Mathematics As “Gate-Keeper” (?): Three Theoretical Perspectives That Aim Toward Empowering All Children with a Key to the Gate DAVID W

____ 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 Dear TME readers, Holly Garrett Anthony 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 Associate Editors veteran and budding scholars in mathematics education. The articles range in topic and thus invite all Ginger Rhodes those vested in mathematics education to read on. Margaret Sloan Both David Stinson and Amy Hackenberg direct our attention toward equity and social justice in Erik Tillema 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 Publication their roles in empowering students. Hackenberg’s review of Burton’s edited book, Which Way Social Stephen Bismarck Justice in Mathematics Education? is both critical and engaging. She artfully draws connections across Laurel Bleich chapters and applauds the picture of social justice painted by the diversity of voices therein. Dennis Hembree Two invited pieces, one by Chandra Orrill and the other by Sybilla Beckmann, ask mathematics Advisors educators to step outside themselves and reexamine features of PhD programs and elementary Denise S. Mewborn textbooks. Orrill’s title question invites mathematics educators to consider what we value in classroom Nicholas Oppong 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 James W. Wilson 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. MESA Officers Finally, Bharath Sriraman and Melissa Freiberg offer insights into the creativity of 2004-2005 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 President of five well-published mathematicians in the production of mathematics. Freiberg reminds us of the Zelha Tunç-Pekkan daily challenge of mathematics educators—providing preservice teachers rich classroom Vice-President experiences—and details the organization, coordination, and evaluation of Family Math Fun Nights in Natasha Brewley elementary schools. It has been my goal thus far to entice you to read what follows, but I now want to focus your Secretary attention on the work of TME. I invite and encourage TME readers to support our journal by getting Amy J. Hackenberg 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. Treasurer Last I would like to comment that publication of Volume 14 Number 1 has been a rewarding Ginger Rhodes process—at times challenging—but always worthwhile. I have grown as an editor, writer, and scholar. NCTM I appreciate the opportunity to work with authors and editors and look forward to continued work this Representative Fall. I extend my thanks to all of the people who make TME possible: reviewers, authors, peers, Angel Abney faculty, and especially, the editors. Undergraduate Holly Garrett Anthony Representatives 105 Aderhold Hall [email protected] The University of Georgia www.ugamesa.org Erin Bernstein Athens, GA 30602-7124 Erin Cain Jessica Ivey

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: [email protected] 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

8 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

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 and middle school mathematics in ways that about an article in which Ball (1991) provided mathematics educators would value?” After all, the descriptions of three teachers’ approaches to working Balls, Lamperts, and McClains1 in the literature offer with their students. In Ball’s article, teachers without high-quality mathematics instruction, attend to student PhDs in mathematics or mathematics education thinking, provide opportunities for knowledge struggled to engage their students in developing construction, and introduce students to a variety of meaningful concepts of mathematics. They could not tools they can use later (e.g., visual representations and provide multiple interpretations of problem solving strategies). Further, these researcher- concepts—particularly representations that provided teachers seem to have a gift for promoting student concrete explanations or tie-ins to the real world. They thinking and moving an entire class forward by demonstrated only stepwise approaches to doing scaffolding lessons, questioning students, and creating mathematics, clinging tightly to procedures and a classroom community where learners consider each algorithms, and provided no evidence that they had a other’s work critically and interact meaningfully. The deeper understanding of the mathematics. In stark reality, however, is that not all mathematics teachers contrast, the same Ball article offered a vignette of have PhDs and it is unlikely that most ever will.2 Lampert’s teaching that illustrated a rich mathematical In working through this question both with the experience for students. Lampert provided multiple graduate students with whom I work and in preparation perspectives, introduced multiple representations, and for this editorial, I have developed some ideas both demonstrated a deep understanding of both about researcher-teachers as a “special” group and mathematics and student learning throughout the about why having a PhD might matter. Based on my episode described. thoughts I would like to propose two conjectures about Given the number of articles in the literature researcher-teacher efforts. First, I conjecture that we painting the ‘typical’ mathematical experience as one should consider the way we think about researcher- that is impoverished, and the growing body of teachers versus research on/with teachers. Second, I literature written by PhD researcher-teachers, I propose that certain features of PhD programs can be wondered, “Do you need a PhD to teach elementary applied to teacher professional development and/or undergraduate education to support all teachers in Chandra Hawley Orrill is a Research Scientist in the Learning creating richer mathematics learning experiences for and Performance Support Laboratory at the University of Georgia. Her research interest is teacher professional their students. This editorial explores these two development with an emphasis on teaching in the midst of conjectures in more detail. change. She is also interested in how professional development impacts the opportunities teachers create for student learning. Researcher-Teachers as a Special Group Acknowledgements In order to understand some of the unique qualities The research reported here came from a variety of projects spanning six years. These projects were supported by grants from of the teaching exemplified by researcher-teachers, it is the Andrew W. Mellon Foundation and the Russell Sage worthwhile to consider why they do what they do so Foundation, the National Science Foundation, Georgia’s well. There are a variety of factors that impact both the Teacher Quality Program (formerly Georgia’s Eisenhower way these people teach and the way we, as consumers Higher Education Program), and the Office of the Vice President of research, read about their teaching. First, researcher- for Research at the University of Georgia. Opinions expressed here are my own and do not necessarily reflect those of the teachers teach well because they have significant granting agencies. My thanks to Holly Anthony, Ernise Singleton, knowledge of mathematics and how children learn Peter Rich, Craig Shepherd, Laurel Bleich, and Drew Polly for mathematics. There is no doubt that teachers, with or their ongoing discussions with me about whether a teacher needs without PhDs, who have strong pedagogical a PhD to teach K-8 mathematics. knowledge and strong content knowledge, create richer

2 Do You Need a PhD? learning experiences for their students (e.g., Ball, with the researcher about what good teaching and Lubienski, & Mewborn, 2001). learning look like – after all, they are typically either Further, in the process of earning a PhD, the researcher (e.g., Ball, 1990a and Lampert, 2001) or researcher-teachers presumably develop reflective they are a full member of the research team (e.g., dispositions, grapple with their own epistemological McClain in Bowers, Cobb, & McClain, 1999). The beliefs, and define their visions of learning and importance of this is profound. A researcher-teacher teaching. This produces teachers who critically wants the same (not negotiated or compromised) examine the world around them and who are outcomes as the researcher, because she either is the introspective in ways that are productive for achieving researcher or is a member of the research team. The the classroom environment valued by mathematics researcher-teacher, therefore, attends to those issues education researchers and described in the NCTM and aspects of the classrooms and student learning that Standards (NCTM, 2000). By developing this are the focus of the research. Further, the researcher- disposition, researcher-teachers are in a unique position teacher provides unlimited, or nearly unlimited, rich to make critical changes to the classroom environment access to her thinking for the research effort because, as needs are identified. Too often, regular classroom again, she has a vested interest in capturing that teachers do not have the time or skills to analyze thinking. Thus, teacher and researcher alignment in formal or informal data about their students and their terms of goals, values, and expectations is important. teaching. In fact, many classroom teachers have only One potential disadvantage for researcher-teachers been exposed to the most basic concepts of student worth noting is the potential for bias to confound the learning theory and research. As a result, even if they research. After all, the researcher-teacher has a biased tried to make sense of the data presented in their view of the teaching being studied because it is her classroom, they would be ill-equipped to make own. Further, because she is invested in the research important changes based on those data. and because she is a member of the research team, it is In addition, researcher-teachers have some possible that her teaching is biased to make the pragmatic luxuries that typical teachers do not have. research work. That is, if the researcher is looking for For example, they usually only teach one subject to particular aspects of teaching, such as student-teacher one class per day, while a typical elementary teacher interactions, the researcher-teacher may attend to those might teach four subjects to one class, and a middle interactions more in the course of instruction than she school teacher might teach one or two subjects to four would under other circumstances. Clearly, the impact or five classes each day. This provides the researcher- of this on the research is determined by both the teacher with more time for reflection and refinement. research questions and the data collection and analysis To be fair, researcher-teachers typically do have other techniques used. work responsibilities – they do not simply teach for 50 Research On/With Teachers minutes and “call it a day.” However, their situation is very different from that of a typical classroom teacher. In order to understand the differences between Researcher-teachers have support with the reflection researcher-teacher research and research on or with process from others studying the classroom, and often full-time teachers, it is necessary to explore some of have no additional responsibilities such as conducting the issues involved with doing research on/with parent conferences, developing individualized plans for teachers. Research in regular classrooms differs in certain students, and attending the team meetings some significant ways from the researcher-teacher common in many teachers’ daily experience. While work alluded to in this editorial. To highlight some of this difference should not be viewed or used as an these differences, I offer examples from my own excuse for classroom teachers to avoid improving their experience in working with middle grades mathematics practice, it is undeniable that a researcher-teacher’s job teachers. is fundamentally different from that of the typical One major difference I alluded to is the values a classroom teacher. teacher holds. In the course of my career, I have been In addition to teaching expertise and workload, fortunate to work with several “good” teachers. researcher-teachers have some advantages over However, the ways in which they were “good” were teachers when participating in others’ studies. Unlike direct reflections of their own values and the values of most “typical” teachers, researcher-teachers are, by the system within which they were working. definition, philosophically aligned with and invested in Sometimes, they were good in the eyes of the the goals of the research. They already have agreement administrators with whom they worked because they

Chandra Hawley Orrill 3 kept their students under control. Sometimes they were developed techniques that supported her students in good for my research in that their practice had the achieving acceptable scores on the New York elements I was interested in, thus making it easier for standardized tests. By these standards, she was me to find the kinds of interactions I was looking for in considered “good.” When she used the simulations I their classrooms. Sometimes they were good in that was researching, she maintained the same kinds of they were predisposed to reflective practice allowing approaches, particularly early in the study. She kept me, as a researcher, easier access to their ideas through students on task and directed them to work more observation and interviews. The quality of the teachers, efficiently. Given my goal of understanding how to though, depended on what measure they were held up promote problem solving, her interactions with the against and what measures they, personally, felt they students were inadequate and impoverished. She were trying to align with. typically did not ask the students questions that Another important aspect of working with teachers provided insight into their thinking and she did not is a lack of access to certain aspects of their thinking. allow them to struggle with a problem. Instead, she For example, I have never been able to analyze a data directed them to an efficient approach for solving the set without thinking, at some point, “I wonder what she problem they were working on, which effectively kept was thinking when she did that?” or “Did she not them on task and motivated them to move forward. understand what that student was asking?” While this presented a challenge to me as the Acknowledging this lack of access to a teacher’s researcher, it would not be fair for me to “accuse” her thinking requires researchers to be careful in their of being less than a good teacher when she was clearly analysis of the teacher’s actions and beliefs and to meeting the expectations of the system within which explain how thinking and actions are interpreted. she worked. This is clearly a case in which there was a Further, at times, such limitations require researchers mismatch between what I, the researcher, valued and to analyze situations from their own perspectives as what the teacher and system valued. Had I been well as from the teacher’s perspective to understand a researching my own practice or the practice of a situation. research team member, this tension would have been As a practical example of the influence of removed. researcher and teacher alignment issues, I offer two As a second example, a teacher I have worked with situations from my own work: one addressing the more recently proved a perplexing puzzle for my team “good” teacher issue and the other addressing the need as we considered her teaching. A point of particular to understand the situation from the teacher’s interest was the teacher’s frustration with poor student perspective. My goal in presenting these two examples performance on tests – regardless of what students did is to highlight issues that arise in research with teachers in class, a significant number failed her tests. In my who are not members of the research team. In one analysis of this case, I recognized that this teacher’s study (Orrill, 2001), I worked with two middle school beliefs about teaching and learning significantly teachers (one mathematics and one science) in New differed from my own. Until I realized this, I was York City to understand how to structure professional unable to understand the magnitude of the barrier the development to support uses of computer-based teacher felt she was facing. At the simplest level, she simulations. My goal for the professional development believed that her role as a good teacher was to present was to enhance teachers’ attention to student problem- new material and provide an opportunity for students solving skills in the context of computer-based, to practice that material. The students’ job, in her view, workplace simulations. The mathematics teacher was was to engage in that practice and develop an considered to be “good” by her principal and other understanding from it. Therefore, when students were teachers. In my observations of her classroom, I found not succeeding, she became extremely frustrated since that she taught mathematics in much the same way as she had presented information and provided the “typical” teachers we read about in case study after opportunities for practice. In her worldview, student case study. She offered many procedures but provided success was out of her hands – she had already done inadequate opportunities for students to interact with what she could to support them. As the researcher in the content in ways that would allow them to develop that setting, it was difficult to understand her deep understanding of the mathematical concepts frustration because I was working from a constructivist underlying those procedures. However, this teacher perspective. Specifically, I was looking for an had remarkable skill in classroom management, which environment in which the teacher provided students was highly valued in her school. Further, she had opportunities to develop their own thinking via an

4 Do You Need a PhD? assortment of models, experiences, and collaborative conducting research in others’ classrooms, and having exchanges. Student test failure, for me, was an other similar experiences. This is in stark contrast to indicator that learning was not complete and that the elementary or middle grades teacher who has students needed different opportunities to build and typically had four years of college—with courses connect knowledge. It took considerable analysis for spread across the curriculum—and only limited “life me, as a researcher with a different perspective and experience” to relate to in the courses that help develop different goals, to understand how the teacher these knowledge areas. Second, one of the most understood her role and how she enacted her beliefs powerful outcomes of earning a PhD is the about her role in the classroom. development of a concrete picture of a desired learning My point in these two examples is that in much environment that looks beyond issues of classroom research there are significant and important differences management and logistics to focus on the kinds of in the worldviews of the participants and the learning and teaching that will take place. Third, PhDs researcher. These differences can lead to frustrations in develop a rich, precise vocabulary aligned with that of data collection, hurdles in data analysis, and, in the the standards-writers and the researchers. In becoming worst cases, assessments of the teachers that are simply a researcher, the holder of the PhD becomes active in not fair. For example, in the early 1990’s there were the conversation of the field—meaning that person has many articles written about the implementation of the developed a refined vocabulary and vision that is standards in California (e.g., Ball, 1990b; Cohen, 1990; shared, in some way, by the field. This is not to say Wilson, 1990). In many of these cases, the teachers that there is a definitive definition of K-8 mathematics struggled to implement a set of standards that were education that is shared across the field of mathematics written from a particular perspective that they did not education, rather that there is a shared way of fully understand. This led to implementations that were discussing and thinking about mathematics education far from ideal in the eyes of the researchers who that allows a more consistent enactment of standards understood the initial intent of the standards. Too and practices. often, teachers were presented by researchers as Finally, many researcher-teachers implement or hopeless or inadequate—in contrast, the teachers develop a “special” curriculum. In the case of Lampert reportedly perceived themselves as adhering to these (2001), the teacher was creating open-ended problems new standards. Likely, if the researchers and teachers each day to support mathematical topics. In other had philosophical alignment afforded by the cases, the research team has developed materials for researcher-teacher approach the findings would have the researcher-teachers to implement. Often, these been tremendously different. After all, had these materials are far richer than traditional mathematics studies focused on researcher-teachers, the teacher and textbooks. While there may not be a single disposition the researcher would have had a shared understanding that could be pulled from the process of earning a PhD of the intent of the standards and had a shared vision of that allows researcher-teachers to be successful what their implementation should look like. implementers of non-traditional materials, it is clear that there is something different between PhD-holding PhD Program Features That Could Be Useful In researcher-teachers and other teachers. Likely, part of Teacher Development this ability is related to the knowledge constructs the While not all people who hold PhDs are good researcher-teachers have that allow them to implement teachers, certain habits of mind are developed as part those materials. In my own work, I have found that of the process of earning a PhD that can significantly teachers who are not well-versed in the curricula, who impact the learning environment a teacher designs. lack conceptual knowledge, or who lack the Given the high-quality of teaching exhibited by the pedagogical content knowledge to see connections researcher-teachers referred to in this article, it seems between various mathematical ideas do not know how likely that there are aspects of the PhD program that to utilize these kinds of materials to make the could be adapted for teacher professional development. experiences mathematically rich for their students. First, the researcher-teacher typically has Clearly, some attention to the aspects of earning a PhD developed solid pedagogical knowledge, content that relate to these dispositions would benefit knowledge, and pedagogical content knowledge. This preservice and inservice teachers. comes from having time and encouragement to read about different practices in a focused way, participating in shared discourse with colleagues,

Chandra Hawley Orrill 5 Teacher Development classrooms, success is measured in the number of While it may not be feasible, or even reasonable, to problems students can answer correctly, often within a expect teachers to pursue doctoral degrees, there may specific amount of time. be some characteristics of doctoral education that are How Features of PhD Programs May Change This worthwhile for consideration as components or foci of To enhance the interactions among teachers, professional development and undergraduate programs. students, and materials/content there are a number of To frame this section, I want to draw on the work of elements from doctoral training that may be worth Cohen and Ball (1999) who have argued that the pursuing. First, teachers can use guided reflection as a learning environment is shaped by the interactions of means to step out of the teaching moment to consider three critical elements: teachers, students, and critical aspects of the teaching and learning materials/content. This model assumes that for each environment. Through reflection, teachers have the element a variety of beliefs, values, and backgrounds opportunity to align their beliefs and practices (e.g. work together to create each unique learning Wedman, Espinosa, & Laffey, 1998) and to make their environment. Considering the classroom from this intent more explicit rather than relying on tacit “gut perspective is critical to understanding why the instinct” (e.g., Richardson, 1990). The reflective solutions to the problems highlighted in research on practitioner can learn to look at a learning environment and with teachers are complex. as a whole by considering how students and materials What We See Now are interacting, looking for evidence of conceptual A quick overview of my definition of the “typical” development, and thinking about ways to improve their classroom may be warranted at this point. Based on the own role in the classroom. The researcher-teachers classrooms described in the literature and those I work (Ball, Lampert, and McClain) cited in this article all in, the typical mathematics classroom remains focused reported using reflection regularly as part of their on teachers’ delivering information to the students, practice. typically by working sample problems on the board. Another element of the PhD experience worth Students are responsible for using this information to consideration is the development of solid content and work problems on worksheets or in their books. pedagogical knowledge. Teachers who do not Students are asked to do things like name the fractional understand mathematics cannot be as effective as those portion of a circle that is colored in or to work 20 who do. For example, teachers who do not know how addition or multiplication problems. Many teachers use to use representations to model multiplication of manipulatives or drawn representations to introduce fractions cannot use that pedagogical strategy in their new ideas to their students. However, their intent is to classrooms. Teachers who lack adequate content or provide a concrete example and move the students to pedagogical knowledge cannot know what to do when the abstract activities of arithmetic as quickly as a student suggests an approach to solving a problem possible or to use the manipulative to motivate the that does not work—too often the only approach the students to want to do the arithmetic. Mathematics teacher has is to point out errors to the student and learning in these classrooms is more about developing demonstrate “one more time” the “right” way to work efficient means for working problems than developing the problem. I assert that combining teacher rich understandings of why those methods work. development of content knowledge and pedagogical Referring back to the Cohen and Ball triangle of knowledge with the development of a reflective interactions, the interactions in these classrooms could disposition will lead to the emergence of pedagogical best be characterized by what follows. The teacher content knowledge. By pedagogical content interprets the materials/content and delivers that knowledge, I refer to knowledge that is a combination interpretation to the students. The students look to of knowing what content can be learned/taught with teachers as holders of all information. Teachers are to which pedagogies and knowing when to use each of provide guidance when students are unable to solve a these approaches to teach students. problem, to provide feedback about the “rightness” of Some of the habits of mind developed in a doctoral student work, and to find the errors students have made program in education translate directly into practice in their work. The students interact with the materials without focusing on the entire teacher-student- by working problems. The students may or may not materials interaction triad. For example, one interact with the concepts at a meaningful level – that potentially powerful factor to address is the teacher- depends on the teacher and the activity. In these student interaction. PhD programs in education offer

6 Do You Need a PhD? tremendous opportunities for thinking about this communicate with others in the field and to understand relationship in meaningful ways, and in the researcher- input from the research. Unfortunately, it is not teacher work, attention to this interaction is ubiquitous. practical to expect most teachers to earn a doctoral It is absolutely critical to support teachers in learning degree. The question then becomes, “What elements to listen to students and respond to them in meaningful can we take from earning an advanced degree that will ways. Further, given the poor grounding most teachers help teachers in the classroom?” By incorporating have in learning theory, it may be that developing a these elements into teacher education and professional theoretical understanding of how people learn should development programs, we can greatly improve be a part of this (this is supported in recent research classroom instruction. such as Philipp, Clement, Thanheiser, Schappelle, & REFERENCES Sowder, 2003). Finally, focusing professional development on techniques for questioning that allow Ball, D. L. (1990a). Halves, pieces, and twoths: Constructing representational concepts in teaching fractions. East Lansing, MI: the teacher to access student understanding will National Center for Research on Teacher Education. provide teachers with ways to access student thinking. Ball, D. L. (1990b). Reflection and deflections of policy: The case of Carol Turner. Educational Evaluation and Policy Analysis, 12(3), 247–259. Conclusion Ball, D. L. (1991). Research on teaching mathematics: Making subject- matter knowledge part of the equation. In J. Brophy (Ed.), Advances While it is not realistic to expect that all classroom in research on teaching (Vol. 3, pp. 1–48). Greenwich, CT: JAI Press. teachers will earn doctoral degrees, there are elements Ball, D. L., Lubienski, S. T., & Mewborn, D. S. (2001). Research on that go into the attainment of a PhD that can lead to teaching mathematics: The unsolved problem of teachers' improved classroom teaching. Therefore, it seems mathematical knowledge. In V. Richardson (Ed.), Handbook of research on teaching (4th ed.). Washington, DC: American reasonable to capitalize on what we know about the Educational Research Association. process of getting and having a doctorate versus more Bowers, J., Cobb, P., & McClain, K. (1999). The evolution of mathematical traditional routes to becoming a teacher. practices: A case study. Cognition and Instruction, 17(1), 25–64. Granted, there are aspects of researcher-teachers' Cohen, D. (1990). A revolution in one classroom: The case of Mrs. Oublier. activities that are not addressed simply by considering Educational Evaluation and Policy Analysis, 12(3), 327–345. Cohen, D., & Ball, D. B. (1999). Instruction, capacity, & improvement (No. their educational background or their role in the CPRE-RR-43). Philadelphia, PA: Consortium for Policy Research in research team. For example, high quality materials are Education. extremely important. Further, it is vital that teachers Lampert, M. (2001). Teaching problems and the problems of teaching. New are supported in learning how to interact with those Have, CT: Yale University Press. materials (and the content they are trying to convey) if National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author. we want to raise the bar on teaching and learning. No Orrill, C. H. (2001). Building technology-based learning-centered one can create rich learning experiences around classrooms: The evolution of a professional development framework. materials they do not understand. On the other hand, Educational Technology Research and Development, 49(1), 15–34. researcher-teachers have been able to find ways to Philipp, R. A., Clement, L., Thanheiser, E., Schappelle, B., & Sowder, J. T. (2003). Integrating mathematics and pedagogy: An investigation of capitalize on even the weakest of materials. For the effects on elementary preservice teachers' beliefs and learning of example, Lampert (2001) discusses how she was able mathematics. Paper presented at the Research Presession of the 81st to use the topic ideas from the traditional textbook her Annual Meeting of the National Council of Teachers of Mathematics, San Antonio, TX. Available online: school used to develop rich problems that allowed http://www.sci.sdsu.edu/CRMSE/IMAP/pubs.html. students prolonged and repeated exposure to critical Richardson, V. (1990). Significant and worthwhile change in teaching mathematics content—it is clear that the typical teacher practice. Educational Researcher, 19(7), 10–18. is unable to capitalize on materials in these ways. Wedman, J. M., Espinosa, L. M., & Laffey, J. M. (1998). A process for understanding how a field-based course influences teachers' beliefs Certainly, there is an appropriate place in professional and practices, Paper presented at the Annual Meeting of the American development efforts to support teachers’ use of Educational Research Association, San Diego, CA. materials. Wilson, S. M. (1990). A conflict of interests: The case of Mark Black. While this article has only begun to explore the Educational Evaluation and Policy Analysis, 12(3), 309–326. differences between a typical classroom teacher’s environment and that of a researcher-teacher, it appears 1 I cite examples of each of these researcher-teachers’ work throughout this that researcher-teachers have some advantages over editorial. This list is not exhaustive. 2 other teachers. They are better able to understand and 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 address what is going on in the classroom, as well as PhD programs and what teachers do in their everyday lives. This is not to the material they are expected to work with. assert that earning a PhD is not helpful for a teacher, rather that it is not Researcher-teachers are also better able to likely in the current educational system.

Chandra Hawley Orrill 7 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 experiences as a secondary mathematics teacher, Mathematics Education at The University of Georgia preservice-teacher supervisor, and researcher supported for the 2002–2003 academic year was to assist with a Oakes’s (1985) assertions that often students are four-year Spencer-funded qualitative research project distributed into “ability” groups based on their race, entitled “Learning to Teach Elementary Mathematics.” gender, and class. Nonetheless, my perception after This assistantship presented the opportunity to conduct five observations was that ability grouping according research at elementary schools in two suburban to these attributes was diminishing—at least in these counties—a new experience for me since my prior elementary schools. In other words, the student make- professional experience in education had been within up of each mathematics lesson that I observed the context of secondary mathematics education. My appeared to be representative of the demographics of research duties consisted of organizing, coding, the school. analyzing, and writing-up existing data, as well as However, on my sixth observation, at an collecting new data. This new data included elementary school with 35.8 % Black, 12.8 % Asian, transcribed interviews of preservice and novice 5.3 % Hispanic, 3.5 % Multi-racial, and 0.5 % elementary school teachers and fieldnotes from American Indian1 children, I observed a 3rd grade classroom observations. mathematics lesson that was 94.4% White (at least it By January 2003, I had conducted five was 50% female). The make-up of the classroom was observations in 1st, 2nd, and 3rd grade classrooms at not initially unrepresentative of the school’s two elementary schools with diverse populations. I was racial/ethnic demographics, but became so shortly impressed with the preservice and novice elementary before the start of the mathematics lesson as some teachers’ mathematics pedagogy and ability to interact students left the classroom while others entered. When with their students. Given that my research interest is I questioned why the students were exchanged between equity and social justice in education, I was mindful of classrooms, I was informed that the mathematics the “racial,” ethnic, gender, and class make-up of the lesson was for the “advanced” third graders. Because classroom and how these attributes might help me of my experience in secondary mathematics education, explain the teacher-student interactions I observed. My I am aware that academic tracking is a nationally practiced education policy, and that it even occurs in David W. Stinson is a doctoral candidate in The Department of many districts and schools as early as 5th grade—but Mathematics Education at The University of Georgia. In the fall these were eight-year-old children! Has the structure of of 2004, he will join the faculty of the Middle-Secondary public education begun to decide who is and who is not Education and Instructional Technology Department at Georgia “capable” mathematically in the 3rd grade? Has the State University. His research interests are the sociopolitical and cultural aspects of mathematics and mathematics teaching and structure of public education begun to decide who will learning with an emphasis on equity and social justice in be proletariat and who will be bourgeoisie in the 3rd mathematics education and education in general. grade—with eight-year-old children? How did school

8 Mathematics as “Gate-Keeper” (?) mathematics begin to (re)produce and regulate racial, He wrote: ethnic, gender, and class divisions, becoming a We shall persuade those who are to perform high “gatekeeper”? And (if) school mathematics is a functions in the city to undertake calculation, but gatekeeper, how might mathematics educators ensure not as amateurs. They should persist in their studies that gatekeeping mathematics becomes an inclusive until they reach the level of pure thought, where instrument for empowerment rather than an exclusive they will be able to contemplate the very nature of instrument for stratification? number. The objects of study ought not to be This article provides a two-part discussion centered buying and selling, as if they were preparing to be on the last question. The first part of the discussion merchants or brokers. Instead, it should serve the provides a historical perspective of the concept of purposes of war and lead the soul away from the gatekeeper in mathematics education, verifying that world of appearances toward essence and reality. (p. 219) mathematics is an exclusive instrument for stratification, effectively nullifying the if. The intent of Although Plato believed that mathematics was of this historical perspective is not to debate whether value for all people in everyday transactions, the study mathematics should be a gatekeeper but to provide a of mathematics that would lead some men from perspective that reveals existence of mathematics as a “Hades to the halls of the gods” (p. 215) should be gatekeeper (and instrument for stratification) in the reserved for those that were “naturally skilled in current education structure of the United States. In the calculation” (p. 220); hence, the birth of mathematics discussion, I state why I believe all students are not as the privileged discipline or gatekeeper. provided with a key to the gate. This view of mathematics as a gatekeeper has After arguing that mathematics is a gatekeeper and persisted through time and manifested itself in early inequities are present in the structure of education, I research in the field of mathematics education in the proceed to the second part of the discussion: how United States. In Stanic’s (1986) review of might mathematics educators ensure that gatekeeping mathematics education of the late 19th and early 20th mathematics becomes an inclusive instrument for centuries, he identified the 1890s as establishing empowerment? In this discussion, I first define “mathematics education as a separate and distinct empowerment and empowering mathematics. Then, I professional area” (p. 190), and the 1930s as make note of the “social turn” in mathematics developing the “crisis” (p. 191) in mathematics education research, which provides a framework for education. This crisis—a crisis for mathematics the situated, culturally relevant, and critical educators—was the projected extinction of perspectives of mathematics education that are mathematics as a required subject in the secondary presented. Finally, I argue that these theoretical school curriculum. Drawing on the work of Kliebard perspectives replace characteristics of exclusion and (c.f., Kliebard, 1995), Stanic provided a summary of stratification (of gatekeeping mathematics) with curriculum interest groups that influenced the position characteristics of inclusion and empowerment. I of mathematics in the school curriculum: (a) the conclude the article by challenging the reader to think humanists, who emphasized the traditional disciplines differently. of study found in Western philosophy; (b) the developmentalists, who emphasized the “natural” Mathematics a Gatekeeper: A Historical development of the child; (c) the social efficiency Perspective educators, who emphasized a “scientific” approach that Discourse regarding the “gatekeeper” concept in led to the natural development of social stratification; mathematics can be traced back over 2300 years ago to and (d) the social meliorists, who emphasized Plato’s (trans. 1996) dialogue, The Republic. In the education as a means of working toward social justice. fictitious dialogue between Socrates and Glaucon Stanic (1986) noted that mathematics educators, in regarding education, Plato argued that mathematics general, sided with the humanists, claiming: was “virtually the first thing everyone has to “mathematics should be an important part of the school learn…common to all arts, science, and forms of curriculum” (p. 193). He also argued that the thought” (p. 216). Although Plato believed that all development of the National Council of Teachers of students needed to learn arithmetic—”the trivial Mathematics (NCTM) in 1920 was partly in response business of being able to identify one, two, and three” to the debate that surrounded the position of (p. 216)—he reserved advanced mathematics for those mathematics within the school curriculum. that would serve as philosopher guardians2 of the city.

David W. Stinson 9 The founders of the Council wrote: how, but more importantly, who should be taught mathematics. Mathematics courses have been assailed on every hand. So-called educational reformers have The question of who should be taught mathematics tinkered with the courses, and they, not knowing initially appeared in the debates of the 1920s and the subject and its values, in many cases have centered on “ascertaining who was prepared for the thrown out mathematics altogether or made it study of algebra” (Kilpatrick, 1992, p. 21). These entirely elective. …To help remedy the existing debates led to an increase in grouping students situation the National Council of Teachers of according to their presumed mathematics ability. This Mathematics was organized. (C. M. Austin as “ability” grouping often resulted in excluding female quoted in Stanic, 1986, p. 198) students, poor students, and students of color from the The backdrop to the mathematics education crisis opportunity to enroll in advanced mathematics courses was the tremendous growth in school population that (Oakes, 1985; Oakes, Ormseth, Bell, & Camp, 1990). occurred between 1890 and 1940—a growth of nearly Sixty years after the beginning of the debates, the 20 times (Stanic, 1986). This dramatic increase in the recognition of this unjust exclusion from advanced student population yielded the belief that the overall mathematics courses spurred the NCTM to publish the intellectual capabilities of students had decreased; Curriculum and Evaluation Standards for School consequently, students became characterized as the Mathematics (Standards, 1989) that included “army of incapables” (G. S. Hall as quoted in Stanic, statements similar to the following: 1986, p. 194). Stanic presented the results of this The social injustices of past schooling practices can prevailing belief by citing the 1933 National Survey of no longer be tolerated. Current statistics indicate Secondary Education, which concluded that less than that those who study advanced mathematics are half of the secondary schools required algebra and most often white males. …Creating a just society plane geometry. And, he illustrated mathematics in which women and various ethnic groups enjoy teachers’ perspectives by providing George Counts’ equal opportunities and equitable treatment is no 1926 survey of 416 secondary school teachers. longer an issue. Mathematics has become a critical Eighteen of the 48 mathematics teachers thought that filter for employment and full participation in our fewer pupils should take mathematics, providing a society. We cannot afford to have the majority of our population mathematically illiterate: Equity has contrast to teachers of other academic disciplines who become an economic necessity. (p. 4) believed that “their own subjects should be more largely patronized” (G. S. Counts as quoted in Stanic, In the Standards the NCTM contrasted societal p. 196). Even so, the issues of how mathematics should needs of the industrial age with those of the be positioned in the school curriculum and who should information age, concluding that the educational goals take advanced mathematics courses was not a major of the industrial age no longer met the needs of the national concern until the 1950s. information age. They characterized the information During the 1950s, mathematics education in U.S. age as a dramatic shift in the use of technology which schools began to be attacked from many segments of had “changed the nature of the physical, life, and social society: the business sector and military for graduating sciences; business; industry; and government” (p. 3). students who lacked computational skills, colleges for The Council contended, “The impact of this failing to prepare entering students with mathematics technological shift is no longer an intellectual knowledge adequate for college work, and the public abstraction. It has become an economic reality” (p. 3). for having “watered down” the mathematics The NCTM (1989) believed this shift demanded curriculum as a response to progressivism (Kilpatrick, new societal goals for mathematics education: (a) 1992). The launching of Sputnik in 1957 further mathematically literate workers, (b) lifelong learning, exacerbated these attacks leading to a national demand (c) opportunity for all, and (d) an informed electorate. for rigorous mathematics in secondary schools. This They argued, “Implicit in these goals is a school demand spurred a variety of attempts to reform system organized to serve as an important resource for mathematics education: “the ‘new’ math of the 1960s, all citizens throughout their lives” (p. 3). These goals the ‘back-to-basic’ programs of the 1970s, and the required those responsible for mathematics education ‘problem-solving’ focus of the 1980s” (Johnston, to strip mathematics from its traditional notions of 1997). Within these programs of reform, the questions exclusion and basic computation and develop it into a were not only what mathematics should be taught and dynamic form of an inclusive literacy, particularly given that mathematics had become a critical filter for

10 Mathematics as “Gate-Keeper” (?) full employment and participation within a democratic advanced mathematics provided an advantage in society. Countless other education scholars academics and in the job market—the same argument (Frankenstein, 1995; Moses & Cobb, 2001; Secada, provided by the NCTM and education scholars. 1995; Skovsmose, 1994; Tate, 1995) have made The statistical analyses in the report entitled, Do similar arguments as they recognize the need for all Gatekeeper Courses Expand Educational Options? (U. students to be provided the opportunity to enroll in S. Department of Education, 1999) presented the advanced mathematics courses, arguing that a dynamic following findings: mathematics literacy is a gatekeeper for economic Students who enrolled in algebra as eighth-graders access, full citizenship, and higher education. In the were more likely to reach advanced math courses paragraphs that follow, I highlight quantitative and (e.g., algebra 3, trigonometry, or calculus, etc.) in qualitative studies that substantiate mathematics as a high school than students who did not enroll in gatekeeper. algebra as eighth-graders. The claims that mathematics is a “critical filter” or Students who enrolled in algebra as eighth-graders, gatekeeper to economic access, full citizenship, and and completed an advanced math course during higher education are quantitatively substantiated by high school, were more likely to apply to a four- two reports by the U. S. government: the 1997 White year college than those eighth-grade students who Paper entitled Mathematics Equals Opportunity and did not enroll in algebra as eighth-graders, but who the 1999 follow-up summary of the 1988 National also completed an advanced math course during Education Longitudinal Study (NELS: 88) entitled Do high school. (pp. 1–2) Gatekeeper Courses Expand Education Options? The The summary concluded that not all students who U. S. Department of Education prepared both reports took advanced mathematics courses in high school based on data from the NELS: 88 samples of 24,599 enrolled in a four-year postsecondary school, although eighth graders from 1,052 schools, and the 1992 they were more likely to do so—again confirming follow-up study of 12,053 students. mathematics as a gatekeeper. In Mathematics Equals Opportunity, the following Nicholas Lemann’s (1999) book The Big Test: The statements were made: Secret History of the American Meritocracy provides a In the United States today, mastering mathematics qualitative substantiation that mathematics is a has become more important than ever. Students gatekeeper to economic access, full citizenship, and with a strong grasp of mathematics have an higher education. In Parts I and II of his book, Lemann advantage in academics and in the job market. The presented a detailed historical narrative of the merger 8th grade is a critical point in mathematics between the Educational Testing Service with the education. Achievement at that stage clears the College Board. Leman argued this merger established way for students to take rigorous high school how mathematics would directly and indirectly mathematics and science courses—keys to college categorize Americans—becoming a gatekeeper—for entrance and success in the labor force. the remainder of the 20th and beginning of the 21st Students who take rigorous mathematics and centuries. During World War I, the United States War science courses are much more likely to go to Department (currently known as the Department of college than those who do not. Defense) categorized people using an adapted version Algebra is the “gateway” to advanced mathematics of Binet’s Intelligence Quotient test to determine the and science in high school, yet most students do entering rank and duties of servicemen. This not take it in middle school. categorization evolved into ranking people by Taking rigorous mathematics and science courses “aptitude” through administering standardized tests in in high school appears to be especially important contemporary U. S. education. for low-income students. In Part III of his book, Lemann presented a case- study characterization of contemporary Platonic Despite the importance of low-income students taking rigorous mathematics and science courses, guardians, individuals who unjustly (or not) benefited these students are less likely to take them. (U. S. from the concept of aptitude testing and the ideal of Department of Education, 1997, pp. 5–6) American meritocracy. Lemann argued that because of their ability to demonstrate mathematics proficiency This report, based on statistical analyses, explicitly (among other disciplines) on standardized tests, these stated that algebra was the “gateway” or gatekeeper to individuals found themselves passing through the gates advanced (i.e., rigorous) mathematics courses and that

David W. Stinson 11 to economic access, full citizenship, and higher of the web of capitalist society, and likely to persist as education. long as capitalism survives” (p. 137). The concept of mathematics as providing the key Although Bowles’s statements imply that only the for passing through the gates to economic access, full overthrow of capitalism will emancipate education citizenship, and higher education is located in the core from its inequalities, I believe that developing of Western philosophy. In the United States, school mathematics classrooms that are empowering to all mathematics evolved from a discipline in “crisis” into students might contribute to educational experiences one that would provide the means of “sorting” that are more equitable and just. This development may students. As student enrollment in public schools also assist in the deconstruction of capitalism so that it increased, the opportunity to enroll in advanced might be reconstructed to be more equitable and just. mathematics courses (the key) was limited because The following discussion presents three theoretical some students were characterized as “incapable.” perspectives that I have identified as empowering Female students, poor students, and students of color students. These perspectives aim to assist in more were offered a limited access to quality advanced equitable and just educative experiences for all mathematics education. This limited access was a students: the situated perspective, the culturally motivating factor behind the Standards, and the relevant perspective, and the critical perspective. I subsequent NCTM documents.3 believe these perspectives provide a plausible answer NCTM and education scholars’ argument that to the second question asked above: How do we as mathematics had and continues to have a gatekeeping mathematics educators transform the status quo in the status has been confirmed both quantitatively and mathematics classroom? qualitatively. Given this status, I pose two questions: An Inclusive Empowering Mathematics Education (a) Why does U.S. education not provide all students access to a quality, advanced (mathematics) education To frame the discussion that follows, I provide a that would empower them with economic access and definition of empowerment and empowering full citizenship? and (b) How can we as mathematics mathematics. Freire (1970/2000) framed the notion of educators transform the status quo in the mathematics empowerment within the concept of conscientização, classroom? defined as “learning to perceive social, political and To fully engage in the first question demands a economic contradictions, and to take action against the deconstruction of the concepts of democratic public oppressive elements of reality” (p. 35). He argued that schooling and American meritocracy and an analysis of conscientização leads people not to “destructive the morals and ethics of capitalism. To provide such a fanaticism” but makes it possible “for people to enter deconstruction and analysis is beyond the scope of this the historical process as responsible Subjects4” (p. 36), article. Nonetheless, I believe that Bowles’s enrolling them in a search for self-affirmation. (1971/1977) argument provides a comprehensive, yet Similarly, Lather (1991) defined empowerment as the condensed response to the question of why U. S. ability to perform a critical analysis regarding the education remains unequal without oversimplifying the causes of powerlessness, the ability to identify the complexities of the question. Through a historical structures of oppression, and the ability to act as a analysis of schooling he revealed four components of single subject, group, or both to effect change toward U. S. education: (a) schools evolved not in pursuit of social justice. She claimed that empowerment is a equality, but in response to the developing needs of learning process one undertakes for oneself; “it is not capitalism (e.g., a skilled and educated work force); (b) something done ‘to’ or ‘for’ someone” (Lather, 1991, as the importance of a skilled and educated work force p. 4). In effect, empowerment provides the subject with grew within capitalism so did the importance of the skills and knowledge to make sociopolitical maintaining educational inequality in order to critiques about her or his surroundings and to take reproduce the class structure; (c) from the 1920s to action (or not) against the oppressive elements of those 1970s the class structure in schools showed no signs of surroundings. The emphasis in both definitions is self- diminishment (the same argument can be made for the empowerment with an aim toward sociopolitical 1970s to 2000s); and (d) the inequality in education critique. With this emphasis in mind, I next define had “its root in the very class structures which it serves empowering mathematics. to legitimize and reproduce” (p. 137). He concluded by Ernest (2002) provided three domains of writing: “Inequalities in education are thus seen as part empowering mathematics—mathematical, social, and epistemological—that assist in organizing how I define

12 Mathematics as “Gate-Keeper” (?) empowering mathematics. Mathematical The Situated Perspective empowerment relates to “gaining the power over the The situated perspective is the coupling of language, skills and practices of using mathematics” scholarship from cultural anthropology and cultural (section 1, ¶ 3) (e.g., school mathematics). Social psychology. In the situated perspective, learning empowerment involves using mathematics as a tool for becomes a process of changing participation in sociopolitical critique, gaining power over the social changing communities of practice in which an domains—“the worlds of work, life and social affairs” individual’s resulting knowledge becomes a function (section 1, ¶ 4). And, epistemological empowerment of the environment in which she or he operates. concerns the “individual’s growth of confidence in not Consequently, in the situated perspective, the dualisms only using mathematics, but also a personal sense of of mind and world are viewed as artificial constructs power over the creation and validation of (Boaler, 2000b). Moreover, the situated perspective, in knowledge”(section 1, ¶ 5). Ernest argued, and I agree, contrast to constructivist perspectives, emphasizes that all students gain confidence in their mathematics interactive systems that are larger in scope than the skills and abilities through the use of mathematics in behavioral and cognitive processes of the individual routine and nonroutine ways and that this confidence student. will logically lead to higher levels of mathematics Mathematics knowledge in the situated perspective attainment. All students achieving higher levels of is understood as being co-constituted in a community attainment will assist in leveling the racial, gender, and within a context. It is the community and context in class imbalances that currently persist in advanced which the student learns the mathematics that mathematics courses. Effectively, Ernest’s definition of significantly impacts how the student uses and empowering mathematics echoes the definition of understands the mathematics (Boaler, 2000b). Boaler empowerment stated earlier. (1993) suggested that learning mathematics in contexts Using Ernest’s three domains of empowering assists in providing student motivation and interest and mathematics as a starting point, I selected three enhances transference of skills by linking classroom empowering mathematics perspectives. In doing so, I mathematics with real-world mathematics. She argued, kept in mind Stanic’s (1989) challenge to mathematics however, that learning mathematics in contexts does educators: “If mathematics educators take seriously the not mean learning mathematics ideas and procedures goal of equity, they must question not just the common by inserting them into “real-world” textbook problems view of school mathematics but also their own taken- or by extending mathematics to larger real-world class for-granted assumptions about its nature and worth” (p. projects. Rather, she suggested that the classroom itself 58). I believe that the situated perspective, culturally becomes the context in which mathematics is learned relevant perspective, and critical perspective, in and understood: “If the students’ social and cultural varying degrees, motivate such questioning and values are encouraged and supported in the resonate with the definition I have given of mathematics classroom, through the use of contexts or empowering mathematics. These configurations are through an acknowledgement of personal routes and complex theoretical perspectives derived from multiple direction, then their learning will have more meaning” scholars who sometimes have conflicting working (p. 17). definitions. These perspectives, located in the “social The situated perspective offers different notions of turn” (Lerman, 2000, p. 23) of mathematics education what it means to have mathematics ability, changing research, originate outside the realm of “traditional” the concept from “either one has mathematics ability or mathematics education theory, in that they are rooted not” to an analysis of how the environment coin anthropology, cultural psychology, sociology, and constitutes the mathematics knowledge that is learned sociopolitical critique. In the discussion that follows, I (Boaler, 2000a). Boaler argued that this change in how provide sketches of each theoretical perspective by mathematics ability is assessed in the situated briefly summarizing the viewpoints of key scholars perspective could “move mathematics education away working within the perspective. I then explain how from the discriminatory practices that produce more each perspective holds possibilities in transforming failures than successes toward something considerably gatekeeping mathematics from an exclusive instrument more equitable and supportive of social justice” (p. for stratification into an inclusive instrument for 118). empowerment.

David W. Stinson 13 The Culturally Relevant Perspective solution methods to nonroutine problems, and Working toward social justice is also a component perceiving mathematics as a tool for sociopolitical of the culturally relevant perspective. Ladson-Billings critique (Gutstein, 2003). (1994) developed the “culturally relevant” (p. 17) The Critical Perspective perspective as she studied teachers who were Perceiving mathematics as a tool for sociopolitical successful with African-American children. This critique is also a feature of the critical perspective. This perspective is derived from the work of cultural perspective is rooted in the social and political critique anthropologists who studied the cultural disconnects of the Frankfurt School (circa 1920) whose between (White) teachers and students of color and membership included but was not limited to Max made suggestions about how teachers could “match Horkheimer, Theodor Adorno, Leo Lowenthal, and their teaching styles to the culture and home Franz Neumann. The critical perspective is backgrounds of their students” (Ladson-Billings, 2001, characterized as (a) providing an investigation into the p. 75). Ladson-Billings defined the culturally relevant sources of knowledge, (b) identifying social problems perspective as promoting student achievement and and plausible solutions, and (c) reacting to social success through cultural competence (teachers assist injustices. In providing these most general and students in developing a positive identification with unifying characteristics of a critical education, their home culture) and through sociopolitical Skovsmose (1994) noted, “A critical education cannot consciousness (teachers help students develop a civic be a simple prolongation of existing social and social awareness in order to work toward equity relationships. It cannot be an apparatus for prevailing and social justice). inequalities in society. To be critical, education must Teachers working from a culturally relevant react to social contradictions” (p. 38). perspective (a) demonstrate a belief that children can Skovsmose (1994), drawing from Freire’s be competent regardless of race or social class, (b) (1970/2000) popularization of the concept provide students with scaffolding between what they conscientização and his work in literacy know and what they do not know, (c) focus on empowerment, derived the term “mathemacy” (p. 48). instruction during class rather than busy-work or Skovsmose claimed that since modern society is highly behavior management, (d) extend students’ thinking technological and the core of all modern-day beyond what they already know, and (e) exhibit in- technology is mathematics that mathemacy is a means depth knowledge of students as well as subject matter of empowerment. He stated, “If mathemacy has a role (Ladson-Billings, 1995). Ladson-Billings argued that to play in education, similar to but not identical to the all children “can be successful in mathematics when role of literacy, then mathemacy must be seen as their understanding of it is linked to meaningful composed of different competences: a mathematical, a cultural referents, and when the instruction assumes technological, and a reflective” (p. 48). that all students are capable of mastering the subject In the critical perspective, mathematics knowledge matter” (p. 141). is seen as demonstrating these three competencies Mathematics knowledge in the culturally relevant (Skovsmose, 1994). Mathematical competence is perspective is viewed as a version of demonstrating proficiency in the normally understood ethnomathematics—ethno defined as all culturally skills of school mathematics, reproducing and identifiable groups with their jargons, codes, symbols, mastering various theorems, proofs, and algorithms. myths, and even specific ways of reasoning and Technological competence demonstrates proficiency in inferring; mathema defined as categories of analysis; applying mathematics in model building, using and tics defined as methods or techniques (D’ mathematics in pursuit of different technological aims. Ambrosio, 1985/1997, 1997). In the culturally relevant And, reflective competence achieves mathematics’ mathematics classroom, the teacher builds from the critical dimension, reflecting upon and evaluating the students’ ethno or informal mathematics and orients just and unjust uses of mathematics. Skovsmose the lesson toward their culture and experiences, while contended that mathemacy is a necessary condition for developing the students’ critical thinking skills a politically informed citizenry and efficient labor (Gutstein, Lipman, Hernandez, & de los Reyes, 1997). force, claiming that mathemacy provides a means for The positive results of teaching from a culturally empowerment in organizing and reorganizing social relevant perspective are realized when students and political institutions and their accompanying develop mathematics empowerment: deducing traditions. mathematical generalizations and constructing creative

14 Mathematics as “Gate-Keeper” (?) Transforming Gatekeeping Mathematics perspective, as mathematics is understood as a tool that The preceding sketches demonstrate that these can be used for critique. three theoretical perspectives approach mathematics How do the three aspects of mathematics and and mathematics teaching and learning differently than mathematics teaching and learning relate to each other traditional perspectives. All three perspectives, in in these perspectives and how does this relationship varying degrees, question the taken-for-granted address the three domains of empowering assumptions about mathematics and its nature and mathematics? First, mathematics empowerment is worth, locate the formation of mathematics knowledge achieved because each perspective questions the within the social community, and argue that assumptions that are often taken-for-granted about the mathematics is an indispensable instrument used in nature and worth of mathematics. Although all three perspectives see value in the study of mathematics, sociopolitical critique. In the following paragraphs I 5 explicate the degrees to which the three perspectives including “academic” mathematics, they differ from traditional perspectives in that academic mathematics address these issues. 6 The situated perspective negates the assumption itself is troubled with regards to its contextual that mathematics is a contextually free discipline, existence, its cultural connectedness, and its critical contending that it is the context in which mathematics utility. Second, students achieve social empowerment is learned that determines how it will be used and because all three perspectives argue that students understood. The culturally relevant perspective negates should engage in mathematics contextually and the assumption that mathematics is a culturally free culturally; and, therefore students have the opportunity discipline, recognizing mathematics is not separate to gain confidence in using mathematics in routine and from culture but is a product of culture. The critical nonroutine problems. The advocates for these three perspective redefines the worth of mathematics perspectives argue that as students expand the use of through an acknowledgment and critical examination mathematics into nonroutine problems, they become of the just and, often overlooked, unjust uses of cognizant of how mathematics can be used as a tool for mathematics. sociopolitical critique. Finally students achieve The situated perspective locates mathematics epistemological empowerment because all three knowledge in the social community. In this perspectives trouble academic mathematics that in turn perspective, mathematics is not learned from a may lead students to understand that the concept of a mathematics textbook and then applied to real-world “true” or “politically-free” mathematics is a fiction. contexts, but is negotiated in communities that exist in Students will hopefully understand that mathematics real-world contexts. The culturally relevant perspective knowledge is (and always has been) a contextually and also locates mathematics knowledge in the social culturally (and politically) constructed human community. This perspective argues teachers should endeavor. If students achieve this perspective of begin to build on the collective mathematics mathematics, they will better understand their role as knowledge present in the classroom communities, producers of mathematics knowledge, not just moving toward mathematics found in textbooks. The consumers. Hence, the three domains of empowering critical perspective does not locate mathematics mathematics—mathematical, social, and knowledge in the social community but is oriented epistemological—are achieved in each perspective or towards using mathematics to critique and transform through various combinations of the three perspectives. the social and political communities in which The chief aim of an empowering mathematics is to mathematics exists and has its origins. transform gatekeeping mathematics from a discipline The situated perspective posits that students will of oppressive exclusion into a discipline of begin to understand mathematics as a discipline that is empowering inclusion. (This aim is inclusive of learned in the context of communities. It is in this way mathematics educators and education researchers.) that students may learn how mathematics can be Empowering inclusion is achieved when students (and applied in uncovering the inequities and injustices teachers of mathematics) are presented with the present in communities or can be used for opportunity to learn that the foundations of sociopolitical critique. Similarly, one of the two tenets mathematics can be troubled. This troubling of of the culturally relevant perspective is for the teacher mathematics’ foundations transforms the discourse in to assist students in developing a sociopolitical the mathematics classroom from a discourse of consciousness. Finally, using mathematics as a means transmitting mathematics to a “chosen” few students, for sociopolitical critique is essential to the critical into a discourse of exploring mathematics with all

David W. Stinson 15 students. Empowering inclusion is achieved when school mathematics as gatekeeper get produced and students (and teachers of mathematics) are presented regulated? How does school mathematics as with the opportunity to learn that, similar to literacy, gatekeeper exist? (Bové, 1995). These questions mathemacy is a tool that can be used to reword worlds. transform the discussions around gatekeeper This rewording of worlds (Freire, 1970/2000) with mathematics from discussions that attempt to find mathematics transforms mathematics from a tool used meaning in gatekeeper mathematics to discussions that by a few students in “mathematical” pursuits, into a examine the ethics of gatekeeper mathematics. Implicit tool used by all students in sociopolitical pursuits. in this examination is an analysis of how the structure Finally, empowering inclusion is achieved when of schools and those responsible for that structure are students (and teachers of mathematics) are presented implicated (or not) in reproducing the unethical effects with the opportunity to learn that mathematics of gatekeeping mathematics. knowledge is constructed human knowledge. This Will asking the questions noted above transform returning to the origins of mathematics knowledge gatekeeping mathematics from an exclusive instrument transforms mathematics from an Ideal of the gods for stratification into an inclusive instrument for reproduced by a few students, into a human endeavor empowerment? Will asking these questions stop the produced by all students. “ability” sorting of eight-year-old children? Will asking these questions encourage mathematics teachers Concluding Thoughts (and educators) to adopt the situated, culturally The concept of mathematics as gatekeeper has a relevant, or critical perspectives, perspectives that aim very long and disturbing history. There have been toward empowering all children with a key? Although educators satisfied with the gatekeeping status of I believe that there are no definitive answers to these mathematics and those that have questioned not only questions, I do believe that critically examining (and its gatekeeping status but also its nature and worth. In implementing) the different possibilities for my thinking about mathematics as a gatekeeper and the mathematics teaching and learning found in the possibility of transforming mathematics education, I theoretical perspectives explained in this article often reflect on Foucault’s challenge. He challenged us provides a sensible beginning to transforming to think the un-thought, to think: “how is it that one mathematics education. In closing, I fervently proclaim particular statement appeared rather than another?” the way we use mathematics today in our nation’s (Foucault, 1969/1972, p. 27). With Foucault’s schools must stop! Mathematics should not be used as challenge in mind, I often think what if Plato had said, an instrument for stratification but rather an instrument We shall persuade those who are to perform high for empowerment! functions in the city to undertake ______, but REFERENCES not as amateurs. They should persist in their studies until they reach the level of pure thought, where Boaler, J. (2000a). Exploring situated insights into research and they will be able to contemplate the very nature of learning. Journal for Research in Mathematics Education, 31(1), 113–119. ______…. it should serve the purposes of war and lead the soul away from the world of Boaler, J. (2000b). Mathematics from another world: Traditional communities and the alienation of learners. Journal of appearances toward essence and reality. (trans. Mathematical Behavior, 18(4), 379–397. 1996, p. 219) Boaler, J. (1993). The role of context in the mathematics In the preceding blanks, I insert different human classroom: Do they make mathematics more “real”? For the pursuits, such as writing, speaking, painting, sculpting, Learning of Mathematics, 13(2), 12–17. dancing, and so on, asking: does mathematics really Bové, P. A. (1995). Discourse. In F. Lentricchia & T. McLaughlin lead the soul away from the world of appearances (Eds.), Critical terms for literary study (pp. 50–65). Chicago: University of Chicago Press. toward essence and reality?7 Or could dancing, for example, achieve the same result? 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16 Mathematics as “Gate-Keeper” (?) D’Ambrosio, U. (1997). Foreword. In A. B. Powell & M. Moses, R. P., & Cobb, C. E. (2001). Radical equations: Math Frankenstein (Eds.), Ethnomathematics: Challenging literacy and civil rights. Boston: Beacon Press. Eurocentrism in mathematics education (pp. xv–xxi). Albany: National Council of Teachers of Mathematics. (2000). Principles State University of New York Press. and standards for school mathematics. Reston, VA: Author. Derrida, J. (1997). Of grammatology (Corrected ed.). Baltimore: National Council of Teachers of Mathematics. (1995). Assessment Johns Hopkins University Press. (Original work published standards for school mathematics. Reston, VA: Author. 1974) National Council of Teachers of Mathematics. (1991). Professional Ernest, P. (2002). Empowerment in mathematics education. standards for teaching mathematics. Reston, VA: Author. Philosophy of Mathematics Education, 15. Retrieved January National Council of Teachers of Mathematics. 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Teaching mathematics for understanding: Strategies and activities (Unpublished manuscript). Atlanta: Johns Hopkins University Press. (Original work published Georgia State University. 1974) Kilpatrick, J. (1992). A history of research in mathematics Stanic, G. M. A. (1989). Social inequality, cultural discontinuity, education. In D. A. Grouws (Ed.), Handbook of research on and equity in school mathematics. Peabody Journal of Education, 66(2), 57–71. mathematics teaching and learning (pp. 3–38). New York: Macmillan. Stanic, G. M. A. (1986). The growing crisis in mathematics education in the early twentieth century. Journal for Research Kliebard, H. M. (1995). The struggle for the American curriculum. New York: Routledge. in Mathematics Education, 17(3), 190–205. Tate, W. F. (1995). Economics, equity, and the national Ladson-Billings, G. (2001). The power of pedagogy: Does teaching matter? In W. H. Watkins, J. H. Lewis, & V. Chou (Eds.), mathematics assessment: Are we creating a national toll road? Race and education: The roles of history and society in In W. G. Secada, E. Fennema, & L. Byrd (Eds.), New directions for equity in mathematics education (pp. 191–206). educating African American students (pp. 73–88). Boston: Allyn & Bacon. Cambridge: Cambridge University Press. U.S. Department of Education. (1997). Mathematics equals Ladson-Billings, G. (1995). Making mathematics meaningful in a multicultural context. In W. G. Secada, E. Fennema, & L. opportunity. White Paper prepared for U.S. Secretary of Byrd (Eds.), New directions for equity in mathematics Education Richard W. Riley. Retrieved January 26, 2004, from http://www.ed.gov/pubs/math/mathemat.pdf education (pp. 126–145). Cambridge: Cambridge University Press. U.S. Department of Education. (1999). Do gatekeeper courses Ladson-Billings, G. (1994). The Dreamkeepers: Successful expand education options? National Center for Education teachers of African American children. San Francisco: Jossey- Statistics. Retrieved January 26, 2004, from Bass. http://nces.ed.gov/pubsearch/pubsinfo.asp?pubid=1999303 Lather, P. (1991). Getting smart: Feminist research and pedagogy 1 with/in the postmodern. New York: Routledge. The student racial/ethnic data were based on the 2001-2002 Lemann, N. (1999). The big test: The secret history of the Georgia Public Education Report Card; the racial/ethnic American meritocracy (1st ed.). New York: Farrar Straus and classifications were the State of Georgia’s not this author’s. For Giroux. details of racial/ethnic data on all schools in the State of Georgia Lerman, S. (2000). The social turn in mathematics education see: http://techservices.doe.k12.ga.us/reportcard/ 2 research. In J. Boaler (Ed.), International perspectives on Plato (trans. 1996) in establishing his utopian Republic imagined mathematics education, (pp. 19–44). Westport, CT: Ablex. that the philosopher guardians of the city, identified as the

David W. Stinson 17 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 Mathematics (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.

18 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 does one define creativity? In particular what exactly is field of mathematics as a whole. The constant increase mathematical creativity? Is it the discovery of a new in the number of journals devoted to mathematics theorem by a research mathematician? Does student research bears evidence to the growth of mathematics. discovery of a hitherto known result also constitute Yet what lies at the essence of this growth, the creativity? These are among the areas of exploration in creativity of the mathematician, has not been the this paper. subject of much research. It is usually the case that The Problem Of Defining Creativity most mathematicians are uninterested in analyzing the thought processes that result in mathematical creation Mathematical creativity has been simply described (Ervynck, 1991). The earliest known attempt to study as discernment, or choice (Poincaré, 1948). According mathematical creativity was an extensive questionnaire to Poincaré (1948), to create consists precisely in not published in the French periodical L'Enseigement making useless combinations and in making those Mathematique (1902). This questionnaire and a lecture which are useful and which are only a small minority. on creativity given by the renowned 20th century Poincaré is referring to the fact that the “proper” mathematician Henri Poincaré to the Societé de combination of only a small minority of ideas results in Psychologie inspired his colleague Jacques Hadamard, a creative insight whereas a majority of such another prominent 20th century mathematician, to combinations does not result in a creative outcome. investigate the psychology of mathematical creativity This may seem like a vague characterization of (Hadamard, 1945). Hadamard (1945) undertook an mathematical creativity. One can interpret Poincaré's informal inquiry among prominent mathematicians and "choice" metaphor to mean the ability of the scientists in America, including George Birkhoff, mathematician to choose carefully between questions George Polya, and Albert Einstein, about the mental (or problems) that bear fruition, as opposed to those images used in doing mathematics. Hadamard (1945), that lead to nothing new. But this interpretation does influenced by the Gestalt psychology of his time, not resolve the fact that Poincaré’s definition of theorized that mathematicians’ creative processes creativity overlooks the problem of novelty. In other followed the four-stage Gestalt model (Wallas, 1926) words, characterizing mathematical creativity as the of preparation-incubation-illumination-verification. ability to choose between useful and useless As we will see, the four-stage Gestalt model is a combinations is akin to characterizing the art of characterization of the mathematician's creative sculpting as a process of cutting away the unnecessary! process, but it does not define creativity per se. How Poincaré's (1948) definition of creativity was a result of the circumstances under which he stumbled Bharath Sriraman is an assistant professor of mathematics and upon deep results in Fuchsian functions. The first stage mathematics education at the University of Montana. His in creativity consists of working hard to get an insight publications and research interests are in the areas of cognition, foundational issues, mathematical creativity, problem-solving, into the problem at hand. Poincaré (1948) called this proof, and gifted education. the preliminary period of conscious work. This period is also referred to as the preparatory stage (Hadamard,

Bharath Sriraman 19 1945). In the second, or incubatory, stage (Hadamard, generalize mathematical content. There is also an 1945), the problem is put aside for a period of time and outstanding example of a mathematician (George the mind is occupied with other problems. In the third Polya) attempting to give heuristics to tackle problems stage the solution suddenly appears while the in a manner akin to the methods used by trained mathematician is perhaps engaged in other unrelated mathematicians. Polya (1954) observed that in "trying activities. "This appearance of sudden illumination is a to solve a problem, we consider different aspects of it manifest sign of long, unconscious prior work" in turn, we roll it over and over in our minds; variation (Poincaré, 1948). Hadamard (1945) referred to this as of the problem is essential to our work." Polya (1954) the illuminatory stage. However, the creative process emphasized the use of a variety of heuristics for does not end here. There is a fourth and final stage, solving mathematical problems of varying complexity. which consists of expressing the results in language or In examining the plausibility of a mathematical writing. At this stage, one verifies the result, makes it conjecture, mathematicians use a variety of strategies. precise, and looks for possible extensions through In looking for conspicuous patterns, mathematicians utilization of the result. The “Gestalt model” has some use such heuristics as (1) verifying consequences, (2) shortcomings. First, the model mainly applies to successively verifying several consequences, (3) problems that have been posed a priori by verifying an improbable consequence, (4) inferring mathematicians, thereby ignoring the fascinating from analogy, and (5) deepening the analogy. process by which the actual questions evolve. As is evident in the preceding paragraphs, the Additionally, the model attributes a large portion of problem of defining creativity is by no means an easy what “happens” in the incubatory and illuminatory one. However, psychologists’ renewed interest in the phases to subconscious drives. The first of these phenomenon of creativity has resulted in literature that shortcomings, the problem of how questions are attempts to define and operationalize the word developed, is partially addressed by Ervynck (1991) in “creativity.” Recently psychologists have attempted to his three-stage model. link creativity to measures of intelligence (Sternberg, Ervynck (1991) described mathematical creativity 1985) and to the ability to abstract, generalize in terms of three stages. The first stage (Stage 0) is (Sternberg, 1985), and solve complex problems referred to as the preliminary technical stage, which (Frensch & Sternberg, 1992). Sternberg and Lubart consists of "some kind of technical or practical (2000) define creativity as the ability to produce application of mathematical rules and procedures, unexpected original work that is useful and adaptive. without the user having any awareness of the Mathematicians would raise several arguments with theoretical foundation" (p. 42). The second stage this definition, simply because the results of creative (Stage 1) is that of algorithmic activity, which consists work may not always have implications that are primarily of performing mathematical techniques, such “useful” in terms of applicability in the contemporary as explicitly applying an algorithm repeatedly. The world. A recent example that comes to mind is Andrew third stage (Stage 2) is referred to as creative Wiles’ proof of Fermat’s Last Theorem. The (conceptual, constructive) activity. This is the stage in mathematical community views his work as creative. It which true mathematical creativity occurs and consists was unexpected and original but had no applicability in of non-algorithmic decision making. "The decisions the sense Sternberg and Lubart (2000) suggest. Hence, that have to be taken may be of a widely divergent I think it is sufficient to define creativity as the ability nature and always involve a choice" (p. 43). Although to produce novel or original work, which is compatible Ervynck (1991) tries to describe the process by which a with my personal definition of mathematical creativity mathematician arrives at the questions through his as the process that results in unusual and insightful characterizations of Stage 0 and Stage 1, his solutions to a given problem, irrespective of the level description of mathematical creativity is very similar to of complexity. In the context of this study involving those of Poincaré and Hadamard. In particular his use professional mathematicians, mathematical creativity is of the term “non-algorithmic decision making” is defined as the publication of original results in analogous to Poincaré’s use of the “choice” metaphor. prominent mathematics research journals. The mathematics education literature indicates that The Motivation For Studying Creativity very few attempts have been made to explicitly define mathematical creativity. There are references made to The lack of recent mathematics education literature creativity by the Soviet researcher Krutetskii (1976) in on the subject of creativity was one of the motivations the context of students’ abilities to abstract and for conducting this study. Fifteen years ago Muir

20 Mathematical Creativity (1988) invited mathematicians to complete a modified (intuitionist) viewpoint is that “human mathematical and updated version of the survey that appeared in activity is fundamental in the creation of new L'Enseigement Mathematique (1902) but the results of knowledge and that both mathematical truths and the this endeavor are as yet unknown. The purpose of this existence of mathematical objects must be established study was to gain insight into the nature of by constructive methods" (Ernest, 1991, p. 29). mathematical creativity. I was interested in distilling Contradictions like Russell’s Paradox were a major common attributes of the creative process to see if blow to the absolutist view of mathematical there were any underlying themes that characterized knowledge, for if mathematics is certain and all its mathematical creativity. The specific questions of theorems are certain, how can there be contradictions exploration in this study were: among its theorems? The early constructivists in mathematics were the intuitionists Brouwer and Is the Gestalt model of mathematical creativity still applicable today? Heyting. Constructivists claim that both mathematical truths and the existence of mathematical objects must What are the characteristics of the creative process be established by constructivist methods. in mathematics? The question then is how does a mathematician go Does the study of mathematical creativity have any about conducting mathematics research? Do the implications for the classroom? questions appear out of the blue, or is there a mode of Literature Review thinking or inquiry that leads to meaningful questions and to the methodology for tackling these questions? I Any study on the nature of mathematical creativity contend that the types of questions asked are begs the question as to whether the mathematician determined to a large extent by the culture in which the discovers or invents mathematics. Therefore, this mathematician lives and works. Simply put, it is review begins with a brief description of the four most impossible for an individual to acquire knowledge of popular viewpoints on the nature of mathematics. This the external world without social interaction. is followed by a comprehensive review of According to Ernest (1994) there is no underlying contemporary models of creativity from psychology. metaphor for the wholly isolated individual mind. The Nature of Mathematics Instead, the underlying metaphor is that of persons in conversation, persons who participate in meaningful Mathematicians actively involved in research have linguistic interaction and dialogue (Ernest, 1994). certain beliefs about the ontological nature of Language is the shaper, as well as being the mathematics that influence their approach to research “summative” product, of individual minds (Davis & Hersh, 1981; Sriraman, 2004a). The Platonist (Wittgenstein, 1978). The recent literature in viewpoint is that mathematical objects exist prior to psychology acknowledges these social dimensions of their discovery and that “any meaningful question human activity as being instrumental in the creative about a mathematical object has a definite answer, process. whether we are able to determine it or not” (Davis & Hersh, 1981). According to this view, mathematicians The Notion of Creativity in Psychology do not invent or create mathematics - they discover As stated earlier, research on creativity has been on mathematics. Logicists hold that “all concepts of the fringes of psychology, educational psychology, and mathematics can ultimately be reduced to logical mathematics education. It is only in the last twenty-five concepts” which implies that “all mathematical truths years that there has been a renewed interest in the can be proved from the axioms and rules of inference phenomenon of creativity in the psychology and logic alone” (Ernest, 1991). Formalists do not community. The Handbook of Creativity (Sternberg, believe that mathematics is discovered; they believe 2000), which contains a comprehensive review of all mathematics is simply a game, created by research then available in the field of creativity, mathematicians, based on strings of symbols that have suggests that most of the approaches used in the study no meaning (Davis & Hersh, 1981). of creativity can be subsumed under six categories: Constructivism (incorporating Intuitionism) is one mystical, pragmatic, psychodynamic, psychometric, of the major schools of thought (besides Platonism, cognitive, and social-personality. Each of these Logicism and Formalism) that arose due to the approaches is briefly reviewed. 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 21 The mystical approach as Albert Einstein, but the behaviorists criticized this The mystical approach to studying creativity approach because of the difficulty in measuring suggests that creativity is the result of divine proposed theoretical constructs. inspiration or is a spiritual process. In the history of The psychometric approach mathematics, Blaise Pascal claimed that many of his The psychometric approach to studying creativity mathematical insights came directly from God. The entails quantifying the notion of creativity with the aid renowned 19th century algebraist Leopold Kronecker of paper and pencil tasks. An example of this would be said that “God made the integers, all the rest is the the Torrance Tests of Creative Thinking, developed by work of man” (Gallian, 1994). Kronecker believed that Torrance (1974), that are used by many gifted all other numbers, being the work of man, were to be programs in middle and high schools to identify avoided; and although his radical beliefs did not attract students that are gifted/creative. These tests consist of many supporters, the intuitionists advocated his beliefs several verbal and figural tasks that call for problem- about constructive proofs many years after his death. solving skills and divergent thinking. The test is scored There have been attempts to explore possible for fluency, flexibility, originality (the statistical rarity relationships between mathematicians’ beliefs about of a response), and elaboration (Sternberg, 2000). the nature of mathematics and their creativity (Davis Sternberg (2000) states that there are positive and and Hersh, 1981; Hadamard, 1945; Poincaré, 1948; negative sides to the psychometric approach. On the Sriraman, 2004a). These studies indicate that such a positive side, these tests allow for research with non- relationship does exist. It is commonly believed that eminent people, are easy to administer, and objectively the neo-Platonist view is helpful to the research scored. The negative side is that numerical scores fail mathematician because of the innate belief that the to capture the concept of creativity because they are sought after result/relationship already exists. based on brief paper and pencil tests. Researchers call The pragmatic approach for using more significant productions such as writing The pragmatic approach entails “being concerned samples, drawings, etc., subjectively evaluated by a primarily with developing creativity” (Sternberg, 2000, panel of experts, instead of simply relying on a p. 5), as opposed to understanding it. Polya’s (1954) numerical measure. emphasis on the use of a variety of heuristics for The cognitive approach solving mathematical problems of varying complexity The cognitive approach to the study of creativity is an example of a pragmatic approach. Thus, focuses on understanding the “mental representations heuristics can be viewed as a decision-making and processes underlying human thought” (Sternberg, mechanism which leads the mathematician down a 2000, p. 7). Weisberg (1993) suggests that creativity certain path, the outcome of which may or may not be entails the use of ordinary cognitive processes and fruitful. The popular technique of brainstorming, often results in original and extraordinary products. These used in corporate or other business settings, is another products are the result of cognitive processes acting on example of inducing creativity by seeking as many the knowledge already stored in the memory of the ideas or solutions as possible in a non-critical setting. individual. There is a significant amount of literature in The psychodynamic approach the area of information processing (Birkhoff, 1969; The psychodynamic approach to studying Minsky, 1985) that attempts to isolate and explain creativity is based on the idea that creativity arises cognitive processes in terms of machine metaphors. from the tension between conscious reality and The social-personality approach unconscious drives (Hadamard, 1945; Poincaré, 1948, The social-personality approach to studying Sternberg, 2000, Wallas, 1926; Wertheimer, 1945). creativity focuses on personality and motivational The four-step Gestalt model (preparation-incubation- variables as well as the socio-cultural environment as illumination-verification) is an example of the use of a sources of creativity. Sternberg (2000) states that psychodynamic approach to studying creativity. It numerous studies conducted at the societal level should be noted that the gestalt model has served as indicate that “eminent levels of creativity over large kindling for many contemporary problem-solving spans of time are statistically linked to variables such models (Polya, 1945; Schoenfeld, 1985; Lester, 1985). as cultural diversity, war, availability of role models, Early psychodynamic approaches to creativity were availability of financial support, and competitors in a used to construct case studies of eminent creators such domain” (p. 9).

22 Mathematical Creativity Most of the recent literature on creativity (Csikzentmihalyi, 2000). These three components - (Csikszentmihalyi, 1988, 2000; Gruber & Wallace, individual, domain, and field - are necessary because 2000; Sternberg & Lubart, 1996) suggests that the individual operates from a cultural or symbolic creativity is the result of a confluence of one or more (domain) aspect as well as a social (field) aspect. of the factors from these six aforementioned “The domain is a necessary component of categories. The “confluence” approach to the study of creativity because it is impossible to introduce a creativity has gained credibility, and the research variation without reference to an existing pattern. New literature has numerous confluence theories for better is meaningful only in reference to the old” understanding the process of creativity. A review of the (Csikzentmihalyi, 2000). Thus, creativity occurs when most commonly cited confluence theories of creativity an individual proposes a change in a given domain, and a description of the methodology employed for which is then transmitted by the field through time. data collection and data analysis in this study follow. The personal background of an individual and his position in a domain naturally influence the likelihood Confluence Theories of Creativity of his making a contribution. For example, a The three most commonly cited “confluence” mathematician working at a research university is more approaches to the study of creativity are the “systems likely to produce research papers because of the time approach” (Csikszentmihalyi, 1988, 2000); “the case available for “thinking” as well as the creative study as evolving systems approach” (Gruber & influence of being immersed in a culture where ideas Wallace, 2000), and the “investment theory approach” flourish. It is no coincidence that in the history of (Sternberg & Lubart, 1996). The case study as an science, there are significant contributions from evolving system has the following components. First, it clergymen such as Pascal and Mendel because they views creative work as multi-faceted. So, in had the means and the leisure to “think.” constructing a case study of a creative work, one must Csikszentmihalyi (2000) argues that novel ideas, which distill the facets that are relevant and construct the case could result in significant changes, are unlikely to be study based on the chosen facets. Some facets that can adopted unless they are sanctioned by the experts. be used to construct an evolving system case study are: These “gatekeepers” (experts) constitute the field. For (1) uniqueness of the work; (2) a narrative of what the example, in mathematics, the opinion of a very small creator achieved; (3) systems of belief; (4) multiple number of leading researchers was enough to certify time-scales (construct the time-scales involved in the the validity of Andrew Wiles’ proof of Fermat’s Last production of the creative work); (5) problem solving; Theorem. and (6) contextual frame such as family, schooling, and There are numerous examples in the history of teacher’s influences (Gruber & Wallace, 2000). In mathematics that fall within the systems model. For summary, constructing a case study of a creative work instance, the Bourbaki, a group of mostly French as an evolving system entails incorporating the many mathematicians who began meeting in the 1930s, facets suggested by Gruber & Wallace (2000). One aimed to write a thorough unified account of all could also evaluate a case study involving creative mathematics. The Bourbaki were essentially a group of work by looking for the above mentioned facets. expert mathematicians that tried to unify all of The systems approach mathematics and become the gatekeepers of the field, so to speak, by setting the standard for rigor. Although The systems approach takes into account the social the Bourbakists failed in their attempt, students of the and cultural dimensions of creativity instead of simply Bourbakists, who are editors of certain prominent viewing creativity as an individualistic psychological journals, to this day demand a very high degree of rigor process. The systems approach studies the interaction in submitted articles, thereby serving as gatekeepers of between the individual, domain, and field. The field the field. consists of people who have influence over a domain. A different example is that of the role of proof. For example, editors of mathematics research journals Proof is the social process through which the have influence over the domain of mathematics. The mathematical community validates the mathematician's domain is in a sense a cultural organism that preserves creative work (Hanna, 1991). The Russian logician and transmits creative products to individuals in the Manin (1977) said "A proof becomes a proof after the field. The systems model suggests that creativity is a social act of accepting it as a proof. This is true of process that is observable at the “intersection where mathematics as it is of physics, linguistics, and individuals, domains and fields interact” biology."

Bharath Sriraman 23 In summary, the systems model of creativity years of numerical calculations. Andrew Wiles’ proof suggests that for creativity to occur, a set of rules and of Fermat’s Last Theorem was a seven-year practices must be transmitted from the domain to the undertaking. The Riemann hypothesis states that the individual. The individual then must produce a novel roots of the zeta function (complex numbers z, at variation in the content of the domain, and this which the zeta function equals zero) lie on the line variation must be selected by the field for inclusion in parallel to the imaginary axis and half a unit to the the domain. right of it. This is perhaps the most outstanding unproved conjecture in mathematics with numerous Gruber and Wallace’s case study as evolving implications. The analyst Levinson undertook a systems approach determined calculation on his deathbed that increased In contrast to Csikszentmihalyi’s (2000) argument the credibility of the Riemann-hypothesis. This is calling for a focus on communities in which creativity another example of creative work that falls within manifests itself, Gruber and Wallace (2000) propose a Gruber and Wallace's (2000) model. model that treats each individual as a unique evolving system of creativity and ideas; and, therefore, each The investment theory approach individual’s creative work must be studied on its own. According to the investment theory model, creative This viewpoint of Gruber and Wallace (2000) is a people are like good investors; that is, they buy low belated victory of sorts for the Gestaltists, who and sell high (Sternberg & Lubart, 1996). The context essentially proclaimed the same thing almost a century here is naturally in the realm of ideas. Creative people ago. Gruber and Wallace’s (2000) use of terminology conjure up ideas that are either unpopular or that jibes with current trends in psychology seems to disrespected and invest considerable time convincing make their ideas more acceptable. They propose a other people about the intrinsic worth of these ideas model that calls for “detailed analytic and sometimes (Sternberg & Lubart, 1996). They sell high in the sense narrative descriptions of each case and efforts to that they let other people pursue their ideas while they understand each case as a unique functioning system move on to the next idea. Investment theory claims that (Gruber & Wallace, 2000, p. 93). It is important to note the convergence of six elements constitutes creativity. that the emphasis of this model is not to explain the The six elements are intelligence, knowledge, thinking origins of creativity, nor is it the personality of the styles, personality, motivation, and environment. It is creative individual, but on “how creative work works” important that the reader not mistake the word (p. 94). The questions of concern to Gruber and intelligence for an IQ score. On the contrary, Sternberg Wallace are: (1) What do creative people do when they (1985) suggests a triarchic theory of intelligence that are being creative? and (2) How do creative people consists of synthetic (ability to generate novel, task deploy available resources to accomplish something appropriate ideas), analytic, and practical abilities. unique? In this model creative work is defined as that Knowledge is defined as knowing enough about a which is novel and has value. This definition is particular field to move it forward. Thinking styles are consistent with that used by current researchers in defined as a preference for thinking in original ways of creativity (Csikszentmihalyi, 2000; Sternberg & one’s choosing, the ability to think globally as well as Lubart, 2000). Gruber and Wallace (2000) also claim locally, and the ability to distinguish questions of that creative work is always the result of purposeful importance from those that are not important. behavior and that creative work is usually a long Personality attributes that foster creative functioning undertaking “reckoned in months, years and decades” are the willingness to take risks, overcome obstacles, (p. 94). and tolerate ambiguity. Finally, motivation and an I do not agree with the claim that creative work is environment that is supportive and rewarding are always the result of purposeful behavior. One essential elements of creativity (Sternberg, 1985). counterexample that comes to mind is the discovery of In investment theory, creativity involves the penicillin. The discovery of penicillin could be interaction between a person, task, and environment. attributed purely to chance. On the other hand, there This is, in a sense, a particular case of the systems are numerous examples that support the claim that model (Csikszentmihalyi, 2000). The implication of creative work sometimes entails work that spans years, viewing creativity as the interaction between person, and in mathematical folklore there are numerous task, and environment is that what is considered novel examples of such creative work. For example, Kepler’s or original may vary from one person, task, and laws of planetary motion were the result of twenty environment to another. The investment theory model

24 Mathematical Creativity suggests that creativity is more than a simple sum of Background of the Subjects the attained level of functioning in each of the six Five mathematicians from the mathematical elements. Regardless of the functioning levels in other sciences faculty at a large Ph.D. granting mid-western elements, a certain level or threshold of knowledge is university were selected. These mathematicians were required without which creativity is impossible. High chosen based on their accomplishments and the levels of intelligence and motivation can positively diversity of the mathematical areas in which they enhance creativity, and compensations can occur to worked, measured by counting the number of counteract weaknesses in other elements. For example, published papers in prominent journals, as well as one could be in an environment that is non-supportive noting the variety of mathematical domains in which of creative efforts, but a high level of motivation could they conducted research. Four of the mathematicians possibly overcome this and encourage the pursuit of were tenured full professors, each of whom had been creative endeavors. professional mathematicians for more than 30 years. This concludes the review of three commonly cited One of the mathematicians was considerably younger prototypical confluence theories of creativity, namely but was a tenured associate professor. All interviews the systems approach (Csikszentmihalyi, 2000), which were conducted formally, in a closed door setting, in suggests that creativity is a sociocultural process each mathematician’s office. The interviews were involving the interaction between the individual, audiotaped and transcribed verbatim. domain, and field; Gruber & Wallace’s (2000) model that treats each individual case study as a unique Data Analysis evolving system of creativity; and investment theory Since creativity is an extremely complex construct (Sternberg & Lubart, 1996), which suggests that involving a wide range of interacting behaviors, I creativity is the result of the convergence of six believe it should be studied holistically. The principle elements (intelligence, knowledge, thinking styles, of analytic induction (Patton, 2002) was applied to the personality, motivation, and environment). interview transcripts to discover dominant themes that Having reviewed the research literature on described the behavior under study. According to creativity, the focus is shifted to the methodology Patton (2002), "analytic induction, in contrast to employed for studying mathematical creativity. grounded theory, begins with an analyst's deduced Methodology propositions or theory-derived hypotheses and is a procedure for verifying theories and propositions based The Interview Instrument on qualitative data” (Taylor and Bogdan, 1984, p. 127). The purpose of this study was to gain an insight Following the principles of analytic induction, the data into the nature of mathematical creativity. In an effort was carefully analyzed in order to extract common to determine some of the characteristics of the creative strands. These strands were then compared to process, I was interested in distilling common theoretical constructs in the existing literature with the attributes in the ways mathematicians create explicit purpose of verifying whether the Gestalt model mathematics. Additionally, I was interested in testing was applicable to this qualitative data as well as to the applicability of the Gestalt model. Because the extract themes that characterized the mathematician’s main focus of the study was to ascertain qualitative creative process. If an emerging theme could not be aspects of creativity, a formal interview methodology classified or named because I was unable to grasp its was selected as the primary method of data collection. properties or significance, then theoretical comparisons The interview instrument (Appendix A) was developed were made. Corbin and Strauss (1998) state that “using by modifying questions from questionnaires in comparisons brings out properties, which in turn can be L’Enseigement Mathematique (1902) and Muir (1988). used to examine the incident or object in the data. The The rationale behind using this modified questionnaire specific incidents, objects, or actions that we use when was to allow the mathematicians to express themselves making theoretical comparisons can be derived from freely while responding to questions of a general the literature and experience. It is not that we use nature and to enable me to test the applicability of the experience or literature as data “but rather that we use four-stage Gestalt model of creativity. Therefore, the the properties and dimensions derived from the existing instruments were modified to operationalize comparative incidents to examine the data in front of the Gestalt theory and to encourage the natural flow of us” (p. 80). Themes that emerged were social ideas, thereby forming the basis of a thesis that would interaction, preparation, use of heuristics, imagery, emerge from this exploration. incubation, illumination, verification, intuition, and

Bharath Sriraman 25 proof. Excerpts from interviews that highlight these valued the interaction they had with their graduate characteristics are reconstructed in the next section students. Excerpts of individual responses follow.1 along with commentaries that incorporate the wider Excerpt 1 conversation, and a continuous discussion of connections to the existing literature. A. I've had only one graduate student per semester and she is just finishing up her PhD right now, Results, Commentaries & Discussion and I'd say it has been a very good interaction The mathematicians in this study worked in to see somebody else get interested in the academic environments and regularly fulfilled teaching subject and come up with new ideas, and and committee duties. The mathematicians were free to exploring those ideas with her. choose their areas of research and the problems on B. I have had a couple of students who have sort which they focused. Four of the five mathematicians of started but who haven't continued on to a had worked and published as individuals and as PhD, so I really can't speak to that. But the members of occasional joint ventures with interaction was positive. mathematicians from other universities. Only one of the mathematicians had done extensive collaborative C. Of course, I have a lot of collaborators, these work. All but one of the mathematicians were unable are my former students you know…I am to formally structure their time for research, primarily always all the time working with students, this due to family commitments and teaching is normal situation. responsibilities during the regular school year. All the D. That is difficult to answer (silence)…it is mathematicians found it easier to concentrate on positive because it is good to interact with research in the summers because of lighter or non- other people. It is negative because it can take existent teaching responsibilities during that time. Two a lot of time. As you get older your brain of the mathematicians showed a pre-disposition doesn't work as well as it used to towards mathematics at the early secondary school and…younger people by and large their minds level. The others became interested in mathematics are more open, there is less garbage in there later, during their university education. The already. So, it is exciting to work with younger mathematicians who participated in this study did not people who are in their most creative time. report any immediate family influence that was of When you are older, you have more primary importance in their mathematical experience, when you are younger your mind development. Four of the mathematicians recalled works faster …not as fettered. being influenced by particular teachers, and one reported being influenced by a textbook. The three E. Oh…it is a positive factor I think, because it mathematicians who worked primarily in analysis continues to stimulate ideas …talking about made a conscious effort to obtain a broad overview of things and it also reviews things for you in the mathematics not necessarily of immediate relevance to process, puts things in perspective, and keep their main interests. The two algebraists expressed the big picture. It is helpful really in your own interest in other areas of mathematics but were research to supervise students. primarily active in their chosen field. Commentary on Excerpt 1 Supervision Of Research & Social Interaction The responses of the mathematicians in the As noted earlier, all the mathematicians in this preceding excerpt are focused on research supervision; study were tenured professors in a research university. however, all of the mathematicians acknowledged the In addition to teaching, conducting research, and role of social interaction in general as an important fulfilling committee obligations, many mathematicians aspect that stimulated creative work. Many of the play a big role in mentoring graduate students mathematicians mentioned the advantages of being interested in their areas of research. Research able to e-mail colleagues and going to research supervision is an aspect of creativity because any conferences and other professional meetings. This is interaction between human beings is an ideal setting further explored in the following section, which for the exchange of ideas. During this interaction the focuses on preparation. mathematician is exposed to different perspectives on the subject, and all of the mathematicians in this study

26 Mathematical Creativity Preparation and the Use of Heuristics own. That doesn't mean that I don't When mathematicians are about to investigate a simultaneously try to work on something. new topic, there is usually a body of existing research Commentary on Excerpt 2 in the area of the new topic. One of goals of this study These responses indicate that the mathematician was to find out how creative mathematicians spends a considerable amount of time researching the approached a new topic or a problem. Did they try their context of the problem. This is primarily done by own approach, or did they first attempt to assimilate reading the existing literature and by talking to other what was already known about that topic? Did the mathematicians in the new area. This finding is mathematicians make use of computers to gain insight consistent with the systems model, which suggests that into the problem? What were the various modes of creativity is a dynamic process involving the approaching a new topic or problem? The responses interaction between the individual, domain, and field indicate that a variety of approaches were used. (Csikzentmihalyi, 2000). At this stage, it is reasonable Excerpt 2 to ask whether a mathematician works on a single problem until a breakthrough occurs or does a A. Talk to people who have been doing this topic. mathematician work on several problems concurrently? Learn the types of questions that come up. It was found that each of the mathematicians worked Then I do basic research on the main ideas. I on several problems concurrently, using a back and find that talking to people helps a lot more than forth approach. reading because you get more of a feel for what the motivation is beneath everything. Excerpt 3 B. What might happen for me, is that I may start A. I work on several different problems for a reading something, and, if feel I can do a better protracted period of time… there have been job, then I would strike off on my own. But for times when I have felt, yes, I should be able to the most part I would like to not have to prove this result, then I would concentrate on reinvent a lot that is already there. So, a lot of that thing for a while but they tend to be what has motivated my research has been the several different things that I was thinking desire to understand an area. So, if somebody about a particular stage. has already laid the groundwork then it's B. I probably tend to work on several problems at helpful. Still I think a large part of doing the same time. There are several different research is to read the work that other people questions that I am working have done. on…mm…probably the real question is how C. It is connected with one thing that simply…my often do you change the focus? Do I work on style was that I worked very much and I even two different problems on the same day? And work when I could not work. Simply the that is probably up to whatever comes to mind problems that I solve attract me so much, that in that particular time frame. I might start the question was who will die working on one rather than the other. But I first…mathematics or me? It was never clear would tend to focus on one particular problem who would die. for a period of weeks, then you switch to something else. Probably what happens is that D. Try and find out what is known. I won't say I work on something and I reach a dead end assimilate…try and find out what's known and then I may shift gears and work on a different get an overview, and try and let the problem problem for a while, reach a dead end there speak…mostly by reading because you don't and come back to the original problem, so it’s have that much immediate contact with other back and forth. people in the field. But I find that I get more from listening to talks that other people are C. I must simply think on one thing and not giving than reading. switch so much. E. Well! I have been taught to be a good scholar. D. I find that I probably work on one. There A good scholar attempts to find out what is might be a couple of things floating around but first known about something or other before I am working on one and if I am not getting they spend their time simply going it on their

Bharath Sriraman 27 anywhere, then I might work on the other and work. I have a very geometrically based intuition and then go back. uhh…so very definitely I do a lot of manipulations. E. I usually have couple of things going. When I A. That is a problem because of the particular get stale on one, then I will pick up the other, area I am in. I can't draw any diagrams, things and bounce back and forth. Usually I have one are infinite, so I would love to be able to get that is primarily my focus at a given time, and some kind of a computer diagram to show the I will spend time on it over another; but it is complexity for a particular ring… to have not uncommon for me to have a couple of something like the Julia sets problems going at a given time. Sometimes or…mmm…fractal images, things which are when I am looking for an example that is not infinite but you can focus in closer and closer coming, instead of spending my time beating to see possible relationships. I have thought my head against the wall, looking for that about that with possibilities on the computer. example is not a very good use of time. To think about the most basic ring, you would Working on another helps to generate ideas have to think of the ring of integers and all of that I can bring back to the other problem. the relationships for divisibility, so how do you somehow describe this tree of divisibility for Commentary on Excerpt 3 integers…it is infinite. The preceding excerpt indicates that mathematicians tend to work on more than one B. Science is language, you think through problem at a given time. Do mathematicians switch language. But it is language simply; you put back and forth between problems in a completely together theorems by logic. You first see the random manner, or do they employ and exhaust a theorem in nature…you must see that systematic train of thought about a problem before somewhat is reasonable and then you go and switching to a different problem? Many of the begin and then of course there is big, big, big mathematicians reported using heuristic reasoning, work to just come to some theorem in non- trying to prove something one day and disprove it the linear elliptic equations… next day, looking for both examples and C. A lot of mathematics, whether we are teaching counterexamples, the use of "manipulations" (Polya, or doing, is attaching meaning to what we are 1954) to gain an insight into the problem. This doing and this is going back to the earlier indicates that mathematicians do employ some of the question when you talked about how do you do heuristics made explicit by Polya. It was unclear it, what kind of heuristics do you use? What whether the mathematicians made use of computers to kind of images do you have that you are using? gain an experimental or computational insight into the A lot of doing mathematics is creating these problem. I was also interested in knowing the types of abstract images that connect things and then imagery used by mathematicians in their work. The making sense of them but that doesn't appear mathematicians in this study were queried about this, in proofs either. and the following excerpt gives us an insight into that D. Pictorial, linguistic, kinesthetic...any of them is aspect of mathematical creativity. the point right! Sometimes you think of one, Imagery sometimes another. It really depends on the The mathematicians in this study were asked about problem you are looking at, they are very the kinds of imagery they used to think about much…often I think of functions as very mathematical objects. Their responses are reported kinesthetic, moving things from here to there. here to give the reader a glimpse of the ways Other approaches you are talking about is mathematicians think of mathematical objects. Their going to vary from problem to problem, or responses also highlight the difficulty of explicitly even day to day. Sometimes when I am describing imagery. working on research, I try to view things in as many different ways as possible, to see what is Excerpt 4 really happening. So there are a variety of Yes I do, yes I do, I tend to draw a lot of pictures approaches. when I am doing research, I tend to manipulate things in the air, you know to try to figure out how things

28 Mathematical Creativity Commentary on Excerpt 4 preparation, so that the sub-conscious or Besides revealing the difficulty of describing intuitive side may work on it and the answer mental imagery, all the mathematicians reported that comes back but you can't really tell when. You they did not use computers in their work. This have to be open to this, lay the groundwork, characteristic of the pure mathematician's work is think about it and then these flashes of echoed in Poincaré's (1948) use of the “choice” intuition come and they represent the other metaphor and Ervynck's (1991) use of the term “non- side of the brain communicating with you at algorithmic decision making.” The doubts expressed whatever odd time. by the mathematicians about the incapability of D. I am not sure you can really separate them machines to do their work brings to mind the reported because they are somewhat connected. You words of Garrett Birkhoff, one of the great applied spend a lot of time working on something and mathematicians of our time. In his retirement you are not getting anywhere with it…with the presidential address to the Society for Industrial and deliberate effort, then I think your mind Applied Mathematics, Birkhoff (1969) addressed the continues to work and organize. And maybe role of machines in human creative endeavors. In when the pressure is off the idea comes…but particular, part of this address was devoted to the idea comes because of the hard work. discussing the psychology of the mathematicians (and E. Usually they come after I have worked very hence of mathematics). Birkhoff (1969) said: hard on something or another, but they may The remarkable recent achievements of computers come at an odd moment. They may come into have partially fulfilled an old dream. These my head before I go to bed …What do I do at achievements have led some people to speculate that point? Yes I write it down (laughing). that tomorrow's computers will be even more Sometimes when I am walking somewhere, the "intelligent" than humans, especially in their mind flows back to it (the problem) and says powers of mathematical reasoning...the ability of good mathematicians to sense the significant and to what about that, why don't you try that. That avoid undue repetition seems, however, hard to sort of thing happens. One of the best ideas I computerize; without it, the computer has to pursue had was when I was working on my thesis millions of fruitless paths avoided by experienced …Saturday night, having worked on it quite a human mathematicians. (pp. 430-438) bit, sitting back and saying why don't I think Incubation and Illumination about it again…and ping! There it was…I knew what it was, I could do that. Often ideas Having reported on the role of research supervision are handed to you from the outside, but they and social interaction, the use of heuristics and don't come until you have worked on it long imagery, all of which can be viewed as aspects of the enough. preparatory stage of mathematical creativity, it is natural to ask what occurs next. As the literature Commentary on Excerpt 5 suggests, after the mathematician works hard to gain As is evident in the preceding excerpt, three out of insight into a problem, there is usually a transition the five mathematicians reported experiences period (conscious work on the problem ceases and consistent with the Gestalt model. Mathematician C unconscious work begins), during which the problem is attributed his breakthroughs on problems to his put aside before the breakthrough occurs. The unflinching will to never give up and to divine mathematicians in this study reported experiences that inspiration, echoing the voice of Pascal in a sense. are consistent with the existing literature (Hadamard, However, Mathematician A attributed breakthroughs to 1945; Poincaré, 1948). chance. In other words, making the appropriate Excerpt 5 (psychological) connections by pure chance which eventually result in the sought after result. B. One of the problems is first one does some I think it is necessary to comment about the preparatory work, that has to be the left side unusual view of mathematician A. Chance plays an [of the brain], and then you let it sit. I don't important role in mathematical creativity. Great ideas think you get ideas out of nowhere, you have and insights may be the result of chance such as the to do the groundwork first, okay. This is why discovery of penicillin. Ulam (1976) estimated that people will say, now we have worked on this there is a yearly output of 200,000 theorems in problem, so let us sleep on it. So you do the

Bharath Sriraman 29 mathematics. Chance plays a role in what is considered field, is one instance of a unification of apparently important in mathematical research since only a random fragments because the proof involves algebra, handful of results and techniques survive out of the complex analysis, and number theory. volumes of published research. I wish to draw a Polya (1954) addresses the role of chance in a distinction between chance in the "Darwinian" sense probabilistic sense. It often occurs in mathematics that (as to what survives), and chance in the psychological a series of mathematical trials (involving computation) sense (which results in discovery/invention). The role generate numbers that are close to a Platonic ideal. The of chance is addressed by Muir (1988) as follows. classic example is Euler's investigation of the infinite series 1 + 1/4 + 1/9 + 1/16 +…+ 1/n2 +…. Euler The act of creation of new entities has two aspects: the generation of new possibilities, for which we obtained an approximate numerical value for the sum might attempt a stochastic description, and the of the series using various transformations of the selection of what is valuable from among them. series. The numerical approximation was 1.644934. However the importation of biological metaphors Euler confidently guessed the sum of the series to be to explain cultural evolution is dubious…both π2/6. Although the numerical value obtained by Euler creation and selection are acts of design within a and the value of π2/6 coincided up to seven decimal social context. (p. 33) places, such a coincidence could be attributed to Thus, Muir (1988) rejects the Darwinian chance. However, a simple calculation shows that the explanation. On the other hand, Nicolle (1932) in probability of seven digits coinciding is one in ten Biologie de L'Invention does not acknowledge the role million! Hence, Euler did not attribute this coincidence of unconsciously present prior work in the creative to chance but boldly conjectured that the sum of this process. He attributes breakthroughs to pure chance. series was indeed π2/6 and later proved his conjecture to be true (Polya, 1954, pp. 95-96). By a streak of lightning, the hitherto obscure problem, which no ordinary feeble lamp would Intuition, Verification and Proof have revealed, is at once flooded in light. It is like a creation. Contrary to progressive acquirements, Once illumination has occurred, whether through such an act owes nothing to logic or to reason. The sheer chance, incubation, or divine intervention, act of discovery is an accident. (Hadamard, 1945) mathematicians usually try to verify that their intuitions were correct with the construction of a proof. Nicolle's Darwinian explanation was rejected by The following section discusses how these Hadamard on the grounds that to claim creation occurs mathematicians went about the business of verifying by pure chance is equivalent to asserting that there are their intuitions and the role of formal proof in the effects without causes. Hadamard further argued that creative process. They were asked whether they relied although Poincaré attributed his particular on repeatedly checking a formal proof, used multiple breakthrough in Fuchsian functions to chance, Poincaré converging partial proofs, looked first for coherence did acknowledge that there was a considerable amount with other results in the area, or looked at applications. of previous conscious effort, followed by a period of Most of the mathematicians in this study mentioned unconscious work. Hadamard (1945) further argued that the last thing they looked at was a formal proof. that even if Poincaré's breakthrough was the result of This is consistent with the literature on the role of chance alone, chance alone was insufficient to explain formal proof in mathematics (Polya, 1954; Usiskin, the considerable body of creative work credited to 1987). Most of the mathematicians mentioned the need Poincaré in almost every area of mathematics. The for coherence with other results in the area. The question then is how does (psychological) chance mathematician’s responses to the posed question work? follow. It is my conjecture that the mind throws out fragments (ideas) that are products of past experience. Excerpt 6 Some of these fragments can be juxtaposed and B. I think I would go for repeated checking of the combined in a meaningful way. For example, if one formal proof…but I don't think that that is reads a complicated proof consisting of a thousand really enough. All of the others have to also be steps, a thousand random fragments may not be enough taken into account. I mean, you can believe to construct a meaningful proof. However the mind that something is true although you may not chooses relevant fragments from these random fully understand it. This is the point that was fragments and links them into something meaningful. made in the lecture by … of … University on Wedderburn's Theorem, that a finite division ring is a

30 Mathematical Creativity Dirichlet series. He was saying that we have process. “Mathematics in the making resembles any had a formal proof for some time, but that is other human knowledge in the making. The result of not to say that it is really understood, and what the mathematician’s creative work is demonstrative did he mean by that? Not that the proof wasn't reasoning, a proof; but the proof is discovered by understood, but it was the implications of the plausible reasoning, by guessing” (Polya, 1954). How result that are not understood, their mathematicians approached proof in this study was connections with other results, applications very different from the logical approach found in proof and why things really work. But probably the in most textbooks. The logical approach is an artificial first thing that I would really want to do is reconstruction of discoveries that are being forced into check the formal proof to my satisfaction, so a deductive system, and in this process the intuition that I believe that it is correct although at that that guided the discovery process gets lost. point I really do not understand its Conclusions implications… it is safe to say that it is my surest guide. The goal of this study was to gain an insight into mathematical creativity. As suggested by the literature

C. First you must see it in the nature, something, review, the existing literature on mathematical first you must see that this theorem creativity is relatively sparse. In trying to better corresponds to something in nature, then if you understand the process of creativity, I find that the have this impression, it is something relatively Gestalt model proposed by Hadamard (1945) is still reasonable, then you go to proofs…and of applicable today. This study has attempted to add some course I have also several theorems and proofs detail to the preparation-incubation-illumination- that are wrong, but the major amount of proofs verification model of Gestalt by taking into account the and theorems are right. role of imagery, the role of intuition, the role of social D. The last thing that comes is the formal proof. I interaction, the use of heuristics, and the necessity of look for analogies with other things… How proof in the creative process. your results that you think might be true would The mathematicians worked in a setting that was illuminate other things and would fit in the conducive to prolonged research. There was a general structure. convergence of intelligence, knowledge, thinking styles, personality, motivation and environment that E. Since I work in an area of basic research, it is enabled them to work creatively (Sternberg, 2000; usually coherence with other things, that is Sternberg & Lubart, 1996, 2000). The preparatory probably more than anything else. Yes, one stage of mathematical creativity consists of various could go back and check the proof and that sort approaches used by the mathematician to lay the of thing but usually the applications are yet to groundwork. These include reading the existing come, they aren't there already. Usually what literature, talking to other mathematicians in the guides the choice of the problem is the particular mathematical domain (Csikzentmihalyi, potential for application, part of what 1988; 2000), trying a variety of heuristics (Polya, represents good problems is their potential for 1954), and using a back-and-forth approach of use. So, you certainly look to see if it makes plausible guessing. One of the mathematicians said that sense in the big picture…that is a coherence he first looked to see if the sought after relationships phenomenon. Among those you've given me, corresponded to natural phenomenon. that’s probably the most that fits. All of the mathematicians in this study worked on Commentary on Excerpt 6 more than one problem at a given moment. This is This excerpt indicates that for mathematicians, consistent with the investment theory view of creativity valid proofs have varied degrees of rigor. “Among (Sternberg & Lubart, 1996). The mathematicians mathematicians, rigor varies depending on time and invested an optimal amount of time on a given circumstance, and few proofs in mathematics journals problem, but switched to a different problem if no meet the criteria used by secondary school geometry breakthrough was forthcoming. All the mathematicians in this study considered this as the most important and teachers (each statement of proof is backed by reasons). Generally one increases rigor only when the difficult stage of creativity. The prolonged hard work result does not seem to be correct” (Usiskin, 1987). was followed by a period of incubation where the Proofs are in most cases the final step in this testing problem was put aside, often while the preparatory

Bharath Sriraman 31 stage is repeated for a different problem; and thus, Implications there is a transition in the mind from conscious to It is in the best interest of the field of mathematics unconscious work on the problem. One mathematician education that we identify and nurture creative talent in cited this as the stage at which the "problem begins to the mathematics classroom. "Between the work of a talk to you." Another offered that the intuitive side of student who tries to solve a difficult problem in the brain begins communicating with the logical side at mathematics and a work of invention (creation)…there this stage and conjectured that this communication was is only a difference of degree" (Polya, 1954). not possible at a conscious level. Creativity as a feature of mathematical thinking is not a The transition from incubation to illumination patent of the mathematician! (Krutetskii, 1976); and often occurred when least expected. Many reported the although most studies on creativity have focused on breakthrough occurring as they were going to bed, or eminent individuals (Arnheim, 1962; Gardner, 1993, walking, or sometimes as a result of speaking to 1997; Gruber, 1981), I suggest that contemporary someone else about the problem. One mathematician models from creativity research can be adapted for illustrated this transition with the following: "You talk studying samples of creativity such as are produced by to somebody and they say just something that might high school students. Such studies would reveal more have been very ordinary a month before but if they say about creativity in the classroom to the mathematics it when you are ready for it, and Oh yeah, I can do it education research community. Educators could that way, can’t I! But you have to be ready for it. consider how often mathematical creativity is Opportunity knocks but you have to be able to answer manifested in the school classroom and how teachers the door." might identify creative work. One plausible way to Illumination is followed by the mathematician’s approach these concerns is to reconstruct and evaluate verifying the result. In this study, most of the student work as a unique evolving system of creativity mathematicians looked for coherence of the result with (Gruber & Wallace, 2000) or to incorporate some of other existing results in the area of research. If the the facets suggested by Gruber & Wallace (2000). This result cohered with other results and fit the general necessitates the need to find suitable problems at the structure of the area, only then did the mathematician appropriate levels to stimulate student creativity. try to construct a formal proof. In terms of the A common trait among mathematicians is the mathematician’s beliefs about the nature of reliance on particular cases, isomorphic reformulations, mathematics and its influence on their research, the or analogous problems that simulate the original study revealed that four of the mathematicians leaned problem situations in their search for a solution (Polya, towards Platonism, in contrast to the popular notion 1954; Skemp, 1986). Creating original mathematics that Platonism is an exception today. A detailed requires a very high level of motivation, persistence, discussion of this aspect of the research is beyond the and reflection, all of which are considered indicators of scope of this paper; however, I have found that beliefs creativity (Amabile, 1983; Policastro & Gardner, 2000; regarding the nature of mathematics not only Gardner, 1993). The literature suggests that most influenced how these mathematicians conducted creative individuals tend to be attracted to complexity, research but also were deeply connected to their of which most school mathematics curricula has very theological beliefs (Sriraman, 2004a). little to offer. Classroom practices and math curricula The mathematicians hoped that the results of their rarely use problems with the sort of underlying creative work would be sanctioned by a group of mathematical structure that would necessitate students’ experts in order to get the work included in the domain having a prolonged period of engagement and the (Csikzentmihalyi, 1988, 2000), primarily in the form of independence to formulate solutions. It is my publication in a prominent journal. However, the conjecture that in order for mathematical creativity to acceptance of a mathematical result, the end product of manifest itself in the classroom, students should be creation, does not ensure its survival in the Darwinian given the opportunity to tackle non-routine problems sense (Muir, 1988). The mathematical result may or with complexity and structure - problems which may not be picked up by other mathematicians. If the require not only motivation and persistence but also mathematical community picks it up as a viable result, considerable reflection. This implies that educators then it is likely to undergo mutations and lead to new should recognize the value of allowing students to mathematics. This, however, is determined by chance! reflect on previously solved problems to draw comparisons between various isomorphic problems (English, 1991, 1993; Hung, 2000; Maher & Kiczek,

32 Mathematical Creativity 2000; Maher & Martino, 1997; Maher & Speiser, Hadamard, J. (1945). Essay on the psychology of invention in the 1996; Sriraman, 2003; Sriraman, 2004b). In addition, mathematical field. Princeton, NJ: Princeton University Press. encouraging students to look for similarities in a class Hanna, G. (1991). Mathematical proof. In D. Tall (Ed.). Advanced mathematical thinking (pp. 54 60). Dordrecht: Kluwer. of problems fosters "mathematical" behavior (Polya, − 1954), leading some students to discover sophisticated Hung, D. (2000). Some insights into the generalizations of mathematical meanings. Journal of Mathematical Behavior, mathematical structures and principles in a manner 19, 63–82. akin to the creative processes of professional Krutetskii, V. A. (1976). The psychology of mathematical abilities mathematicians. in school children. (J. Kilpatrick & I. Wirszup, Eds.; J. Teller, Trans.). Chicago: University of Chicago Press. (Original work REFERENCES published 1968) Amabile, T. M. (1983). Social psychology of creativity: A L'Enseigement Mathematique. (1902), 4, 208–211. componential conceptualization. Journal of Personality and L'Enseigement Mathematique. (1904), 6, 376. Social Psychology, 45, 357−376. Lester, F. K. (1985). Methodological considerations in research on Arnheim, R. (1962). Picasso’s guernica. Berkeley: University of mathematical problem solving. In E. A. Silver (Ed.), Teaching California Press. and learning mathematical problem solving: Multiple Birkhoff, G. (1969). Mathematics and psychology. SIAM Review, research perspectives (pp. 41–70). Hillsdale, NJ: Erlbaum. 11, 429−469. Maher, C. A., & Kiczek R. D. (2000). Long term building of Corbin, J., & Strauss, A. (1998). Basics of qualitative research. mathematical ideas related to proof making. Contributions to Thousand Oaks, CA: Sage. Paolo Boero, G. Harel, C. Maher, M. Miyasaki. (organizers) Csikszentmihalyi, M. (1988). Society, culture, and person: A Proof and Proving in Mathematics Education. Paper systems view of creativity. In R. J. Sternberg (Ed.), The nature distributed at ICME9 -TSG 12. Tokyo/Makuhari, Japan. of creativity: Contemporary psychological perspectives (pp. Maher, C. A., & Speiser M. (1997). How far can you go with block 325−339). Cambridge UK: Cambridge University Press. towers? Stephanie's intellectual development. Journal of Csikszentmihalyi, M. (2000). Implications of a systems perspective Mathematical Behavior, 16(2), 125−132. for the study of creativity. In R. J. Sternberg (Ed.), Handbook Maher, C. A., & Martino A. M. (1996). The development of the of creativity (pp. 313−338). Cambridge UK: Cambridge idea of mathematical proof: A 5-year case study. Journal for University Press. Research in Mathematics Education, 27(2), 194−214. Davis, P. J., & Hersh, R. (1981). The mathematical experience. Manin, Y. I. (1977). A course in mathematical logic. New York: New York: Houghton Mifflin. Springer-Verlag. English, L. D. (1991). Young children's combinatoric strategies. Minsky, M. (1985). The society of mind. New York: Simon & Educational Studies in Mathematics, 22, 451−474. Schuster. English, L. D. (1993). Children's strategies in solving two- and Muir, A. (1988). The psychology of mathematical creativity. three-dimensional combinatorial problems. Journal for Mathematical Intelligencer, 10(1), 33−37. Research in Mathematics Education, 24(3), 255−273. Nicolle, C. (1932). Biologie de l'invention, Paris: Alcan. Ernest, P. (1991). The philosophy of mathematics education, Patton, M. Q. (2002). Qualitative research and evaluation methods. Briston, PA: Falmer. Thousand Oaks, CA: Sage. Ernest, P. (1994). Conversation as a metaphor for mathematics and Policastro, E., & Gardner, H. (2000). From case studies to robust learning. Proceedings of the British Society for Research into generalizations: An approach to the study of creativity. In R. J. Learning Mathematics Day Conference, Manchester Sternberg (Ed.), Handbook of creativity (pp. 213−225). Metropolitan University (pp. 58−63). Nottingham: BSRLM. Cambridge, UK: Cambridge University Press. Ervynck, G. (1991). Mathematical creativity. In D. Tall (Ed.), Poincaré, H. (1948). Science and method. New York: Dover. Advanced mathematical thinking (pp. 42−53). Dordrecht: Polya, G. (1945). How to solve it. Princeton, NJ: Princeton Kluwer. University Press. Frensch, P., & Sternberg, R. (1992). Complex problem solving: Principles and mechanisms. New Jersey: Erlbaum. Polya, G. (1954). Mathematics and plausible reasoning: Induction and analogy in mathematics (Vol. II). Princeton, NJ: Princeton Gallian, J. A. (1994). Contemporary abstract algebra. Lexington, University Press. MA: Heath. Schoenfeld, A. H. (1985). Mathematical problem solving. New Gardner, H. (1997). Extraordinary minds. New York: Basic Books. York: Academic Press. Gardner, H. (1993). Frames of mind. New York: Basic Books. Skemp, R. (1986). The psychology of learning mathematics. Gruber, H. E. (1981). Darwin on man. Chicago: University of Middlesex, UK: Penguin Books. Chicago Press. Sriraman, B. (2003). Mathematical giftedness, problem solving, Gruber, H. E., & Wallace, D. B. (2000). The case study method and and the ability to formulate generalizations. The Journal of evolving systems approach for understanding unique creative Secondary Gifted Education. XIV(3), 151−165. people at work. In R. J. Sternberg (Ed.), Handbook of Sriraman, B. (2004a). The influence of Platonism on mathematics creativity (pp. 93-115). Cambridge UK: Cambridge University research and theological beliefs. Theology and Science, 2(1), Press. 131−147.

Bharath Sriraman 33 Sriraman, B. (2004b). Discovering a mathematical principle: The APPENDIX A: Interview Protocol case of Matt. Mathematics in School (UK), 3(2), 25−31. The interview instrument was developed by modifying questions Sternberg, R. J. (1985). Human abilities: An information from questionnaires in L’Enseigement Mathematique (1902) and processing approach. New York: W. H. Freeman. Muir (1988). Sternberg, R. J. (2000). Handbook of creativity. Cambridge, UK: 1. Describe your place of work and your role within it. Cambridge University Press. 2. Are you free to choose the mathematical problems you tackle Sternberg, R. J., & Lubart, T. I. (1996). Investing in creativity. or are they determined by your work place? American Psychologist, 51, 677−688. 3. Do you work and publish mainly as an individual or as part of a group? Sternberg, R. J., & Lubart, T. I. (2000). The concept of creativity: 4. Is supervision of research a positive or negative factor in your Prospects and paradigms. In R. J. Sternberg (Ed.), Handbook work? of creativity (pp. 93 115). Cambridge, UK: Cambridge − 5. Do you structure your time for mathematics? University Press. 6. What are your favorite leisure activities apart from Taylor, S. J., & Bogdan, R. (1984). Introduction to qualitative mathematics? research methods: The search for meanings. New York: John 7. Do you recall any immediate family influences, teachers, Wiley & Sons. colleagues or texts, of primary importance in your Torrance, E. P. (1974). Torrance tests of creative thinking: Norms- mathematical development? technical manual. Lexington, MA: Ginn. 8. In which areas were you initially self-educated? In which areas do you work now? If different, what have been the Ulam, S. (1976). Adventures of a mathematician. New York: Scribners. reasons for changing? 9. Do you strive to obtain a broad overview of mathematics not Usiskin, Z. P. (1987). Resolving the continuing dilemmas in school of immediate relevance to your area of research? geometry. In M. M. Lindquist, & A. P. Shulte (Eds.), Learning 10. Do you make a distinction between thought processes in and teaching geometry, K-12: 1987 yearbook (pp. 17−31). learning and research? Reston, VA: National Council of Teachers of Mathematics. 11. When you are about to begin a new topic, do you prefer to Wallas, G. (1926). The art of thought. New York: Harcourt, Brace assimilate what is known first or do you try your own & Jovanovich. approach? Weisberg, R. W. (1993). Creativity: Beyond the myth of genius. 12. Do you concentrate on one problem for a protracted period of New York: Freeman. time or on several problems at the same time? 13. Have your best ideas been the result of prolonged deliberate Wertheimer, M. (1945). Productive thinking. New York: Harper. effort or have they occurred when you were engaged in other Wittgenstein, L. (1978). Remarks on the foundations of unrelated tasks? mathematics (Rev. Ed.).Cambridge: Massachusetts Institute of 14. How do you form an intuition about the truth of a proposition? Technology Press. 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).

34 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 entertaining family experience centered on academics necessity of parent awareness and involvement in with very little expense to or preparation by the school. schools, there are still limited connections among Finally, FMFNs offer a unique opportunity for schools, parents, and higher education institutions. It is professional interaction among university and school especially important for parents and teachers to be faculties and staff. aware of the premises and types of activities that At the University of Wisconsin-Whitewater, we support effective mathematics learning as advocated by require the FMFN project for students enrolled in the the National Council of Teachers of Mathematics Math for Elementary Teachers content courses and (NCTM, 2000). However, many parents did not grow provide it as an optional project for students enrolled in up learning in ways the NCTM advocates; they see the elementary mathematics methods course. Since hands-on activities as a fun “waste of time” rather than students take the Math for Elementary Teachers an avenue for providing conceptual underpinnings for courses in their freshman or sophomore year, FMFN is mathematics. Teachers must realize that fun hands-on a good way to get preservice teachers thinking about activities, though motivating for students, must also the content they are going to teach. Also, the have mathematical integrity in order to be included in experience supports the developmental view of the curriculum. To facilitate both parents and teachers mathematics learning presented in the content course reaching these goals, our university presents what we and provides an experiential background for students in call Family Math Fun Night (FMFN) at area the methods courses. The preservice teachers use elementary schools. activities from the Family Math books we keep on Numerous schools and districts report using some reserve, and we encourage students to devise or find variation of Family Math to help parents understand activities from other sources. The preservice teachers their children’s mathematics curricula better (Wood, in the methods courses are especially encouraged to 1991, 1992; Carlson, 1991; Pagni, 2002; Kyle, examine professional journals and databases in McIntyre, & Moore, 2001). Our program is a variation preparation for their projects. of Stenmark, Thompson, and Cossey’s (1986) Family Parent-teacher groups at schools provide a small Math. In contrast to their Family Math, we choose to amount of funding (usually about $25) to purchase have our preservice teachers present activities at stickers, pencils, erasers, etc. for prizes; though some elementary schools. This provides our preservice pre-service teachers buy their own, and many pre- teachers with an opportunity to have a positive, early service teacher groups do not give out prizes at all. The experience in schools and allows them to test ideas lack of prizes does not seem to affect the popularity of about mathematics education they have learned in their the activities for most children. For past FMFNs, we university classes. Also, FMFNs provide an 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 Melissa Freiberg is an associate professor in the Department of Curriculum and Instruction at the University of Whitewater- that a theme often emerges. For example, we have had Wisconsin. She has a PhD in Urban Education with an emphasis FMFNs whose activities revolve around sports and in teacher education. Her research interests are teacher FMFNs whose activities relate to voting. induction and hands-on learning. Reflecting on our version of FMFN raises points of interest that are worth sharing: (1) the types of

Melissa R. Freiberg 35 activities that are presented at the events and what incorporate drill and practice are usually presented in determines their quality, (2) what considerations are the context of a game. For example, one student group necessary for coordinating a FMFN, and (3) what can used a plastic bowling set to practice: addition and be learned as a result of the experiences. In the subtraction facts with younger children, how to keep a following sections, I will attend to each of these running total with slightly older children, and how to categories. identify fractions and percents for upper elementary children. Other examples of drill and practice activities Types of Activities are educational video games in which correct responses For each school site, the university organizers help students reach a goal (fuel for the spacecraft, provide two activities in addition to those the pre- money to buy souvenirs, moving closer to a target, service teachers present. The first activity uses jars etc.). These activities allow children to pick the containing snacks that are taken to the school a week difficulty of the task and move through different levels prior to the FMFN. Jars of varying shape are used for of calculation, building their self-confidence and different age levels. Each class within an age level knowledge. We encourage preservice teachers to estimates the number of snacks in the jar and records broaden their activities to include topics such as its estimate. During the FMFN, individual students or geometry, estimation, logic, patterns, graph parents can make estimates and enter them for a interpretation, and computation since all are important particular class. The class with the closest estimate to review. Board games are yet another way to support receives the snacks. This activity serves two purposes. drill and practice activity. The board is laid out on the The first purpose is to generate interest in and floor so that students walk around it landing on spaces. awareness of the event and encourage participation. When a student is on a space, he or she is asked a The second purpose is to support NCTM’s efforts mathematics question that varies depending on the age (NCTM, 2000) by emphasizing estimation skills. The of the student. second activity provided by the university requires a school representative to greet children and parents at Problem Solving the door and ask them to add a sticker to his or her Examples of problem solving activities are games birth month on a pre-designed bar graph. This helps from which preservice teachers create adaptations. take attendance for the evening and also helps children Preservice teachers like to challenge themselves with see the process of data collection and how a graph games that incorporate mathematical ideas and skills evolves from the process. and then adapt them to the skill level of the children. Preservice teachers design all other activities, and Adaptations of games such as Yahtzee® Equations® or their activities must involve mathematics concepts 24® help children plan and carry out different covered in their math classes (Math For the strategies. Memory games, similar to Concentration® Elementary Teachers I—numeration, whole number are used to match fractions to decimals, operations to and fraction operations, problem solving; Math For the results, or various representations of numbers. These Elementary Teacher II—geometry, measurement, games1 are inexpensive to produce, easy to explain, probability, and statistics). The types of projects the and easily adaptable for different ages and grade levels. preservice teachers choose to present usually fall into A second example of a problem solving activity is the categories of drill and practice, problem solving, or asking children to identify or copy patterns in beads, estimation. I will discuss types of activities that fall pictures, tessellations, or shapes. Bead stringing is into each category and then discuss two exceptional commonly used to demonstrate patterns. The youngest activities that do not fall into any of the three children describe and extend simple patterns while categories. somewhat older children choose a preset pattern and string beads to illustrate the pattern. The oldest group Drill and Practice of students designs bead strings that contain multiple Although students are charged (and monitored) to patterns such as combining patterns of color with do more than BINGO or flash cards as the essence of patterns of shape or size. This activity is more the activity, drill and practice may be part of the expensive because children keep the materials they use activity. Pre-service teachers’ initial attempts at to make the bracelets or necklaces. creating these activities are generally weak but with coaching or feedback, they develop more thought- provoking activities. Rich activities designed to

36 Family Math Estimation preservice teachers a clear understanding of In addition to the introductory snack estimation expectations, and to detail past problems we have activity, almost every FMFN has at least one faced. Since incorporating FMFN into our curriculum, preservice teacher designed activity that asks children we have identified objectives and assessments assuring to estimate capacity, weight, area, and/or quantity. One that FMFN activities are mathematically sound (see popular activity requires children to estimate through Appendix A and Form A). The most frequent problems the use of indirect measurement. In this activity, there we encounter revolve around logistics such as are approximately 15 objects to measure and the coordinating transportation to schools, advertising the characteristics of objects vary in difficulty according to event in the community, and setting up the school children’s differing abilities. space. The following steps are used to conduct our In recent years, we have seen a growing number of FMFN events and might be helpful for those who want activities that use estimation to help students develop to organize similar work: probability concepts. These activities illustrate our 1. Contact schools that might be interested in hosting preservice teachers’ increased awareness of the the event. We contact school districts through direct importance of estimation and probability as well as mailings or use various connections our department has their increased confidence in students’ abilities to do to area schools. After several years of conducting three such activities. In these activities, children are asked FMFNs each semester, most schools contact us to how frequently an event happens or how close an schedule the event. estimated answer is to the correct solution. 2. Information about FMFN is given to our preservice Exceptional Activities teachers with their class syllabus. The preservice Two exceptional activities from the past do not fall teachers are allowed to choose the topic around which into any of the above categories. They are exceptional they will make their activity (within guidelines because they are unique and demonstrate the creativity mentioned earlier). Groups may be made up of students of the preservice teachers who made them. The first from different classes requiring FMFN or from classes was presented in one of the first FMFNs we ran. that offer it as an optional activity. Preservice teachers, with the help of the students, used 3. The preservice teachers turn in a description of their math symbols to represent letters of each child's name activities (see Form A and Evaluation Form) indicating on a nametag. Children were then told to see if they how it will be adjusted for various ages/grades, how could figure out other people’s names by equating the parents will be involved, and how they will assess the letter of a name with a math symbol. For example, success of their activities. This allows the faculty to Anne's name might be + φ ≠ = (add, null set, not equal, assess the activities for mathematical integrity and equal) and she would then know the math symbol that avoid redundancies in activities. It also gives students a corresponded to the letter “a”, “n”, and “e” and could foundation for writing their reflections on the event use this to deduce the names of other people. (See number 8). The second exceptional activity had three pictures made up of geometric shapes. Children were given a 4. We assign our preservice teachers to specific dates paper shape and asked to match their paper shape with and schools based on preferences and class schedules. the shape in one of the pictures. The youngest children Groups are usually made up of three to four people and had shapes that were congruent to shapes in the about twenty groups are assigned to each school. picture, while older students were asked to find shapes 5. We confirm who is assigned to each school and similar to their shape but that differed in size, color, or allow groups to indicate special set-up needs (see Form orientation. The preservice teachers prompted children B). The preservice teachers indicate if they are able to to name the shape and describe its attributes. This provide transportation to the schools so car pools can activity proved quite challenging for children but was be established. extremely popular. 6. A faculty member or preservice teacher visits each Coordinating an FMFN school to determine space and resource availability, to In organizing FMFNs, we have discovered that discuss the role of the school staff, and to give communication among all the parties involved is suggestions for advertising the event. We suggest the essential. We have developed guidelines and a timeline school connect FMFN with a regularly scheduled PTA to facilitate communication, to give schools and meeting. Sending reminders home with school

Melissa R. Frieberg 37 children, having the event on the school calendar, and One of the most rewarding results of this writing an article in the school or local newsletter experience for the university faculty and staff is the explaining the event are ways that have been effective opportunity to work collaboratively across departments in bringing FMFN to the attention of parents. and colleges. College of Education faculty/staff who teach the elementary mathematics methods courses 7. The night of the event, university and school assist faculty and staff from the Mathematics personnel monitor the preservice teacher groups and Department in planning, implementing, assessing, and the families attending. At the close of the event, we revising the program. Additionally, the experience announce the winning class for the estimation exercise provides an opportunity for the Mathematics and leave activity kits at the school for classroom use. Department members to visit local elementary schools 8. Each university preservice teacher group turns in a with teachers and children. Education faculty and staff written reflection of impressions of the event. This who do regular supervision of student teachers in report is not only helpful in assessing the university schools get to see students' abilities to teach to a students' learning, but also helps us identify problems variety of ages and abilities, which requires flexibility that might need to be addressed in the future. This and instant adaptations that might be missed in single report focuses on the content and success of the grade level settings. activity, how students handled problems and questions Teachers, administrators, and parents are effusive that arose, how students interacted with parents and in their praise for the event. The university students teachers, and how they collaborated with their groups. mention that they often have classroom teachers 9. An individual report is also required from each waiting “like vultures” to pick up the activity at the end preservice teacher. This report is focused on how the of the night. Alternately, classroom teachers give university students ideas for improving or adapting the student felt the group process worked, what was learned about mathematics, and a self-reflection about projects for different children’s needs or abilities. one's ability as a teacher. Administrators find that the turnout for this event is higher than for other school sponsored programs and, Conclusions interestingly, draws more fathers. We average about In the introduction, we stated that we found this 200 participants at each event, even in schools where activity to be beneficial to university preservice there are fewer than 300 students. teachers, university faculty and staff, school staff, Parents have a varied level of involvement in parents, and especially children. Although this paper is activities from merely standing and waiting to sitting not intended to present a research study on FMFN, we down and participating with their children in the believe that we have seen beneficial results for those activity. Many times parents mention that they are involved. surprised at how well their children performed on a The university students have consistently, and given task or how well they thought through a almost unanimously, responded positively to their problem. In rare instances parents appear to be participation in FMFN both in their reports and in class impatient or negative with respect to their children’s discussions. Even students who described themselves efforts, and the university students get their first chance as poor math students found the experience to be to try out their mediating skills. Although not a benefit enjoyable and uplifting. They appreciated the chance to to children, parent outbursts do give university students work with a small group of elementary students. As an opportunity to see how parents influence children’s one student said, "I found that helping them [the learning. children] out with solving a problem was an interesting Most importantly, it appears that the elementary and rewarding experience...this is what teaching is all children who attend FMFN come away satisfied. about." Many university students were especially University and school faculty have observed that surprised and buoyed by the fact that they were able to students almost universally leave the event feeling adjust questions, offer hints and assistance, or explain successful and empowered in math. Certainly children mathematical ideas more easily than they anticipated. fearful in math are less likely to attend, but we have They also learned how to share responsibilities, ask for watched children start out very tentatively and soon help, and make changes to their activities as needed. find themselves immersed in an activity. Virtually all Too often preservice teachers believe these things are a the children at each event try every activity, but they sign of weakness rather than a sign of collaboration. return to certain activities—and these are rarely the FMFN helps change that perception. easiest activities. This behavior indicates that students

38 Family Math are motivated by activities that challenge them and Family Math Fun Night Project Requirements make them think rather than simple mastery. Your grade for this project is based on 80 points. The numbers In conclusion, we have found that all of us have following the due dates below indicate the points that can be earned gained from the experiences. As university instructors on each portion of the project. we continually need to listen to our students in order to Jan 31 (5 points) Form A – Group Membership and Activity Idea adapt and refine the expectations and requirements for Hand in one copy of Form A to each instructor of members of your FMFN. As prospective mathematics teachers, our group. Your Group ID Code will be assigned when returned. students have the chance to devise and carry out February 17 for District #1 (School A), February 19 for all others activities in a low-stress, supportive atmosphere. (20 points) Activity Description Schools and teachers are provided with examples Typed descriptions of your FMFN project should include: of activities that complement classroom instruction. Names of group members with leader indicated, Group ID Code, Parents see how their children’s active involvement in name of activity, date and location of presentation. Procedures and/or instructions you will be giving for the activity. activities enhances their learning, and parents may What the child is to do and learn from your activity? Include come away with a better understanding of the sample problems and activities for each level. mathematics curriculum. Finally, children always seem If adults accompany children at the event, how will the adult to walk away feeling successful and eager to move on participate in your activity? to the next level in mathematics. If prizes are used, how will they be awarded? Who will supply the prizes? REFERENCES How will you evaluate different aspects of your activity? Refer to the attached evaluation sheet used by faculty and questions listed Carlson, C. G. (1991). Getting parents involved in their children’s below. education. Education Digest, 57(10), 10–12. Feb 24 (School A), Feb 26 (School B), March 5 (School C), March Kyle, D. W., McIntyre, E., & Moore, G. H. (2001). Connecting 12 (School D) Form B – Needs List mathematics instruction with the families of young children. Teaching Children Mathematics, 8, 80–86. Hand in one copy of Form B to each instructor of members of your group. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author. Mar 7 (School A), March 17 (School B), March 31 (School C), April 7 (School D) (5 pts) FMFN Evaluation Form Pagni, D. (2002). Mathematics outside of schools. Teaching Children Mathematics, 9, 75–78. Hand in two copies of FMFN Evaluation Form to your group leader’s instructor with answers completed for the questions on the Stenmark, J. K., Thompson, V., & Cossey, R. (1986). Family math. right side of the form. Berkeley: University of California-Berkeley, Mar 11 (School A), March 20 (School B), April 3 (School C), April Wood, J. (Ed.). (1992). Variations on a theme: Family math night. 10 (School D) (30 points) FMFN Event Curriculum review, 32(2), 10. Retrieved May 17, 2004, from Galileo database (ISSN 0147-2453; No. 9705276559). Run your activity (6:30-8:00 p.m. in School A & B, 6:00 to 7:30 in School D) and have fun. Wood, J. (Ed.). (1991). “Family math” teaches English as well as Arrive at school 30 to 60 minutes prior to start. math. Curriculum review, 31(1), 21. Retrieved May 17, 2004, Set up your activity. from Galileo database (ISSN 0147-2453; No. 9705223330). Try to find time to visit and play the activities of other groups Internet site for student information on FMFN: during the evening. http://facstaff.uww.edu/whitmorr/whitmore/FMFN.html Mar 19 (School A), Apr. 4 (School B), April 11 (School C), April 16 (School D) (20 points) Individual Evaluation 1 Equations® is a game in which a specific number of cards are Sorry, no evaluations will be returned until all evaluations have been graded. drawn. The cards have whole numbers on them, and students are to arrange the cards and determine operations that will create an Your individual evaluation of the group learning activity (2 to 3 equation. 24® is a similar game in which each card has four whole pages) should include: numbers that, when using different operations on the numbers, will Your name, Group ID Code, your activity name, school attended, equal 24. Concentration® is a game in which a set of cards is placed group members’ names and their instructor, if other than your face down in an array, and players take turns turning up two cards instructor, and method(s) used to evaluate your activity. at a time looking for pairs. In commercially made games these are Did your group work well together? Why or why not? How well usually identical pictures; however, in educational games these may did you work within your group? What part of the project did you be two equivalent numbers using different symbols or do? representations. 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?

Melissa R. Frieberg 39 Did things go as planned during FMFN? What did you not If the table has attached benches, our group will need only __ anticipate? additional chairs. How did you modify/adjust your activity during the evening to Our group would also like the following to be supplied by the host meet the needs of the students/parents? Include specific examples school: of difficulties and adaptations. Our group would prefer to be located (please check one and give What would you do differently if you did a similar activity again? your reasoning in the space to the right) What did you learn about yourself and the grade(s) you are ___so we can hang things on a wall behind us planning to teach? Is teaching at this level still your goal? Why or ___in a corner of the room why not? ___in the center of the room ___it doesn't matter Grammar and other English mechanics will count. ___near a power source Form A Our group (please check one) ___doesn't plan to use prizes Group Membership and Activity Idea (Spring 2003) ___will supply its own prizes Due: Friday, January 31, 2003 ___is counting on having the school supply prizes Value: 5 point Assigned Group ID Code: Appendix A Please turn in one copy of this form to each teacher of a member of This semester you will work with elementary students and their your group. (Group ID Code will be assigned after you submit parents/guardians in a project called Family Math Fun Night Form A. Use it on all subsequent submissions.) (FMFN). This project is designed to show children and parents that Materials will be returned the group via the leader. mathematics is an essential part of their everyday life and can be FUN!! Most importantly, it provides the opportunity for you to be Group Leader’s Name, Phone, Email Address, Course/Section, involved with elementary children as they do mathematics in Teacher's Name, Other Members’ Names: enjoyable problem solving activities. Brief Activity Description: As a member of a group, you will be presenting an activity for Indicate your choice for FMFN presentation. Consider evening Family Math Fun Night (FMFN) at one of four elementary schools: classes, sports schedules, previous commitments, and work School A (PK-5, 300 students); School B (K-5, 280 students; and schedules of all members of the group in making your selections. If School C (K-3, 400 students). All children from these schools and your group requires a particular time, please explain the their families will be invited to attend from 6:30 - 8:00 p.m. (6:00 to circumstances. You will not be allowed to switch assignments after 7:30 in one school). The fourth elementary presentation is from 1:30 they have been made unless you can find a group able to exchange to 3:00 at School D. We will run all events like a carnival having with you. booths (tables) set up with various activities. There will not be a whole group presentation. You should plan to be at your school at ___Our group has no preference of night presentation; any night least one half hour early. This will allow you time to set up your will work for us. activity, and to visit and enjoy the activities of other groups before ___Our group would prefer the following nights: (Please circle first the children and parents arrive. You should be cleaned-up and out of and second choices, and give reasons in space to the right.) the school 30 minutes after the closing time. Does your group have transportation for FMFN? yes no FORMING GROUPS. Who will design the activities for this carnival? Your group will select, make, and present your activity at Could your group provide transportation for others the FMFN. Form a group of four; a group with 3 or 5 students must be evening of FMFN? yes no approved by your instructor(s). Group members may be from any section of the course you are taking. As you are selecting groups, Form B think about class and work schedules for all of the members of your Family Math Fun Night: Needs List 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 Instructor(s) Group ID Code the activity. You may indicate your group's preference for evening Due: of presentation. VERY IMPORTANT: After groups have been Please turn in one copy of this form to each teacher of a member of assigned an evening, you will not be able to change assignments your group. unless you can find a group willing to switch with you. Name of Activity: SELECTING AN ACTIVITY. Your group should select an Brief description of activity: activity that is accessible and meaningful to the full range of Group Leader: students in attendance. If you are unsure what is taught at various Other group members: grade levels, do some research in the LMC on the lower level of the Things you may need for your activity: library. Your activity should be fun and challenging for students Tables - Limit your project to one table. These may be lunch tables and parents and need not be competitive. It should involve problem with attached benches solving, not merely mechanics or facts. Flash card type drill is not Chairs - remember most elementary teachers do not sit down usually fun, and is not appropriate for a FMFN activity. Be sure to Tape, scissors, pencils, paper, scrap paper, markers, etc - please involve parents in your activity; parents should be doing not just bring your own !! watching. “Helping by giving hints and encouragement” is not Our group will need to have (please indicate how many) sufficient adult involvement. Your activity will need to be planned Table (zero or one): Chairs: 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

40 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 Evaluation Form latest cartoon characters may attract elementary students, but may Submit two copies to your group leader’s instructor be an infringement of copyright. Invent a clone! Be creative! Don't Activity Date: Instructor(s): just take an activity from a book or off a shelf; put something of Activity Name: yourself into it. Don’t just use the activity you, or a friend, used last Group Members: semester. Math 148 and 149 students should develop an activity Faculty evaluators will use the following portion (and rate between that involves math topics they will be covering in class. Realizing 1 and 5). that there are many connections between the mathematics in the Math content: (Problem solving, concept development, more than two courses, this does not exclude presenting a topic from your course in an activity that also uses a topic from the other course. mechanics) Adaptability of project: (Grade level, special needs, mental, written Take this opportunity to develop an activity you could use in your and manipulative capabilities) future classroom. Please do not use TWISTER activities. The book Family Math has been placed on reserve (2 hour, no Materials: (Quality, durability and economy of materials) Appeal and Creativity: (Attract and retain participants) overnight) in the library. You will need to ask for it by name at the Interaction: (With students and adults, where possible) main circulation desk. This book has over 100 Family Math activities. You may wish to use one of these, combine a couple, Professionalism: (Dress, group demeanor, setup on time, modify one, or come up with an idea on your own. You could also enthusiasm) Total Points (out of 30): check Teaching Children Mathematics, other periodicals, and the Average number of points: Internet for ideas. Make this a fun learning experience for you! WHAT YOU WILL NEED. Your group must have a sign with (Based on evaluations) the name of your activity. You may need to make some equipment Groups are to provide the following information in the space to be used at your booth such as markers, counters, game board, provided: etc. Other things such as pencils, scissors, ruler, scrap paper, and Describe your activity’s math content and how you emphasized it. manipulatives are also useful. The LMC has some equipment that How did you adapt your activity to meet all students’ capabilities? can be checked out. If they cannot meet your needs, your instructor Describe the quality, durability and economy of your may have some ideas. You may also want to have copies of materials. 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 41 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 problem- solving 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 used in Singapore can include problems that are quite Mathematics Texts? complex and advanced. Children can reasonably be What is special about the elementary mathematics expected to solve these problems given the problem- texts used in Singapore? These texts look very solving and sense-making tools they have been different from major elementary school mathematics exposed to. texts used in the U.S. The presentation of mathematics Thus the strong performance of Singapore’s in Singapore’s elementary texts is direct and brief. children in mathematics may be due in part to the way Words are used sparingly, but even so, problems mathematics is presented in their textbooks, including sometimes have complex sentence structures. The page the way simple pictures and diagrams are used to layout is clean and uncluttered. Perhaps the most communicate mathematical ideas and to provide sense- striking feature is the heavy use of pictures and making aids for solving problems. If so, then teachers, diagrams to present material succinctly—although mathematics educators, and instructional designers in pictures are never used for embellishment. Simple the U.S. will benefit from studying the presentation of pictures and diagrams accompany many problems, and mathematics in Singapore’s textbooks, so that they can the same types of pictures and diagrams are used help children in the U.S. improve their understanding repeatedly, as supports for different types of problems, of mathematics and their ability to solve problems. and across grade levels. These simple pictures and Using Strip Diagrams to Solve Story diagrams are not mere procedural aids designed to help Problems children produce speedy solutions without One of the most interesting aspects of the understanding. Rather, the pictures and diagrams elementary school mathematics texts and workbooks appear to be designed to help children make sense of used in Singapore (Curriculum Planning and problems and to use solution strategies that can be Development Division, Ministry of Education, justified on solid conceptual grounds. Because of this Singapore, 1999, hereafter referred to as Primary pictorial, sense-making approach, the elementary texts Mathematics and Primary Mathematics Workbook) is Sybilla Beckmann is a mathematician at the University of the repeated use of a few simple types of diagrams to Georgia who has a strong interest in education. She has aid in solving problems. Starting in volume 3A, which developed three mathematics content courses for prospective is used in the first half of 3rd grade, simple “strip elementary teachers and has written a textbook, Mathematics for diagrams” accompany a variety of story problems. Elementary Teachers, published by Addison-Wesley, for use in Consider the following 3rd grade subtraction story such courses. In the 2004/2005 academic year, she will teach a class of 6th grade mathematics daily at a local public middle problem: school.

Sybilla Beckman 42 Mary made 686 biscuits. She sold some of them. If A farmer has 7 ducks. He has 5 times as many 298 were left over, how many biscuits did she sell? chickens as ducks….How many more chickens (Primary Mathematics volume 3A, page 20, than ducks does he have? (Primary Mathematics problem 4) volume 3A, page 46, problem 4) The problem is accompanied by a strip diagram like (Note: The first part of the problem asks how many the one shown in Figure 1. 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: Although the strip diagrams will not always help Meilin saved $184. She saved $63 more than Betty. children carry out the required calculations (for How much did Betty save? (Primary Mathematics example, we don’t see how to carry out the subtraction volume 3A, page 21, problem 7) $184 – $63 from Figure 2), they are clearly designed to help children decide which operations to use. Instead This problem is accompanied by a strip diagram like of relying on superficial and unreliable clues like key the one in Figure 2. 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 Figure 2: How Much Did Betty Save? problems. Other problems that could be solved with the aid of a strip diagram do not have an accompanying These two problems are examples of some of the diagram and do not mention drawing a diagram. more difficult types of subtraction story problems for Fraction problems, such as the following 4th grade children. The first problem is difficult because we must problem, are naturally modeled with strip diagrams take an unknown number of biscuits away from the such as the accompanying diagram in Figure 4: initial number of biscuits. This problem is of the type David spent 2/5 of his money on a storybook. The change-take-from, unknown change (see Fuson, 2003, storybook cost $20. How much money did he have for a discussion of the classification of addition and at first? (Primary Mathematics volume 4A, page subtraction story problems). The second problem is 62, problem 11) difficult because it includes the phrase “$63 more Without a diagram, the problem becomes much than,” which may prompt children to add $63 rather more difficult to solve. We could formulate it with the than subtract it. This problem is of type compare, equation (2/5)x = 20 where x stands for David’s inconsistent (see Fuson, 2003). The term inconsistent is original amount of money, which we can solve by used because the phrase “more than” is inconsistent dividing 20 by 2/5. Notice that the diagram can help us with the required subtraction. Other linguistically see why we should divide fractions by multiplying by difficult problems, including those that involve a the reciprocal of the divisor. When we solve the multiplicative comparison with a phrase such as “N problem with the aid of the diagram, we first divide times as many as”, are common in Primary $20 by 2, and then we multiply the result by 5. In other Mathematics and are often supported with a strip words, we multiply $20 by 5/2, the reciprocal of 2/5. diagram. Consider the following 3rd grade problem, which is supported with a diagram like the one in Figure 3:

Sybilla Beckman 43 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 Figure 5: Raju and Samy Split Some Money of 4th grade, algebra story problems begin to appear. Consider the following problems: Notice that the manipulations we perform with 1. 300 children are divided into two groups. There strip diagrams usually correspond to the algebraic are 50 more children in the first group than in the manipulations we perform in solving the problem second group. How many children are there in the algebraically. For example, to solve the previous Raju second group? (Primary Mathematics volume 4A, and Samy problem, we could let S be Samy’s initial page 40, problem 8) amount of money. Then, 2S + 100 = 410 2. The difference between two numbers is 2184. If the bigger number is 3 times the smaller number, as we also see in Figure 5. When we solve the problem find the sum of the two numbers. (Primary algebraically, we subtract 100 from 410 and then Mathematics volume 4A, page 40, problem 9) divide the resulting 310 by 2, just as we do when we solve the problem with the aid of the strip diagram. 3. 3000 exercise books are arranged into 3 piles. Strip diagrams make it possible for children who The fist pile has 10 more books than the second pile. The number of books in the second pile is have not studied algebra to attempt remarkably twice the number of books in the third pile. How complex problems, such as the following two, which many books are there in the third pile? (Primary are accompanied by diagrams like the ones in Figure 6 Mathematics volume 4A, page 41, problem 10) and Figure 7 respectively: These problems are readily formulated and solved Encik Hassan gave 2/5 of his money to his wife algebraically with equations, but since the text has not and spent 1/2 of the remainder. If he had $300 left, introduced equations with variables, the children are how much money did he have at first? (Primary presumably expected to draw diagrams to help them Mathematics volume 5A, page 59, problem 6) solve these problems. Notice that from an algebraic Raju had 3 times as much money as Gopal. After point of view, the second problem is most naturally Raju spent $60 and Gopal spent $10, they each had formulated with two linear equations in two unknowns, an equal amount of money left. How much money and yet 4th graders can solve this problem. did Raju have at first? (Primary Mathematics The 5th grade Primary Mathematics texts and volume 6B, page 67, problem 1) 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:

44 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%) These problems are similar to problems in Primary Figure 6: How Much Money Did Encik Hassan Have at Mathematics. The strong performance of Singapore 8th First? 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 Figure 7: How Much Did Raju Have at First? on the following item in the content domain “Data Representation, Analysis and Probability” classified as “Knowing”: Performance of 8th Graders on TIMSS If a fair coin is tossed, the probability that it will In light of the complex problems that children in land heads up is 1/2. In four successive tosses, a Singapore are taught how to solve in elementary fair coin lands heads up each time. What is likely school, the strong performance of Singapore’s 8th to happen when the coin is tossed a fifth time? graders on the TIMSS assessment is not surprising. A. It is more likely to land tails up than heads up. Among the released TIMSS 8th grade assessment items in the content domain “Fractions and Number B. It is more likely to land heads up than tails up. Sense” classified as “Investigating and Solving C. It is equally likely to land heads up or tails up. Problems,” Singapore 8th graders scored higher than D. More information is needed to answer the U.S. 8th graders on all items. These released items question. included the following problems (see NCES, 2003): (Problem F08, page 74. Overall percent correct, Laura had $240. She spent 5/8 of it. How much United States: 62%, Singapore: 48%) money did she have left? (Problem R14, page 29. Overall percent correct, Singapore: 78%, United The mathematics texts used in Singapore through States: 25%). 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.

Sybilla Beckman 45 Conclusion REFERENCES The mathematics textbooks used in elementary Curriculum Planning and Development Division, Ministry of Education, Singapore (1999, 2000). Primary Mathematics (3rd schools in Singapore show how to represent quantities ed.) volumes 1A–6B. Singapore: Times Media Private with drawings of strips. With the aid of these simple Limited. Note: additional copyright dates listed on books in strip diagrams, children can use straightforward this series are 1981, 1982, 1983, 1984, 1985, 1992, 1993, reasoning to solve many challenging story problems 1994, 1995, 1996, 1997, thus 8th graders who took the 1999 conceptually. The TIMSS 8th grade assessment shows TIMSS assessment used an edition of these books. that 8th graders in Singapore are effective problem Curriculum Planning and Development Division, Ministry of Education, Singapore (1999, 2000). Primary Mathematics solvers and are much better problem solvers than U.S. Workbook (3rd ed.) volumes 1A–6B. Singapore: Times Media 8th graders. Although cultural factors probably also Private Limited. affect the strong mathematics performance of children Fuson, K. C. (2003). Developing Mathematical Power in Whole in Singapore, children in the U.S. could probably Number Operations. In J. Kilpatrick, W. G. Martin, and D. strengthen their problem-solving abilities by learning Schifter, (Eds.), A Research companion to principles and Singapore’s methods and by being exposed to more standards for school mathematics (pp. 68–94). Reston,VA: National Council of Teachers of Mathematics. challenging and linguistically complex story problems National Center for Education Statistics (2003). Trends in early in their mathematics education. 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

46 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 Education (ICME9) in Tokyo, Japan, in 2000. Perhaps book reflects a “shift in focus from equity to a more this context explains why approximately half of the inclusive perspective that embraces social justice as a chapters focus primarily on gender, while other contested area of investigation within mathematics chapters include issues related to differences in race, education” (p. xv). What’s interesting is that the class, language, and thinking styles. Burton notes that question in the title lacks a verb—is the question this book, as the fourth publication of IOWME, “which are ways to social justice in mathematics “reflects the development of the group’s interests that education?” Or more tentatively, “which ways might have evolved over 16 years from a sharp focus on bring about social justice in mathematics education?” gender issues to its present wider interest in social Or perhaps the focus is more on research, either up to justice” (p. xiii). now or in the future: “which ways have research on In the introduction Burton also outlines the process social justice in mathematics education taken? Or by which the book developed. After a general call for “which ways could (should?) research on social justice papers, an international review panel of mathematics in mathematics education take?” Each of the thirteen educators reviewed submissions. Chapter authors were chapters in the volume addresses at least one of those then paired to give feedback to each other on their four questions. Overall, this book responds to its title work in order to promote dialogue as well as “cross- question through diverse voices that call for expanding referencing possibilities” (p. xv). As perhaps is always work on gender issues into broader sociocultural, the case in an edited book without summary pieces to political, and technological contexts; rethinking and highlight connections between chapters, the cross- refining key notions such as equity, citizenship, and referencing of concepts in this volume could be difference; and considering how to conduct studies that expanded. Burton does a nice job of drawing some reach beyond school and university boundaries toward connections in her introduction, but otherwise such families, communities, and policy-makers. resonance is largely left to the reader. Fortunately, as I The collection is the third volume in the hope to demonstrate in this review, there is ample International Perspectives on Mathematics Education opportunity to draw connections between chapters (and series for which Burton has served as series editor.1 In also occasionally to wish that an author had heeded her introduction she describes the origin of the book in another author’s points or ideas!) the activities of the International Organization of Organization of the Book Women in Mathematics Education (IOWME) at the Ninth International Congress of Mathematics The thirteen chapters in the book are organized into three sections. The four chapters in the first Amy Hackenberg is at work on her doctoral dissertation on the section focus on definitional work, conceptual emergence of sixth graders’ algebraic reasoning from their frameworks, and reviews of and recommendations for quantitative reasoning in the context of mathematically caring research, thereby “setting the scene” (p. 1). The authors teacher-student relations. In addition to her fascination with of this section are from Australia (Brew), Germany mathematical learning and the orchestration of it, she is compelled by issues of social justice, the nature and (Jungwirth), the United Kingdom (Povey), and the consequences of social interaction, and the relationship between United States (Hart). The second section consists of the “social” and the “psychological” in mathematics education. seven chapters primarily about studies that take place in classrooms and address the question “what does

47 Book Review social justice mean in classrooms?” (p. 101). The nature of equity and justice within different contexts: a authors of this section are from Australia (Forgasz, typology of gender-sensitive teaching, previous and Leder, and Thomas; Zevenbergen), Germany (Ferri current research on equity and justice in mathematics and Kaiser), Malawi (Chamdimba), the United States education, citizenship education in the United and Peru (Secada, Cueto, and Andrade), and the United Kingdom, and statistical analyses of gender differences Kingdom (Mendick; Wiliam). The last section includes in mathematics education. two chapters focused specifically on “computers and Jungwirth describes a typology of gender-sensitive mathematics learning” (p. 261) with regard to social teaching that consists of three types distinguished by justice. The authors (Wood, Viskic, and Petocz; Vale) modifications made according to gender, the degree to come from Australia and Eastern Europe, but all now which gender groups are identified and treated as practice mathematics education in Australia. monolithic, and corresponding conceptions of equity. The placement of chapters within this organization In Type I teaching, teachers are “gender-blind” and is a little puzzling. Wiliam’s illuminating chapter on make no modifications according to gender since they the construction of statistical differences and its believe that boys and girls can do math equally well. In implications is included in the second section on Type II teaching, teachers adjust practices based on classroom studies, but since it grapples with definitions gender but tend to treat students of a single gender as and conceptual ideas (and is not a classroom study), it monolithic (i.e., tend to essentialize.) Jungwirth might have been better placed in the first more believes that in the third (and implicitly most theoretically-oriented section. Brew’s chapter, a study advanced) type, the concept of equity “no longer about reasons that mothers return to study applies…Equity here refers to the individual, with mathematics, is included in the first section but seems respect to learning arrangements and, somewhat to fit better in the second, despite the fact that the study qualified, to outcomes” (p. 16). Teachers engaging in does not take place in mathematics classrooms. Type III teaching attend to individual differences Support for changing the placement of Brew’s chapter within gender groups and tailor teaching to individuals. is provided by the position of Mendick’s: Her report of Although Jungwirth’s typology offers a conceptual young British men’s choices to study mathematics framework for examining the equitable implications of beyond compulsory schooling is only peripherally teachers’ orientations toward mathematics teaching and located in classrooms and was still placed in the second mathematics classrooms, her dichotomizing of groups section. and individuals is problematic. For example, in their The other weak organizational aspect of the book attention to individuals, might not Type III teachers is the inclusion of only two chapters in the third section create classrooms in which mathematics could be on computers and mathematics learning. One wonders devoid of women, which Jungwirth sees as if there were intentions for a more substantial section considerably less evolved than even Type I teaching? but some papers did not make the publication deadline. The problem seems to be in characterizing equity In any case, because both chapters in this section report based on group-individual dichotomies—to adhere too on studies set in classrooms, it seems that they could strongly to group identities can result in essentializing, have been included in the second section—or that while to focus primarily on the individual can leave out perhaps two sections about studies might have been trends and broad characteristics of groups that are warranted, one that focused directly on studies in important considerations in work toward equity and mathematics classrooms and one that included research social justice (cf. Lubienski, 2003). on mathematics education outside of immediate These issues are reflected in Hart’s review of classroom contexts. scholarship on equity and justice in mathematics education over the last 25 years. Her chapter is notable Conceptually-Oriented Chapters: What Is Equity? for explicit discussion about different ways researchers What Is Social Justice? have used equity and justice (and equality); for her Organizational difficulties aside, I focus first on clearly stated choice to use equity to mean justice; and the more conceptually-oriented chapters, which are for her formulation of calls for future research. In contained in the first three chapters of the first section particular, she calls for research on pedagogies that of the book as well as in Wiliam’s chapter from the contribute to justice; self-study of educators’ own second section. These authors engage in definitional practices; and more research that explores student and conceptual work that forms a foundation for motivation, socialization, identity, and agency with research on social justice. All four authors ponder the respect to mathematics. Hart highlights Martin’s

48 Book Review (2000) study on factors contributing to failure and may seem counterintuitive (and certainly differs from success of African American students in mathematics typical U.S. selection processes!), Wiliam makes a as an exemplar for future research because of its compelling argument that is worth reading. multilevel framework for analyzing mathematics Chapters on Studies in or Surrounding socialization and identity. Although her points about Mathematics Classrooms his work are well taken, the considerable space she gives to this relatively recent study seems odd given In these chapters—Brew’s chapter from the first her aims to review 25 years of research. section as well as the other 8 chapters in the book—the Povey continues Jungwirth’s and Hart’s diverse voices in the volume become quite apparent, definitional work by considering the complex and not only because of the different geographical locations contested notion of citizenship in relation to social or ethnic heritages of the authors but because of the justice and mathematics education. She describes how diverse ways in which the authors focus on issues of recent mandates for citizenship education in England social justice in relation to mathematics classrooms and reinforce a conservative perspective by focusing on mathematical study. These nine chapters can also be political and legal citizenship (the right to vote, for loosely grouped as exemplifying, supporting, example), without questioning the nature and character informing, or aligning with the more conceptually- of social citizenship, let alone its connections to “the oriented chapters. (mathematics) education of future citizens” (p. 52). In particular, two chapters that focus specifically Povey believes that for citizenship to be a useful on teaching practices in relation to social justice may concept in democratizing mathematics classrooms the exemplify and inform Jungwith’s typology. The concept “will have to be more plural, more active, and authors of these chapters attend to how teachers more concerned with participation in the here and approach students who belong to disadvantaged now” (p. 56). groups. Chamdimba, whose research took place in the Perhaps the strongest chapter of these four (and southern African country of Malawi, studied the year one of the strongest in the collection) is Wiliam’s on 11 students of a Malawian teacher who agreed to use the construction of statistical differences in cooperative learning to potentially promote a “learner- mathematical assessments. He demonstrates that in friendly classroom climate” (p. 156) for girls. As a gender research in mathematics education, effect sizes researcher, Chamdimba might exemplify a Type II of standardized differences between male and female orientation out of her concerns over Malawian girls’ test scores are relatively small, and the variability lack of representation and achievement in mathematics within a gender is greater than between genders. Based and subsequent Malawian women’s lack of bargaining on this analysis, Wiliam concludes that differences power as a group for social and economic resources in between genders depend on what counts as the country. Chamdimba’s conclusion that female mathematics on assessments. In particular, what counts students experienced largely positive effects might as mathematics may be maintained because it supports help Jungwirth refine her typology so that recognizing patriarchal hegemony. students as part of disenfranchised groups and acting As an implication of his argument, Wiliam on that recognition to address the group is seen as proposes “random justice” (p. 202) to produce equity legitimate and useful (i.e., not necessarily less evolved in selection based on test scores. Wiliam calls the than Type III teaching.) However, Chamdimba’s study percentage of the population that reaches a certain is also subject to scrutiny over whether a particular standard (for, say, entrance to medical school) a classroom structure can bring about improvements in recruitment population. Usually, selecting from a all Malawian females’ educational, social, and recruitment population (i.e., creating a selection economic status. population) involves choosing a small top percentage Perhaps a better example of the subtlety involved of it. This mode of selection perpetuates selecting more in the group-individual distinctions with regard to males than females, largely because males show social justice is found in Zevenbergen’s study. greater variability in their test scores compared to Zevenbergen used Bourdieu’s tools as a frame for females (males produce more highs and lows.) Wiliam understanding teachers’ beliefs about students from proposes that a random sample of the recruitment socially disadvantaged backgrounds in the South-East population that sustains the gender (or racial, class, Queensland region of Australia. Eight of the 9 teachers etc.) make-up of it is “the only fair way” (p. 204) of interviewed expressed views of students as deficient creating a selection population. Although this proposal due to poverty and cultural practices. Stretching

Amy J. Hackenberg 49 Jungwirth’s typology beyond gender-sensitivity, the Finally, the remaining three chapters in the book ninth teacher had more of a Type III orientation in her connect with Povey’s chapter in exploring a particular respect for these students as individuals. However, by contested and complex concept or relate to Wiliam’s expressing an understanding of how parents’ lack of work on considering the construction of difference. cultural capital prevented them from challenging the Brew’s study entails rethinking aspects of the complex ways in which schools (under)served their children, concept of mothering in the context of mathematical this teacher did not ignore these students as belonging learning of both mothers and their children. By to a disadvantaged group. This teacher’s ability to including voices of the children in the study, Brew is understand and value students as both individuals and able to show the fluid roles of care-taking between part of a group might allow Jungwirth to amplify and studying mothers and their children (e.g., children further articulate her typology. sometimes acted as carers for their mothers) and “the These two chapters and three others exhibit work pivotal role that children can play…in providing not that aligns with Hart’s call for research on pedagogies only a consistent motivating factor but also enhancing that contribute to social justice and on one’s own their mother’s intellectual development” (p. 94). teaching in relation to social justice. Vale’s two case What Povey does for citizenship and Brew does for studies of computer-intensive mathematics learning in mothering, Mendick does for masculinity in the two junior secondary mathematics classrooms focus on context of doing mathematics. In a very strong and how teachers’ practices with technology impede (but thoughtful chapter, she describes stories of three young might facilitate) more just classroom environments. British men who have opted to study mathematics in Vale’s work is complemented by the three university their A-levels even though they do not enjoy it. classroom studies presented by Wood, Viskic, and Mendick’s smart use of a poststructuralist perspective Petocz. In studying their own computer-intensive that deconstructs the classic opposition between teaching of differential equations, statistics, and structure and agency allows her to argue that taking up preparatory mathematics classes, these three mathematics is a way for the men to “do masculinity” researchers found positive attitudes toward the use of in a variety of ways: to prove their intelligence to technology across gender. Finally, Ferri and Kaiser’s employers and others as well as to secure a future in comparative case study on the styles of mathematical labor market. The stories of the three males prompt the thinking of year 9 and 10 students (ages 15-16) has question: “why is maths a more powerful proof of implications for developing pedagogies that recognize ability than other subjects?” (p. 182). To respond, differences other than due to gender, race, or class, and Mendick contrasts the men’s stories with young that thereby contribute to justice and diversity in women’s stories (part of her larger research project.) classrooms. This artful move is not intended to draw However, Secada, Cueto, and Andrade’s large- dichotomies between how men and women “do maths” scale, comprehensive study of the conditions of differently—Mendick cautions against such simplistic schooling for fourth and fifth-grade children who speak conclusions and notes that some females use Aymara, Quechua, and Spanish in Peru may be the mathematics the way these three males do. Instead the strongest example of work toward Hart’s contrast allows her to demonstrate and deepen her recommendation of multilevel frameworks in research theorizing of masculinity as a relational configuration on social justice. These researchers intended to create a of a practice, as well as to argue for more complexity “policy-relevant study” (p. 106). To do so they in gender reform work. Thus for her, “maths and articulated their conceptions of equity as distributive gender are mutually constitutive; maths reform work is social justice (opportunity to learn mathematics is a gender reform work” (p. 184). By examining gender in social good and should not be related to accidents of this way, like Wiliam, she calls into question birth) and socially enlightened self-interest (it is in differences between males and females in relation to everyone’s interest for everyone to do well so as not to mathematics and supports his contention that what cause great cost to society). In addition, the researchers counts as mathematics (and, Mendick would add, as took as a premise that equity must come with both high masculine and feminine) is the basis for these quality and equality (i.e., lowering the bar does not differences. foster equity). Thus they contribute to definitional Differences between males and females are also work while formulating “practical” conclusions and the subject of the chapter by Fogasz, Leder, and recommendations for Peruvian governmental policy. Thomas. They used a new survey instrument to capture the beliefs of over 800 grade 7–10 Australian students

50 Book Review regarding gender stereotyping of mathematics. Their REFERENCES findings revealed interesting reversals of expected (stereotyped) beliefs. For example, their participants Conlin, M. (2003, May 26). The new gender gap. Business Week online. Retrieved September 1, 2003, from believed that boys are more likely than girls to give up http://www.businessweek.com when they find a problem too difficult, and that girls Lubienski, S. T. (2003). Celebrating diversity and denying are more likely than boys to like math and find it disparities: A critical assessment. Educational Researcher, interesting. However, through an examination of 32(8), 30–38. participation rates and achievement levels of male and Martin, D. B. (2000). Mathematics success and failure among female grade 12 mathematics students from 1994 to African-American youth: The roles of sociohistorical context, 1999 in Victoria, Australia, the researchers refute community forces, school influence, and individual agency. recent, media-hyped contentions (see, e.g., Conlin, Mahwah, NJ: Lawrence Erlbaum. 2003; Weaver-Hightower, 2003) that males are now Weaver-Hightower, M. (2003). The “boy turn” in research on gender and education. Review of Educational Research, 73(4). disadvantaged in mathematics. Frankly, Fogasz and 471–498. 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. 1 Nevertheless, their work supports the notion that The first volume was Multiple Perspectives on Mathematics Teaching and Learning (2000) edited by Jo Boaler; the second mathematics may be maintained as a male domain volume was Researching Mathematics Classrooms: A Critical despite certain advances of females. Examination of Methodology (2002) edited by Simon Goodchild Overall, I agree with Burton that the chapters in and Lyn English. 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 51 CONFERENCES 2004…

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

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

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

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

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

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

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

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

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

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

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

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

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

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

52 Book Review The Mathematics Education Student Association is an official affiliate of the National Council of Teachers of Mathematics. MESA is an integral part of The University of Georgia’s mathematics education community and is dedicated to serving all students. Membership is open to all UGA students, as well as other members of the mathematics education community. Visit MESA online at http://www.ugamesa.org

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53 The Mathematics Educator (ISSN 1062-9017) is a semiannual publication of the Mathematics Education Student Association (MESA) at The University of Georgia. The purpose of the journal is to promote the interchange of ideas among the mathematics education community locally, nationally, and internationally. The Mathematics Educator presents a variety of viewpoints on a broad spectrum of issues related to mathematics education. The Mathematics Educator is abstracted in Zentralblatt für Didaktik der Mathematik (International Reviews on Mathematical Education). The Mathematics Educator encourages the submission of a variety of types of manuscripts from students and other professionals in mathematics education including: • reports of research (including experiments, case studies, surveys, philosophical studies, and historical studies), curriculum projects, or classroom experiences; • commentaries on issues pertaining to research, classroom experiences, or public policies in mathematics education; • literature reviews; • theoretical analyses; • critiques of general articles, research reports, books, or software; • mathematical problems; • translations of articles previously published in other languages; • abstracts of or entire articles that have been published in journals or proceedings that may not be easily available. The Mathematics Educator strives to provide a forum for collaboration of mathematics educators with varying levels of professional experience. The work presented should be well conceptualized; should be theoretically grounded; and should promote the interchange of stimulating, exploratory, and innovative ideas among learners, teachers, and researchers. Guidelines for Manuscripts: • Manuscripts should be double-spaced with one-inch margins and 12-point font, and be a maximum of 25 pages (including references and footnotes). An abstract should be included and references should be listed at the end of the manuscript. The manuscript, abstract, and references should conform to the Publication Manual of the American Psychological Association, Fifth Edition (APA 5th). • An electronic copy is required. (A hard copy should be available upon request.) The electronic copy may be in Word, Rich Text, or PDF format. The electronic copy should be submitted via an email attachment to [email protected]. Author name, work address, telephone number, fax, and email address must appear on the cover sheet. The editors of The Mathematics Educator use a blind review process therefore no author identification should appear on the manuscript after the cover sheet. Also note on the cover sheet if the manuscript is based on dissertation research, a funded project, or a paper presented at a professional meeting. • Pictures, tables, and figures should be camera ready and in a format compatible with Word 95 or later. Original figures, tables, and graphs should appear embedded in the document and conform to APA 5th - both in electronic and hard copy forms. To Become a Reviewer: Contact the Editor at the postal or email address below. Please indicate if you have special interests in reviewing articles that address certain topics such as curriculum change, student learning, teacher education, or technology. Postal Address: Electronic address: The Mathematics Educator [email protected] 105 Aderhold Hall The University of Georgia Athens, GA 30602-712

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