematicians’ creative processes followed the four-stage Gestalt model (Wallas, 1926) of preparation-incubation-illumination-verification.

As we will see, the four-stage Gestalt model is a characterization of the mathematician's creative process, but it does not define creativity per se. How Bharath Sriraman is an assistant professor of mathematics and mathematics education at the University of Montana. His publications and research interests are in the areas of cognition, foundational issues, mathematical creativity, problem-solving, proof, and gifted education.

Bharath Sriraman

does one define creativity? In particular what exactly is mathematical creativity? Is it the discovery of a new theorem by a research mathematician? Does student discovery of a hitherto known result also constitute creativity? These are among the areas of exploration in this paper.

The Problem Of Defining Creativity Mathematical creativity has been simply described as discernment, or choice (Poincaré, 1948). According to Poincaré (1948), to create consists precisely in not making useless combinations and in making those which 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 in a creative insight whereas a majority of such combinations does not result in a creative outcome.

This may seem like a vague characterization of mathematical creativity. One can interpret Poincaré's "choice" metaphor to mean the ability of the mathematician to choose carefully between questions (or problems) that bear fruition, as opposed to those that lead to nothing new. But this interpretation does not resolve the fact that Poincaré’s definition of creativity overlooks the problem of novelty. In other words, characterizing mathematical creativity as the ability to choose between useful and useless combinations is akin to characterizing the art of sculpting as a process of cutting away the unnecessary!

Poincaré's (1948) definition of creativity was a result of the circumstances under which he stumbled upon deep results in Fuchsian functions. The first stage in creativity consists of working hard to get an insight into the problem at hand. Poincaré (1948) called this the preliminary period of conscious work. This period is also referred to as the preparatory stage (Hadamard, 19

1945). In the second, or incubatory, stage (Hadamard, 1945), the problem is put aside for a period of time and the mind is occupied with other problems. In the third stage the solution suddenly appears while the mathematician is perhaps engaged in other unrelated activities. "This appearance of sudden illumination is a manifest sign of long, unconscious prior work" (Poincaré, 1948). Hadamard (1945) referred to this as the illuminatory stage. However, the creative process does not end here. There is a fourth and final stage, which consists of expressing the results in language or writing. At this stage, one verifies the result, makes it precise, and looks for possible extensions through utilization of the result. The “Gestalt model” has some shortcomings. First, the model mainly applies to problems that have been posed a priori by mathematicians, thereby ignoring the fascinating process by which the actual questions evolve.

Additionally, the model attributes a large portion of what “happens” in the incubatory and illuminatory phases to subconscious drives. The first of these shortcomings, the problem of how questions are developed, is partially addressed by Ervynck (1991) in his three-stage model.

Ervynck (1991) described mathematical creativity in terms of three stages. The first stage (Stage 0) is referred to as the preliminary technical stage, which consists of "some kind of technical or practical application of mathematical rules and procedures, without the user having any awareness of the theoretical foundation" (p. 42). The second stage (Stage 1) is that of algorithmic activity, which consists primarily of performing mathematical techniques, such as explicitly applying an algorithm repeatedly. The third stage (Stage 2) is referred to as c r e a t i v e (conceptual, constructive) activity. This is the stage in which true mathematical creativity occurs and consists of non-algorithmic decision making. "The decisions that have to be taken may be of a widely divergent nature and always involve a choice" (p. 43). Although Ervynck (1991) tries to describe the process by which a mathematician arrives at the questions through his characterizations of Stage 0 and Stage 1, his description of mathematical creativity is very similar to those of Poincaré and Hadamard. In particular his use of the term “non-algorithmic decision making” is analogous to Poincaré’s use of the “choice” metaphor.

The mathematics education literature indicates that very few attempts have been made to explicitly define mathematical creativity. There are references made to creativity by the Soviet researcher Krutetskii (1976) in the context of students’ abilities to abstract and 20

generalize mathematical content. There is also an outstanding example of a mathematician (George Polya) attempting to give heuristics to tackle problems in a manner akin to the methods used by trained mathematicians. Polya (1954) observed that in "trying to solve a problem, we consider different aspects of it in turn, we roll it over and over in our minds; variation of the problem is essential to our work." Polya (1954) emphasized the use of a variety of heuristics for solving mathematical problems of varying complexity.

In examining the plausibility of a mathematical conjecture, mathematicians use a variety of strategies.

In looking for conspicuous patterns, mathematicians use such heuristics as (1) verifying consequences, (2) successively verifying several consequences, (3) verifying an improbable consequence, (4) inferring from analogy, and (5) deepening the analogy.

As is evident in the preceding paragraphs, the problem of defining creativity is by no means an easy one. However, psychologists’ renewed interest in the phenomenon of creativity has resulted in literature that attempts to define and operationalize the word “creativity.” Recently psychologists have attempted to link 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 produce unexpected original work that is useful and adaptive.

Mathematicians would raise several arguments with this definition, simply because the results of creative work may not always have implications that are “useful” in terms of applicability in the contemporary world. A recent example that comes to mind is Andrew Wiles’ proof of Fermat’s Last Theorem. The mathematical community views his work as creative. It was unexpected and original but had no applicability in the sense Sternberg and Lubart (2000) suggest. Hence, I think it is sufficient to define creativity as the ability to produce novel or original work, which is compatible with my personal definition of mathematical creativity as the process that results in unusual and insightful solutions to a given problem, irrespective of the level of complexity. In the context of this study involving professional mathematicians, mathematical creativity is defined as the publication of original results in prominent mathematics research journals.

The Motivation For Studying Creativity The lack of recent mathematics education literature on the subject of creativity was one of the motivations for conducting this study. Fifteen years ago Muir Mathematical Creativity

(1988) invited mathematicians to complete a modified and updated version of the survey that appeared in L'Enseigement Mathematique (1902) but the results of this endeavor are as yet unknown. The purpose of this study was to gain insight into the nature of mathematical creativity. I was interested in distilling common attributes of the creative process to see if there were any underlying themes that characterized mathematical creativity. The specific questions of exploration in this study were: Is the Gestalt model of mathematical creativity still applicable today?

What are the characteristics of the creative process in mathematics?

Does the study of mathematical creativity have any implications for the classroom?

Literature Review Any study on the nature of mathematical creativity begs the question as to whether the mathematician discovers or invents mathematics. Therefore, this review begins with a brief description of the four most popular viewpoints on the nature of mathematics. This is followed by a comprehensive review of contemporary models of creativity from psychology.

The Nature of Mathematics Mathematicians actively involved in research have certain beliefs about the ontological nature of mathematics that influence their approach to research (Davis & Hersh, 1981; Sriraman, 2004a). The Platonist viewpoint is that mathematical objects exist prior to their discovery and that “any meaningful question about a mathematical object has a definite answer, whether we are able to determine it or not” (Davis & Hersh, 1981). According to this view, mathematicians do not invent or create mathematics - they discover mathematics. Logicists hold that “all concepts of mathematics can ultimately be reduced to logical concepts” which implies that “all mathematical truths can be proved from the axioms and rules of inference and logic alone” (Ernest, 1991). Formalists do not believe that mathematics is discovered; they believe mathematics is simply a game, created by mathematicians, based on strings of symbols that have no meaning (Davis & Hersh, 1981).

Constructivism (incorporating Intuitionism) is one of the major schools of thought (besides Platonism, Logicism and Formalism) that arose due to the contradictions that emerged in the development of the theory of sets and the theory of functions during the early part of the 20th century. The constructivist Bharath Sriraman

(intuitionist) viewpoint is that “human mathematical activity is fundamental in the creation of new knowledge and that both mathematical truths and the existence of mathematical objects must be established by constructive methods" (Ernest, 1991, p. 29).

Contradictions like Russell’s Paradox were a major blow to the absolutist view of mathematical knowledge, for if mathematics is certain and all its theorems are certain, how can there be contradictions among its theorems? The early constructivists in mathematics were the intuitionists Brouwer and Heyting. Constructivists claim that both mathematical truths and the existence of mathematical objects must be established by constructivist methods.

The question then is how does a mathematician go about conducting mathematics research? Do the questions appear out of the blue, or is there a mode of thinking or inquiry that leads to meaningful questions and to the methodology for tackling these questions? I contend that the types of questions asked are determined to a large extent by the culture in which the mathematician lives and works. Simply put, it is impossible for an individual to acquire knowledge of the external world without social interaction.

According to Ernest (1994) there is no underlying metaphor for the wholly isolated individual mind.

Instead, the underlying metaphor is that of persons in conversation, persons who participate in meaningful linguistic interaction and dialogue (Ernest, 1994).

Language is the shaper, as well as being the “summative” product, of individual minds (Wittgenstein, 1978). The recent literature in psychology acknowledges these social dimensions of human activity as being instrumental in the creative process.

The Notion of Creativity in Psychology As stated earlier, research on creativity has been on the fringes of psychology, educational psychology, and mathematics education. It is only in the last twenty-five years that there has been a renewed interest in the phenomenon of creativity in the psychology community. The Handbook of Creativity (Sternberg, 2000), which contains a comprehensive review of all research then available in the field of creativity, suggests that most of the approaches used in the study of creativity can be subsumed under six categories: mystical, pragmatic, psychodynamic, psychometric, cognitive, and social-personality. Each of these approaches is briefly reviewed.

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The mystical approach The mystical approach to studying creativity suggests that creativity is the result of divine inspiration or is a spiritual process. In the history of mathematics, Blaise Pascal claimed that many of his mathematical insights came directly from God. The renowned 19th century algebraist Leopold Kronecker said that “God made the integers, all the rest is the work of man” (Gallian, 1994). Kronecker believed that all other numbers, being the work of man, were to be avoided; and although his radical beliefs did not attract many supporters, the intuitionists advocated his beliefs about constructive proofs many years after his death.

There have been attempts to explore possible relationships between mathematicians’ beliefs about the nature of mathematics and their creativity (Davis and Hersh, 1981; Hadamard, 1945; Poincaré, 1948; Sriraman, 2004a). These studies indicate that such a relationship does exist. It is commonly believed that the neo-Platonist view is helpful to the research mathematician because of the innate belief that the sought after result/relationship already exists.

The pragmatic approach The 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 for solving mathematical problems of varying complexity is an example of a pragmatic approach. Thus, heuristics can be viewed as a decision-making mechanism which leads the mathematician down a certain path, the outcome of which may or may not be fruitful. The popular technique of brainstorming, often used in corporate or other business settings, is another example of inducing creativity by seeking as many ideas or solutions as possible in a non-critical setting.

The psychodynamic approach The psychodynamic approach to studying creativity is based on the idea that creativity arises from the tension between conscious reality and unconscious drives (Hadamard, 1945; Poincaré, 1948, Sternberg, 2000, Wallas, 1926; Wertheimer, 1945).

The four-step Gestalt model (preparation-incubationillumination-verification) is an example of the use of a psychodynamic approach to studying creativity. It should be noted that the gestalt model has served as kindling for many contemporary problem-solving models (Polya, 1945; Schoenfeld, 1985; Lester, 1985).

Early psychodynamic approaches to creativity were used to construct case studies of eminent creators such 22

as Albert Einstein, but the behaviorists criticized this approach because of the difficulty in measuring proposed theoretical constructs.

The psychometric approach The psychometric approach to studying creativity entails quantifying the notion of creativity with the aid of paper and pencil tasks. An example of this would be the Torrance Tests of Creative Thinking, developed by Torrance (1974), that are used by many gifted programs in middle and high schools to identify students that are gifted/creative. These tests consist of several verbal and figural tasks that call for problemsolving skills and divergent thinking. The test is scored for fluency, flexibility, originality (the statistical rarity of a response), and elaboration (Sternberg, 2000).

Sternberg (2000) states that there are positive and negative sides to the psychometric approach. On the positive side, these tests allow for research with noneminent people, are easy to administer, and objectively scored. The negative side is that numerical scores fail to capture the concept of creativity because they are based on brief paper and pencil tests. Researchers call for using more significant productions such as writing samples, drawings, etc., subjectively evaluated by a panel of experts, instead of simply relying on a numerical measure.

The cognitive approach The cognitive approach to the study of creativity focuses on understanding the “mental representations and processes underlying human thought” (Sternberg, 2000, p. 7). Weisberg (1993) suggests that creativity entails the use of ordinary cognitive processes and results in original and extraordinary products. These products are the result of cognitive processes acting on the knowledge already stored in the memory of the individual. There is a significant amount of literature in the area of information processing (Birkhoff, 1969; Minsky, 1985) that attempts to isolate and explain cognitive processes in terms of machine metaphors.

The social-personality approach The social-personality approach to studying creativity focuses on personality and motivational variables as well as the socio-cultural environment as sources of creativity. Sternberg (2000) states that numerous studies conducted at the societal level indicate that “eminent levels of creativity over large spans of time are statistically linked to variables such as cultural diversity, war, availability of role models, availability of financial support, and competitors in a domain” (p. 9).

Mathematical Creativity

Most of the recent literature on creativity (Csikszentmihalyi, 1988, 2000; Gruber & Wallace, 2000; Sternberg & Lubart, 1996) suggests that creativity is the result of a confluence of one or more of the factors from these six aforementioned categories. The “confluence” approach to the study of creativity has gained credibility, and the research literature has numerous confluence theories for better understanding the process of creativity. A review of the most commonly cited confluence theories of creativity and a description of the methodology employed for data collection and data analysis in this study follow.

Confluence Theories of Creativity The three most commonly cited “confluence” approaches to the study of creativity are the “systems approach” (Csikszentmihalyi, 1988, 2000); “the case study as evolving systems approach” (Gruber & Wallace, 2000), and the “investment theory approach” (Sternberg & Lubart, 1996). The case study as an evolving system has the following components. First, it views creative work as multi-faceted. So, in constructing a case study of a creative work, one must distill the facets that are relevant and construct the case study based on the chosen facets. Some facets that can be used to construct an evolving system case study are: (1) uniqueness of the work; (2) a narrative of what the creator achieved; (3) systems of belief; (4) multiple time-scales (construct the time-scales involved in the production of the creative work); (5) problem solving; and (6) contextual frame such as family, schooling, and teacher’s influences (Gruber & Wallace, 2000). In summary, constructing a case study of a creative work as an evolving system entails incorporating the many facets suggested by Gruber & Wallace (2000). One could also evaluate a case study involving creative work by looking for the above mentioned facets.

The systems approach The systems approach takes into account the social and cultural dimensions of creativity instead of simply viewing creativity as an individualistic psychological process. The systems approach studies the interaction between the individual, domain, and field. The field consists of people who have influence over a domain.

For example, editors of mathematics research journals have influence over the domain of mathematics. The domain is in a sense a cultural organism that preserves and transmits creative products to individuals in the field. The systems model suggests that creativity is a process that is observable at the “intersection where individuals, domains and fields interact” Bharath Sriraman

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

“The domain is a necessary component of creativity because it is impossible to introduce a variation without reference to an existing pattern. New is meaningful only in reference to the old” (Csikzentmihalyi, 2000). Thus, creativity occurs when an individual proposes a change in a given domain, which is then transmitted by the field through time.

The personal background of an individual and his position in a domain naturally influence the likelihood of his making a contribution. For example, a mathematician working at a research university is more likely to produce research papers because of the time available for “thinking” as well as the creative influence of being immersed in a culture where ideas flourish. It is no coincidence that in the history of science, there are significant contributions from clergymen such as Pascal and Mendel because they had the means and the leisure to “think.”

Csikszentmihalyi (2000) argues that novel ideas, which could result in significant changes, are unlikely to be adopted unless they are sanctioned by the experts.

These “gatekeepers” (experts) constitute the field. For example, in mathematics, the opinion of a very small number of leading researchers was enough to certify the validity of Andrew Wiles’ proof of Fermat’s Last Theorem.

There are numerous examples in the history of mathematics that fall within the systems model. For instance, the Bourbaki, a group of mostly French mathematicians who began meeting in the 1930s, aimed to write a thorough unified account of all mathematics. The Bourbaki were essentially a group of expert mathematicians that tried to unify all of mathematics and become the gatekeepers of the field, so to speak, by setting the standard for rigor. Although the Bourbakists failed in their attempt, students of the Bourbakists, who are editors of certain prominent journals, to this day demand a very high degree of rigor in submitted articles, thereby serving as gatekeepers of the field.

A different example is that of the role of proof.

Proof is the social process through which the mathematical community validates the mathematician's creative work (Hanna, 1991). The Russian logician Manin (1977) said "A proof becomes a proof after the social act of accepting it as a proof. This is true of mathematics as it is of physics, linguistics, and biology."

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In summary, the systems model of creativity suggests that for creativity to occur, a set of rules and practices must be transmitted from the domain to the individual. The individual then must produce a novel variation in the content of the domain, and this variation must be selected by the field for inclusion in the domain.

Gruber and Wallace’s case study as evolving systems approach In contrast to Csikszentmihalyi’s (2000) argument calling for a focus on communities in which creativity manifests itself, Gruber and Wallace (2000) propose a model that treats each individual as a unique evolving system of creativity and ideas; and, therefore, each individual’s creative work must be studied on its own.

This viewpoint of Gruber and Wallace (2000) is a belated victory of sorts for the Gestaltists, who essentially proclaimed the same thing almost a century ago. Gruber and Wallace’s (2000) use of terminology that jibes with current trends in psychology seems to make their ideas more acceptable. They propose a model that calls for “detailed analytic and sometimes narrative descriptions of each case and efforts to understand each case as a unique functioning system (Gruber & Wallace, 2000, p. 93). It is important to note that the emphasis of this model is not to explain the origins of creativity, nor is it the personality of the creative individual, but on “how creative work works” (p. 94). The questions of concern to Gruber and Wallace are: (1) What do creative people do when they are being creative? and (2) How do creative people deploy available resources to accomplish something unique? In this model creative work is defined as that which is novel and has value. This definition is consistent with that used by current researchers in creativity (Csikszentmihalyi, 2000; Sternberg & Lubart, 2000). Gruber and Wallace (2000) also claim that creative work is always the result of purposeful behavior and that creative work is usually a long undertaking “reckoned in months, years and decades” (p. 94).

I do not agree with the claim that creative work is always the result of purposeful behavior. One counterexample that comes to mind is the discovery of penicillin. The discovery of penicillin could be attributed purely to chance. On the other hand, there are numerous examples that support the claim that creative work sometimes entails work that spans years, and in mathematical folklore there are numerous examples of such creative work. For example, Kepler’s laws of planetary motion were the result of twenty 24

years of numerical calculations. Andrew Wiles’ proof of Fermat’s Last Theorem was a seven-year undertaking. The Riemann hypothesis states that the roots of the zeta function (complex numbers z, at which the zeta function equals zero) lie on the line parallel to the imaginary axis and half a unit to the right of it. This is perhaps the most outstanding unproved conjecture in mathematics with numerous implications. The analyst Levinson undertook a determined calculation on his deathbed that increased the credibility of the Riemann-hypothesis. This is another example of creative work that falls within Gruber and Wallace's (2000) model.

The investment theory approach According to the investment theory model, creative people are like good investors; that is, they buy low and sell high (Sternberg & Lubart, 1996). The context here is naturally in the realm of ideas. Creative people conjure up ideas that are either unpopular or disrespected and invest considerable time convincing other people about the intrinsic worth of these ideas (Sternberg & Lubart, 1996). They sell high in the sense that they let other people pursue their ideas while they move on to the next idea. Investment theory claims that the convergence of six elements constitutes creativity.

The six elements are intelligence, knowledge, thinking styles, personality, motivation, and environment. It is important that the reader not mistake the word intelligence for an IQ score. On the contrary, Sternberg (1985) suggests a triarchic theory of intelligence that consists of synthetic (ability to generate novel, task appropriate ideas), analytic, and practical abilities.

Knowledge is defined as knowing enough about a particular field to move it forward. Thinking styles are defined as a preference for thinking in original ways of one’s choosing, the ability to think globally as well as locally, and the ability to distinguish questions of importance from those that are not important.

Personality attributes that foster creative functioning are the willingness to take risks, overcome obstacles, and tolerate ambiguity. Finally, motivation and an environment that is supportive and rewarding are essential elements of creativity (Sternberg, 1985).

In investment theory, creativity involves the interaction between a person, task, and environment.

This is, in a sense, a particular case of the systems model (Csikszentmihalyi, 2000). The implication of viewing creativity as the interaction between person, task, and environment is that what is considered novel or original may vary from one person, task, and environment to another. The investment theory model Mathematical Creativity

suggests that creativity is more than a simple sum of the attained level of functioning in each of the six elements. Regardless of the functioning levels in other elements, a certain level or threshold of knowledge is required without which creativity is impossible. High levels of intelligence and motivation can positively enhance creativity, and compensations can occur to counteract weaknesses in other elements. For example, one could be in an environment that is non-supportive of creative efforts, but a high level of motivation could possibly overcome this and encourage the pursuit of creative endeavors.

This concludes the review of three commonly cited prototypical confluence theories of creativity, namely the systems approach (Csikszentmihalyi, 2000), which suggests that creativity is a sociocultural process involving the interaction between the individual, domain, and field; Gruber & Wallace’s (2000) model that treats each individual case study as a unique evolving system of creativity; and investment theory (Sternberg & Lubart, 1996), which suggests that creativity is the result of the convergence of six elements (intelligence, knowledge, thinking styles, personality, motivation, and environment).

Having reviewed the research literature on creativity, the focus is shifted to the methodology employed for studying mathematical creativity.

Methodology The Interview Instrument The purpose of this study was to gain an insight into the nature of mathematical creativity. In an effort to determine some of the characteristics of the creative process, I was interested in distilling common attributes in the ways mathematicians create mathematics. Additionally, I was interested in testing the applicability of the Gestalt model. Because the main focus of the study was to ascertain qualitative aspects of creativity, a formal interview methodology was selected as the primary method of data collection.

The interview instrument (Appendix A) was developed by modifying questions from questionnaires in L’Enseigement Mathematique (1902) and Muir (1988).

The rationale behind using this modified questionnaire was to allow the mathematicians to express themselves freely while responding to questions of a general nature and to enable me to test the applicability of the four-stage Gestalt model of creativity. Therefore, the existing instruments were modified to operationalize the Gestalt theory and to encourage the natural flow of ideas, thereby forming the basis of a thesis that would emerge from this exploration.

Bharath Sriraman

Background of the Subjects Five mathematicians from the mathematical sciences faculty at a large Ph.D. granting mid-western university were selected. These mathematicians were chosen based on their accomplishments and the diversity of the mathematical areas in which they worked, measured by counting the number of published papers in prominent journals, as well as noting the variety of mathematical domains in which they conducted research. Four of the mathematicians were tenured full professors, each of whom had been professional mathematicians for more than 30 years.

One of the mathematicians was considerably younger but was a tenured associate professor. All interviews were conducted formally, in a closed door setting, in each mathematician’s office. The interviews were audiotaped and transcribed verbatim.

Data Analysis Since creativity is an extremely complex construct involving a wide range of interacting behaviors, I believe it should be studied holistically. The principle of analytic induction (Patton, 2002) was applied to the interview transcripts to discover dominant themes that described the behavior under study. According to Patton (2002), "analytic induction, in contrast to grounded theory, begins with an analyst's deduced propositions or theory-derived hypotheses and is a procedure for verifying theories and propositions based on qualitative data” (Taylor and Bogdan, 1984, p. 127).

Following the principles of analytic induction, the data was carefully analyzed in order to extract common strands. These strands were then compared to theoretical constructs in the existing literature with the explicit purpose of verifying whether the Gestalt model was applicable to this qualitative data as well as to extract themes that characterized the mathematician’s creative process. If an emerging theme could not be classified or named because I was unable to grasp its properties or significance, then theoretical comparisons were made. Corbin and Strauss (1998) state that “using comparisons brings out properties, which in turn can be used to examine the incident or object in the data. The specific incidents, objects, or actions that we use when making theoretical comparisons can be derived from the literature and experience. It is not that we use experience or literature as data “but rather that we use the properties and dimensions derived from the comparative incidents to examine the data in front of us” (p. 80). Themes that emerged were social interaction, preparation, use of heuristics, imagery, incubation, illumination, verification, intuition, and 25

proof. Excerpts from interviews that highlight these characteristics are reconstructed in the next section along with commentaries that incorporate the wider conversation, and a continuous discussion of connections to the existing literature.

Results, Commentaries & Discussion The mathematicians in this study worked in academic environments and regularly fulfilled teaching and committee duties. The mathematicians were free to choose their areas of research and the problems on which they focused. Four of the five mathematicians had worked and published as individuals and as members of occasional joint ventures with mathematicians from other universities. Only one of the mathematicians had done extensive collaborative work. All but one of the mathematicians were unable to formally structure their time for research, primarily due to family commitments and teaching responsibilities during the regular school year. All the mathematicians found it easier to concentrate on research in the summers because of lighter or nonexistent teaching responsibilities during that time. Two of the mathematicians showed a pre-disposition towards mathematics at the early secondary school level. The others became interested in mathematics later, during their university education. The mathematicians who participated in this study did not report any immediate family influence that was of primary importance in their mathematical development. Four of the mathematicians recalled being influenced by particular teachers, and one reported being influenced by a textbook. The three mathematicians who worked primarily in analysis made a conscious effort to obtain a broad overview of mathematics not necessarily of immediate relevance to their main interests. The two algebraists expressed interest in other areas of mathematics but were primarily active in their chosen field.

Supervision Of Research & Social Interaction As noted earlier, all the mathematicians in this study were tenured professors in a research university.

In addition to teaching, conducting research, and fulfilling committee obligations, many mathematicians play a big role in mentoring graduate students interested in their areas of research. Research supervision is an aspect of creativity because any interaction between human beings is an ideal setting for the exchange of ideas. During this interaction the mathematician is exposed to different perspectives on the subject, and all of the mathematicians in this study 26

valued the interaction they had with their graduate students. Excerpts of individual responses follow.1 Excerpt 1 A. I've had only one graduate student per semester and she is just finishing up her PhD right now, and I'd say it has been a very good interaction to see somebody else get interested in the subject and come up with new ideas, and exploring those ideas with her.

B. I have had a couple of students who have sort of started but who haven't continued on to a PhD, so I really can't speak to that. But the interaction was positive.

C. Of course, I have a lot of collaborators, these are my former students you know…I am always all the time working with students, this is normal situation.

D. That is difficult to answer (silence)…it is positive because it is good to interact with other people. It is negative because it can take a lot of time. As you get older your brain doesn't work as well as it used to and…younger people by and large their minds are more open, there is less garbage in there already. So, it is exciting to work with younger people who are in their most creative time.

When you are older, you have more experience, when you are younger your mind works faster …not as fettered.

E. Oh…it is a positive factor I think, because it continues to stimulate ideas …talking about things and it also reviews things for you in the process, puts things in perspective, and keep the big picture. It is helpful really in your own research to supervise students.

Commentary on Excerpt 1 The responses of the mathematicians in the preceding excerpt are focused on research supervision; however, all of the mathematicians acknowledged the role of social interaction in general as an important aspect that stimulated creative work. Many of the mathematicians mentioned the advantages of being able to e-mail colleagues and going to research conferences and other professional meetings. This is further explored in the following section, which focuses on preparation.

Mathematical Creativity

Preparation and the Use of Heuristics When mathematicians are about to investigate a new topic, there is usually a body of existing research in the area of the new topic. One of goals of this study was to find out how creative mathematicians approached a new topic or a problem. Did they try their own approach, or did they first attempt to assimilate what was already known about that topic? Did the mathematicians make use of computers to gain insight into the problem? What were the various modes of approaching a new topic or problem? The responses indicate 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. I find that talking to people helps a lot more than reading because you get more of a feel for what the motivation is beneath everything.

B. What might happen for me, is that I may start reading something, and, if feel I can do a better job, then I would strike off on my own. But for the most part I would like to not have to reinvent a lot that is already there. So, a lot of what has motivated my research has been the desire to understand an area. So, if somebody has already laid the groundwork then it's helpful. Still I think a large part of doing research is to read the work that other people have done.

C. It is connected with one thing that simply…my style was that I worked very much and I even work when I could not work. Simply the problems that I solve attract me so much, that the question was who will die first…mathematics or me? It was never clear who would die.

D. Try and find out what is known. I won't say assimilate…try and find out what's known and get an overview, and try and let the problem speak…mostly by reading because you don't have that much immediate contact with other people in the field. But I find that I get more from listening to talks that other people are giving than reading.

E. Well! I have been taught to be a good scholar.

A good scholar attempts to find out what is first known about something or other before they spend their time simply going it on their Bharath Sriraman

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

Commentary on Excerpt 2 These responses indicate that the mathematician spends a considerable amount of time researching the context of the problem. This is primarily done by reading the existing literature and by talking to other mathematicians in the new area. This finding is consistent with the systems model, which suggests that creativity is a dynamic process involving the interaction between the individual, domain, and field (Csikzentmihalyi, 2000). At this stage, it is reasonable to ask whether a mathematician works on a single problem until a breakthrough occurs or does a mathematician work on several problems concurrently?

It was found that each of the mathematicians worked on several problems concurrently, using a back and forth approach.

Excerpt 3 A. I work on several different problems for a protracted period of time… there have been times when I have felt, yes, I should be able to prove this result, then I would concentrate on that thing for a while but they tend to be several different things that I was thinking about a particular stage.

B. I probably tend to work on several problems at the same time. There are several different questions that I am working on…mm…probably the real question is how often do you change the focus? Do I work on two different problems on the same day? And that is probably up to whatever comes to mind in that particular time frame. I might start working on one rather than the other. But I would tend to focus on one particular problem for a period of weeks, then you switch to something else. Probably what happens is that I work on something and I reach a dead end then I may shift gears and work on a different problem for a while, reach a dead end there and come back to the original problem, so it’s back and forth.

C . I must simply think on one thing and not switch so much.

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

27

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

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

E. I usually have couple of things going. When I get stale on one, then I will pick up the other, and bounce back and forth. Usually I have one that is primarily my focus at a given time, and I will spend time on it over another; but it is not uncommon for me to have a couple of problems going at a given time. Sometimes when I am looking for an example that is not coming, instead of spending my time beating my head against the wall, looking for that example is not a very good use of time.

Working on another helps to generate ideas that I can bring back to the other problem.

A. That is a problem because of the particular area I am in. I can't draw any diagrams, things are infinite, so I would love to be able to get some kind of a computer diagram to show the complexity for a particular ring… to have something like the Julia sets or…mmm…fractal images, things which are infinite but you can focus in closer and closer to see possible relationships. I have thought about that with possibilities on the computer.

To think about the most basic ring, you would have to think of the ring of integers and all of the relationships for divisibility, so how do you somehow describe this tree of divisibility for integers…it is infinite.

Commentary on Excerpt 3 The preceding excerpt indicates that mathematicians tend to work on more than one problem at a given time. Do mathematicians switch back and forth between problems in a completely random manner, or do they employ and exhaust a systematic train of thought about a problem before switching to a different problem? Many of the mathematicians reported using heuristic reasoning, trying to prove something one day and disprove it the next day, looking for both examples and counterexamples, the use of "manipulations" (Polya, 1954) to gain an insight into the problem. This indicates that mathematicians do employ some of the heuristics made explicit by Polya. It was unclear whether the mathematicians made use of computers to gain an experimental or computational insight into the problem. I was also interested in knowing the types of imagery used by mathematicians in their work. The mathematicians in this study were queried about this, and the following excerpt gives us an insight into that aspect of mathematical creativity.

Imagery The mathematicians in this study were asked about the kinds of imagery they used to think about mathematical objects. Their responses are reported here to give the reader a glimpse of the ways mathematicians think of mathematical objects. Their responses also highlight the difficulty of explicitly describing imagery.

Excerpt 4 Yes I do, yes I do, I tend to draw a lot of pictures when I am doing research, I tend to manipulate things in the air, you know to try to figure out how things 28

B . Science is language, you think through language. But it is language simply; you put together theorems by logic. You first see the theorem in nature…you must see that somewhat is reasonable and then you go and begin and then of course there is big, big, big work to just come to some theorem in nonlinear elliptic equations…

C. A lot of mathematics, whether we are teaching or doing, is attaching meaning to what we are doing and this is going back to the earlier question when you talked about how do you do it, what kind of heuristics do you use? What kind of images do you have that you are using?

A lot of doing mathematics is creating these abstract images that connect things and then making sense of them but that doesn't appear in proofs either.

D. Pictorial, linguistic, kinesthetic...any of them is the point right! Sometimes you think of one, sometimes another. It really depends on the problem you are looking at, they are very much…often I think of functions as very kinesthetic, moving things from here to there.

Other approaches you are talking about is going to vary from problem to problem, or even day to day. Sometimes when I am working on research, I try to view things in as many different ways as possible, to see what is really happening. So there are a variety of approaches.

Mathematical Creativity

Commentary on Excerpt 4 Besides revealing the difficulty of describing mental imagery, all the mathematicians reported that they did not use computers in their work. This characteristic of the pure mathematician's work is echoed in Poincaré's (1948) use of the “choice” metaphor and Ervynck's (1991) use of the term “nonalgorithmic decision making.” The doubts expressed by the mathematicians about the incapability of machines to do their work brings to mind the reported words of Garrett Birkhoff, one of the great applied mathematicians of our time. In his retirement presidential address to the Society for Industrial and Applied Mathematics, Birkhoff (1969) addressed the role of machines in human creative endeavors. In particular, part of this address was devoted to discussing the psychology of the mathematicians (and hence of mathematics). Birkhoff (1969) said: The remarkable recent achievements of computers have partially fulfilled an old dream. These achievements have led some people to speculate that tomorrow's computers will be even more "intelligent" than humans, especially in their powers of mathematical reasoning...the ability of good mathematicians to sense the significant and to avoid undue repetition seems, however, hard to computerize; without it, the computer has to pursue millions of fruitless paths avoided by experienced human mathematicians. (pp. 430-438)

Incubation and Illumination Having reported on the role of research supervision and social interaction, the use of heuristics and imagery, all of which can be viewed as aspects of the preparatory stage of mathematical creativity, it is natural to ask what occurs next. As the literature suggests, after the mathematician works hard to gain insight into a problem, there is usually a transition period (conscious work on the problem ceases and unconscious work begins), during which the problem is put aside before the breakthrough occurs. The mathematicians in this study reported experiences that are consistent with the existing literature (Hadamard, 1945; Poincaré, 1948).

Excerpt 5 B. 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't think you get ideas out of nowhere, you have to do the groundwork first, okay. This is why people will say, now we have worked on this problem, so let us sleep on it. So you do the Bharath Sriraman

preparation, so that the sub-conscious or intuitive side may work on it and the answer comes back but you can't really tell when. You have to be open to this, lay the groundwork, think about it and then these flashes of intuition come and they represent the other side of the brain communicating with you at whatever odd time.

D. I am not sure you can really separate them because they are somewhat connected. You spend a lot of time working on something and you are not getting anywhere with it…with the deliberate effort, then I think your mind continues to work and organize. And maybe when the pressure is off the idea comes…but the idea comes because of the hard work.

E. Usually they come after I have worked very hard on something or another, but they may come at an odd moment. They may come into my head before I go to bed …What do I do at that point? Yes I write it down (laughing).

Sometimes when I am walking somewhere, the mind flows back to it (the problem) and says what about that, why don't you try that. That sort of thing happens. One of the best ideas I had was when I was working on my thesis …Saturday night, having worked on it quite a bit, sitting back and saying why don't I think about it again…and ping! There it was…I knew what it was, I could do that. Often ideas are handed to you from the outside, but they don't come until you have worked on it long enough.

Commentary on Excerpt 5 As is evident in the preceding excerpt, three out of the five mathematicians reported experiences consistent with the Gestalt model. Mathematician C attributed his breakthroughs on problems to his unflinching will to never give up and to divine inspiration, echoing the voice of Pascal in a sense.

However, Mathematician A attributed breakthroughs to chance. In other words, making the appropriate (psychological) connections by pure chance which eventually result in the sought after result.

I think it is necessary to comment about the unusual view of mathematician A. Chance plays an important role in mathematical creativity. Great ideas and insights may be the result of chance such as the discovery of penicillin. Ulam (1976) estimated that there is a yearly output of 200,000 theorems in 29

mathematics. Chance plays a role in what is considered important in mathematical research since only a handful of results and techniques survive out of the volumes of published research. I wish to draw a distinction between chance in the "Darwinian" sense (as to what survives), and chance in the psychological sense (which results in discovery/invention). The role of 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 we might attempt a stochastic description, and the selection of what is valuable from among them.

However the importation of biological metaphors to explain cultural evolution is dubious…both creation and selection are acts of design within a social context. (p. 33)

Thus, Muir (1988) rejects the Darwinian explanation. On the other hand, Nicolle (1932) in Biologie de L'Invention does not acknowledge the role of unconsciously present prior work in the creative process. He attributes breakthroughs to pure chance.

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

Nicolle's Darwinian explanation was rejected by Hadamard on the grounds that to claim creation occurs by pure chance is equivalent to asserting that there are effects without causes. Hadamard further argued that although Poincaré attributed his particular breakthrough in Fuchsian functions to chance, Poincaré did acknowledge that there was a considerable amount of previous conscious effort, followed by a period of unconscious work. Hadamard (1945) further argued that even if Poincaré's breakthrough was the result of chance alone, chance alone was insufficient to explain the considerable body of creative work credited to Poincaré in almost every area of mathematics. The question then is how does (psychological) chance work?

It is my conjecture that the mind throws out fragments (ideas) that are products of past experience.

Some of these fragments can be juxtaposed and combined in a meaningful way. For example, if one reads a complicated proof consisting of a thousand steps, a thousand random fragments may not be enough to construct a meaningful proof. However the mind chooses relevant fragments from these random fragments and links them into something meaningful.

Wedderburn's Theorem, that a finite division ring is a 30

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

Polya (1954) addresses the role of chance in a probabilistic sense. It often occurs in mathematics that a series of mathematical trials (involving computation) generate numbers that are close to a Platonic ideal. The classic example is Euler's investigation of the infinite series 1 + 1/4 + 1/9 + 1/16 +…+ 1/n2 +…. Euler obtained an approximate numerical value for the sum of the series using various transformations of the series. The numerical approximation was 1.644934.

Euler confidently guessed the sum of the series to be π2/6. Although the numerical value obtained by Euler and the value of π 2/6 coincided up to seven decimal places, such a coincidence could be attributed to chance. However, a simple calculation shows that the probability of seven digits coinciding is one in ten million! Hence, Euler did not attribute this coincidence to chance but boldly conjectured that the sum of this series was indeed π2/6 and later proved his conjecture to be true (Polya, 1954, pp. 95-96).

Intuition, Verification and Proof Once illumination has occurred, whether through sheer chance, incubation, or divine intervention, mathematicians usually try to verify that their intuitions were correct with the construction of a proof.

The following section discusses how these mathematicians went about the business of verifying their intuitions and the role of formal proof in the creative process. They were asked whether they relied on repeatedly checking a formal proof, used multiple converging partial proofs, looked first for coherence with other results in the area, or looked at applications.

Most of the mathematicians in this study mentioned that the last thing they looked at was a formal proof.

This is consistent with the literature on the role of formal proof in mathematics (Polya, 1954; Usiskin, 1987). Most of the mathematicians mentioned the need for coherence with other results in the area. The mathematician’s responses to the posed question follow.

Excerpt 6 B. I think I would go for repeated checking of the formal proof…but I don't think that that is really enough. All of the others have to also be taken into account. I mean, you can believe that something is true although you may not fully understand it. This is the point that was made in the lecture by … of … University on Mathematical Creativity

Dirichlet series. He was saying that we have had a formal proof for some time, but that is not to say that it is really understood, and what did he mean by that? Not that the proof wasn't understood, but it was the implications of the result that are not understood, their connections with other results, applications and why things really work. But probably the first thing that I would really want to do is check the formal proof to my satisfaction, so that I believe that it is correct although at that point I really do not understand its implications… it is safe to say that it is my surest guide.

C. First you must see it in the nature, something, first you must see that this theorem corresponds to something in nature, then if you have this impression, it is something relatively reasonable, then you go to proofs…and of course I have also several theorems and proofs that are wrong, but the major amount of proofs and theorems are right.

D. The last thing that comes is the formal proof. I look for analogies with other things… How your results that you think might be true would illuminate other things and would fit in the general structure.

E. Since I work in an area of basic research, it is usually coherence with other things, that is probably more than anything else. Yes, one could go back and check the proof and that sort of thing but usually the applications are yet to come, they aren't there already. Usually what guides the choice of the problem is the potential for application, part of what represents good problems is their potential for use. So, you certainly look to see if it makes sense in the big picture…that is a coherence phenomenon. Among those you've given me, that’s probably the most that fits.

Commentary on Excerpt 6 This excerpt indicates that for mathematicians, valid proofs have varied degrees of rigor. “Among mathematicians, rigor varies depending on time and circumstance, and few proofs in mathematics journals meet the criteria used by secondary school geometry teachers (each statement of proof is backed by reasons). Generally one increases rigor only when the result does not seem to be correct” (Usiskin, 1987).

Proofs are in most cases the final step in this testing Bharath Sriraman

process. “Mathematics in the making resembles any other human knowledge in the making. The result of the mathematician’s creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing” (Polya, 1954). How mathematicians approached proof in this study was very different from the logical approach found in proof in most textbooks. The logical approach is an artificial reconstruction of discoveries that are being forced into a deductive system, and in this process the intuition that guided the discovery process gets lost.

Conclusions The goal of this study was to gain an insight into mathematical creativity. As suggested by the literature review, the existing literature on mathematical creativity is relatively sparse. In trying to better understand the process of creativity, I find that the Gestalt model proposed by Hadamard (1945) is still applicable today. This study has attempted to add some detail to the preparation-incubation-illuminationverification model of Gestalt by taking into account the role of imagery, the role of intuition, the role of social interaction, the use of heuristics, and the necessity of proof in the creative process.

The mathematicians worked in a setting that was conducive to prolonged research. There was a convergence of intelligence, knowledge, thinking styles, personality, motivation and environment that enabled them to work creatively (Sternberg, 2000; Sternberg & Lubart, 1996, 2000). The preparatory stage of mathematical creativity consists of various approaches used by the mathematician to lay the groundwork. These include reading the existing literature, talking to other mathematicians in the particular mathematical domain (Csikzentmihalyi, 1988; 2000), trying a variety of heuristics (Polya, 1954), and using a back-and-forth approach of plausible guessing. One of the mathematicians said that he first looked to see if the sought after relationships corresponded to natural phenomenon.

All of the mathematicians in this study worked on more than one problem at a given moment. This is consistent with the investment theory view of creativity (Sternberg & Lubart, 1996). The mathematicians invested an optimal amount of time on a given problem, but switched to a different problem if no breakthrough was forthcoming. All the mathematicians in this study considered this as the most important and difficult stage of creativity. The prolonged hard work was followed by a period of incubation where the problem was put aside, often while the preparatory 31

stage is repeated for a different problem; and thus, there is a transition in the mind from conscious to unconscious work on the problem. One mathematician cited this as the stage at which the "problem begins to talk to you." Another offered that the intuitive side of the brain begins communicating with the logical side at this stage and conjectured that this communication was not possible at a conscious level.

The transition from incubation to illumination often occurred when least expected. Many reported the breakthrough occurring as they were going to bed, or walking, or sometimes as a result of speaking to someone else about the problem. One mathematician illustrated this transition with the following: "You talk to somebody and they say just something that might have been very ordinary a month before but if they say it when you are ready for it, and Oh yeah, I can do it that way, can’t I! But you have to be ready for it.

Opportunity knocks but you have to be able to answer the door."

Illumination is followed by the mathematician’s verifying the result. In this study, most of the mathematicians looked for coherence of the result with other existing results in the area of research. If the result cohered with other results and fit the general structure of the area, only then did the mathematician try to construct a formal proof. In terms of the mathematician’s beliefs about the nature of mathematics and its influence on their research, the study revealed that four of the mathematicians leaned towards Platonism, in contrast to the popular notion that Platonism is an exception today. A detailed discussion of this aspect of the research is beyond the scope of this paper; however, I have found that beliefs regarding the nature of mathematics not only influenced how these mathematicians conducted research but also were deeply connected to their theological beliefs (Sriraman, 2004a).

The mathematicians hoped that the results of their creative work would be sanctioned by a group of experts in order to get the work included in the domain (Csikzentmihalyi, 1988, 2000), primarily in the form of publication in a prominent journal. However, the acceptance of a mathematical result, the end product of creation, does not ensure its survival in the Darwinian sense (Muir, 1988). The mathematical result may or may not be picked up by other mathematicians. If the mathematical community picks it up as a viable result, then it is likely to undergo mutations and lead to new mathematics. This, however, is determined by chance!

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Implications It is in the best interest of the field of mathematics education that we identify and nurture creative talent in the mathematics classroom. "Between the work of a student who tries to solve a difficult problem in mathematics and a work of invention (creation)…there is only a difference of degree" (Polya, 1954).

Creativity as a feature of mathematical thinking is not a patent of the mathematician! (Krutetskii, 1976); and although most studies on creativity have focused on eminent individuals (Arnheim, 1962; Gardner, 1993, 1997; Gruber, 1981), I suggest that contemporary models from creativity research can be adapted for studying samples of creativity such as are produced by high school students. Such studies would reveal more about creativity in the classroom to the mathematics education research community. Educators could consider how often mathematical creativity is manifested in the school classroom and how teachers might identify creative work. One plausible way to approach these concerns is to reconstruct and evaluate student work as a unique evolving system of creativity (Gruber & Wallace, 2000) or to incorporate some of the facets suggested by Gruber & Wallace (2000). This necessitates the need to find suitable problems at the appropriate levels to stimulate student creativity.

A common trait among mathematicians is the reliance on particular cases, isomorphic reformulations, or analogous problems that simulate the original problem situations in their search for a solution (Polya, 1954; Skemp, 1986). Creating original mathematics requires a very high level of motivation, persistence, and reflection, all of which are considered indicators of creativity (Amabile, 1983; Policastro & Gardner, 2000; Gardner, 1993). The literature suggests that most creative individuals tend to be attracted to complexity, of which most school mathematics curricula has very little to offer. Classroom practices and math curricula rarely use problems with the sort of underlying mathematical structure that would necessitate students’ having a prolonged period of engagement and the independence to formulate solutions. It is my conjecture that in order for mathematical creativity to manifest itself in the classroom, students should be given the opportunity to tackle non-routine problems with complexity and structure - problems which require not only motivation and persistence but also considerable reflection. This implies that educators should recognize the value of allowing students to reflect on previously solved problems to draw comparisons between various isomorphic problems (English, 1991, 1993; Hung, 2000; Maher & Kiczek, Mathematical Creativity

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

REFERENCES Amabile, T. M. (1983). Social psychology of creativity: A componential conceptualization. Journal of Personality and Social Psychology, 45, 357−376.

Arnheim, R. (1962). Picasso’s guernica. Berkeley: University of California 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: A systems view of creativity. In R. J. Sternberg (Ed.), The nature of creativity: Contemporary psychological perspectives (pp.

325−339). Cambridge UK: Cambridge University Press.

Csikszentmihalyi, M. (2000). Implications of a systems perspective for the study of creativity. In R. J. Sternberg (Ed.), Handbook of creativity (pp. 313−338). Cambridge UK: Cambridge University Press.

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

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

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

Krutetskii, V. A. (1976). The psychology of mathematical abilities in school children. (J. Kilpatrick & I. Wirszup, Eds.; J. Teller, Trans.). Chicago: University of Chicago Press. (Original work published 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.), Teaching and learning mathematical problem solving: Multiple research perspectives (pp. 41–70). Hillsdale, NJ: Erlbaum.

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

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

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

Davis, P. J., & Hersh, R. (1981). The mathematical experience.

New York: Houghton Mifflin.

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

English, L. D. (1991). Young children's combinatoric strategies.

Educational Studies in Mathematics, 22, 451−474.

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

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

Muir, A. (1988). The psychology of mathematical creativity.

Mathematical Intelligencer, 10(1), 33−37.

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

Patton, M. Q. (2002). Qualitative research and evaluation methods.

Thousand Oaks, CA: Sage.

Ernest, P. (1994). Conversation as a metaphor for mathematics and learning. Proceedings of the British Society for Research into Learning Mathematics Day Conference, Manchester Metropolitan University (pp. 58−63). Nottingham: BSRLM.

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.

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

Poincaré, H. (1948). Science and method. New York: Dover.

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 creative people at work. In R. J. Sternberg (Ed.), Handbook of creativity (pp. 93-115). Cambridge UK: Cambridge University Press.

Bharath Sriraman

Nicolle, C. (1932). Biologie de l'invention, Paris: Alcan.

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: Princeton University Press.

Schoenfeld, A. H. (1985). Mathematical problem solving. New York: 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 of Secondary Gifted Education. XIV(3), 151−165.

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

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

Sternberg, R. J. (1985). Human abilities: An information processing 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.), Handbook of creativity (pp. 93−115). Cambridge, UK: Cambridge University Press.

Taylor, S. J., & Bogdan, R. (1984). Introduction to qualitative research methods: The search for meanings. New York: John Wiley & Sons.

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

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

Usiskin, Z. P. (1987). Resolving the continuing dilemmas in school geometry. In M. M. Lindquist, & A. P. Shulte (Eds.), Learning and 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 of Technology Press.

APPENDIX A: Interview Protocol The interview instrument was developed by modifying questions from questionnaires in L’Enseigement Mathematique (1902) and Muir (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 your mathematical development?

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

9. Do you strive to obtain a broad overview of mathematics not of immediate relevance to your area of research?

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

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

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

13. Have your best ideas been the result of prolonged deliberate effort or have they occurred when you were engaged in other unrelated 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 theological issues have been omitted in this protocol. The discussion resulting from these questions are reported in Sriraman (2004a).

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Mathematical Creativity

The Mathematics Educator 2004, Vol. 14, No. 1, 35–41

Getting Everyone Involved in Family Math Melissa R. Freiberg Teachers from the departments of Mathematics and Computer Science, and Curriculum and Instruction at the University of Wisconsin-Whitewater collaboratively developed and implemented an evening math event, Family Math Fun Night, at local elementary schools. As an assignment, preservice elementary teachers developed hands-on mathematical activities, adaptable for different ages and abilities, to engage children and parents. The pre-service elementary teachers presented a variety of activities at each school site to small groups of families and school personnel. This paper outlines the purpose, structure, and benefits of the project for all its participants.

In an age when we continually hear about the necessity of parent awareness and involvement in schools, there are still limited connections among schools, parents, and higher education institutions. It is especially important for parents and teachers to be aware of the premises and types of activities that support effective mathematics learning as advocated by the National Council of Teachers of Mathematics (NCTM, 2000). However, many parents did not grow up learning in ways the NCTM advocates; they see hands-on activities as a fun “waste of time” rather than an avenue for providing conceptual underpinnings for mathematics. Teachers must realize that fun hands-on activities, though motivating for students, must also have mathematical integrity in order to be included in the curriculum. To facilitate both parents and teachers reaching these goals, our university presents what we call Family Math Fun Night (FMFN) at area elementary schools.

Numerous schools and districts report using some variation of Family Math to help parents understand their children’s mathematics curricula better (Wood, 1991, 1992; Carlson, 1991; Pagni, 2002; Kyle, McIntyre, & Moore, 2001). Our program is a variation of Stenmark, Thompson, and Cossey’s (1986) Family Math. In contrast to their Family Math, we choose to have our preservice teachers present activities at elementary schools. This provides our preservice teachers with an opportunity to have a positive, early experience in schools and allows them to test ideas about mathematics education they have learned in their university classes. Also, FMFNs provide an Melissa Freiberg is an associate professor in the Department of Curriculum and Instruction at the University of WhitewaterWisconsin. She has a PhD in Urban Education with an emphasis in teacher education. Her research interests are teacher induction and hands-on learning.

Melissa R. Freiberg

entertaining family experience centered on academics with very little expense to or preparation by the school.

Finally, FMFNs offer a unique opportunity for professional interaction among university and school faculties and staff.

At the University of Wisconsin-Whitewater, we require the FMFN project for students enrolled in the Math for Elementary Teachers content courses and provide it as an optional project for students enrolled in the elementary mathematics methods course. Since students take the Math for Elementary Teachers courses in their freshman or sophomore year, FMFN is a good way to get preservice teachers thinking about the content they are going to teach. Also, the experience supports the developmental view of mathematics learning presented in the content course and provides an experiential background for students in the methods courses. The preservice teachers use activities from the Family Math books we keep on reserve, and we encourage students to devise or find activities from other sources. The preservice teachers in the methods courses are especially encouraged to examine professional journals and databases in preparation for their projects.

Parent-teacher groups at schools provide a small amount of funding (usually about $25) to purchase stickers, pencils, erasers, etc. for prizes; though some pre-service teachers buy their own, and many preservice teacher groups do not give out prizes at all. The lack of prizes does not seem to affect the popularity of the activities for most children. For past FMFNs, we have received small grants from NASA to devise activities that have a space theme. We have not designated a theme for the event since, but have found that a theme often emerges. For example, we have had FMFNs whose activities revolve around sports and FMFNs whose activities relate to voting.

Reflecting on our version of FMFN raises points of interest that are worth sharing: (1) the types of 35

activities that are presented at the events and what determines their quality, (2) what considerations are necessary for coordinating a FMFN, and (3) what can be learned as a result of the experiences. In the following sections, I will attend to each of these categories.

Types of Activities For each school site, the university organizers provide two activities in addition to those the preservice teachers present. The first activity uses jars containing snacks that are taken to the school a week prior to the FMFN. Jars of varying shape are used for different age levels. Each class within an age level estimates the number of snacks in the jar and records its estimate. During the FMFN, individual students or parents can make estimates and enter them for a particular class. The class with the closest estimate receives the snacks. This activity serves two purposes.

The first purpose is to generate interest in and awareness of the event and encourage participation.

The second purpose is to support NCTM’s efforts (NCTM, 2000) by emphasizing estimation skills. The second activity provided by the university requires a school representative to greet children and parents at the door and ask them to add a sticker to his or her birth month on a pre-designed bar graph. This helps take attendance for the evening and also helps children see the process of data collection and how a graph evolves from the process.

Preservice teachers design all other activities, and their activities must involve mathematics concepts covered in their math classes (Math For the Elementary Teachers I—numeration, whole number and fraction operations, problem solving; Math For the Elementary Teacher II—geometry, measurement, probability, and statistics). The types of projects the preservice teachers choose to present usually fall into the categories of drill and practice, problem solving, or estimation. I will discuss types of activities that fall into each category and then discuss two exceptional activities that do not fall into any of the three categories.

Drill and Practice Although students are charged (and monitored) to do more than BINGO or flash cards as the essence of the activity, drill and practice may be part of the activity. Pre-service teachers’ initial attempts at creating these activities are generally weak but with coaching or feedback, they develop more thoughtprovoking activities. Rich activities designed to 36

incorporate drill and practice are usually presented in the context of a game. For example, one student group used a plastic bowling set to practice: addition and subtraction facts with younger children, how to keep a running total with slightly older children, and how to identify fractions and percents for upper elementary children. Other examples of drill and practice activities are educational video games in which correct responses help students reach a goal (fuel for the spacecraft, money to buy souvenirs, moving closer to a target, etc.). These activities allow children to pick the difficulty of the task and move through different levels of calculation, building their self-confidence and knowledge. We encourage preservice teachers to broaden their activities to include topics such as geometry, estimation, logic, patterns, graph interpretation, and computation since all are important to review. Board games are yet another way to support drill and practice activity. The board is laid out on the floor so that students walk around it landing on spaces.

When a student is on a space, he or she is asked a mathematics question that varies depending on the age of the student.

Problem Solving Examples of problem solving activities are games from which preservice teachers create adaptations.

Preservice teachers like to challenge themselves with games that incorporate mathematical ideas and skills and then adapt them to the skill level of the children.

Adaptations of games such as Yahtzee® Equations® or 24® help children plan and carry out different strategies. Memory games, similar to Concentration® are used to match fractions to decimals, operations to results, or various representations of numbers. These games1 are inexpensive to produce, easy to explain, and easily adaptable for different ages and grade levels.

A second example of a problem solving activity is asking children to identify or copy patterns in beads, pictures, tessellations, or shapes. Bead stringing is commonly used to demonstrate patterns. The youngest children describe and extend simple patterns while somewhat older children choose a preset pattern and string beads to illustrate the pattern. The oldest group of students designs bead strings that contain multiple patterns such as combining patterns of color with patterns of shape or size. This activity is more expensive because children keep the materials they use to make the bracelets or necklaces.

Family Math

Estimation In addition to the introductory snack estimation activity, almost every FMFN has at least one preservice teacher designed activity that asks children to estimate capacity, weight, area, and/or quantity. One popular activity requires children to estimate through the use of indirect measurement. In this activity, there are approximately 15 objects to measure and the characteristics of objects vary in difficulty according to children’s differing abilities.

In recent years, we have seen a growing number of activities that use estimation to help students develop probability concepts. These activities illustrate our preservice teachers’ increased awareness of the importance of estimation and probability as well as their increased confidence in students’ abilities to do such activities. In these activities, children are asked how frequently an event happens or how close an estimated answer is to the correct solution.

Exceptional Activities Two exceptional activities from the past do not fall into any of the above categories. They are exceptional because they are unique and demonstrate the creativity of the preservice teachers who made them. The first was presented in one of the first FMFNs we ran.

Preservice teachers, with the help of the students, used math symbols to represent letters of each child's name on a nametag. Children were then told to see if they could figure out other people’s names by equating the letter 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 that corresponded to the letter “a”, “n”, and “e” and could use this to deduce the names of other people.

The second exceptional activity had three pictures made up of geometric shapes. Children were given a paper shape and asked to match their paper shape with the shape in one of the pictures. The youngest children had shapes that were congruent to shapes in the picture, while older students were asked to find shapes similar to their shape but that differed in size, color, or orientation. The preservice teachers prompted children to name the shape and describe its attributes. This activity proved quite challenging for children but was extremely popular.

Coordinating an FMFN In organizing FMFNs, we have discovered that communication among all the parties involved is essential. We have developed guidelines and a timeline to facilitate communication, to give schools and Melissa R. Frieberg

preservice teachers a clear understanding of expectations, and to detail past problems we have faced. Since incorporating FMFN into our curriculum, we have identified objectives and assessments assuring that FMFN activities are mathematically sound (see Appendix A and Form A). The most frequent problems we encounter revolve around logistics such as coordinating transportation to schools, advertising the event in the community, and setting up the school space. The following steps are used to conduct our FMFN events and might be helpful for those who want to organize similar work: 1. Contact schools that might be interested in hosting the event. We contact school districts through direct mailings or use various connections our department has to area schools. After several years of conducting three FMFNs each semester, most schools contact us to schedule the event.

2. Information about FMFN is given to our preservice teachers with their class syllabus. The preservice teachers are allowed to choose the topic around which they will make their activity (within guidelines mentioned earlier). Groups may be made up of students from different classes requiring FMFN or from classes that offer it as an optional activity.

3. The preservice teachers turn in a description of their activities (see Form A and Evaluation Form) indicating how it will be adjusted for various ages/grades, how parents will be involved, and how they will assess the success of their activities. This allows the faculty to assess the activities for mathematical integrity and avoid redundancies in activities. It also gives students a foundation for writing their reflections on the event (See number 8).

4. We assign our preservice teachers to specific dates and schools based on preferences and class schedules.

Groups are usually made up of three to four people and about twenty groups are assigned to each school.

5. We confirm who is assigned to each school and allow groups to indicate special set-up needs (see Form B). The preservice teachers indicate if they are able to provide transportation to the schools so car pools can be established.

6. A faculty member or preservice teacher visits each school to determine space and resource availability, to discuss the role of the school staff, and to give suggestions for advertising the event. We suggest the school connect FMFN with a regularly scheduled PTA meeting. Sending reminders home with school 37

children, having the event on the school calendar, and writing an article in the school or local newsletter explaining the event are ways that have been effective in bringing FMFN to the attention of parents.

7. The night of the event, university and school personnel monitor the preservice teacher groups and the families attending. At the close of the event, we announce the winning class for the estimation exercise and leave activity kits at the school for classroom use.

8. Each university preservice teacher group turns in a written reflection of impressions of the event. This report is not only helpful in assessing the university students' learning, but also helps us identify problems that might need to be addressed in the future. This report focuses on the content and success of the activity, how students handled problems and questions that arose, how students interacted with parents and teachers, and how they collaborated with their groups.

9. An individual report is also required from each preservice teacher. This report is focused on how the student felt the group process worked, what was learned about mathematics, and a self-reflection about one's ability as a teacher.

Conclusions In the introduction, we stated that we found this activity to be beneficial to university preservice teachers, university faculty and staff, school staff, parents, and especially children. Although this paper is not intended to present a research study on FMFN, we believe that we have seen beneficial results for those involved.

The university students have consistently, and almost unanimously, responded positively to their participation in FMFN both in their reports and in class discussions. Even students who described themselves as poor math students found the experience to be enjoyable and uplifting. They appreciated the chance to work with a small group of elementary students. As one student said, "I found that helping them [the children] out with solving a problem was an interesting and rewarding experience...this is what teaching is all about." Many university students were especially surprised and buoyed by the fact that they were able to adjust questions, offer hints and assistance, or explain mathematical ideas more easily than they anticipated.

They also learned how to share responsibilities, ask for help, and make changes to their activities as needed.

Too often preservice teachers believe these things are a sign of weakness rather than a sign of collaboration.

FMFN helps change that perception.

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One of the most rewarding results of this experience for the university faculty and staff is the opportunity to work collaboratively across departments and colleges. College of Education faculty/staff who teach the elementary mathematics methods courses assist faculty and staff from the Mathematics Department in planning, implementing, assessing, and revising the program. Additionally, the experience provides an opportunity for the Mathematics Department members to visit local elementary schools with teachers and children. Education faculty and staff who do regular supervision of student teachers in schools get to see students' abilities to teach to a variety of ages and abilities, which requires flexibility and instant adaptations that might be missed in single grade level settings.

Teachers, administrators, and parents are effusive in their praise for the event. The university students mention that they often have classroom teachers waiting “like vultures” to pick up the activity at the end of the night. Alternately, classroom teachers give university students ideas for improving or adapting the projects for different children’s needs or abilities.

Administrators find that the turnout for this event is higher than for other school sponsored programs and, interestingly, draws more fathers. We average about 200 participants at each event, even in schools where there are fewer than 300 students.

Parents have a varied level of involvement in activities from merely standing and waiting to sitting down and participating with their children in the activity. Many times parents mention that they are surprised at how well their children performed on a given task or how well they thought through a problem. In rare instances parents appear to be impatient or negative with respect to their children’s efforts, and the university students get their first chance to try out their mediating skills. Although not a benefit to children, parent outbursts do give university students an opportunity to see how parents influence children’s learning.

Most importantly, it appears that the elementary children who attend FMFN come away satisfied.

University and school faculty have observed that students almost universally leave the event feeling successful and empowered in math. Certainly children fearful in math are less likely to attend, but we have watched children start out very tentatively and soon find themselves immersed in an activity. Virtually all the children at each event try every activity, but they return to certain activities—and these are rarely the easiest activities. This behavior indicates that students Family Math

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

In conclusion, we have found that all of us have gained from the experiences. As university instructors we continually need to listen to our students in order to adapt and refine the expectations and requirements for FMFN. As prospective mathematics teachers, our students have the chance to devise and carry out activities in a low-stress, supportive atmosphere.

Schools and teachers are provided with examples of activities that complement classroom instruction.

Parents see how their children’s active involvement in activities enhances their learning, and parents may come away with a better understanding of the mathematics curriculum. Finally, children always seem to walk away feeling successful and eager to move on to the next level in mathematics.

REFERENCES Carlson, 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). Principles and standards for school mathematics. Reston, VA: Author.

Pagni, D. (2002). Mathematics outside of schools. Teaching Children 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, from Galileo database (ISSN 0147-2453; No. 9705276559).

Wood, J. (Ed.). (1991). “Family math” teaches English as well as math. 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 1

Equations® is a game in which a specific number of cards are drawn. The cards have whole numbers on them, and students are to arrange the cards and determine operations that will create an equation. 24® is a similar game in which each card has four whole numbers that, when using different operations on the numbers, will equal 24. Concentration® is a game in which a set of cards is placed face down in an array, and players take turns turning up two cards at a time looking for pairs. In commercially made games these are usually identical pictures; however, in educational games these may be two equivalent numbers using different symbols or representations.

Melissa R. Frieberg

Family Math Fun Night Project Requirements Your grade for this project is based on 80 points. The numbers following the due dates below indicate the points that can be earned on each portion of the project.

Jan 31 (5 points) Form A – Group Membership and Activity Idea Hand in one copy of Form A to each instructor of members of your group. Your Group ID Code will be assigned when returned.

February 17 for District #1 (School A), February 19 for all others (20 points) Activity Description Typed 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? Include sample problems and activities for each level.

If adults accompany children at the event, how will the adult participate in your activity?

If prizes are used, how will they be awarded? Who will supply the prizes?

How will you evaluate different aspects of your activity? Refer to the attached evaluation sheet used by faculty and questions listed below.

Feb 24 (School A), Feb 26 (School B), March 5 (School C), March 12 (School D) Form B – Needs List Hand in one copy of Form B to each instructor of members of your group.

Mar 7 (School A), March 17 (School B), March 31 (School C), April 7 (School D) (5 pts) FMFN Evaluation Form Hand in two copies of FMFN Evaluation Form to your group leader’s instructor with answers completed for the questions on the right side of the form.

Mar 11 (School A), March 20 (School B), April 3 (School C), April 10 (School D) (30 points) FMFN Event Run your activity (6:30-8:00 p.m. in School A & B, 6:00 to 7:30 in School 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 groups during the evening.

Mar 19 (School A), Apr. 4 (School B), April 11 (School C), April 16 (School D) (20 points) Individual Evaluation Sorry, no evaluations will be returned until all evaluations have been graded.

Your individual evaluation of the group learning activity (2 to 3 pages) should include: Your name, Group ID Code, your activity name, school attended, group members’ names and their instructor, if other than your instructor, and method(s) used to evaluate your activity.

Did your group work well together? Why or why not? How well did you work within your group? What part of the project did you do?

Briefly state what the math concepts were that you were integrating into your activity. Was this activity an effective means to convey these concepts to the student? How could your activity be adapted for use in a classroom?

What strategies did you see students use? What strategies did you use to help them succeed?

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Did things go as planned during FMFN? What did you not anticipate?

How did you modify/adjust your activity during the evening to meet the needs of the students/parents? Include specific examples of 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 are planning to teach? Is teaching at this level still your goal? Why or why not?

Grammar and other English mechanics will count.

Form A Group Membership and Activity Idea (Spring 2003) Due: Friday, January 31, 2003

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 host school: Our group would prefer to be located (please check one and give your 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

Value: 5 point

Appendix A

Assigned Group ID Code: Please turn in one copy of this form to each teacher of a member of your group. (Group ID Code will be assigned after you submit Form 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 evening classes, sports schedules, previous commitments, and work schedules of all members of the group in making your selections. If your group requires a particular time, please explain the circumstances. You will not be allowed to switch assignments after they have been made unless you can find a group able to exchange with you. ___Our group has no preference of night presentation; any night will work for us. ___Our group would prefer the following nights: (Please circle first and second choices, and give reasons in space to the right.)

Does your group have transportation for FMFN? yes

no

Could your group provide transportation for others the evening of FMFN? yes no

Form B Family Math Fun Night: Needs List Instructor(s) Group ID Code Due: Please turn in one copy of this form to each teacher of a member of your 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 tables with attached benches Chairs - remember most elementary teachers do not sit down Tape, scissors, pencils, paper, scrap paper, markers, etc - please bring your own !!

Our group will need to have (please indicate how many) Table (zero or one): Chairs:

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This 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 that mathematics is an essential part of their everyday life and can be FUN!! Most importantly, it provides the opportunity for you to be involved with elementary children as they do mathematics in enjoyable problem solving activities.

As a member of a group, you will be presenting an activity for Family Math Fun Night (FMFN) at one of four elementary schools: School A (PK-5, 300 students); School B (K-5, 280 students; and School C (K-3, 400 students). All children from these schools and their families will be invited to attend from 6:30 - 8:00 p.m. (6:00 to 7:30 in one school). The fourth elementary presentation is from 1:30 to 3:00 at School D. We will run all events like a carnival having booths (tables) set up with various activities. There will not be a whole group presentation. You should plan to be at your school at least one half hour early. This will allow you time to set up your activity, and to visit and enjoy the activities of other groups before the children and parents arrive. You should be cleaned-up and out of the school 30 minutes after the closing time.

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

SELECTING AN ACTIVITY. Your group should select an activity that is accessible and meaningful to the full range of students in attendance. If you are unsure what is taught at various grade levels, do some research in the LMC on the lower level of the library. Your activity should be fun and challenging for students and parents and need not be competitive. It should involve problem solving, not merely mechanics or facts. Flash card type drill is not usually fun, and is not appropriate for a FMFN activity. Be sure to involve parents in your activity; parents should be doing not just watching. “Helping by giving hints and encouragement” is not sufficient adult involvement. Your activity will need to be planned with space limitations in mind. Plan on setting up on one six to eight-foot table. Please also realize there will be about twenty activities in a gym-sized room; consider how your activity and its Family Math

sounds and lights will affect others. You are not to present an activity with music, popping balloons or other distractions for neighboring groups. Be aware of copyright laws! For example, the latest cartoon characters may attract elementary students, but may be an infringement of copyright. Invent a clone! Be creative! Don't just take an activity from a book or off a shelf; put something of yourself into it. Don’t just use the activity you, or a friend, used last semester. Math 148 and 149 students should develop an activity that involves math topics they will be covering in class. Realizing that there are many connections between the mathematics in the two courses, this does not exclude presenting a topic from your course in an activity that also uses a topic from the other course.

Take this opportunity to develop an activity you could use in your future classroom. Please do not use TWISTER activities.

The book Family Math has been placed on reserve (2 hour, no overnight) in the library. You will need to ask for it by name at the main circulation desk. This book has over 100 Family Math activities. You may wish to use one of these, combine a couple, modify one, or come up with an idea on your own. You could also check Teaching Children Mathematics, other periodicals, and the Internet for ideas. Make this a fun learning experience for you!

WHAT YOU WILL NEED. Your group must have a sign with the name of your activity. You may need to make some equipment to be used at your booth such as markers, counters, game board, etc. Other things such as pencils, scissors, ruler, scrap paper, and manipulatives are also useful. The LMC has some equipment that can be checked out. If they cannot meet your needs, your instructor may have some ideas. You may also want to have copies of handouts, problems, or puzzles available for parents/teachers to take home. Remember these are activities for the children and parents, 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 your FMFN. The PTO's of the various schools have given us some money with which to purchase small prizes - pencils, erasers, stickers, etc. These will be divided among the groups requesting them. There will not be a large number of prizes per group. Please limit the candy your group plans to use; not all children are allowed candy, especially after supper. Many groups in the past have presented very successful activities without prizes. Do not spend a lot of money purchasing prizes. The students should be having fun doing math -- NOT seeing who can accumulate the most/best prizes!

EVALUATION. Three-quarters of your grade will be assigned through group work. If your group contains members from more than one class, some written work must be submitted to each instructor involved. Your group will supply two copies of the FMFN Evaluation Form a few days prior to your activity night. A copy of this form is attached. On the night of your presentation, faculty attending FMFN will evaluate your project. A week after your FMFN, a typed individual reflective evaluation is to be submitted. Select a method to help you evaluate your activity. You may get written evaluations from students and parents; keep a journal of student/parent reactions during the evening, etc.

March 11, the first FMFN, is only SEVEN weeks away! It is time to get started selecting a group and an activity NOW. The deadline for forming groups and selecting an activity is January 31st.

Melissa R. Frieberg

Evaluation Form Submit two copies to your group leader’s instructor Activity Date: Instructor(s): Activity Name: Group Members: Faculty evaluators will use the following portion (and rate between 1 and 5).

Math content: (Problem solving, concept development, more than mechanics) Adaptability of project: (Grade level, special needs, mental, written and 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 space provided: 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 your materials.

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The Mathematics Educator 2004, Vol. 14, No. 1, 42–46

Solving Algebra and Other Story Problems with Simple Diagrams: a Method Demonstrated in Grade 4–6 Texts Used in Singapore Sybilla 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 involve social aspects. However, the mathematics texts used in Singapore present some interesting, accessible problemsolving methods, which help children solve problems in ways that are sensible and intuitive. Could the texts used in Singapore be a significant factor in children’s mathematics achievement? There are some reasons to believe so. In this article, I give reasons for studying the way mathematics is presented in the elementary mathematics texts used in Singapore; show some of the mathematics problems presented in these texts and the simple diagrams that accompany these problems as sense-making aids; and present data from TIMSS indicating that children in Singapore are proficient problem solvers who far outperform U.S. children in problem-solving.

Why Study the Methods of Singapore’s Mathematics Texts?

What is special about the elementary mathematics texts used in Singapore? These texts look very different from major elementary school mathematics texts used in the U.S. The presentation of mathematics in Singapore’s elementary texts is direct and brief.

Words are used sparingly, but even so, problems sometimes have complex sentence structures. The page layout is clean and uncluttered. Perhaps the most striking feature is the heavy use of pictures and diagrams to present material succinctly—although pictures are never used for embellishment. Simple pictures and diagrams accompany many problems, and the same types of pictures and diagrams are used repeatedly, as supports for different types of problems, and across grade levels. These simple pictures and diagrams are not mere procedural aids designed to help children produce speedy solutions without understanding. Rather, the pictures and diagrams appear to be designed to help children make sense of problems and to use solution strategies that can be justified on solid conceptual grounds. Because of this pictorial, sense-making approach, the elementary texts Sybilla Beckmann is a mathematician at the University of Georgia who has a strong interest in education. She has developed three mathematics content courses for prospective elementary teachers and has written a textbook, Mathematics for Elementary Teachers, published by Addison-Wesley, for use in such courses. In the 2004/2005 academic year, she will teach a class of 6th grade mathematics daily at a local public middle school.

Sybilla Beckman

used in Singapore can include problems that are quite complex and advanced. Children can reasonably be expected to solve these problems given the problemsolving and sense-making tools they have been exposed to.

Thus the strong performance of Singapore’s children in mathematics may be due in part to the way mathematics is presented in their textbooks, including the way simple pictures and diagrams are used to communicate mathematical ideas and to provide sensemaking aids for solving problems. If so, then teachers, mathematics educators, and instructional designers in the U.S. will benefit from studying the presentation of mathematics in Singapore’s textbooks, so that they can help children in the U.S. improve their understanding of mathematics and their ability to solve problems.

Using Strip Diagrams to Solve Story Problems One of the most interesting aspects of the elementary school mathematics texts and workbooks used in Singapore (Curriculum Planning and Development Division, Ministry of Education, Singapore, 1999, hereafter referred to as Primary Mathematics and Primary Mathematics Workbook) is the repeated use of a few simple types of diagrams to aid in solving problems. Starting in volume 3A, which is used in the first half of 3rd grade, simple “strip diagrams” accompany a variety of story problems.

Consider the following 3rd grade subtraction story problem:

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Mary made 686 biscuits. She sold some of them. If 298 were left over, how many biscuits did she sell?

(Primary Mathematics volume 3A, page 20, problem 4)

A farmer has 7 ducks. He has 5 times as many chickens as ducks….How many more chickens than ducks does he have? (Primary Mathematics volume 3A, page 46, problem 4)

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

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

Figure 1: How Many Biscuits Were Sold?

Figure 3: How Many More Chickens Than Ducks?

On the next page in volume 3A is the following problem: Meilin saved $184. She saved $63 more than Betty.

How much did Betty save? (Primary Mathematics volume 3A, page 21, problem 7)

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

Figure 2: How Much Did Betty Save?

These two problems are examples of some of the more difficult types of subtraction story problems for children. The first problem is difficult because we must take an unknown number of biscuits away from the initial number of biscuits. This problem is of the type change-take-from, unknown change (see Fuson, 2003, for a discussion of the classification of addition and subtraction story problems). The second problem is difficult because it includes the phrase “$63 more than,” which may prompt children to add $63 rather than subtract it. This problem is of type compare, inconsistent (see Fuson, 2003). The term inconsistent is used because the phrase “more than” is inconsistent with the required subtraction. Other linguistically difficult problems, including those that involve a multiplicative comparison with a phrase such as “N times as many as”, are common in P r i m a r y Mathematics and are often supported with a strip diagram. Consider the following 3rd grade problem, which is supported with a diagram like the one in Figure 3:

Sybilla Beckman

Although the strip diagrams will not always help children carry out the required calculations (for example, we don’t see how to carry out the subtraction $184 – $63 from Figure 2), they are clearly designed to help children decide which operations to use. Instead of relying on superficial and unreliable clues like key words, the simple visual diagram can help children understand why the appropriate operations make sense.

The diagram prompts children to choose the appropriate operations on solid conceptual grounds.

From volume 3A onward, strip diagrams regularly accompany some of the addition, subtraction, multiplication, division, fraction, and decimal story problems. Other problems that could be solved with the aid of a strip diagram do not have an accompanying diagram and do not mention drawing a diagram.

Fraction problems, such as the following 4th grade problem, are naturally modeled with strip diagrams such as the accompanying diagram in Figure 4: David spent 2/5 of his money on a storybook. The storybook cost $20. How much money did he have at first? (Primary Mathematics volume 4A, page 62, problem 11)

Without a diagram, the problem becomes much more difficult to solve. We could formulate it with the equation (2/5)x = 20 where x stands for David’s original amount of money, which we can solve by dividing 20 by 2/5. Notice that the diagram can help us see why we should divide fractions by multiplying by the reciprocal of the divisor. When we solve the problem with the aid of the diagram, we first divide $20 by 2, and then we multiply the result by 5. In other words, we multiply $20 by 5/2, the reciprocal of 2/5.

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Raju and Samy shared $410 between them. Raju received $100 more than Samy. How much money did Samy receive? (Primary Mathematics volume 5A, page 23, problem 1)

Figure 4: How Much Money Did David Have?

The problems presented previously are arithmetic problems, even though we could also formulate and solve these problems algebraically with equations. But starting with volume 4A, which is used in the first half of 4th grade, algebra story problems begin to appear.

Consider the following problems: 1. 300 children are divided into two groups. There are 50 more children in the first group than in the second group. How many children are there in the second group? (Primary Mathematics volume 4A, page 40, problem 8) 2. The difference between two numbers is 2184. If the bigger number is 3 times the smaller number, find the sum of the two numbers. (P r i m a r y Mathematics volume 4A, page 40, problem 9) 3. 3000 exercise books are arranged into 3 piles.

The fist pile has 10 more books than the second pile. The number of books in the second pile is twice the number of books in the third pile. How many books are there in the third pile? (Primary Mathematics volume 4A, page 41, problem 10)

These problems are readily formulated and solved algebraically with equations, but since the text has not introduced equations with variables, the children are presumably expected to draw diagrams to help them solve these problems. Notice that from an algebraic point of view, the second problem is most naturally formulated with two linear equations in two unknowns, and yet 4th graders can solve this problem.

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

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Figure 5: Raju and Samy Split Some Money

Notice that the manipulations we perform with strip diagrams usually correspond to the algebraic manipulations we perform in solving the problem algebraically. For example, to solve the previous Raju and Samy problem, we could let S be Samy’s initial amount of money. Then, 2S + 100 = 410

as we also see in Figure 5. When we solve the problem algebraically, we subtract 100 from 410 and then divide the resulting 310 by 2, just as we do when we solve the problem with the aid of the strip diagram.

Strip diagrams make it possible for children who have not studied algebra to attempt remarkably complex problems, such as the following two, which are accompanied by diagrams like the ones in Figure 6 and Figure 7 respectively: Encik Hassan gave 2/5 of his money to his wife and spent 1/2 of the remainder. If he had $300 left, how much money did he have at first? (Primary Mathematics volume 5A, page 59, problem 6) Raju had 3 times as much money as Gopal. After Raju spent $60 and Gopal spent $10, they each had an equal amount of money left. How much money did Raju have at first? (Primary Mathematics volume 6B, page 67, problem 1)

Solving Problems with Simple Diagrams

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

A. 36 B. 48 C. 60 D. 96 (Problem N16, page 19. Overall percent correct, Singapore: 81%, United States: 41%) Figure 6: How Much Money Did Encik Hassan Have at First?

Figure 7: How Much Did Raju Have at First?

Performance of 8th Graders on TIMSS In light of the complex problems that children in Singapore are taught how to solve in elementary school, the strong performance of Singapore’s 8th graders on the TIMSS assessment is not surprising.

Among the released TIMSS 8th grade assessment items in the content domain “Fractions and Number Sense” classified as “Investigating and Solving Problems,” Singapore 8th graders scored higher than U.S. 8th graders on all items. These released items included the following problems (see NCES, 2003): Laura had $240. She spent 5/8 of it. How much money did she have left? (Problem R14, page 29.

Overall percent correct, Singapore: 78%, United States: 25%).

Sybilla Beckman

These problems are similar to problems in Primary Mathematics. The strong performance of Singapore 8th graders on these problems indicates that the instruction children receive in solving these kinds of problems is effective. Similarly, among the released TIMSS 8th grade assessment items in the content domain “Algebra” classified as “Investigating and Solving Problems,” Singapore 8th graders scored higher than U.S. 8th graders on all items.

But the strong problem-solving abilities of Singapore’s 8th graders in fractions and number sense and in algebra does not necessarily result in factual knowledge in other mathematical domains in which the children have not had instruction. For example, U.S. 8th graders scored higher than Singapore 8th graders on the following item in the content domain “Data Representation, Analysis and Probability” classified as “Knowing”: If a fair coin is tossed, the probability that it will land heads up is 1/2. In four successive tosses, a fair coin lands heads up each time. What is likely to 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 the question.

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

The mathematics texts used in Singapore through 8th grade do not address probability. Thus the difference in performance in fraction, number sense, and algebra problem-solving versus knowledge about probability can reasonably be attributed to effective instruction.

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Conclusion The mathematics textbooks used in elementary schools in Singapore show how to represent quantities with drawings of strips. With the aid of these simple strip diagrams, children can use straightforward reasoning to solve many challenging story problems conceptually. The TIMSS 8th grade assessment shows that 8th graders in Singapore are effective problem solvers and are much better problem solvers than U.S. 8th graders. Although cultural factors probably also affect the strong mathematics performance of children in Singapore, children in the U.S. could probably strengthen their problem-solving abilities by learning Singapore’s methods and by being exposed to more challenging and linguistically complex story problems early in their mathematics education.

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REFERENCES Curriculum Planning and Development Division, Ministry of Education, Singapore (1999, 2000). Primary Mathematics (3rd ed.) volumes 1A–6B. Singapore: Times Media Private Limited. Note: additional copyright dates listed on books in this series are 1981, 1982, 1983, 1984, 1985, 1992, 1993, 1994, 1995, 1996, 1997, thus 8th graders who took the 1999 TIMSS assessment used an edition of these books.

Curriculum Planning and Development Division, Ministry of Education, Singapore (1999, 2000). Primary Mathematics Workbook (3rd ed.) volumes 1A–6B. Singapore: Times Media Private Limited.

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

National Center for Education Statistics (2003). Trends in international mathematics and science study. Retrieved May 3, 2004, from http://nces.ed.gov/timss/results.asp and from http://nces.ed.gov/timss/educators.asp

Solving Problems with Simple Diagrams

The Mathematics Educator 2004, Vol. 14, No. 1, 47–51

Book Review…

Diverse Voices Call for Rethinking and Refining Notions of Equity Amy 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 this book reflects a “shift in focus from equity to a more inclusive perspective that embraces social justice as a contested area of investigation within mathematics education” (p. xv). What’s interesting is that the question in the title lacks a verb—is the question “which are ways to social justice in mathematics education?” Or more tentatively, “which ways might bring about social justice in mathematics education?”

Or perhaps the focus is more on research, either up to now or in the future: “which ways have research on social justice in mathematics education taken? Or “which ways could (should?) research on social justice in mathematics education take?” Each of the thirteen chapters in the volume addresses at least one of those four questions. Overall, this book responds to its title question through diverse voices that call for expanding work on gender issues into broader sociocultural, political, and technological contexts; rethinking and refining key notions such as equity, citizenship, and difference; and considering how to conduct studies that reach beyond school and university boundaries toward families, communities, and policy-makers.

The collection is the third volume in the International Perspectives on Mathematics Education series for which Burton has served as series editor.1 In her introduction she describes the origin of the book in the activities of the International Organization of Women in Mathematics Education (IOWME) at the Ninth International Congress of Mathematics Amy Hackenberg is at work on her doctoral dissertation on the emergence of sixth graders’ algebraic reasoning from their quantitative reasoning in the context of mathematically caring teacher-student relations. In addition to her fascination with mathematical learning and the orchestration of it, she is compelled by issues of social justice, the nature and consequences of social interaction, and the relationship between the “social” and the “psychological” in mathematics education.

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

In the introduction Burton also outlines the process by which the book developed. After a general call for papers, an international review panel of mathematics educators reviewed submissions. Chapter authors were then paired to give feedback to each other on their work in order to promote dialogue as well as “crossreferencing possibilities” (p. xv). As perhaps is always the case in an edited book without summary pieces to highlight connections between chapters, the crossreferencing of concepts in this volume could be expanded. Burton does a nice job of drawing some connections in her introduction, but otherwise such resonance is largely left to the reader. Fortunately, as I hope to demonstrate in this review, there is ample opportunity to draw connections between chapters (and also occasionally to wish that an author had heeded another author’s points or ideas!)

Organization of the Book The thirteen chapters in the book are organized into three sections. The four chapters in the first section focus on definitional work, conceptual frameworks, and reviews of and recommendations for research, thereby “setting the scene” (p. 1). The authors of this section are from Australia (Brew), Germany (Jungwirth), the United Kingdom (Povey), and the United States (Hart). The second section consists of seven chapters primarily about studies that take place in classrooms and address the question “what does Book Review

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

The placement of chapters within this organization is a little puzzling. Wiliam’s illuminating chapter on the construction of statistical differences and its implications is included in the second section on classroom studies, but since it grapples with definitions and conceptual ideas (and is not a classroom study), it might have been better placed in the first more theoretically-oriented section. Brew’s chapter, a study about reasons that mothers return to study mathematics, is included in the first section but seems to fit better in the second, despite the fact that the study does not take place in mathematics classrooms.

Support for changing the placement of Brew’s chapter is provided by the position of Mendick’s: Her report of young British men’s choices to study mathematics beyond compulsory schooling is only peripherally located in classrooms and was still placed in the second section.

The other weak organizational aspect of the book is the inclusion of only two chapters in the third section on computers and mathematics learning. One wonders if there were intentions for a more substantial section but some papers did not make the publication deadline.

In any case, because both chapters in this section report on studies set in classrooms, it seems that they could have been included in the second section—or that perhaps two sections about studies might have been warranted, one that focused directly on studies in mathematics classrooms and one that included research on mathematics education outside of immediate classroom contexts.

Conceptually-Oriented Chapters: What Is Equity?

What Is Social Justice?

Organizational difficulties aside, I focus first on the more conceptually-oriented chapters, which are contained in the first three chapters of the first section of the book as well as in Wiliam’s chapter from the second section. These authors engage in definitional and conceptual work that forms a foundation for research on social justice. All four authors ponder the 48

nature of equity and justice within different contexts: a typology of gender-sensitive teaching, previous and current research on equity and justice in mathematics education, citizenship education in the United Kingdom, and statistical analyses of gender differences in mathematics education.

Jungwirth describes a typology of gender-sensitive teaching that consists of three types distinguished by modifications made according to gender, the degree to which gender groups are identified and treated as monolithic, and corresponding conceptions of equity.

In Type I teaching, teachers are “gender-blind” and make no modifications according to gender since they believe that boys and girls can do math equally well. In Type II teaching, teachers adjust practices based on gender but tend to treat students of a single gender as monolithic (i.e., tend to essentialize.) Jungwirth believes that in the third (and implicitly most advanced) type, the concept of equity “no longer applies…Equity here refers to the individual, with respect to learning arrangements and, somewhat qualified, to outcomes” (p. 16). Teachers engaging in Type III teaching attend to individual differences within gender groups and tailor teaching to individuals.

Although Jungwirth’s typology offers a conceptual framework for examining the equitable implications of teachers’ orientations toward mathematics teaching and mathematics classrooms, her dichotomizing of groups and individuals is problematic. For example, in their attention to individuals, might not Type III teachers create classrooms in which mathematics could be devoid of women, which Jungwirth sees as considerably less evolved than even Type I teaching?

The problem seems to be in characterizing equity based on group-individual dichotomies—to adhere too strongly to group identities can result in essentializing, while to focus primarily on the individual can leave out trends and broad characteristics of groups that are important considerations in work toward equity and social justice (cf. Lubienski, 2003).

These issues are reflected in Hart’s review of scholarship on equity and justice in mathematics education over the last 25 years. Her chapter is notable for explicit discussion about different ways researchers have used equity and justice (and equality); for her clearly stated choice to use equity to mean justice; and for her formulation of calls for future research. In particular, she calls for research on pedagogies that contribute to justice; self-study of educators’ own practices; and more research that explores student motivation, socialization, identity, and agency with respect to mathematics. Hart highlights Martin’s Book Review

(2000) study on factors contributing to failure and success of African American students in mathematics as an exemplar for future research because of its multilevel framework for analyzing mathematics socialization and identity. Although her points about his work are well taken, the considerable space she gives to this relatively recent study seems odd given her aims to review 25 years of research.

Povey continues Jungwirth’s and Hart’s definitional work by considering the complex and contested notion of citizenship in relation to social justice and mathematics education. She describes how recent mandates for citizenship education in England reinforce a conservative perspective by focusing on political and legal citizenship (the right to vote, for example), without questioning the nature and character of social citizenship, let alone its connections to “the (mathematics) education of future citizens” (p. 52).

Povey believes that for citizenship to be a useful concept in democratizing mathematics classrooms the concept “will have to be more plural, more active, and more concerned with participation in the here and now” (p. 56).

Perhaps the strongest chapter of these four (and one of the strongest in the collection) is Wiliam’s on the construction of statistical differences in mathematical assessments. He demonstrates that in gender research in mathematics education, effect sizes of standardized differences between male and female test scores are relatively small, and the variability within a gender is greater than between genders. Based on this analysis, Wiliam concludes that differences between genders depend on what counts as mathematics on assessments. In particular, what counts as mathematics may be maintained because it supports patriarchal hegemony.

As an implication of his argument, Wiliam proposes “random justice” (p. 202) to produce equity in selection based on test scores. Wiliam calls the percentage of the population that reaches a certain standard (for, say, entrance to medical school) a recruitment population. Usually, selecting from a recruitment population (i.e., creating a selection population) involves choosing a small top percentage of it. This mode of selection perpetuates selecting more males than females, largely because males show greater variability in their test scores compared to females (males produce more highs and lows.) Wiliam proposes that a random sample of the recruitment population that sustains the gender (or racial, class, etc.) make-up of it is “the only fair way” (p. 204) of creating a selection population. Although this proposal Amy J. Hackenberg

may seem counterintuitive (and certainly differs from typical U.S. selection processes!), Wiliam makes a compelling argument that is worth reading.

Chapters on Studies in or Surrounding Mathematics Classrooms In these chapters—Brew’s chapter from the first section as well as the other 8 chapters in the book—the diverse voices in the volume become quite apparent, not only because of the different geographical locations or ethnic heritages of the authors but because of the diverse ways in which the authors focus on issues of social justice in relation to mathematics classrooms and mathematical study. These nine chapters can also be loosely grouped as exemplifying, supporting, informing, or aligning with the more conceptuallyoriented chapters.

In particular, two chapters that focus specifically on teaching practices in relation to social justice may exemplify and inform Jungwith’s typology. The authors of these chapters attend to how teachers approach students who belong to disadvantaged groups. Chamdimba, whose research took place in the southern African country of Malawi, studied the year 11 students of a Malawian teacher who agreed to use cooperative learning to potentially promote a “learnerfriendly classroom climate” (p. 156) for girls. As a researcher, Chamdimba might exemplify a Type II orientation out of her concerns over Malawian girls’ lack of representation and achievement in mathematics and subsequent Malawian women’s lack of bargaining power as a group for social and economic resources in the country. Chamdimba’s conclusion that female students experienced largely positive effects might help Jungwirth refine her typology so that recognizing students as part of disenfranchised groups and acting on that recognition to address the group is seen as legitimate and useful (i.e., not necessarily less evolved than Type III teaching.) However, Chamdimba’s study is also subject to scrutiny over whether a particular classroom structure can bring about improvements in all Malawian females’ educational, social, and economic status.

Perhaps a better example of the subtlety involved in the group-individual distinctions with regard to social justice is found in Zevenbergen’s study.

Zevenbergen used Bourdieu’s tools as a frame for understanding teachers’ beliefs about students from socially disadvantaged backgrounds in the South-East Queensland region of Australia. Eight of the 9 teachers interviewed expressed views of students as deficient due to poverty and cultural practices. Stretching 49

Jungwirth’s typology beyond gender-sensitivity, the ninth teacher had more of a Type III orientation in her respect for these students as individuals. However, by expressing an understanding of how parents’ lack of cultural capital prevented them from challenging the ways in which schools (under)served their children, this teacher did not ignore these students as belonging to a disadvantaged group. This teacher’s ability to understand and value students as both individuals and part of a group might allow Jungwirth to amplify and further articulate her typology.

These two chapters and three others exhibit work that aligns with Hart’s call for research on pedagogies that contribute to social justice and on one’s own teaching in relation to social justice. Vale’s two case studies of computer-intensive mathematics learning in two junior secondary mathematics classrooms focus on how teachers’ practices with technology impede (but might facilitate) more just classroom environments.

Vale’s work is complemented by the three university classroom studies presented by Wood, Viskic, and Petocz. In studying their own computer-intensive teaching of differential equations, statistics, and preparatory mathematics classes, these three researchers found positive attitudes toward the use of technology across gender. Finally, Ferri and Kaiser’s comparative case study on the styles of mathematical thinking of year 9 and 10 students (ages 15-16) has implications for developing pedagogies that recognize differences other than due to gender, race, or class, and that thereby contribute to justice and diversity in classrooms.

However, Secada, Cueto, and Andrade’s largescale, comprehensive study of the conditions of schooling for fourth and fifth-grade children who speak Aymara, Quechua, and Spanish in Peru may be the strongest example of work toward Hart’s recommendation of multilevel frameworks in research on social justice. These researchers intended to create a “policy-relevant study” (p. 106). To do so they articulated their conceptions of equity as distributive social justice (opportunity to learn mathematics is a social good and should not be related to accidents of birth) and socially enlightened self-interest (it is in everyone’s interest for everyone to do well so as not to cause great cost to society). In addition, the researchers took as a premise that equity must come with both high quality and equality (i.e., lowering the bar does not foster equity). Thus they contribute to definitional work while formulating “practical” conclusions and recommendations for Peruvian governmental policy.

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Finally, the remaining three chapters in the book connect with Povey’s chapter in exploring a particular contested and complex concept or relate to Wiliam’s work on considering the construction of difference.

Brew’s study entails rethinking aspects of the complex concept of mothering in the context of mathematical learning of both mothers and their children. By including voices of the children in the study, Brew is able to show the fluid roles of care-taking between studying mothers and their children (e.g., children sometimes acted as carers for their mothers) and “the pivotal role that children can play…in providing not only a consistent motivating factor but also enhancing their mother’s intellectual development” (p. 94).

What Povey does for citizenship and Brew does for mothering, Mendick does for masculinity in the context of doing mathematics. In a very strong and thoughtful chapter, she describes stories of three young British men who have opted to study mathematics in their A-levels even though they do not enjoy it.

Mendick’s smart use of a poststructuralist perspective that deconstructs the classic opposition between structure and agency allows her to argue that taking up mathematics is a way for the men to “do masculinity” in a variety of ways: to prove their intelligence to employers and others as well as to secure a future in labor market. The stories of the three males prompt the question: “why is maths a more powerful proof of ability than other subjects?” (p. 182). To respond, Mendick contrasts the men’s stories with young women’s stories (part of her larger research project.)

This artful move is not intended to draw dichotomies between how men and women “do maths” differently—Mendick cautions against such simplistic conclusions and notes that some females use mathematics the way these three males do. Instead the contrast allows her to demonstrate and deepen her theorizing of masculinity as a relational configuration of a practice, as well as to argue for more complexity in gender reform work. Thus for her, “maths and gender are mutually constitutive; maths reform work is gender reform work” (p. 184). By examining gender in this way, like Wiliam, she calls into question differences between males and females in relation to mathematics and supports his contention that what counts as mathematics (and, Mendick would add, as masculine and feminine) is the basis for these differences.

Differences between males and females are also the subject of the chapter by Fogasz, Leder, and Thomas. They used a new survey instrument to capture the beliefs of over 800 grade 7–10 Australian students Book Review

regarding gender stereotyping of mathematics. Their findings revealed interesting reversals of expected (stereotyped) beliefs. For example, their participants believed that boys are more likely than girls to give up when they find a problem too difficult, and that girls are more likely than boys to like math and find it interesting. However, through an examination of participation rates and achievement levels of male and female grade 12 mathematics students from 1994 to 1999 in Victoria, Australia, the researchers refute recent, media-hyped contentions (see, e.g., Conlin, 2003; Weaver-Hightower, 2003) that males are now disadvantaged in mathematics. Frankly, Fogasz and colleagues might have benefited from Wiliam’s advice on examining effect size—it is hard to know how much significance to give to the differences they found.

Nevertheless, their work supports the notion that mathematics may be maintained as a male domain despite certain advances of females.

Overall, I agree with Burton that the chapters in this volume achieve the goal of providing “an introduction for new researchers as well as stimulation for those seeking to develop their thinking in new or unfamiliar directions” (p. xiii). Although the organization is a bit puzzling and some chapters are clearly stronger than others, the book is a useful read for researchers in mathematics education. More important, the diversity of voices—and the connections that readers can draw among this diversity—gives a complex and layered picture of how resources, sociocultural contexts, governmental policy, teacher and student practices, human preferences and expectations, and researchers’ theorizing and interpretations, all contribute to “…who does, and who does not, become a learner of mathematics” (p. xviii).

Amy J. Hackenberg

REFERENCES Conlin, M. (2003, May 26). The new gender gap. Business Week online. Retrieved September 1, 2003, from http://www.businessweek.com Lubienski, S. T. (2003). Celebrating diversity and denying disparities: A critical assessment. Educational Researcher, 32(8), 30–38.

Martin, D. B. (2000). Mathematics success and failure among African-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 on gender and education. Review of Educational Research, 73(4).

471–498.

1

The first volume was Multiple Perspectives on Mathematics Teaching and Learning (2000) edited by Jo Boaler; the second volume was Researching Mathematics Classrooms: A Critical Examination of Methodology (2002) edited by Simon Goodchild and Lyn English.

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CONFERENCES 2004…

CMESG/GCEDM Canadian Mathematics Education Study Group http://plato.acadiau.ca/courses/educ/reid/cmesg/cmesg.html

Universite Laval Quebec, Canada

May 28–June 1

HIC The 3rd Annual Hawaii International Conference on Statistics, Mathematics and Related Fields http://www.hicstatistics.org/index.htm

Honolulu, Hawaii

June 9–12

EDGE Symposium Graduate School Experience for Women in Mathematics: From Assessment to Action http://www.edgeforwomen.org/symposium.html

Atlanta, Georgia

June 25–26

AMESA Tenth Annual National Congress http://www.sun.ac.za/MATHED/AMESA/AMESA2004/Index.htm

Potchefstroom, South Africa

July 1–4

ICOTS7 International Conference on Teaching Statistics http://www.maths.otago.ac.nz/icots7/layout.php

Salvador, Brazil

July 2–7

ICME – 10 The 10th International Congress on Mathematics Education http://www.icme-10.dk

Copenhagen, Denmark

July 4–11

HPM History & Pedagogy of Mathematics Conference http://www-conference.slu.se/hpm/about/

Uppsala, Sweden

July 12–17

PME-28 International Group for the Psychology of Mathematics Education http://home.hia.no/~annebf/pme28/

Bergen, Norway

July 14–18

JSM of the ASA Joint Statistical Meetings of the American Statistical Association http://www.amstat.org/meetings

Toronto, Canada

August 8–12

CABRI 2004 Third CabriGeometry International Conference http://italia2004.cabriworld.com/redazione/cabrieng2004

Rome, Italy

September 9–12

GCTM GCTM Annual Conference http://www.gctm.org/georgia_mathematics_conference.htm

Rock Eagle, Georgia

PME-NA North American Chapter International Group for the Psychology of Mathematics Education http://www.pmena.org

Toronto, Canada

October 21–24

SSMA School Science and Mathematics Association http://www.ssma.org

College Park, Georgia

October 21–23

AAMT 2005 Australian Association of Mathematics Teachers http://www.aamt.edu.au/mmv

Sydney, Australia

January 17–20 2005

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October 14–16

Book Review

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In this Issue, Guest Editorial… Do You Need a PhD to Teach K–8 Mathematics in Ways Respected by the Mathematics Education Community?

CHANDRA HAWLEY ORRILL Mathematics as “Gate-Keeper” (?): Three Theoretical Perspectives that Aim Toward Empowering All Children With a Key to the Gate DAVID W. STINSON The Characteristics of Mathematical Creativity BHARATH SRIRAMAN Getting Everyone Involved in Family Math MELISSA R. FREIBERG In Focus… Solving Algebra and Other Story Problems with Simple Diagrams: a Method Demonstrated in Grade 4–6 Texts Used in Singapore SYBILLA BECKMANN Book Review… Diverse Voices Call for Rethinking and Refining Notions of Equity AMY J. HACKENBERG